+ All Categories
Home > Documents > An Efficient Compact Difference Method for Temporal ...

An Efficient Compact Difference Method for Temporal ...

Date post: 03-Dec-2021
Category:

Documents
Upload: others
View: 3 times
Download: 0 times
Download Report this document
Share this document with a friend
10
Research Article An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations Lei Ren and Lei Liu School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu , China Correspondence should be addressed to Lei Ren; [email protected] Received 11 June 2019; Accepted 25 July 2019; Published 21 August 2019 Academic Editor: Andrei D. Mironov Copyright © 2019 Lei Ren and Lei Liu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2− in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the 2-1 approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results. 1. Introduction e Black-Scholes model, proposed in 1973 by Black and Scholes [1] and Merton [2], gives a theoretical estimate of the price of European-style options. Until Now, some of Black-Scholes models involving the fractional derivatives have emerged. In [3], Wyss priced a European call option by a time-fractional Black-Scholes model. In [4], Liang et al. derive a biparameter fractional Black-Merton-Scholes equation and obtain the explicit option pricing formulas for the European call option and put option, individually. An explicit closed-form analytical solution for barrier options under a generalized time-fractional Black-Scholes model by using eigenfunction expansion method together with the Laplace transform is derived in [5]. In [6], a discrete implicit numerical scheme with a spatially second-order accuracy and a temporally 2− order accuracy is constructed; the stability and convergence of the proposed numerical scheme are analysed using Fourier analysis. In [7], H.Zhang et al. use some numerical technique to price a European double- knock-out barrier option, and then the characteristics of the three fractional Black-Scholes models are analysed through comparison with the classical Black-Scholes model. More recently, a numerical scheme of fourth-order in space and 2− in time is derived in [8]; the solvability and convergence of the proposed numerical scheme are proved rigorously using a Fourier analysis. Some computationally efficient numerical methods have been proposed for solving fractional differ- ential equation, for example, which include finite difference methods, finite element methods, finite volume methods, spectral methods, and meshless methods [9–26]. In this paper, we continue the work of R.H.De Staelen et al. [8]. e class of equations is given by (, ) + 1 2 2 2 2 (, ) 2 + ( − ) (, ) = (, ) , (, ) ∈ ( , ) × (0, ) (1) with the following boundary (barrier) and final conditions ( , ) = (), ( , ) = (), ∈ (0, ] , (2) and its initial condition (, ) = () , ∈ [ , ], (3) where is the risk free rate, is the dividend rate, and >0 is the volatility of the returns. e functions and Hindawi Advances in Mathematical Physics Volume 2019, Article ID 3263589, 9 pages https://doi.org/10.1155/2019/3263589
Transcript
Page 1: An Efficient Compact Difference Method for Temporal ...

Research ArticleAn Efficient Compact Difference Method forTemporal Fractional Subdiffusion Equations

Lei Ren and Lei Liu

School of Mathematics and Statistics Shangqiu Normal University Shangqiu 476000 China

Correspondence should be addressed to Lei Ren renqihanhotmailcom

Received 11 June 2019 Accepted 25 July 2019 Published 21 August 2019

Academic Editor Andrei D Mironov

Copyright copy 2019 Lei Ren and Lei LiuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation Anumerical scheme for the equation has been derived to obtain 2 minus 120572 in time and fourth-order in space We improve the results byconstructing a compact scheme of second-order in time while keeping fourth-order in space Based on the 1198712-1120590 approximationformula and a fourth-order compact finite difference approximation the stability of the constructed scheme and its convergence ofsecond-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method Applications usingtwo model problems demonstrate the theoretical results

1 Introduction

The Black-Scholes model proposed in 1973 by Black andScholes [1] and Merton [2] gives a theoretical estimate ofthe price of European-style options Until Now some ofBlack-Scholes models involving the fractional derivativeshave emerged In [3] Wyss priced a European call optionby a time-fractional Black-Scholes model In [4] Liang etal derive a biparameter fractional Black-Merton-Scholesequation and obtain the explicit option pricing formulas forthe European call option and put option individually Anexplicit closed-form analytical solution for barrier optionsunder a generalized time-fractional Black-Scholes model byusing eigenfunction expansion method together with theLaplace transform is derived in [5] In [6] a discrete implicitnumerical scheme with a spatially second-order accuracyand a temporally 2 minus 120572 order accuracy is constructed thestability and convergence of the proposed numerical schemeare analysed using Fourier analysis In [7] HZhang et aluse some numerical technique to price a European double-knock-out barrier option and then the characteristics of thethree fractional Black-Scholes models are analysed throughcomparison with the classical Black-Scholes model Morerecently a numerical scheme of fourth-order in space and 2minus120572 in time is derived in [8] the solvability and convergence ofthe proposed numerical scheme are proved rigorously using

a Fourier analysis Some computationally efficient numericalmethods have been proposed for solving fractional differ-ential equation for example which include finite differencemethods finite element methods finite volume methodsspectral methods and meshless methods [9ndash26]

In this paper we continue the work of RHDe Staelen etal [8] The class of equations is given by

120597120572119862 (119878 119905)120597119905120572 + 1212059021198782 1205972119862 (119878 119905)1205971198782 + (119903 minus 119863) 119878120597119862 (119878 119905)120597119878

= 119903119862 (119878 119905) (119878 119905) isin (119861119889 119861119906) times (0 119879)(1)

with the following boundary (barrier) and final conditions

119862 (119861119889 119905) = 119875 (119905) 119862 (119861119906 119905) = 119876 (119905)

119905 isin (0 119879] (2)

and its initial condition

119862 (119878 119879) = 119881 (119878) 119878 isin [119861119889 119861119906] (3)

where 119903 is the risk free rate 119863 is the dividend rate and120590 gt 0 is the volatility of the returns The functions 119875 and

HindawiAdvances in Mathematical PhysicsVolume 2019 Article ID 3263589 9 pageshttpsdoiorg10115520193263589

2 Advances in Mathematical Physics

119876 are the rebates paid when the corresponding barrier ishit The terminal playoff of the option is 119881(119878) The fractionalderivative in (1) is a Caputo derivative defined as

120597120572119862 (119878 119905)120597119905120572

=

1Γ (1 minus 120572) int1199050

120597119862120597120578 (119878 120578) (119905 minus 120578)minus120572 d120578 0 lt 120572 lt 1120597119862 (119878 119905)120597119905 120572 = 1

(4)

As described in [8] we consider the transform problem of (1)

1198620D120572

119905119880 (119909 119905) = 1198861205972119880 (119909 119905)1205971199092 + 119887120597119880 (119909 119905)120597119909 minus 119888119880 (119909 119905)+ 119891 (119909 119905)

(119909 119905) isin (0infin) times (0 119879) 119880 (119887119889 119905) = 119901 (119905) 119880 (119887119906 119905) = 119902 (119905)

119905 isin (0 119879] 119880 (119909 0) = 120593 (119909) 119909 isin [119887119889 119887119906]

(5)

The rest of the paper is organized as follows in Section 2an efficient implicit numerical scheme with second-orderaccuracy in time and fourth-order accuracy in space isconstructed The analysis of the stability and convergence arepresented in Section 3 In Section 4 numerical examples aregiven to illustrate the accuracy of the presented scheme and tosupport our theoretical results Concluding remarks are givenin the last section

2 Construction of the Compact FiniteDifference Scheme

In order to simplify the computation and analysis of thefollowing compact finite difference scheme for Black-Scholesmodel we use an indirect approach by introducing a suitabletransformation

According to some simple calculations we transformequation (5) into

1198620D120572

119905 119881(119909 119905) = 1198861205972119881 (119909 119905)1205971199092 minus 119889119881 (119909 119905) + 119892 (119909 119905) (119909 119905) isin (0infin) times (0 119879)

119881 (119887119889 119905) = 119901lowast (119905) 119881 (119887119906 119905) = 119902lowast (119905)

119905 isin (0 119879] 119881 (119909 0) = 120593lowast (119909) 119909 isin [119887119889 119887119906]

(6)

where119901lowast (119905) = 119901 (119905) 119902lowast (119905) = 119896 (119887119906) 119902 (119905) 120593lowast (119909) = 119896 (119909) 120593 (119909)

(7)

It is clear that 119880(119909 119905) is a solution of (5) if and only if 119881(119909 119905)is a solution of (6)

In order to construct the compact finite difference schemefor the problem (5) we consider the above equivalent form(6)

Let 120591 = 119879119873 be the time step and ℎ = (119887119906minus119887119889)119872 = 119871119872be the spatial step where119872119873 are positive integers

Since the grid function V = V119894 | 0 le 119894 le 119872 we thendefine difference operators as follows

120575119909V119894minus12 = 1ℎ (V119894 minus V119894minus1) 1205752119909V119894 = 1ℎ2 (V119894+1 minus 2V119894 + V119894minus1)

H119909V119894 = V119894 + ℎ2121205752119909V119894(8)

We also define1198860 = 1205901minus1205721198870 = 0119886119896 = (119896 + 120590)1minus120572 minus (119896 minus 1205722)1minus120572 (119896 ge 1) 119887119896 = 12 minus 120572 ((119896 + 120590)2minus120572 minus (119896 minus 1205722)2minus120572)

minus 12 ((119896 + 120590)1minus120572 + (119896 minus 1205722 )1minus120572) (119896 ge 1)

(9)

where 120590 = 1 minus 1205722 and

119888119896119899 =

1198860 119896 = 0 119899 = 1119886119896 + 119887119896+1 minus 119887119896 0 le 119896 le 119899 minus 2 119899 ge 2119886119899minus1 minus 119887119899minus1 119896 = 119899 minus 1 119899 ge 2

(10)

Lemma 1 It holds (see [27])1 minus 1205722 (119896 + 120590)minus120572 lt 119886119896 minus 119887119896 lt (119896 + 120590)1minus120572 minus (119896 minus 1205722)1minus120572

(119896 ge 1) (11)

In order to discretize (6) into a compact finite differencesystem we introduce the following lemmas

Lemma 2 Assuming V(119905) isin C3[0 119879] we have1198620D120572

119905 V (119905119899minus1205722) = 1120583119899sum119896=1

119888119899minus119896119899 (V (119905119896) minus V (119905119896minus1))+ O (1205913minus120572)

(12)

where 120583 = 120591120572Γ(2 minus 120572)

Advances in Mathematical Physics 3

Proof From Lemma 2 of [9] we can obtain the proof oflemma

Lemma 3 Assuming V(119905) isin C2[0 119879] When 119899 ge 1 we obtainV (119905119899minus1205722) = 1205722 V (119905119899minus1) + (1 minus 1205722) V (119905119899) + O (1205912) (13)

Proof According to some simple calculations the prooffollows from Taylor expansions of the function V(119905) at thepoint 119905119899minus1205722 for 119905 = 119905119899minus1 and 119905 = 119905119899

Since the above lemmas we then discretize (6) into acompact finite difference scheme In order to analyse wedefine

120575119905V119899minus12 = 1120591 (V119899 minus V119899minus1) (1 le 119899 le 119873) V1198991205722 = 1205722 V119899minus1 + (1 minus 1205722) V119899 (1 le 119899 le 119873)

(14)

We also define the grid functions as follows

119881119899119894 = 119881 (119909119894 119905119899) 119882119899119894 = 120597119881 (119909119894 119905119899)120597119905 119885119899119894 = 1205972119881 (119909119894 119905119899)1205971199092 119892119899119894 = 119892 (119909119894 119905119899)

119892119899minus1205722119894 = 119892 (119909119894 119905119899minus1205722) 119901lowast119899 = 119901lowast (119905119899) 119902lowast119899 = 119902lowast (119905119899) 120593lowast119894 = 120593lowast (119909119894)

(15)

For the second-order spatial derivative 119885119899119894 we adopt thefollowing fourth-order compact approximation (see [28])

H119909119885119899119894 = 1205752119909V (119909119894) + 119874 (ℎ4) 1 le 119894 le 119872 minus 1 (16)

We consider equation (6) at the point (119909119894 119905119899minus1205722) we canobtain1198620D120572

119905119881 (119909119894 119905119899minus1205722) = 119886119885119899minus1205722119894 minus 119889119881119899minus1205722119894 + 119892119899minus1205722119894 (17)

From Lemmas 2 and 3 we have

1120583119899sum119896=1

119888119899minus119896119899 (119881119896119894 minus 119881119896minus1119894 )= 1198861198851198991205722119894 minus 1198891198811198991205722119894 + 119892119899minus1205722119894 + (119877120572119905 )119899119894

0 le 119894 le 119872 1 le 119899 le 119873(18)

where

(119877120572119905 )119899119894 = (119886 minus 119889)O (1205912) + O (1205913minus120572) 0 le 119894 le 119872 1 le 119899 le 119873 (19)

We applyH119909 to equation (18) then we have

1120583119899sum119896=1

119888119899minus119896119899H119909 (119881119896119894 minus 119881119896minus1119894 )= 11988612057521199091198811198991205722119894 minus 119889H1199091198811198991205722119894 +H119909119892119899minus1205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873(20)

where(119877120572119905119909)119899119894 = H119909 (119877120572119905 )119899119894 + 119886 (119877120572119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873 (21)

and10038161003816100381610038161003816(119877120572119905119909)119899119894 10038161003816100381610038161003816 le 119862119877 (1205912 + ℎ4) 1 le 119894 le 119872 minus 1 1 le 119899 le 119873 (22)

If we omit (119877120572119905119909)119899119894 then we have the compact finite differencescheme

1120583119899sum119896=1

119888119899minus119896119899H119909 (V119896119894 minus V119896minus1119894 )= 1198861205752119909V1198991205722119894 minus 119889H119909V1198991205722119894 +H119909119892119899minus1205722119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873V (119887119889 119905) = 119901lowast (119905) V (119887119906 119905) = 119902lowast (119905)

119905 isin (0 119879] V (119909 0) = 120593lowast (119909) 119909 isin [119887119889 119887119906]

(23)

3 Stability and Convergence of the ProposedCompact Difference Scheme

Theorem 4 e compact difference scheme (23) is uniquelysolvable

Proof The compact difference scheme (23) can be written inmatrix form

AV119899 = b119899minus1 (24)

where

b119899minus1 = 119899minus1sum119896=0

120577119896V119896 120577119896 isin R (25)

The tridiagonal coefficient matrixA = (119886119894119895) yields10038161003816100381610038161198861198941198941003816100381610038161003816 = 56 ((119886(120572)0 + 119887(120572)1 )

120583 ) + 119886 (2 minus 120572)ℎ2 + 5 (2 minus 120572)12 119889sum119895 =119894

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816100381610038161003816100381616 ((119886(120572)0 + 119887(120572)1 )

120583 ) minus 119886 (2 minus 120572)ℎ2 + (2 minus 120572)12 11988910038161003816100381610038161003816100381610038161003816100381610038161003816

(26)

4 Advances in Mathematical Physics

It is easy to see that the tridiagonal coefficient matrix A isstrictly diagonally dominantTherefore the coefficientmatrixis nonsingular and hence invertible

Next we consider the stability and convergence analysisof the compact difference scheme (23)

Letting Ω = 119906 | 119906 = (1199060 1199061 119906119872) 1199060 = 119906119872 = 0for grid functions 119906 V isin Ω we define the inner product andnorm as follows

(119906 V) = ℎ119872minus1sum119894=1

119906119894V119894119906 = (119906 119906)12

119906infin = max0le119894le119872

10038161003816100381610038161199061198941003816100381610038161003816 (120575119909119906 120575119909V) = ℎ119872minus1sum

119894=0

120575119909119906119894+12120575119909V119894+12|119906|1 = (120575119909119906 120575119909119906]12 1199061 = (1199062 + |119906|21)12

(27)

According to simple calculations we obtain

(1205752119909119906 V) = minus (120575119909119906 120575119909V) ℎ 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 le 2 |119906|1

ℎ |119906|1 le 2 119906 (28)

In order to analyse we introduce the discrete innerproduct and norm

⟨119906 V⟩ = (H119909119906 minus1205752119909V) = (120575119909119906 120575119909V] minus ℎ212 (1205752119909119906 1205752119909V) 119906120576 = ⟨119906 119906⟩12

(29)

Based on above inner product and norm we have thefollowing lemmas

Lemma 5 (see [29]) Suppose 119906 isin Ω we obtain

1003817100381710038171003817H11990911990610038171003817100381710038172 le 1199062 le 3119871216 1199062120576 1199062infin le 31198718 1199062120576 11990621 le 3 (8 + 1198712)

16 1199062120576 (30)

Lemma 6 (see [27]) Suppose 119906 isin Ω we obtain

119906 le 11987128 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 1199062120576 le 3119871216 100381710038171003817100381710038171205752119909119906100381710038171003817100381710038172

(31)

Lemma 7 (see [9]) Suppose 119906 isin Ω we obtain

( 119899sum119896=1

119888119899minus119896119899H119909 (119906119896 minus 119906119896minus1) minus12057521199091199061198991205722)

ge 12119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) 1 le 119899 le 119873(32)

In the next we then analyse the stability and convergenceof the scheme (23)

Theorem8 (stability) Let V119899 = (V1198990 V1198991 V119899119872) be the solutionof the compact difference scheme (23) with V1198990 = V119899119872 = 0Assume that one of the conditions 1 le 4(4120576 minus 1)11988631198891205761198712 holdsfor some positive constant 120576 gt 14

en it holds

1003817100381710038171003817V11989910038171003817100381710038172120576 le 10038171003817100381710038171003817V0100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 1 le 119899 le 119873

(33)

Proof We take the inner product of equation (23) withminus1205752119909V1198991205722 yield1120583 ( 119899sum119896=1

119888119899minus119896119899H119909 (V119896 minus V119896minus1) minus1205752119909V1198991205722)= minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 minus 119889 (H119909 (V1198991205722) 1205752119909V1198991205722)

minus (H119909119892119899minus1205722 1205752119909V1198991205722) 1 le 119899 le 119873(34)

Using Lemma 7

12120583119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817V119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817V119896minus1100381710038171003817100381710038172120576)le minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576

minus (H119909119892119899minus1205722 12057521199091199061198991205722) 1 le 119899 le 119873(35)

When 1 le 4(4120576 minus 1)11988631198891205761198712 for some positive constant120576 gt 14 we have from the Cauchy-Schwarz inequality andLemmas 6 that

119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 le 3119889119871216 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

le (119886 minus 1198864120576) 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172(36)

minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 + 1198864120576 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

(37)

By (35) and the Cauchy-Schwarz inequality

minus 119886 1003817100381710038171003817100381712057521199091199061198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(38)

Advances in Mathematical Physics 5

Substituting (38) into (35) leads to

119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) le 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (39)

The above inequality can be rewritten as

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 + 119888119899minus1119899 100381710038171003817100381710038171199060100381710038171003817100381710038172lowast+ 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(40)

Since by the definition of 119888119899minus1119899120583119888119899minus1119899 =

120583119886119899minus1 minus 119887119899minus1 lt 2Γ (1 minus 120572)119879120572 (41)

we have from (40) that

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576le 119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576+ 119888119899minus1119899 (100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172)

(42)

Letting

119864 = 100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (43)

and assuming 1199061198962120576 le 119864(0 le 119896 le 119899 minus 1) we obtain1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le

119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 119864 + 119888119899minus1119899119864 = 1198880119899119864 (44)

and we have the needed estimates

Letting 119890119899119894 = 119881119899119894 minus V119899119894 we get the following error equation

1120583119899sum119896=1

119888119899minus119896119899H119909 (119890119896119894 minus 119890119896minus1119894 )= 11988612057521199091198901198991205722119894 minus 119889H1199091198901198991205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873119890 (119887119889 119905) = 0119890 (119887119906 119905) = 0

119905 isin (0 119879] 119890 (119909 0) = 0 119909 isin [119887119889 119887119906]

(45)

Since the above error equation (45) we now obtain thefollowing convergence results

Theorem 9 (convergence) Let 119881119899119894 denote the value of thesolution V(119909 119905) of (23) at the mesh point (119909119894 119905119899) and letV119899 = (V1198990 V1198991 V119899119872) be the solution of the compact differencescheme (23) en when 1 le 4(4120576 minus 1)11988631198891205761198712 it holds1003817100381710038171003817119880119899 minus 1199061198991003817100381710038171003817120576 le 1198621 (1205912 + ℎ4) 1 le 119899 le 119873 (46)

where

1198621 = (4Γ (1 minus 120572) 1198791205721198711198622119877119886 )12 (47)

Proof It follows fromTheorem 8 that

100381710038171003817100381711989011989910038171003817100381710038172120576 le 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817(119877120572119905119909)119899119894 100381710038171003817100381710038172 1 le 119899 le 119873 (48)

Applying (22) we get100381710038171003817100381711989011989910038171003817100381710038172120576 le 11986221 (1205912 + ℎ4)2 (49)

The estimate (46) is proved

Remark 10 The constraint condition 1 le 4(4120576 minus 1)11988631198891205761198712in Theorems 8 and 9 is only for the analysis of the stabilityand convergence of the compact difference scheme (23)Thiscondition is easily verifiable for practical problems

4 Numerical Experiment

For demonstrating the efficiency of the compact differencescheme (23) we make two numerical experiments of it

Suppose 119881119899119894 = V(119909119894 119905119899) be the value of the solution V(119909 119905)of the problem (1)ndash(3) at the mesh point (119909119894 119905119899) From (22)we can see that1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] le 1198622 (1205912 + ℎ4) ] = 1 2infin (50)

where1198622 is a positive constant independent In order to checkthis accuracy of the compact difference scheme we computethe following norm errors

E] (120591 ℎ) = max0le119899le119873

1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] (] = 1 2infin) (51)

The temporal convergence order and the spatial convergenceorder are denoted by

Ot] (120591 ℎ) = log2 (E] (2120591 ℎ)

E] (120591 ℎ) ) Os

] (120591 ℎ) = log2 (E] (120591 2ℎ)E] (120591 ℎ) )

(] = 1 2infin) (52)

Example 1 We first consider a problem which is governedby equation (1) in [0 1] times [0 1] with 119903 = 005 120590 = 025 119886 =12059022 119887 = 119903 minus 119886 119888 = 119903 and

119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572)1199092 (1 minus 119909) minus (119905 + 1)2

sdot [119886 (2 minus 6119909) + 119887 (2119909 minus 31199092) minus 1198881199092 (1 minus 119909)]) (53)

6 Advances in Mathematical Physics

Table 1 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 1100)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 110 37361endash05 37314endash05 62608endash05

120 93611endash06 19968 93492endash06 19968 15684endash05 19970140 23429endash06 19984 23399endash06 19984 39251endash06 19985180 58603endash07 19992 58528endash07 19992 98176endash07 199931160 14654endash07 19997 14635endash07 19997 24549endash07 199971320 36634endash08 20001 36587endash08 20001 61372endash08 20000

12 110 67788endash05 67702endash05 11393endash04120 16994endash05 19960 16972endash05 19960 28555endash05 19964140 42543endash06 19980 42489endash06 19980 71480endash06 19981180 10643endash06 19990 10630endash06 19990 17882endash06 199911160 26617endash07 19995 26583endash07 19995 44718endash07 199961320 66548endash08 19999 66463endash08 19999 11181endash07 19999

34 110 88226endash05 88110endash05 14950endash04120 22098endash05 19973 22069endash05 19973 37435endash05 19976140 55299endash06 19986 55226endash06 19986 93672endash06 19987180 13832endash06 19993 13813endash06 19993 23429endash06 199931160 34587endash07 19996 34542endash07 19996 58585endash07 199971320 86475endash08 19999 86361endash08 19999 14647endash07 19999

Table 2 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 110000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 11190endash04 64607endash05 91369endash05

14 54155endash06 43690 42667endash06 39205 54429endash06 4069318 29041endash07 42209 26840endash07 39907 33922endash07 40041116 17125endash08 40840 16758endash08 40015 21165endash08 40024132 10264endash09 40604 10207endash09 40372 12889endash09 40375

12 12 10340endash04 59701endash05 84430endash0514 50151endash06 43659 39472endash06 39189 49995endash06 4077918 26907endash07 42202 24841endash07 39900 31142endash07 40048116 15831endash08 40871 15487endash08 40036 19400endash08 40047132 92534endash10 40967 92011endash10 40731 11538endash09 40716

34 12 93459endash05 53959endash05 76309endash0514 45477endash06 43611 35734endash06 39165 44770endash06 4091218 24420endash07 42190 22506endash07 39889 27865endash07 40060116 14335endash08 40905 14014endash08 40053 17372endash08 40036132 82043endash10 41270 81562endash10 41028 10150endash09 40971

The boundary and initial conditions are given by (2) and (3)with

119880 (119909 0) = 1199092 (1 minus 119909) 119880 (0 119905) = 119880 (1 119905) = 0 (54)

It is easy to check that119880(119909 119905) = (119905+1)21199092(1minus119909) is the solutionof this problem

For different 120572 we let the spatial step ℎ = 1100 Table 1gives the errors E](120591 ℎ) (] = 1 2infin) and the temporalconvergence orders Ot

](120591 ℎ) (] = 1 2infin) of the computedsolution 119880119899119894 for 120572 = 14 12 34 and different time step 120591From the table we can see that the computed solution 119880119899119894

has the second-order temporal accuracy For comparison thecorresponding temporal convergence orders Ot

](120591 ℎ) (] =infin) given in [8] has only 2minus120572 order thus it is far less accuratethan the compact difference scheme (23) given in this paper

Next we compute the spatial convergence order of thecompact difference scheme (23) Table 2 presents the errorsE](120591 ℎ) (] = 1 2infin) and the spatial convergence orderso119903119889119890119903119874s

](120591 ℎ) (] = 1 2infin) The table demonstrates that thecompact difference scheme (23) has the fourth-order spatialaccuracy

Example 2 In this example we test the error and theconvergence order of the compact difference scheme (23)

Advances in Mathematical Physics 7

Table 3 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2 (120591 = 1100)120572 ℎ E1(120591 ℎ) Os

1(120591 ℎ) E2(120591 ℎ) Os2(120591 ℎ) Einfin(120591 ℎ) Os

infin(120591 ℎ)14 110 95681endash05 95622endash05 13266endash04

120 23985endash05 19961 23971endash05 19961 33254endash05 19961140 60044endash06 19981 60007endash06 19981 83244endash06 19981180 15021endash06 19990 15012endash06 19990 20825endash06 199911160 37564endash07 19996 37541endash07 19996 52078endash07 199961320 93916endash08 19999 93859endash08 19999 13020endash07 19999

12 110 17283endash04 17272endash04 24033endash04120 43358endash05 19950 43331endash05 19950 60286endash05 19951140 10858endash05 19975 10852endash05 19975 15097endash05 19976180 27169endash06 19987 27153endash06 19987 37774endash06 199881160 67952endash07 19994 67910endash07 19994 94474endash07 199941320 16991endash07 19998 16980endash07 19998 23622endash07 19998

34 110 22075endash04 22061endash04 30894endash04120 55326endash05 19964 55291endash05 19964 77418endash05 19966140 13849endash05 19981 13841endash05 19981 19378endash05 19983180 34646endash06 19991 34624endash06 19991 48476endash06 199911160 86641endash07 19995 86587endash07 19995 12123endash06 199961320 21663endash07 19998 21650endash07 19998 30311endash07 19998

Table 4 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2 (ℎ = 115000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 12909endash04 74532endash05 10540endash04

14 62469endash06 43691 49219endash06 39206 62789endash06 4069318 33499endash07 42210 30961endash07 39907 39130endash07 40042116 19745endash08 40846 19323endash08 40021 24403endash08 40031132 11750endash09 40707 11685endash09 40475 14745endash09 40488

12 12 11929endash04 68871endash05 97399endash0514 57849endash06 43660 45533endash06 39189 57675endash06 4077918 31036endash07 42203 28654endash07 39901 35923endash07 40050116 18246endash08 40883 17849endash08 40048 22357endash08 40061132 10507endash09 41182 10447endash09 40946 13056endash09 40980

34 12 10782endash04 62247endash05 88031endash0514 52457endash06 43613 41220endash06 39166 51648endash06 4091218 28166endash07 42191 25959endash07 39891 32142endash07 40062116 16515endash08 40921 16146endash08 40070 20016endash08 40053132 92416endash10 41595 91874endash10 41354 11316endash09 41447

Consider equation (1) in the domain [0 1] times [0 1] with 119903 =05 119886 = 1 119887 = 119903 minus 119886 119888 = 119903 and119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572) (1199093 + 1199092 + 1)

minus (119905 + 1)2sdot [119886 (2 + 6119909) + 119887 (2119909 + 31199092) minus 119888 (1199093 + 1199092 + 1)])

(55)

The boundary and initial conditions are given by (2) and (3)with

1206010 (119905) = 1199093 + 1199092 + 1

119880 (0 119905) = (119905 + 1)2 119880 (1 119905) = 3 (119905 + 1)2

(56)

It is clear that 119880(119909 119905) = (119905 + 1)2(1199093 + 1199092 + 1) is the exactanalytical solution of this problem

Apply the compact difference scheme (23) to solve theabove problem Table 3 presents the errors E](120591 ℎ) (] =1 2infin) and the temporal convergence orders Ot

](120591 ℎ) (] =1 2infin) we can see that the computed solution 119880119899119894 has thesecond-order temporal accuracy

From Table 4 we can obtain the errors E](120591 ℎ) (] = 1 2infin) and the spatial convergence ordersOs](120591 ℎ) (] = 1 2infin)

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 2: An Efficient Compact Difference Method for Temporal ...

2 Advances in Mathematical Physics

119876 are the rebates paid when the corresponding barrier ishit The terminal playoff of the option is 119881(119878) The fractionalderivative in (1) is a Caputo derivative defined as

120597120572119862 (119878 119905)120597119905120572

=

1Γ (1 minus 120572) int1199050

120597119862120597120578 (119878 120578) (119905 minus 120578)minus120572 d120578 0 lt 120572 lt 1120597119862 (119878 119905)120597119905 120572 = 1

(4)

As described in [8] we consider the transform problem of (1)

1198620D120572

119905119880 (119909 119905) = 1198861205972119880 (119909 119905)1205971199092 + 119887120597119880 (119909 119905)120597119909 minus 119888119880 (119909 119905)+ 119891 (119909 119905)

(119909 119905) isin (0infin) times (0 119879) 119880 (119887119889 119905) = 119901 (119905) 119880 (119887119906 119905) = 119902 (119905)

119905 isin (0 119879] 119880 (119909 0) = 120593 (119909) 119909 isin [119887119889 119887119906]

(5)

The rest of the paper is organized as follows in Section 2an efficient implicit numerical scheme with second-orderaccuracy in time and fourth-order accuracy in space isconstructed The analysis of the stability and convergence arepresented in Section 3 In Section 4 numerical examples aregiven to illustrate the accuracy of the presented scheme and tosupport our theoretical results Concluding remarks are givenin the last section

2 Construction of the Compact FiniteDifference Scheme

In order to simplify the computation and analysis of thefollowing compact finite difference scheme for Black-Scholesmodel we use an indirect approach by introducing a suitabletransformation

According to some simple calculations we transformequation (5) into

1198620D120572

119905 119881(119909 119905) = 1198861205972119881 (119909 119905)1205971199092 minus 119889119881 (119909 119905) + 119892 (119909 119905) (119909 119905) isin (0infin) times (0 119879)

119881 (119887119889 119905) = 119901lowast (119905) 119881 (119887119906 119905) = 119902lowast (119905)

119905 isin (0 119879] 119881 (119909 0) = 120593lowast (119909) 119909 isin [119887119889 119887119906]

(6)

where119901lowast (119905) = 119901 (119905) 119902lowast (119905) = 119896 (119887119906) 119902 (119905) 120593lowast (119909) = 119896 (119909) 120593 (119909)

(7)

It is clear that 119880(119909 119905) is a solution of (5) if and only if 119881(119909 119905)is a solution of (6)

In order to construct the compact finite difference schemefor the problem (5) we consider the above equivalent form(6)

Let 120591 = 119879119873 be the time step and ℎ = (119887119906minus119887119889)119872 = 119871119872be the spatial step where119872119873 are positive integers

Since the grid function V = V119894 | 0 le 119894 le 119872 we thendefine difference operators as follows

120575119909V119894minus12 = 1ℎ (V119894 minus V119894minus1) 1205752119909V119894 = 1ℎ2 (V119894+1 minus 2V119894 + V119894minus1)

H119909V119894 = V119894 + ℎ2121205752119909V119894(8)

We also define1198860 = 1205901minus1205721198870 = 0119886119896 = (119896 + 120590)1minus120572 minus (119896 minus 1205722)1minus120572 (119896 ge 1) 119887119896 = 12 minus 120572 ((119896 + 120590)2minus120572 minus (119896 minus 1205722)2minus120572)

minus 12 ((119896 + 120590)1minus120572 + (119896 minus 1205722 )1minus120572) (119896 ge 1)

(9)

where 120590 = 1 minus 1205722 and

119888119896119899 =

1198860 119896 = 0 119899 = 1119886119896 + 119887119896+1 minus 119887119896 0 le 119896 le 119899 minus 2 119899 ge 2119886119899minus1 minus 119887119899minus1 119896 = 119899 minus 1 119899 ge 2

(10)

Lemma 1 It holds (see [27])1 minus 1205722 (119896 + 120590)minus120572 lt 119886119896 minus 119887119896 lt (119896 + 120590)1minus120572 minus (119896 minus 1205722)1minus120572

(119896 ge 1) (11)

In order to discretize (6) into a compact finite differencesystem we introduce the following lemmas

Lemma 2 Assuming V(119905) isin C3[0 119879] we have1198620D120572

119905 V (119905119899minus1205722) = 1120583119899sum119896=1

119888119899minus119896119899 (V (119905119896) minus V (119905119896minus1))+ O (1205913minus120572)

(12)

where 120583 = 120591120572Γ(2 minus 120572)

Advances in Mathematical Physics 3

Proof From Lemma 2 of [9] we can obtain the proof oflemma

Lemma 3 Assuming V(119905) isin C2[0 119879] When 119899 ge 1 we obtainV (119905119899minus1205722) = 1205722 V (119905119899minus1) + (1 minus 1205722) V (119905119899) + O (1205912) (13)

Proof According to some simple calculations the prooffollows from Taylor expansions of the function V(119905) at thepoint 119905119899minus1205722 for 119905 = 119905119899minus1 and 119905 = 119905119899

Since the above lemmas we then discretize (6) into acompact finite difference scheme In order to analyse wedefine

120575119905V119899minus12 = 1120591 (V119899 minus V119899minus1) (1 le 119899 le 119873) V1198991205722 = 1205722 V119899minus1 + (1 minus 1205722) V119899 (1 le 119899 le 119873)

(14)

We also define the grid functions as follows

119881119899119894 = 119881 (119909119894 119905119899) 119882119899119894 = 120597119881 (119909119894 119905119899)120597119905 119885119899119894 = 1205972119881 (119909119894 119905119899)1205971199092 119892119899119894 = 119892 (119909119894 119905119899)

119892119899minus1205722119894 = 119892 (119909119894 119905119899minus1205722) 119901lowast119899 = 119901lowast (119905119899) 119902lowast119899 = 119902lowast (119905119899) 120593lowast119894 = 120593lowast (119909119894)

(15)

For the second-order spatial derivative 119885119899119894 we adopt thefollowing fourth-order compact approximation (see [28])

H119909119885119899119894 = 1205752119909V (119909119894) + 119874 (ℎ4) 1 le 119894 le 119872 minus 1 (16)

We consider equation (6) at the point (119909119894 119905119899minus1205722) we canobtain1198620D120572

119905119881 (119909119894 119905119899minus1205722) = 119886119885119899minus1205722119894 minus 119889119881119899minus1205722119894 + 119892119899minus1205722119894 (17)

From Lemmas 2 and 3 we have

1120583119899sum119896=1

119888119899minus119896119899 (119881119896119894 minus 119881119896minus1119894 )= 1198861198851198991205722119894 minus 1198891198811198991205722119894 + 119892119899minus1205722119894 + (119877120572119905 )119899119894

0 le 119894 le 119872 1 le 119899 le 119873(18)

where

(119877120572119905 )119899119894 = (119886 minus 119889)O (1205912) + O (1205913minus120572) 0 le 119894 le 119872 1 le 119899 le 119873 (19)

We applyH119909 to equation (18) then we have

1120583119899sum119896=1

119888119899minus119896119899H119909 (119881119896119894 minus 119881119896minus1119894 )= 11988612057521199091198811198991205722119894 minus 119889H1199091198811198991205722119894 +H119909119892119899minus1205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873(20)

where(119877120572119905119909)119899119894 = H119909 (119877120572119905 )119899119894 + 119886 (119877120572119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873 (21)

and10038161003816100381610038161003816(119877120572119905119909)119899119894 10038161003816100381610038161003816 le 119862119877 (1205912 + ℎ4) 1 le 119894 le 119872 minus 1 1 le 119899 le 119873 (22)

If we omit (119877120572119905119909)119899119894 then we have the compact finite differencescheme

1120583119899sum119896=1

119888119899minus119896119899H119909 (V119896119894 minus V119896minus1119894 )= 1198861205752119909V1198991205722119894 minus 119889H119909V1198991205722119894 +H119909119892119899minus1205722119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873V (119887119889 119905) = 119901lowast (119905) V (119887119906 119905) = 119902lowast (119905)

119905 isin (0 119879] V (119909 0) = 120593lowast (119909) 119909 isin [119887119889 119887119906]

(23)

3 Stability and Convergence of the ProposedCompact Difference Scheme

Theorem 4 e compact difference scheme (23) is uniquelysolvable

Proof The compact difference scheme (23) can be written inmatrix form

AV119899 = b119899minus1 (24)

where

b119899minus1 = 119899minus1sum119896=0

120577119896V119896 120577119896 isin R (25)

The tridiagonal coefficient matrixA = (119886119894119895) yields10038161003816100381610038161198861198941198941003816100381610038161003816 = 56 ((119886(120572)0 + 119887(120572)1 )

120583 ) + 119886 (2 minus 120572)ℎ2 + 5 (2 minus 120572)12 119889sum119895 =119894

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816100381610038161003816100381616 ((119886(120572)0 + 119887(120572)1 )

120583 ) minus 119886 (2 minus 120572)ℎ2 + (2 minus 120572)12 11988910038161003816100381610038161003816100381610038161003816100381610038161003816

(26)

4 Advances in Mathematical Physics

It is easy to see that the tridiagonal coefficient matrix A isstrictly diagonally dominantTherefore the coefficientmatrixis nonsingular and hence invertible

Next we consider the stability and convergence analysisof the compact difference scheme (23)

Letting Ω = 119906 | 119906 = (1199060 1199061 119906119872) 1199060 = 119906119872 = 0for grid functions 119906 V isin Ω we define the inner product andnorm as follows

(119906 V) = ℎ119872minus1sum119894=1

119906119894V119894119906 = (119906 119906)12

119906infin = max0le119894le119872

10038161003816100381610038161199061198941003816100381610038161003816 (120575119909119906 120575119909V) = ℎ119872minus1sum

119894=0

120575119909119906119894+12120575119909V119894+12|119906|1 = (120575119909119906 120575119909119906]12 1199061 = (1199062 + |119906|21)12

(27)

According to simple calculations we obtain

(1205752119909119906 V) = minus (120575119909119906 120575119909V) ℎ 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 le 2 |119906|1

ℎ |119906|1 le 2 119906 (28)

In order to analyse we introduce the discrete innerproduct and norm

⟨119906 V⟩ = (H119909119906 minus1205752119909V) = (120575119909119906 120575119909V] minus ℎ212 (1205752119909119906 1205752119909V) 119906120576 = ⟨119906 119906⟩12

(29)

Based on above inner product and norm we have thefollowing lemmas

Lemma 5 (see [29]) Suppose 119906 isin Ω we obtain

1003817100381710038171003817H11990911990610038171003817100381710038172 le 1199062 le 3119871216 1199062120576 1199062infin le 31198718 1199062120576 11990621 le 3 (8 + 1198712)

16 1199062120576 (30)

Lemma 6 (see [27]) Suppose 119906 isin Ω we obtain

119906 le 11987128 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 1199062120576 le 3119871216 100381710038171003817100381710038171205752119909119906100381710038171003817100381710038172

(31)

Lemma 7 (see [9]) Suppose 119906 isin Ω we obtain

( 119899sum119896=1

119888119899minus119896119899H119909 (119906119896 minus 119906119896minus1) minus12057521199091199061198991205722)

ge 12119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) 1 le 119899 le 119873(32)

In the next we then analyse the stability and convergenceof the scheme (23)

Theorem8 (stability) Let V119899 = (V1198990 V1198991 V119899119872) be the solutionof the compact difference scheme (23) with V1198990 = V119899119872 = 0Assume that one of the conditions 1 le 4(4120576 minus 1)11988631198891205761198712 holdsfor some positive constant 120576 gt 14

en it holds

1003817100381710038171003817V11989910038171003817100381710038172120576 le 10038171003817100381710038171003817V0100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 1 le 119899 le 119873

(33)

Proof We take the inner product of equation (23) withminus1205752119909V1198991205722 yield1120583 ( 119899sum119896=1

119888119899minus119896119899H119909 (V119896 minus V119896minus1) minus1205752119909V1198991205722)= minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 minus 119889 (H119909 (V1198991205722) 1205752119909V1198991205722)

minus (H119909119892119899minus1205722 1205752119909V1198991205722) 1 le 119899 le 119873(34)

Using Lemma 7

12120583119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817V119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817V119896minus1100381710038171003817100381710038172120576)le minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576

minus (H119909119892119899minus1205722 12057521199091199061198991205722) 1 le 119899 le 119873(35)

When 1 le 4(4120576 minus 1)11988631198891205761198712 for some positive constant120576 gt 14 we have from the Cauchy-Schwarz inequality andLemmas 6 that

119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 le 3119889119871216 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

le (119886 minus 1198864120576) 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172(36)

minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 + 1198864120576 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

(37)

By (35) and the Cauchy-Schwarz inequality

minus 119886 1003817100381710038171003817100381712057521199091199061198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(38)

Advances in Mathematical Physics 5

Substituting (38) into (35) leads to

119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) le 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (39)

The above inequality can be rewritten as

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 + 119888119899minus1119899 100381710038171003817100381710038171199060100381710038171003817100381710038172lowast+ 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(40)

Since by the definition of 119888119899minus1119899120583119888119899minus1119899 =

120583119886119899minus1 minus 119887119899minus1 lt 2Γ (1 minus 120572)119879120572 (41)

we have from (40) that

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576le 119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576+ 119888119899minus1119899 (100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172)

(42)

Letting

119864 = 100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (43)

and assuming 1199061198962120576 le 119864(0 le 119896 le 119899 minus 1) we obtain1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le

119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 119864 + 119888119899minus1119899119864 = 1198880119899119864 (44)

and we have the needed estimates

Letting 119890119899119894 = 119881119899119894 minus V119899119894 we get the following error equation

1120583119899sum119896=1

119888119899minus119896119899H119909 (119890119896119894 minus 119890119896minus1119894 )= 11988612057521199091198901198991205722119894 minus 119889H1199091198901198991205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873119890 (119887119889 119905) = 0119890 (119887119906 119905) = 0

119905 isin (0 119879] 119890 (119909 0) = 0 119909 isin [119887119889 119887119906]

(45)

Since the above error equation (45) we now obtain thefollowing convergence results

Theorem 9 (convergence) Let 119881119899119894 denote the value of thesolution V(119909 119905) of (23) at the mesh point (119909119894 119905119899) and letV119899 = (V1198990 V1198991 V119899119872) be the solution of the compact differencescheme (23) en when 1 le 4(4120576 minus 1)11988631198891205761198712 it holds1003817100381710038171003817119880119899 minus 1199061198991003817100381710038171003817120576 le 1198621 (1205912 + ℎ4) 1 le 119899 le 119873 (46)

where

1198621 = (4Γ (1 minus 120572) 1198791205721198711198622119877119886 )12 (47)

Proof It follows fromTheorem 8 that

100381710038171003817100381711989011989910038171003817100381710038172120576 le 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817(119877120572119905119909)119899119894 100381710038171003817100381710038172 1 le 119899 le 119873 (48)

Applying (22) we get100381710038171003817100381711989011989910038171003817100381710038172120576 le 11986221 (1205912 + ℎ4)2 (49)

The estimate (46) is proved

Remark 10 The constraint condition 1 le 4(4120576 minus 1)11988631198891205761198712in Theorems 8 and 9 is only for the analysis of the stabilityand convergence of the compact difference scheme (23)Thiscondition is easily verifiable for practical problems

4 Numerical Experiment

For demonstrating the efficiency of the compact differencescheme (23) we make two numerical experiments of it

Suppose 119881119899119894 = V(119909119894 119905119899) be the value of the solution V(119909 119905)of the problem (1)ndash(3) at the mesh point (119909119894 119905119899) From (22)we can see that1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] le 1198622 (1205912 + ℎ4) ] = 1 2infin (50)

where1198622 is a positive constant independent In order to checkthis accuracy of the compact difference scheme we computethe following norm errors

E] (120591 ℎ) = max0le119899le119873

1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] (] = 1 2infin) (51)

The temporal convergence order and the spatial convergenceorder are denoted by

Ot] (120591 ℎ) = log2 (E] (2120591 ℎ)

E] (120591 ℎ) ) Os

] (120591 ℎ) = log2 (E] (120591 2ℎ)E] (120591 ℎ) )

(] = 1 2infin) (52)

Example 1 We first consider a problem which is governedby equation (1) in [0 1] times [0 1] with 119903 = 005 120590 = 025 119886 =12059022 119887 = 119903 minus 119886 119888 = 119903 and

119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572)1199092 (1 minus 119909) minus (119905 + 1)2

sdot [119886 (2 minus 6119909) + 119887 (2119909 minus 31199092) minus 1198881199092 (1 minus 119909)]) (53)

6 Advances in Mathematical Physics

Table 1 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 1100)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 110 37361endash05 37314endash05 62608endash05

120 93611endash06 19968 93492endash06 19968 15684endash05 19970140 23429endash06 19984 23399endash06 19984 39251endash06 19985180 58603endash07 19992 58528endash07 19992 98176endash07 199931160 14654endash07 19997 14635endash07 19997 24549endash07 199971320 36634endash08 20001 36587endash08 20001 61372endash08 20000

12 110 67788endash05 67702endash05 11393endash04120 16994endash05 19960 16972endash05 19960 28555endash05 19964140 42543endash06 19980 42489endash06 19980 71480endash06 19981180 10643endash06 19990 10630endash06 19990 17882endash06 199911160 26617endash07 19995 26583endash07 19995 44718endash07 199961320 66548endash08 19999 66463endash08 19999 11181endash07 19999

34 110 88226endash05 88110endash05 14950endash04120 22098endash05 19973 22069endash05 19973 37435endash05 19976140 55299endash06 19986 55226endash06 19986 93672endash06 19987180 13832endash06 19993 13813endash06 19993 23429endash06 199931160 34587endash07 19996 34542endash07 19996 58585endash07 199971320 86475endash08 19999 86361endash08 19999 14647endash07 19999

Table 2 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 110000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 11190endash04 64607endash05 91369endash05

14 54155endash06 43690 42667endash06 39205 54429endash06 4069318 29041endash07 42209 26840endash07 39907 33922endash07 40041116 17125endash08 40840 16758endash08 40015 21165endash08 40024132 10264endash09 40604 10207endash09 40372 12889endash09 40375

12 12 10340endash04 59701endash05 84430endash0514 50151endash06 43659 39472endash06 39189 49995endash06 4077918 26907endash07 42202 24841endash07 39900 31142endash07 40048116 15831endash08 40871 15487endash08 40036 19400endash08 40047132 92534endash10 40967 92011endash10 40731 11538endash09 40716

34 12 93459endash05 53959endash05 76309endash0514 45477endash06 43611 35734endash06 39165 44770endash06 4091218 24420endash07 42190 22506endash07 39889 27865endash07 40060116 14335endash08 40905 14014endash08 40053 17372endash08 40036132 82043endash10 41270 81562endash10 41028 10150endash09 40971

The boundary and initial conditions are given by (2) and (3)with

119880 (119909 0) = 1199092 (1 minus 119909) 119880 (0 119905) = 119880 (1 119905) = 0 (54)

It is easy to check that119880(119909 119905) = (119905+1)21199092(1minus119909) is the solutionof this problem

For different 120572 we let the spatial step ℎ = 1100 Table 1gives the errors E](120591 ℎ) (] = 1 2infin) and the temporalconvergence orders Ot

](120591 ℎ) (] = 1 2infin) of the computedsolution 119880119899119894 for 120572 = 14 12 34 and different time step 120591From the table we can see that the computed solution 119880119899119894

has the second-order temporal accuracy For comparison thecorresponding temporal convergence orders Ot

](120591 ℎ) (] =infin) given in [8] has only 2minus120572 order thus it is far less accuratethan the compact difference scheme (23) given in this paper

Next we compute the spatial convergence order of thecompact difference scheme (23) Table 2 presents the errorsE](120591 ℎ) (] = 1 2infin) and the spatial convergence orderso119903119889119890119903119874s

](120591 ℎ) (] = 1 2infin) The table demonstrates that thecompact difference scheme (23) has the fourth-order spatialaccuracy

Example 2 In this example we test the error and theconvergence order of the compact difference scheme (23)

Advances in Mathematical Physics 7

Table 3 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2 (120591 = 1100)120572 ℎ E1(120591 ℎ) Os

1(120591 ℎ) E2(120591 ℎ) Os2(120591 ℎ) Einfin(120591 ℎ) Os

infin(120591 ℎ)14 110 95681endash05 95622endash05 13266endash04

120 23985endash05 19961 23971endash05 19961 33254endash05 19961140 60044endash06 19981 60007endash06 19981 83244endash06 19981180 15021endash06 19990 15012endash06 19990 20825endash06 199911160 37564endash07 19996 37541endash07 19996 52078endash07 199961320 93916endash08 19999 93859endash08 19999 13020endash07 19999

12 110 17283endash04 17272endash04 24033endash04120 43358endash05 19950 43331endash05 19950 60286endash05 19951140 10858endash05 19975 10852endash05 19975 15097endash05 19976180 27169endash06 19987 27153endash06 19987 37774endash06 199881160 67952endash07 19994 67910endash07 19994 94474endash07 199941320 16991endash07 19998 16980endash07 19998 23622endash07 19998

34 110 22075endash04 22061endash04 30894endash04120 55326endash05 19964 55291endash05 19964 77418endash05 19966140 13849endash05 19981 13841endash05 19981 19378endash05 19983180 34646endash06 19991 34624endash06 19991 48476endash06 199911160 86641endash07 19995 86587endash07 19995 12123endash06 199961320 21663endash07 19998 21650endash07 19998 30311endash07 19998

Table 4 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2 (ℎ = 115000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 12909endash04 74532endash05 10540endash04

14 62469endash06 43691 49219endash06 39206 62789endash06 4069318 33499endash07 42210 30961endash07 39907 39130endash07 40042116 19745endash08 40846 19323endash08 40021 24403endash08 40031132 11750endash09 40707 11685endash09 40475 14745endash09 40488

12 12 11929endash04 68871endash05 97399endash0514 57849endash06 43660 45533endash06 39189 57675endash06 4077918 31036endash07 42203 28654endash07 39901 35923endash07 40050116 18246endash08 40883 17849endash08 40048 22357endash08 40061132 10507endash09 41182 10447endash09 40946 13056endash09 40980

34 12 10782endash04 62247endash05 88031endash0514 52457endash06 43613 41220endash06 39166 51648endash06 4091218 28166endash07 42191 25959endash07 39891 32142endash07 40062116 16515endash08 40921 16146endash08 40070 20016endash08 40053132 92416endash10 41595 91874endash10 41354 11316endash09 41447

Consider equation (1) in the domain [0 1] times [0 1] with 119903 =05 119886 = 1 119887 = 119903 minus 119886 119888 = 119903 and119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572) (1199093 + 1199092 + 1)

minus (119905 + 1)2sdot [119886 (2 + 6119909) + 119887 (2119909 + 31199092) minus 119888 (1199093 + 1199092 + 1)])

(55)

The boundary and initial conditions are given by (2) and (3)with

1206010 (119905) = 1199093 + 1199092 + 1

119880 (0 119905) = (119905 + 1)2 119880 (1 119905) = 3 (119905 + 1)2

(56)

It is clear that 119880(119909 119905) = (119905 + 1)2(1199093 + 1199092 + 1) is the exactanalytical solution of this problem

Apply the compact difference scheme (23) to solve theabove problem Table 3 presents the errors E](120591 ℎ) (] =1 2infin) and the temporal convergence orders Ot

](120591 ℎ) (] =1 2infin) we can see that the computed solution 119880119899119894 has thesecond-order temporal accuracy

From Table 4 we can obtain the errors E](120591 ℎ) (] = 1 2infin) and the spatial convergence ordersOs](120591 ℎ) (] = 1 2infin)

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 3: An Efficient Compact Difference Method for Temporal ...

Advances in Mathematical Physics 3

Proof From Lemma 2 of [9] we can obtain the proof oflemma

Lemma 3 Assuming V(119905) isin C2[0 119879] When 119899 ge 1 we obtainV (119905119899minus1205722) = 1205722 V (119905119899minus1) + (1 minus 1205722) V (119905119899) + O (1205912) (13)

Proof According to some simple calculations the prooffollows from Taylor expansions of the function V(119905) at thepoint 119905119899minus1205722 for 119905 = 119905119899minus1 and 119905 = 119905119899

Since the above lemmas we then discretize (6) into acompact finite difference scheme In order to analyse wedefine

120575119905V119899minus12 = 1120591 (V119899 minus V119899minus1) (1 le 119899 le 119873) V1198991205722 = 1205722 V119899minus1 + (1 minus 1205722) V119899 (1 le 119899 le 119873)

(14)

We also define the grid functions as follows

119881119899119894 = 119881 (119909119894 119905119899) 119882119899119894 = 120597119881 (119909119894 119905119899)120597119905 119885119899119894 = 1205972119881 (119909119894 119905119899)1205971199092 119892119899119894 = 119892 (119909119894 119905119899)

119892119899minus1205722119894 = 119892 (119909119894 119905119899minus1205722) 119901lowast119899 = 119901lowast (119905119899) 119902lowast119899 = 119902lowast (119905119899) 120593lowast119894 = 120593lowast (119909119894)

(15)

For the second-order spatial derivative 119885119899119894 we adopt thefollowing fourth-order compact approximation (see [28])

H119909119885119899119894 = 1205752119909V (119909119894) + 119874 (ℎ4) 1 le 119894 le 119872 minus 1 (16)

We consider equation (6) at the point (119909119894 119905119899minus1205722) we canobtain1198620D120572

119905119881 (119909119894 119905119899minus1205722) = 119886119885119899minus1205722119894 minus 119889119881119899minus1205722119894 + 119892119899minus1205722119894 (17)

From Lemmas 2 and 3 we have

1120583119899sum119896=1

119888119899minus119896119899 (119881119896119894 minus 119881119896minus1119894 )= 1198861198851198991205722119894 minus 1198891198811198991205722119894 + 119892119899minus1205722119894 + (119877120572119905 )119899119894

0 le 119894 le 119872 1 le 119899 le 119873(18)

where

(119877120572119905 )119899119894 = (119886 minus 119889)O (1205912) + O (1205913minus120572) 0 le 119894 le 119872 1 le 119899 le 119873 (19)

We applyH119909 to equation (18) then we have

1120583119899sum119896=1

119888119899minus119896119899H119909 (119881119896119894 minus 119881119896minus1119894 )= 11988612057521199091198811198991205722119894 minus 119889H1199091198811198991205722119894 +H119909119892119899minus1205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873(20)

where(119877120572119905119909)119899119894 = H119909 (119877120572119905 )119899119894 + 119886 (119877120572119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873 (21)

and10038161003816100381610038161003816(119877120572119905119909)119899119894 10038161003816100381610038161003816 le 119862119877 (1205912 + ℎ4) 1 le 119894 le 119872 minus 1 1 le 119899 le 119873 (22)

If we omit (119877120572119905119909)119899119894 then we have the compact finite differencescheme

1120583119899sum119896=1

119888119899minus119896119899H119909 (V119896119894 minus V119896minus1119894 )= 1198861205752119909V1198991205722119894 minus 119889H119909V1198991205722119894 +H119909119892119899minus1205722119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873V (119887119889 119905) = 119901lowast (119905) V (119887119906 119905) = 119902lowast (119905)

119905 isin (0 119879] V (119909 0) = 120593lowast (119909) 119909 isin [119887119889 119887119906]

(23)

3 Stability and Convergence of the ProposedCompact Difference Scheme

Theorem 4 e compact difference scheme (23) is uniquelysolvable

Proof The compact difference scheme (23) can be written inmatrix form

AV119899 = b119899minus1 (24)

where

b119899minus1 = 119899minus1sum119896=0

120577119896V119896 120577119896 isin R (25)

The tridiagonal coefficient matrixA = (119886119894119895) yields10038161003816100381610038161198861198941198941003816100381610038161003816 = 56 ((119886(120572)0 + 119887(120572)1 )

120583 ) + 119886 (2 minus 120572)ℎ2 + 5 (2 minus 120572)12 119889sum119895 =119894

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816100381610038161003816100381616 ((119886(120572)0 + 119887(120572)1 )

120583 ) minus 119886 (2 minus 120572)ℎ2 + (2 minus 120572)12 11988910038161003816100381610038161003816100381610038161003816100381610038161003816

(26)

4 Advances in Mathematical Physics

It is easy to see that the tridiagonal coefficient matrix A isstrictly diagonally dominantTherefore the coefficientmatrixis nonsingular and hence invertible

Next we consider the stability and convergence analysisof the compact difference scheme (23)

Letting Ω = 119906 | 119906 = (1199060 1199061 119906119872) 1199060 = 119906119872 = 0for grid functions 119906 V isin Ω we define the inner product andnorm as follows

(119906 V) = ℎ119872minus1sum119894=1

119906119894V119894119906 = (119906 119906)12

119906infin = max0le119894le119872

10038161003816100381610038161199061198941003816100381610038161003816 (120575119909119906 120575119909V) = ℎ119872minus1sum

119894=0

120575119909119906119894+12120575119909V119894+12|119906|1 = (120575119909119906 120575119909119906]12 1199061 = (1199062 + |119906|21)12

(27)

According to simple calculations we obtain

(1205752119909119906 V) = minus (120575119909119906 120575119909V) ℎ 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 le 2 |119906|1

ℎ |119906|1 le 2 119906 (28)

In order to analyse we introduce the discrete innerproduct and norm

⟨119906 V⟩ = (H119909119906 minus1205752119909V) = (120575119909119906 120575119909V] minus ℎ212 (1205752119909119906 1205752119909V) 119906120576 = ⟨119906 119906⟩12

(29)

Based on above inner product and norm we have thefollowing lemmas

Lemma 5 (see [29]) Suppose 119906 isin Ω we obtain

1003817100381710038171003817H11990911990610038171003817100381710038172 le 1199062 le 3119871216 1199062120576 1199062infin le 31198718 1199062120576 11990621 le 3 (8 + 1198712)

16 1199062120576 (30)

Lemma 6 (see [27]) Suppose 119906 isin Ω we obtain

119906 le 11987128 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 1199062120576 le 3119871216 100381710038171003817100381710038171205752119909119906100381710038171003817100381710038172

(31)

Lemma 7 (see [9]) Suppose 119906 isin Ω we obtain

( 119899sum119896=1

119888119899minus119896119899H119909 (119906119896 minus 119906119896minus1) minus12057521199091199061198991205722)

ge 12119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) 1 le 119899 le 119873(32)

In the next we then analyse the stability and convergenceof the scheme (23)

Theorem8 (stability) Let V119899 = (V1198990 V1198991 V119899119872) be the solutionof the compact difference scheme (23) with V1198990 = V119899119872 = 0Assume that one of the conditions 1 le 4(4120576 minus 1)11988631198891205761198712 holdsfor some positive constant 120576 gt 14

en it holds

1003817100381710038171003817V11989910038171003817100381710038172120576 le 10038171003817100381710038171003817V0100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 1 le 119899 le 119873

(33)

Proof We take the inner product of equation (23) withminus1205752119909V1198991205722 yield1120583 ( 119899sum119896=1

119888119899minus119896119899H119909 (V119896 minus V119896minus1) minus1205752119909V1198991205722)= minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 minus 119889 (H119909 (V1198991205722) 1205752119909V1198991205722)

minus (H119909119892119899minus1205722 1205752119909V1198991205722) 1 le 119899 le 119873(34)

Using Lemma 7

12120583119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817V119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817V119896minus1100381710038171003817100381710038172120576)le minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576

minus (H119909119892119899minus1205722 12057521199091199061198991205722) 1 le 119899 le 119873(35)

When 1 le 4(4120576 minus 1)11988631198891205761198712 for some positive constant120576 gt 14 we have from the Cauchy-Schwarz inequality andLemmas 6 that

119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 le 3119889119871216 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

le (119886 minus 1198864120576) 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172(36)

minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 + 1198864120576 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

(37)

By (35) and the Cauchy-Schwarz inequality

minus 119886 1003817100381710038171003817100381712057521199091199061198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(38)

Advances in Mathematical Physics 5

Substituting (38) into (35) leads to

119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) le 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (39)

The above inequality can be rewritten as

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 + 119888119899minus1119899 100381710038171003817100381710038171199060100381710038171003817100381710038172lowast+ 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(40)

Since by the definition of 119888119899minus1119899120583119888119899minus1119899 =

120583119886119899minus1 minus 119887119899minus1 lt 2Γ (1 minus 120572)119879120572 (41)

we have from (40) that

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576le 119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576+ 119888119899minus1119899 (100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172)

(42)

Letting

119864 = 100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (43)

and assuming 1199061198962120576 le 119864(0 le 119896 le 119899 minus 1) we obtain1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le

119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 119864 + 119888119899minus1119899119864 = 1198880119899119864 (44)

and we have the needed estimates

Letting 119890119899119894 = 119881119899119894 minus V119899119894 we get the following error equation

1120583119899sum119896=1

119888119899minus119896119899H119909 (119890119896119894 minus 119890119896minus1119894 )= 11988612057521199091198901198991205722119894 minus 119889H1199091198901198991205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873119890 (119887119889 119905) = 0119890 (119887119906 119905) = 0

119905 isin (0 119879] 119890 (119909 0) = 0 119909 isin [119887119889 119887119906]

(45)

Since the above error equation (45) we now obtain thefollowing convergence results

Theorem 9 (convergence) Let 119881119899119894 denote the value of thesolution V(119909 119905) of (23) at the mesh point (119909119894 119905119899) and letV119899 = (V1198990 V1198991 V119899119872) be the solution of the compact differencescheme (23) en when 1 le 4(4120576 minus 1)11988631198891205761198712 it holds1003817100381710038171003817119880119899 minus 1199061198991003817100381710038171003817120576 le 1198621 (1205912 + ℎ4) 1 le 119899 le 119873 (46)

where

1198621 = (4Γ (1 minus 120572) 1198791205721198711198622119877119886 )12 (47)

Proof It follows fromTheorem 8 that

100381710038171003817100381711989011989910038171003817100381710038172120576 le 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817(119877120572119905119909)119899119894 100381710038171003817100381710038172 1 le 119899 le 119873 (48)

Applying (22) we get100381710038171003817100381711989011989910038171003817100381710038172120576 le 11986221 (1205912 + ℎ4)2 (49)

The estimate (46) is proved

Remark 10 The constraint condition 1 le 4(4120576 minus 1)11988631198891205761198712in Theorems 8 and 9 is only for the analysis of the stabilityand convergence of the compact difference scheme (23)Thiscondition is easily verifiable for practical problems

4 Numerical Experiment

For demonstrating the efficiency of the compact differencescheme (23) we make two numerical experiments of it

Suppose 119881119899119894 = V(119909119894 119905119899) be the value of the solution V(119909 119905)of the problem (1)ndash(3) at the mesh point (119909119894 119905119899) From (22)we can see that1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] le 1198622 (1205912 + ℎ4) ] = 1 2infin (50)

where1198622 is a positive constant independent In order to checkthis accuracy of the compact difference scheme we computethe following norm errors

E] (120591 ℎ) = max0le119899le119873

1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] (] = 1 2infin) (51)

The temporal convergence order and the spatial convergenceorder are denoted by

Ot] (120591 ℎ) = log2 (E] (2120591 ℎ)

E] (120591 ℎ) ) Os

] (120591 ℎ) = log2 (E] (120591 2ℎ)E] (120591 ℎ) )

(] = 1 2infin) (52)

Example 1 We first consider a problem which is governedby equation (1) in [0 1] times [0 1] with 119903 = 005 120590 = 025 119886 =12059022 119887 = 119903 minus 119886 119888 = 119903 and

119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572)1199092 (1 minus 119909) minus (119905 + 1)2

sdot [119886 (2 minus 6119909) + 119887 (2119909 minus 31199092) minus 1198881199092 (1 minus 119909)]) (53)

6 Advances in Mathematical Physics

Table 1 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 1100)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 110 37361endash05 37314endash05 62608endash05

120 93611endash06 19968 93492endash06 19968 15684endash05 19970140 23429endash06 19984 23399endash06 19984 39251endash06 19985180 58603endash07 19992 58528endash07 19992 98176endash07 199931160 14654endash07 19997 14635endash07 19997 24549endash07 199971320 36634endash08 20001 36587endash08 20001 61372endash08 20000

12 110 67788endash05 67702endash05 11393endash04120 16994endash05 19960 16972endash05 19960 28555endash05 19964140 42543endash06 19980 42489endash06 19980 71480endash06 19981180 10643endash06 19990 10630endash06 19990 17882endash06 199911160 26617endash07 19995 26583endash07 19995 44718endash07 199961320 66548endash08 19999 66463endash08 19999 11181endash07 19999

34 110 88226endash05 88110endash05 14950endash04120 22098endash05 19973 22069endash05 19973 37435endash05 19976140 55299endash06 19986 55226endash06 19986 93672endash06 19987180 13832endash06 19993 13813endash06 19993 23429endash06 199931160 34587endash07 19996 34542endash07 19996 58585endash07 199971320 86475endash08 19999 86361endash08 19999 14647endash07 19999

Table 2 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 110000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 11190endash04 64607endash05 91369endash05

14 54155endash06 43690 42667endash06 39205 54429endash06 4069318 29041endash07 42209 26840endash07 39907 33922endash07 40041116 17125endash08 40840 16758endash08 40015 21165endash08 40024132 10264endash09 40604 10207endash09 40372 12889endash09 40375

12 12 10340endash04 59701endash05 84430endash0514 50151endash06 43659 39472endash06 39189 49995endash06 4077918 26907endash07 42202 24841endash07 39900 31142endash07 40048116 15831endash08 40871 15487endash08 40036 19400endash08 40047132 92534endash10 40967 92011endash10 40731 11538endash09 40716

34 12 93459endash05 53959endash05 76309endash0514 45477endash06 43611 35734endash06 39165 44770endash06 4091218 24420endash07 42190 22506endash07 39889 27865endash07 40060116 14335endash08 40905 14014endash08 40053 17372endash08 40036132 82043endash10 41270 81562endash10 41028 10150endash09 40971

The boundary and initial conditions are given by (2) and (3)with

119880 (119909 0) = 1199092 (1 minus 119909) 119880 (0 119905) = 119880 (1 119905) = 0 (54)

It is easy to check that119880(119909 119905) = (119905+1)21199092(1minus119909) is the solutionof this problem

For different 120572 we let the spatial step ℎ = 1100 Table 1gives the errors E](120591 ℎ) (] = 1 2infin) and the temporalconvergence orders Ot

](120591 ℎ) (] = 1 2infin) of the computedsolution 119880119899119894 for 120572 = 14 12 34 and different time step 120591From the table we can see that the computed solution 119880119899119894

has the second-order temporal accuracy For comparison thecorresponding temporal convergence orders Ot

](120591 ℎ) (] =infin) given in [8] has only 2minus120572 order thus it is far less accuratethan the compact difference scheme (23) given in this paper

Next we compute the spatial convergence order of thecompact difference scheme (23) Table 2 presents the errorsE](120591 ℎ) (] = 1 2infin) and the spatial convergence orderso119903119889119890119903119874s

](120591 ℎ) (] = 1 2infin) The table demonstrates that thecompact difference scheme (23) has the fourth-order spatialaccuracy

Example 2 In this example we test the error and theconvergence order of the compact difference scheme (23)

Advances in Mathematical Physics 7

Table 3 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2 (120591 = 1100)120572 ℎ E1(120591 ℎ) Os

1(120591 ℎ) E2(120591 ℎ) Os2(120591 ℎ) Einfin(120591 ℎ) Os

infin(120591 ℎ)14 110 95681endash05 95622endash05 13266endash04

120 23985endash05 19961 23971endash05 19961 33254endash05 19961140 60044endash06 19981 60007endash06 19981 83244endash06 19981180 15021endash06 19990 15012endash06 19990 20825endash06 199911160 37564endash07 19996 37541endash07 19996 52078endash07 199961320 93916endash08 19999 93859endash08 19999 13020endash07 19999

12 110 17283endash04 17272endash04 24033endash04120 43358endash05 19950 43331endash05 19950 60286endash05 19951140 10858endash05 19975 10852endash05 19975 15097endash05 19976180 27169endash06 19987 27153endash06 19987 37774endash06 199881160 67952endash07 19994 67910endash07 19994 94474endash07 199941320 16991endash07 19998 16980endash07 19998 23622endash07 19998

34 110 22075endash04 22061endash04 30894endash04120 55326endash05 19964 55291endash05 19964 77418endash05 19966140 13849endash05 19981 13841endash05 19981 19378endash05 19983180 34646endash06 19991 34624endash06 19991 48476endash06 199911160 86641endash07 19995 86587endash07 19995 12123endash06 199961320 21663endash07 19998 21650endash07 19998 30311endash07 19998

Table 4 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2 (ℎ = 115000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 12909endash04 74532endash05 10540endash04

14 62469endash06 43691 49219endash06 39206 62789endash06 4069318 33499endash07 42210 30961endash07 39907 39130endash07 40042116 19745endash08 40846 19323endash08 40021 24403endash08 40031132 11750endash09 40707 11685endash09 40475 14745endash09 40488

12 12 11929endash04 68871endash05 97399endash0514 57849endash06 43660 45533endash06 39189 57675endash06 4077918 31036endash07 42203 28654endash07 39901 35923endash07 40050116 18246endash08 40883 17849endash08 40048 22357endash08 40061132 10507endash09 41182 10447endash09 40946 13056endash09 40980

34 12 10782endash04 62247endash05 88031endash0514 52457endash06 43613 41220endash06 39166 51648endash06 4091218 28166endash07 42191 25959endash07 39891 32142endash07 40062116 16515endash08 40921 16146endash08 40070 20016endash08 40053132 92416endash10 41595 91874endash10 41354 11316endash09 41447

Consider equation (1) in the domain [0 1] times [0 1] with 119903 =05 119886 = 1 119887 = 119903 minus 119886 119888 = 119903 and119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572) (1199093 + 1199092 + 1)

minus (119905 + 1)2sdot [119886 (2 + 6119909) + 119887 (2119909 + 31199092) minus 119888 (1199093 + 1199092 + 1)])

(55)

The boundary and initial conditions are given by (2) and (3)with

1206010 (119905) = 1199093 + 1199092 + 1

119880 (0 119905) = (119905 + 1)2 119880 (1 119905) = 3 (119905 + 1)2

(56)

It is clear that 119880(119909 119905) = (119905 + 1)2(1199093 + 1199092 + 1) is the exactanalytical solution of this problem

Apply the compact difference scheme (23) to solve theabove problem Table 3 presents the errors E](120591 ℎ) (] =1 2infin) and the temporal convergence orders Ot

](120591 ℎ) (] =1 2infin) we can see that the computed solution 119880119899119894 has thesecond-order temporal accuracy

From Table 4 we can obtain the errors E](120591 ℎ) (] = 1 2infin) and the spatial convergence ordersOs](120591 ℎ) (] = 1 2infin)

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 4: An Efficient Compact Difference Method for Temporal ...

4 Advances in Mathematical Physics

It is easy to see that the tridiagonal coefficient matrix A isstrictly diagonally dominantTherefore the coefficientmatrixis nonsingular and hence invertible

Next we consider the stability and convergence analysisof the compact difference scheme (23)

Letting Ω = 119906 | 119906 = (1199060 1199061 119906119872) 1199060 = 119906119872 = 0for grid functions 119906 V isin Ω we define the inner product andnorm as follows

(119906 V) = ℎ119872minus1sum119894=1

119906119894V119894119906 = (119906 119906)12

119906infin = max0le119894le119872

10038161003816100381610038161199061198941003816100381610038161003816 (120575119909119906 120575119909V) = ℎ119872minus1sum

119894=0

120575119909119906119894+12120575119909V119894+12|119906|1 = (120575119909119906 120575119909119906]12 1199061 = (1199062 + |119906|21)12

(27)

According to simple calculations we obtain

(1205752119909119906 V) = minus (120575119909119906 120575119909V) ℎ 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 le 2 |119906|1

ℎ |119906|1 le 2 119906 (28)

In order to analyse we introduce the discrete innerproduct and norm

⟨119906 V⟩ = (H119909119906 minus1205752119909V) = (120575119909119906 120575119909V] minus ℎ212 (1205752119909119906 1205752119909V) 119906120576 = ⟨119906 119906⟩12

(29)

Based on above inner product and norm we have thefollowing lemmas

Lemma 5 (see [29]) Suppose 119906 isin Ω we obtain

1003817100381710038171003817H11990911990610038171003817100381710038172 le 1199062 le 3119871216 1199062120576 1199062infin le 31198718 1199062120576 11990621 le 3 (8 + 1198712)

16 1199062120576 (30)

Lemma 6 (see [27]) Suppose 119906 isin Ω we obtain

119906 le 11987128 10038171003817100381710038171003817120575211990911990610038171003817100381710038171003817 1199062120576 le 3119871216 100381710038171003817100381710038171205752119909119906100381710038171003817100381710038172

(31)

Lemma 7 (see [9]) Suppose 119906 isin Ω we obtain

( 119899sum119896=1

119888119899minus119896119899H119909 (119906119896 minus 119906119896minus1) minus12057521199091199061198991205722)

ge 12119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) 1 le 119899 le 119873(32)

In the next we then analyse the stability and convergenceof the scheme (23)

Theorem8 (stability) Let V119899 = (V1198990 V1198991 V119899119872) be the solutionof the compact difference scheme (23) with V1198990 = V119899119872 = 0Assume that one of the conditions 1 le 4(4120576 minus 1)11988631198891205761198712 holdsfor some positive constant 120576 gt 14

en it holds

1003817100381710038171003817V11989910038171003817100381710038172120576 le 10038171003817100381710038171003817V0100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 1 le 119899 le 119873

(33)

Proof We take the inner product of equation (23) withminus1205752119909V1198991205722 yield1120583 ( 119899sum119896=1

119888119899minus119896119899H119909 (V119896 minus V119896minus1) minus1205752119909V1198991205722)= minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 minus 119889 (H119909 (V1198991205722) 1205752119909V1198991205722)

minus (H119909119892119899minus1205722 1205752119909V1198991205722) 1 le 119899 le 119873(34)

Using Lemma 7

12120583119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817V119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817V119896minus1100381710038171003817100381710038172120576)le minus119886 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576

minus (H119909119892119899minus1205722 12057521199091199061198991205722) 1 le 119899 le 119873(35)

When 1 le 4(4120576 minus 1)11988631198891205761198712 for some positive constant120576 gt 14 we have from the Cauchy-Schwarz inequality andLemmas 6 that

119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 le 3119889119871216 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

le (119886 minus 1198864120576) 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172(36)

minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 + 1198864120576 100381710038171003817100381710038171205752119909V1198991205722100381710038171003817100381710038172

(37)

By (35) and the Cauchy-Schwarz inequality

minus 119886 1003817100381710038171003817100381712057521199091199061198991205722100381710038171003817100381710038172 + 119889 10038171003817100381710038171003817V1198991205722100381710038171003817100381710038172120576 minus (H119909119892119899minus1205722 12057521199091199061198991205722)le 120576119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(38)

Advances in Mathematical Physics 5

Substituting (38) into (35) leads to

119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) le 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (39)

The above inequality can be rewritten as

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 + 119888119899minus1119899 100381710038171003817100381710038171199060100381710038171003817100381710038172lowast+ 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(40)

Since by the definition of 119888119899minus1119899120583119888119899minus1119899 =

120583119886119899minus1 minus 119887119899minus1 lt 2Γ (1 minus 120572)119879120572 (41)

we have from (40) that

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576le 119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576+ 119888119899minus1119899 (100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172)

(42)

Letting

119864 = 100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (43)

and assuming 1199061198962120576 le 119864(0 le 119896 le 119899 minus 1) we obtain1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le

119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 119864 + 119888119899minus1119899119864 = 1198880119899119864 (44)

and we have the needed estimates

Letting 119890119899119894 = 119881119899119894 minus V119899119894 we get the following error equation

1120583119899sum119896=1

119888119899minus119896119899H119909 (119890119896119894 minus 119890119896minus1119894 )= 11988612057521199091198901198991205722119894 minus 119889H1199091198901198991205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873119890 (119887119889 119905) = 0119890 (119887119906 119905) = 0

119905 isin (0 119879] 119890 (119909 0) = 0 119909 isin [119887119889 119887119906]

(45)

Since the above error equation (45) we now obtain thefollowing convergence results

Theorem 9 (convergence) Let 119881119899119894 denote the value of thesolution V(119909 119905) of (23) at the mesh point (119909119894 119905119899) and letV119899 = (V1198990 V1198991 V119899119872) be the solution of the compact differencescheme (23) en when 1 le 4(4120576 minus 1)11988631198891205761198712 it holds1003817100381710038171003817119880119899 minus 1199061198991003817100381710038171003817120576 le 1198621 (1205912 + ℎ4) 1 le 119899 le 119873 (46)

where

1198621 = (4Γ (1 minus 120572) 1198791205721198711198622119877119886 )12 (47)

Proof It follows fromTheorem 8 that

100381710038171003817100381711989011989910038171003817100381710038172120576 le 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817(119877120572119905119909)119899119894 100381710038171003817100381710038172 1 le 119899 le 119873 (48)

Applying (22) we get100381710038171003817100381711989011989910038171003817100381710038172120576 le 11986221 (1205912 + ℎ4)2 (49)

The estimate (46) is proved

Remark 10 The constraint condition 1 le 4(4120576 minus 1)11988631198891205761198712in Theorems 8 and 9 is only for the analysis of the stabilityand convergence of the compact difference scheme (23)Thiscondition is easily verifiable for practical problems

4 Numerical Experiment

For demonstrating the efficiency of the compact differencescheme (23) we make two numerical experiments of it

Suppose 119881119899119894 = V(119909119894 119905119899) be the value of the solution V(119909 119905)of the problem (1)ndash(3) at the mesh point (119909119894 119905119899) From (22)we can see that1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] le 1198622 (1205912 + ℎ4) ] = 1 2infin (50)

where1198622 is a positive constant independent In order to checkthis accuracy of the compact difference scheme we computethe following norm errors

E] (120591 ℎ) = max0le119899le119873

1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] (] = 1 2infin) (51)

The temporal convergence order and the spatial convergenceorder are denoted by

Ot] (120591 ℎ) = log2 (E] (2120591 ℎ)

E] (120591 ℎ) ) Os

] (120591 ℎ) = log2 (E] (120591 2ℎ)E] (120591 ℎ) )

(] = 1 2infin) (52)

Example 1 We first consider a problem which is governedby equation (1) in [0 1] times [0 1] with 119903 = 005 120590 = 025 119886 =12059022 119887 = 119903 minus 119886 119888 = 119903 and

119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572)1199092 (1 minus 119909) minus (119905 + 1)2

sdot [119886 (2 minus 6119909) + 119887 (2119909 minus 31199092) minus 1198881199092 (1 minus 119909)]) (53)

6 Advances in Mathematical Physics

Table 1 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 1100)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 110 37361endash05 37314endash05 62608endash05

120 93611endash06 19968 93492endash06 19968 15684endash05 19970140 23429endash06 19984 23399endash06 19984 39251endash06 19985180 58603endash07 19992 58528endash07 19992 98176endash07 199931160 14654endash07 19997 14635endash07 19997 24549endash07 199971320 36634endash08 20001 36587endash08 20001 61372endash08 20000

12 110 67788endash05 67702endash05 11393endash04120 16994endash05 19960 16972endash05 19960 28555endash05 19964140 42543endash06 19980 42489endash06 19980 71480endash06 19981180 10643endash06 19990 10630endash06 19990 17882endash06 199911160 26617endash07 19995 26583endash07 19995 44718endash07 199961320 66548endash08 19999 66463endash08 19999 11181endash07 19999

34 110 88226endash05 88110endash05 14950endash04120 22098endash05 19973 22069endash05 19973 37435endash05 19976140 55299endash06 19986 55226endash06 19986 93672endash06 19987180 13832endash06 19993 13813endash06 19993 23429endash06 199931160 34587endash07 19996 34542endash07 19996 58585endash07 199971320 86475endash08 19999 86361endash08 19999 14647endash07 19999

Table 2 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 110000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 11190endash04 64607endash05 91369endash05

14 54155endash06 43690 42667endash06 39205 54429endash06 4069318 29041endash07 42209 26840endash07 39907 33922endash07 40041116 17125endash08 40840 16758endash08 40015 21165endash08 40024132 10264endash09 40604 10207endash09 40372 12889endash09 40375

12 12 10340endash04 59701endash05 84430endash0514 50151endash06 43659 39472endash06 39189 49995endash06 4077918 26907endash07 42202 24841endash07 39900 31142endash07 40048116 15831endash08 40871 15487endash08 40036 19400endash08 40047132 92534endash10 40967 92011endash10 40731 11538endash09 40716

34 12 93459endash05 53959endash05 76309endash0514 45477endash06 43611 35734endash06 39165 44770endash06 4091218 24420endash07 42190 22506endash07 39889 27865endash07 40060116 14335endash08 40905 14014endash08 40053 17372endash08 40036132 82043endash10 41270 81562endash10 41028 10150endash09 40971

The boundary and initial conditions are given by (2) and (3)with

119880 (119909 0) = 1199092 (1 minus 119909) 119880 (0 119905) = 119880 (1 119905) = 0 (54)

It is easy to check that119880(119909 119905) = (119905+1)21199092(1minus119909) is the solutionof this problem

For different 120572 we let the spatial step ℎ = 1100 Table 1gives the errors E](120591 ℎ) (] = 1 2infin) and the temporalconvergence orders Ot

](120591 ℎ) (] = 1 2infin) of the computedsolution 119880119899119894 for 120572 = 14 12 34 and different time step 120591From the table we can see that the computed solution 119880119899119894

has the second-order temporal accuracy For comparison thecorresponding temporal convergence orders Ot

](120591 ℎ) (] =infin) given in [8] has only 2minus120572 order thus it is far less accuratethan the compact difference scheme (23) given in this paper

Next we compute the spatial convergence order of thecompact difference scheme (23) Table 2 presents the errorsE](120591 ℎ) (] = 1 2infin) and the spatial convergence orderso119903119889119890119903119874s

](120591 ℎ) (] = 1 2infin) The table demonstrates that thecompact difference scheme (23) has the fourth-order spatialaccuracy

Example 2 In this example we test the error and theconvergence order of the compact difference scheme (23)

Advances in Mathematical Physics 7

Table 3 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2 (120591 = 1100)120572 ℎ E1(120591 ℎ) Os

1(120591 ℎ) E2(120591 ℎ) Os2(120591 ℎ) Einfin(120591 ℎ) Os

infin(120591 ℎ)14 110 95681endash05 95622endash05 13266endash04

120 23985endash05 19961 23971endash05 19961 33254endash05 19961140 60044endash06 19981 60007endash06 19981 83244endash06 19981180 15021endash06 19990 15012endash06 19990 20825endash06 199911160 37564endash07 19996 37541endash07 19996 52078endash07 199961320 93916endash08 19999 93859endash08 19999 13020endash07 19999

12 110 17283endash04 17272endash04 24033endash04120 43358endash05 19950 43331endash05 19950 60286endash05 19951140 10858endash05 19975 10852endash05 19975 15097endash05 19976180 27169endash06 19987 27153endash06 19987 37774endash06 199881160 67952endash07 19994 67910endash07 19994 94474endash07 199941320 16991endash07 19998 16980endash07 19998 23622endash07 19998

34 110 22075endash04 22061endash04 30894endash04120 55326endash05 19964 55291endash05 19964 77418endash05 19966140 13849endash05 19981 13841endash05 19981 19378endash05 19983180 34646endash06 19991 34624endash06 19991 48476endash06 199911160 86641endash07 19995 86587endash07 19995 12123endash06 199961320 21663endash07 19998 21650endash07 19998 30311endash07 19998

Table 4 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2 (ℎ = 115000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 12909endash04 74532endash05 10540endash04

14 62469endash06 43691 49219endash06 39206 62789endash06 4069318 33499endash07 42210 30961endash07 39907 39130endash07 40042116 19745endash08 40846 19323endash08 40021 24403endash08 40031132 11750endash09 40707 11685endash09 40475 14745endash09 40488

12 12 11929endash04 68871endash05 97399endash0514 57849endash06 43660 45533endash06 39189 57675endash06 4077918 31036endash07 42203 28654endash07 39901 35923endash07 40050116 18246endash08 40883 17849endash08 40048 22357endash08 40061132 10507endash09 41182 10447endash09 40946 13056endash09 40980

34 12 10782endash04 62247endash05 88031endash0514 52457endash06 43613 41220endash06 39166 51648endash06 4091218 28166endash07 42191 25959endash07 39891 32142endash07 40062116 16515endash08 40921 16146endash08 40070 20016endash08 40053132 92416endash10 41595 91874endash10 41354 11316endash09 41447

Consider equation (1) in the domain [0 1] times [0 1] with 119903 =05 119886 = 1 119887 = 119903 minus 119886 119888 = 119903 and119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572) (1199093 + 1199092 + 1)

minus (119905 + 1)2sdot [119886 (2 + 6119909) + 119887 (2119909 + 31199092) minus 119888 (1199093 + 1199092 + 1)])

(55)

The boundary and initial conditions are given by (2) and (3)with

1206010 (119905) = 1199093 + 1199092 + 1

119880 (0 119905) = (119905 + 1)2 119880 (1 119905) = 3 (119905 + 1)2

(56)

It is clear that 119880(119909 119905) = (119905 + 1)2(1199093 + 1199092 + 1) is the exactanalytical solution of this problem

Apply the compact difference scheme (23) to solve theabove problem Table 3 presents the errors E](120591 ℎ) (] =1 2infin) and the temporal convergence orders Ot

](120591 ℎ) (] =1 2infin) we can see that the computed solution 119880119899119894 has thesecond-order temporal accuracy

From Table 4 we can obtain the errors E](120591 ℎ) (] = 1 2infin) and the spatial convergence ordersOs](120591 ℎ) (] = 1 2infin)

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 5: An Efficient Compact Difference Method for Temporal ...

Advances in Mathematical Physics 5

Substituting (38) into (35) leads to

119899sum119896=1

119888119899minus119896119899 (10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 minus 10038171003817100381710038171003817119906119896minus1100381710038171003817100381710038172120576) le 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (39)

The above inequality can be rewritten as

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576 + 119888119899minus1119899 100381710038171003817100381710038171199060100381710038171003817100381710038172lowast+ 2120576120583119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172

(40)

Since by the definition of 119888119899minus1119899120583119888119899minus1119899 =

120583119886119899minus1 minus 119887119899minus1 lt 2Γ (1 minus 120572)119879120572 (41)

we have from (40) that

1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576le 119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 10038171003817100381710038171003817119906119896100381710038171003817100381710038172120576+ 119888119899minus1119899 (100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172)

(42)

Letting

119864 = 100381710038171003817100381710038171199060100381710038171003817100381710038172120576 + 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817H119909119892119899minus1205722100381710038171003817100381710038172 (43)

and assuming 1199061198962120576 le 119864(0 le 119896 le 119899 minus 1) we obtain1198880119899 100381710038171003817100381711990611989910038171003817100381710038172120576 le

119899minus1sum119896=1

(119888119899minus119896minus1119899 minus 119888119899minus119896119899) 119864 + 119888119899minus1119899119864 = 1198880119899119864 (44)

and we have the needed estimates

Letting 119890119899119894 = 119881119899119894 minus V119899119894 we get the following error equation

1120583119899sum119896=1

119888119899minus119896119899H119909 (119890119896119894 minus 119890119896minus1119894 )= 11988612057521199091198901198991205722119894 minus 119889H1199091198901198991205722119894 + (119877120572119905119909)119899119894

1 le 119894 le 119872 minus 1 1 le 119899 le 119873119890 (119887119889 119905) = 0119890 (119887119906 119905) = 0

119905 isin (0 119879] 119890 (119909 0) = 0 119909 isin [119887119889 119887119906]

(45)

Since the above error equation (45) we now obtain thefollowing convergence results

Theorem 9 (convergence) Let 119881119899119894 denote the value of thesolution V(119909 119905) of (23) at the mesh point (119909119894 119905119899) and letV119899 = (V1198990 V1198991 V119899119872) be the solution of the compact differencescheme (23) en when 1 le 4(4120576 minus 1)11988631198891205761198712 it holds1003817100381710038171003817119880119899 minus 1199061198991003817100381710038171003817120576 le 1198621 (1205912 + ℎ4) 1 le 119899 le 119873 (46)

where

1198621 = (4Γ (1 minus 120572) 1198791205721198711198622119877119886 )12 (47)

Proof It follows fromTheorem 8 that

100381710038171003817100381711989011989910038171003817100381710038172120576 le 4120576Γ (1 minus 120572) 119879120572119886 max1le119899le119873

10038171003817100381710038171003817(119877120572119905119909)119899119894 100381710038171003817100381710038172 1 le 119899 le 119873 (48)

Applying (22) we get100381710038171003817100381711989011989910038171003817100381710038172120576 le 11986221 (1205912 + ℎ4)2 (49)

The estimate (46) is proved

Remark 10 The constraint condition 1 le 4(4120576 minus 1)11988631198891205761198712in Theorems 8 and 9 is only for the analysis of the stabilityand convergence of the compact difference scheme (23)Thiscondition is easily verifiable for practical problems

4 Numerical Experiment

For demonstrating the efficiency of the compact differencescheme (23) we make two numerical experiments of it

Suppose 119881119899119894 = V(119909119894 119905119899) be the value of the solution V(119909 119905)of the problem (1)ndash(3) at the mesh point (119909119894 119905119899) From (22)we can see that1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] le 1198622 (1205912 + ℎ4) ] = 1 2infin (50)

where1198622 is a positive constant independent In order to checkthis accuracy of the compact difference scheme we computethe following norm errors

E] (120591 ℎ) = max0le119899le119873

1003817100381710038171003817119881119899 minus V1198991003817100381710038171003817] (] = 1 2infin) (51)

The temporal convergence order and the spatial convergenceorder are denoted by

Ot] (120591 ℎ) = log2 (E] (2120591 ℎ)

E] (120591 ℎ) ) Os

] (120591 ℎ) = log2 (E] (120591 2ℎ)E] (120591 ℎ) )

(] = 1 2infin) (52)

Example 1 We first consider a problem which is governedby equation (1) in [0 1] times [0 1] with 119903 = 005 120590 = 025 119886 =12059022 119887 = 119903 minus 119886 119888 = 119903 and

119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572)1199092 (1 minus 119909) minus (119905 + 1)2

sdot [119886 (2 minus 6119909) + 119887 (2119909 minus 31199092) minus 1198881199092 (1 minus 119909)]) (53)

6 Advances in Mathematical Physics

Table 1 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 1100)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 110 37361endash05 37314endash05 62608endash05

120 93611endash06 19968 93492endash06 19968 15684endash05 19970140 23429endash06 19984 23399endash06 19984 39251endash06 19985180 58603endash07 19992 58528endash07 19992 98176endash07 199931160 14654endash07 19997 14635endash07 19997 24549endash07 199971320 36634endash08 20001 36587endash08 20001 61372endash08 20000

12 110 67788endash05 67702endash05 11393endash04120 16994endash05 19960 16972endash05 19960 28555endash05 19964140 42543endash06 19980 42489endash06 19980 71480endash06 19981180 10643endash06 19990 10630endash06 19990 17882endash06 199911160 26617endash07 19995 26583endash07 19995 44718endash07 199961320 66548endash08 19999 66463endash08 19999 11181endash07 19999

34 110 88226endash05 88110endash05 14950endash04120 22098endash05 19973 22069endash05 19973 37435endash05 19976140 55299endash06 19986 55226endash06 19986 93672endash06 19987180 13832endash06 19993 13813endash06 19993 23429endash06 199931160 34587endash07 19996 34542endash07 19996 58585endash07 199971320 86475endash08 19999 86361endash08 19999 14647endash07 19999

Table 2 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 110000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 11190endash04 64607endash05 91369endash05

14 54155endash06 43690 42667endash06 39205 54429endash06 4069318 29041endash07 42209 26840endash07 39907 33922endash07 40041116 17125endash08 40840 16758endash08 40015 21165endash08 40024132 10264endash09 40604 10207endash09 40372 12889endash09 40375

12 12 10340endash04 59701endash05 84430endash0514 50151endash06 43659 39472endash06 39189 49995endash06 4077918 26907endash07 42202 24841endash07 39900 31142endash07 40048116 15831endash08 40871 15487endash08 40036 19400endash08 40047132 92534endash10 40967 92011endash10 40731 11538endash09 40716

34 12 93459endash05 53959endash05 76309endash0514 45477endash06 43611 35734endash06 39165 44770endash06 4091218 24420endash07 42190 22506endash07 39889 27865endash07 40060116 14335endash08 40905 14014endash08 40053 17372endash08 40036132 82043endash10 41270 81562endash10 41028 10150endash09 40971

The boundary and initial conditions are given by (2) and (3)with

119880 (119909 0) = 1199092 (1 minus 119909) 119880 (0 119905) = 119880 (1 119905) = 0 (54)

It is easy to check that119880(119909 119905) = (119905+1)21199092(1minus119909) is the solutionof this problem

For different 120572 we let the spatial step ℎ = 1100 Table 1gives the errors E](120591 ℎ) (] = 1 2infin) and the temporalconvergence orders Ot

](120591 ℎ) (] = 1 2infin) of the computedsolution 119880119899119894 for 120572 = 14 12 34 and different time step 120591From the table we can see that the computed solution 119880119899119894

has the second-order temporal accuracy For comparison thecorresponding temporal convergence orders Ot

](120591 ℎ) (] =infin) given in [8] has only 2minus120572 order thus it is far less accuratethan the compact difference scheme (23) given in this paper

Next we compute the spatial convergence order of thecompact difference scheme (23) Table 2 presents the errorsE](120591 ℎ) (] = 1 2infin) and the spatial convergence orderso119903119889119890119903119874s

](120591 ℎ) (] = 1 2infin) The table demonstrates that thecompact difference scheme (23) has the fourth-order spatialaccuracy

Example 2 In this example we test the error and theconvergence order of the compact difference scheme (23)

Advances in Mathematical Physics 7

Table 3 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2 (120591 = 1100)120572 ℎ E1(120591 ℎ) Os

1(120591 ℎ) E2(120591 ℎ) Os2(120591 ℎ) Einfin(120591 ℎ) Os

infin(120591 ℎ)14 110 95681endash05 95622endash05 13266endash04

120 23985endash05 19961 23971endash05 19961 33254endash05 19961140 60044endash06 19981 60007endash06 19981 83244endash06 19981180 15021endash06 19990 15012endash06 19990 20825endash06 199911160 37564endash07 19996 37541endash07 19996 52078endash07 199961320 93916endash08 19999 93859endash08 19999 13020endash07 19999

12 110 17283endash04 17272endash04 24033endash04120 43358endash05 19950 43331endash05 19950 60286endash05 19951140 10858endash05 19975 10852endash05 19975 15097endash05 19976180 27169endash06 19987 27153endash06 19987 37774endash06 199881160 67952endash07 19994 67910endash07 19994 94474endash07 199941320 16991endash07 19998 16980endash07 19998 23622endash07 19998

34 110 22075endash04 22061endash04 30894endash04120 55326endash05 19964 55291endash05 19964 77418endash05 19966140 13849endash05 19981 13841endash05 19981 19378endash05 19983180 34646endash06 19991 34624endash06 19991 48476endash06 199911160 86641endash07 19995 86587endash07 19995 12123endash06 199961320 21663endash07 19998 21650endash07 19998 30311endash07 19998

Table 4 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2 (ℎ = 115000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 12909endash04 74532endash05 10540endash04

14 62469endash06 43691 49219endash06 39206 62789endash06 4069318 33499endash07 42210 30961endash07 39907 39130endash07 40042116 19745endash08 40846 19323endash08 40021 24403endash08 40031132 11750endash09 40707 11685endash09 40475 14745endash09 40488

12 12 11929endash04 68871endash05 97399endash0514 57849endash06 43660 45533endash06 39189 57675endash06 4077918 31036endash07 42203 28654endash07 39901 35923endash07 40050116 18246endash08 40883 17849endash08 40048 22357endash08 40061132 10507endash09 41182 10447endash09 40946 13056endash09 40980

34 12 10782endash04 62247endash05 88031endash0514 52457endash06 43613 41220endash06 39166 51648endash06 4091218 28166endash07 42191 25959endash07 39891 32142endash07 40062116 16515endash08 40921 16146endash08 40070 20016endash08 40053132 92416endash10 41595 91874endash10 41354 11316endash09 41447

Consider equation (1) in the domain [0 1] times [0 1] with 119903 =05 119886 = 1 119887 = 119903 minus 119886 119888 = 119903 and119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572) (1199093 + 1199092 + 1)

minus (119905 + 1)2sdot [119886 (2 + 6119909) + 119887 (2119909 + 31199092) minus 119888 (1199093 + 1199092 + 1)])

(55)

The boundary and initial conditions are given by (2) and (3)with

1206010 (119905) = 1199093 + 1199092 + 1

119880 (0 119905) = (119905 + 1)2 119880 (1 119905) = 3 (119905 + 1)2

(56)

It is clear that 119880(119909 119905) = (119905 + 1)2(1199093 + 1199092 + 1) is the exactanalytical solution of this problem

Apply the compact difference scheme (23) to solve theabove problem Table 3 presents the errors E](120591 ℎ) (] =1 2infin) and the temporal convergence orders Ot

](120591 ℎ) (] =1 2infin) we can see that the computed solution 119880119899119894 has thesecond-order temporal accuracy

From Table 4 we can obtain the errors E](120591 ℎ) (] = 1 2infin) and the spatial convergence ordersOs](120591 ℎ) (] = 1 2infin)

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 6: An Efficient Compact Difference Method for Temporal ...

6 Advances in Mathematical Physics

Table 1 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 1100)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 110 37361endash05 37314endash05 62608endash05

120 93611endash06 19968 93492endash06 19968 15684endash05 19970140 23429endash06 19984 23399endash06 19984 39251endash06 19985180 58603endash07 19992 58528endash07 19992 98176endash07 199931160 14654endash07 19997 14635endash07 19997 24549endash07 199971320 36634endash08 20001 36587endash08 20001 61372endash08 20000

12 110 67788endash05 67702endash05 11393endash04120 16994endash05 19960 16972endash05 19960 28555endash05 19964140 42543endash06 19980 42489endash06 19980 71480endash06 19981180 10643endash06 19990 10630endash06 19990 17882endash06 199911160 26617endash07 19995 26583endash07 19995 44718endash07 199961320 66548endash08 19999 66463endash08 19999 11181endash07 19999

34 110 88226endash05 88110endash05 14950endash04120 22098endash05 19973 22069endash05 19973 37435endash05 19976140 55299endash06 19986 55226endash06 19986 93672endash06 19987180 13832endash06 19993 13813endash06 19993 23429endash06 199931160 34587endash07 19996 34542endash07 19996 58585endash07 199971320 86475endash08 19999 86361endash08 19999 14647endash07 19999

Table 2 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1 (ℎ = 110000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 11190endash04 64607endash05 91369endash05

14 54155endash06 43690 42667endash06 39205 54429endash06 4069318 29041endash07 42209 26840endash07 39907 33922endash07 40041116 17125endash08 40840 16758endash08 40015 21165endash08 40024132 10264endash09 40604 10207endash09 40372 12889endash09 40375

12 12 10340endash04 59701endash05 84430endash0514 50151endash06 43659 39472endash06 39189 49995endash06 4077918 26907endash07 42202 24841endash07 39900 31142endash07 40048116 15831endash08 40871 15487endash08 40036 19400endash08 40047132 92534endash10 40967 92011endash10 40731 11538endash09 40716

34 12 93459endash05 53959endash05 76309endash0514 45477endash06 43611 35734endash06 39165 44770endash06 4091218 24420endash07 42190 22506endash07 39889 27865endash07 40060116 14335endash08 40905 14014endash08 40053 17372endash08 40036132 82043endash10 41270 81562endash10 41028 10150endash09 40971

The boundary and initial conditions are given by (2) and (3)with

119880 (119909 0) = 1199092 (1 minus 119909) 119880 (0 119905) = 119880 (1 119905) = 0 (54)

It is easy to check that119880(119909 119905) = (119905+1)21199092(1minus119909) is the solutionof this problem

For different 120572 we let the spatial step ℎ = 1100 Table 1gives the errors E](120591 ℎ) (] = 1 2infin) and the temporalconvergence orders Ot

](120591 ℎ) (] = 1 2infin) of the computedsolution 119880119899119894 for 120572 = 14 12 34 and different time step 120591From the table we can see that the computed solution 119880119899119894

has the second-order temporal accuracy For comparison thecorresponding temporal convergence orders Ot

](120591 ℎ) (] =infin) given in [8] has only 2minus120572 order thus it is far less accuratethan the compact difference scheme (23) given in this paper

Next we compute the spatial convergence order of thecompact difference scheme (23) Table 2 presents the errorsE](120591 ℎ) (] = 1 2infin) and the spatial convergence orderso119903119889119890119903119874s

](120591 ℎ) (] = 1 2infin) The table demonstrates that thecompact difference scheme (23) has the fourth-order spatialaccuracy

Example 2 In this example we test the error and theconvergence order of the compact difference scheme (23)

Advances in Mathematical Physics 7

Table 3 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2 (120591 = 1100)120572 ℎ E1(120591 ℎ) Os

1(120591 ℎ) E2(120591 ℎ) Os2(120591 ℎ) Einfin(120591 ℎ) Os

infin(120591 ℎ)14 110 95681endash05 95622endash05 13266endash04

120 23985endash05 19961 23971endash05 19961 33254endash05 19961140 60044endash06 19981 60007endash06 19981 83244endash06 19981180 15021endash06 19990 15012endash06 19990 20825endash06 199911160 37564endash07 19996 37541endash07 19996 52078endash07 199961320 93916endash08 19999 93859endash08 19999 13020endash07 19999

12 110 17283endash04 17272endash04 24033endash04120 43358endash05 19950 43331endash05 19950 60286endash05 19951140 10858endash05 19975 10852endash05 19975 15097endash05 19976180 27169endash06 19987 27153endash06 19987 37774endash06 199881160 67952endash07 19994 67910endash07 19994 94474endash07 199941320 16991endash07 19998 16980endash07 19998 23622endash07 19998

34 110 22075endash04 22061endash04 30894endash04120 55326endash05 19964 55291endash05 19964 77418endash05 19966140 13849endash05 19981 13841endash05 19981 19378endash05 19983180 34646endash06 19991 34624endash06 19991 48476endash06 199911160 86641endash07 19995 86587endash07 19995 12123endash06 199961320 21663endash07 19998 21650endash07 19998 30311endash07 19998

Table 4 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2 (ℎ = 115000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 12909endash04 74532endash05 10540endash04

14 62469endash06 43691 49219endash06 39206 62789endash06 4069318 33499endash07 42210 30961endash07 39907 39130endash07 40042116 19745endash08 40846 19323endash08 40021 24403endash08 40031132 11750endash09 40707 11685endash09 40475 14745endash09 40488

12 12 11929endash04 68871endash05 97399endash0514 57849endash06 43660 45533endash06 39189 57675endash06 4077918 31036endash07 42203 28654endash07 39901 35923endash07 40050116 18246endash08 40883 17849endash08 40048 22357endash08 40061132 10507endash09 41182 10447endash09 40946 13056endash09 40980

34 12 10782endash04 62247endash05 88031endash0514 52457endash06 43613 41220endash06 39166 51648endash06 4091218 28166endash07 42191 25959endash07 39891 32142endash07 40062116 16515endash08 40921 16146endash08 40070 20016endash08 40053132 92416endash10 41595 91874endash10 41354 11316endash09 41447

Consider equation (1) in the domain [0 1] times [0 1] with 119903 =05 119886 = 1 119887 = 119903 minus 119886 119888 = 119903 and119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572) (1199093 + 1199092 + 1)

minus (119905 + 1)2sdot [119886 (2 + 6119909) + 119887 (2119909 + 31199092) minus 119888 (1199093 + 1199092 + 1)])

(55)

The boundary and initial conditions are given by (2) and (3)with

1206010 (119905) = 1199093 + 1199092 + 1

119880 (0 119905) = (119905 + 1)2 119880 (1 119905) = 3 (119905 + 1)2

(56)

It is clear that 119880(119909 119905) = (119905 + 1)2(1199093 + 1199092 + 1) is the exactanalytical solution of this problem

Apply the compact difference scheme (23) to solve theabove problem Table 3 presents the errors E](120591 ℎ) (] =1 2infin) and the temporal convergence orders Ot

](120591 ℎ) (] =1 2infin) we can see that the computed solution 119880119899119894 has thesecond-order temporal accuracy

From Table 4 we can obtain the errors E](120591 ℎ) (] = 1 2infin) and the spatial convergence ordersOs](120591 ℎ) (] = 1 2infin)

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 7: An Efficient Compact Difference Method for Temporal ...

Advances in Mathematical Physics 7

Table 3 The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2 (120591 = 1100)120572 ℎ E1(120591 ℎ) Os

1(120591 ℎ) E2(120591 ℎ) Os2(120591 ℎ) Einfin(120591 ℎ) Os

infin(120591 ℎ)14 110 95681endash05 95622endash05 13266endash04

120 23985endash05 19961 23971endash05 19961 33254endash05 19961140 60044endash06 19981 60007endash06 19981 83244endash06 19981180 15021endash06 19990 15012endash06 19990 20825endash06 199911160 37564endash07 19996 37541endash07 19996 52078endash07 199961320 93916endash08 19999 93859endash08 19999 13020endash07 19999

12 110 17283endash04 17272endash04 24033endash04120 43358endash05 19950 43331endash05 19950 60286endash05 19951140 10858endash05 19975 10852endash05 19975 15097endash05 19976180 27169endash06 19987 27153endash06 19987 37774endash06 199881160 67952endash07 19994 67910endash07 19994 94474endash07 199941320 16991endash07 19998 16980endash07 19998 23622endash07 19998

34 110 22075endash04 22061endash04 30894endash04120 55326endash05 19964 55291endash05 19964 77418endash05 19966140 13849endash05 19981 13841endash05 19981 19378endash05 19983180 34646endash06 19991 34624endash06 19991 48476endash06 199911160 86641endash07 19995 86587endash07 19995 12123endash06 199961320 21663endash07 19998 21650endash07 19998 30311endash07 19998

Table 4 The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2 (ℎ = 115000)120572 120591 E1(120591 ℎ) Ot

1(120591 ℎ) E2(120591 ℎ) Ot2(120591 ℎ) Einfin(120591 ℎ) Ot

infin(120591 ℎ)14 12 12909endash04 74532endash05 10540endash04

14 62469endash06 43691 49219endash06 39206 62789endash06 4069318 33499endash07 42210 30961endash07 39907 39130endash07 40042116 19745endash08 40846 19323endash08 40021 24403endash08 40031132 11750endash09 40707 11685endash09 40475 14745endash09 40488

12 12 11929endash04 68871endash05 97399endash0514 57849endash06 43660 45533endash06 39189 57675endash06 4077918 31036endash07 42203 28654endash07 39901 35923endash07 40050116 18246endash08 40883 17849endash08 40048 22357endash08 40061132 10507endash09 41182 10447endash09 40946 13056endash09 40980

34 12 10782endash04 62247endash05 88031endash0514 52457endash06 43613 41220endash06 39166 51648endash06 4091218 28166endash07 42191 25959endash07 39891 32142endash07 40062116 16515endash08 40921 16146endash08 40070 20016endash08 40053132 92416endash10 41595 91874endash10 41354 11316endash09 41447

Consider equation (1) in the domain [0 1] times [0 1] with 119903 =05 119886 = 1 119887 = 119903 minus 119886 119888 = 119903 and119891 (119909 119905) = ( 21199052minus120572Γ (3 minus 120572) + 21199051minus120572Γ (2 minus 120572) (1199093 + 1199092 + 1)

minus (119905 + 1)2sdot [119886 (2 + 6119909) + 119887 (2119909 + 31199092) minus 119888 (1199093 + 1199092 + 1)])

(55)

The boundary and initial conditions are given by (2) and (3)with

1206010 (119905) = 1199093 + 1199092 + 1

119880 (0 119905) = (119905 + 1)2 119880 (1 119905) = 3 (119905 + 1)2

(56)

It is clear that 119880(119909 119905) = (119905 + 1)2(1199093 + 1199092 + 1) is the exactanalytical solution of this problem

Apply the compact difference scheme (23) to solve theabove problem Table 3 presents the errors E](120591 ℎ) (] =1 2infin) and the temporal convergence orders Ot

](120591 ℎ) (] =1 2infin) we can see that the computed solution 119880119899119894 has thesecond-order temporal accuracy

From Table 4 we can obtain the errors E](120591 ℎ) (] = 1 2infin) and the spatial convergence ordersOs](120591 ℎ) (] = 1 2infin)

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 8: An Efficient Compact Difference Method for Temporal ...

8 Advances in Mathematical Physics

These numerical results demonstrate that the accuracy of thecompact difference scheme (23) is fourth-order

5 Concluding Remarks

In this paper a high-order compact finite difference methodfor a class of time-fractional Black-Scholes equations ispresented and analysed We apply the 1198712-1120590 approximationformula to the Caputo derivative then we construct a fourth-order compact finite difference approximation for the spatialderivative We have analysed the solvability stability andconvergence of the constructed scheme and provided theoptimal error estimates The constructed scheme has thesecond-order temporal accuracy and the fourth-order spatialaccuracy which improves the temporal accuracy of themethod given in [8]

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of ChinaNo 11401363 the Education Foundationof Henan Province No 19A110030 the Foundation for theTraining of Young Key Teachers in Colleges and Universitiesin Henan Province No 2018GGJS134

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoTheory of rational option pricingrdquo Bell Journalof Economics and Management Science vol 4 pp 141ndash183 1973

[3] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus and Applied Analysis vol 3 no 1 pp 51ndash61 2000

[4] J-R Liang J Wang W-J Zhang W-Y Qiu and F-Y RenldquoOption pricing of a bi-fractional Black-Merton-Scholesmodelwith the Hurst exponent H in [121]rdquo Applied MathematicsLetters vol 23 no 8 pp 859ndash863 2010

[5] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[6] H Zhang F Liu I Turner and Q Yang ldquoNumerical solutionof the time fractional Black-Scholesmodel governing Europeanoptionsrdquo Computers amp Mathematics with Applications vol 71no 9 pp 1772ndash1783 2016

[7] H Zhang F Liu I Turner and S Chen ldquoThe numericalsimulation of the tempered fractional BlackndashScholes equationfor European double barrier optionrdquo Applied MathematicalModelling vol 40 no 11-12 pp 5819ndash5834 2016

[8] R De Staelen and A Hendy ldquoNumerically pricing doublebarrier options in a time-fractional BlackndashScholes modelrdquo

Computers amp Mathematics with Applications vol 74 no 6 pp1166ndash1175 2017

[9] A A Alikhanov ldquoA new difference scheme for the time frac-tional diffusion equationrdquo Journal of Computational Physics vol280 pp 424ndash438 2015

[10] A Bueno-Orovio D Kay V Grau B Rodriguez and KBurrage ldquoFractional diffusion models of cardiac electricalpropagation role of structural heterogeneity in dispersion ofrepolarizationrdquo Journal of the Royal Society Interface vol 11article no 352 pp 20140352-20140352 2014

[11] K Burrage N Hale and D Kay ldquoAn efficient implicit FEMscheme for fractional-in-space reaction-diffusion equationsrdquoSIAM Journal on Scientific Computing vol 34 no 4 pp A2145ndashA2172 2012

[12] S Chen F Liu P Zhuang and V Anh ldquoFinite differenceapproximations for the fractional Fokker-Planck equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp256ndash273 2009

[13] Y Dimitrov ldquoNumerical approximations for fractional differen-tial equationsrdquo Journal of Fractional Calculus and Applicationsvol 5 pp 1ndash45 2014

[14] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Computa-tional Physics vol 230 no 3 pp 586ndash595 2011

[15] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[16] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[17] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[18] T A M Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[19] Y Luchko andA Punzi ldquoModeling anomalous heat transport ingeothermal reservoirs via fractional diffusion equationsrdquo GEM- International Journal on Geomathematics vol 1 no 2 pp 257ndash276 2011

[20] C Li and F Zeng Numerical Methods for Fractional CalculusChapman and HallCRC Boca Raton FL USA 2015

[21] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquoPhysics Reportsvol 339 no 1 pp 1ndash77 2000

[22] A A Samarskii e eory of Difference Schemes MarcelDekker New York NY USA 2001

[23] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014

[24] Y Zhang D A Benson and D M Reeves ldquoTime and spacenonlocalities underlying fractional-derivative models distinc-tion and literature review of field applicationsrdquo Advances inWater Resources vol 32 no 4 pp 561ndash581 2009

[25] L Zhao and W Deng ldquoA series of high-order quasi-compactschemes for space fractional diffusion equations based onthe superconvergent approximations for fractional derivativesrdquoNumericalMethods for Partial Differential Equations vol 31 no5 pp 1345ndash1381 2015

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 9: An Efficient Compact Difference Method for Temporal ...

Advances in Mathematical Physics 9

[26] Y Guo ldquoSolvability for a nonlinear fractional differential equa-tionrdquoBulletin of the AustralianMathematical Society vol 80 no1 pp 125ndash138 2009

[27] Y Wang and L Ren ldquoEfficient compact finite difference meth-ods for a class of time-fractional convectionndashreactionndashdiffusionequations with variable coefficientsrdquo International Journal ofComputer Mathematics vol 96 no 2 pp 264ndash297 2018

[28] Y N Zhang Z Z Sun and H W Wu ldquoError estimates ofCrank-Nicolson-type difference schemes for the subdiffusionequationrdquo SIAM Journal on Numerical Analysis vol 49 pp2302ndash2322 2011

[29] Y Wang ldquoA compact finite difference method for solving aclass of time fractional convection-subdiffusion equationsrdquo BITNumerical Mathematics vol 55 no 4 pp 1187ndash1217 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 10: An Efficient Compact Difference Method for Temporal ...

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Recommended
Algorithms for Fast Gradient Temporal Difference Learning · Algorithms for Fast Gradient Temporal Difference Learning ... of the classical TD methods have been proposed. ... the

Algorithms for Fast Gradient Temporal Difference Learning · Algorithms for Fast Gradient Temporal Difference Learning ... of the classical TD methods have been proposed. ... the

Documents

Temporal-Difference Learning - TU Chemnitz · Temporal-Difference Learning 18 Random Walk Example! • All episodes start in the center state C! • proceed either left or right by

Temporal-Difference Learning - TU Chemnitz · Temporal-Difference Learning 18 Random Walk Example! • All episodes start in the center state C! • proceed either left or right by

Documents

Temporal Difference Learning of Position Evaluation in the Game of ...

Temporal Difference Learning of Position Evaluation in the Game of ...

Documents

Pathologies of Temporal Difference Methods in Approximate ...web.mit.edu/dimitrib/www/TD_Pathologies.pdf · Pathologies of Temporal Difference Methods in Approximate Dynamic Programming

Pathologies of Temporal Difference Methods in Approximate ...web.mit.edu/dimitrib/www/TD_Pathologies.pdf · Pathologies of Temporal Difference Methods in Approximate Dynamic Programming

Documents

Scattered node compact finite difference-type formulas generated ...

Scattered node compact finite difference-type formulas generated ...

Documents

A Reduced High-order Compact Finite Difference Scheme Based … · 2019-05-27 · A Reduced High-order Compact Finite Difference Scheme Based on Proper Orthogonal Decomposition for

A Reduced High-order Compact Finite Difference Scheme Based … · 2019-05-27 · A Reduced High-order Compact Finite Difference Scheme Based on Proper Orthogonal Decomposition for

Documents

High-order compact finite difference methods

High-order compact finite difference methods

Documents

Chapter 6: Temporal Difference Learningweb.stanford.edu/.../rich_sutton_slides/11-12-TD.pdf · 98 CHAPTER 6. TEMPORAL-DIFFERENCE LEARNING one-step TD, because it is a special case

Chapter 6: Temporal Difference Learningweb.stanford.edu/.../rich_sutton_slides/11-12-TD.pdf · 98 CHAPTER 6. TEMPORAL-DIFFERENCE LEARNING one-step TD, because it is a special case

Documents

An Application of Temporal Difference Learning to Draughts · trained by the method of temporal difference learning. It achieves a high level of play simply by self play, with minimal

An Application of Temporal Difference Learning to Draughts · trained by the method of temporal difference learning. It achieves a high level of play simply by self play, with minimal

Documents

True Online Temporal-Difference Learning

True Online Temporal-Difference Learning

Documents

Actuator Capabilities - Compact Automation€¦ · Actuator Capabilities The Compact Difference... At Compact we make the customer central to everything we do. CommonApplications:

Actuator Capabilities - Compact Automation€¦ · Actuator Capabilities The Compact Difference... At Compact we make the customer central to everything we do. CommonApplications:

Documents

Lecture 18: Temporal-Difference Learning

Lecture 18: Temporal-Difference Learning

Documents

Lecture 10: Q-Learning, Function Approximation, Temporal ...katselis.web.engr.illinois.edu/ECE586/Lecture10.pdfLecture 10: Q-Learning, Function Approximation, Temporal Difference Learning

Lecture 10: Q-Learning, Function Approximation, Temporal ...katselis.web.engr.illinois.edu/ECE586/Lecture10.pdfLecture 10: Q-Learning, Function Approximation, Temporal Difference Learning

Documents

Can Temporal-Difference and Q-Learning Learn ...

Can Temporal-Difference and Q-Learning Learn ...

Documents

Reinforcement Learning Introduction Passive Reinforcement Learning Temporal Difference Learning Active Reinforcement Learning Applications Summary.

Reinforcement Learning Introduction Passive Reinforcement Learning Temporal Difference Learning Active Reinforcement Learning Applications Summary.

Documents

An Analysis Of Temporal-difference Learning With …jnt/Papers/J063-97-bvr-td.pdf · Abstract— We discuss the temporal-difference learning algo-rithm, as applied to approximating

An Analysis Of Temporal-difference Learning With …jnt/Papers/J063-97-bvr-td.pdf · Abstract— We discuss the temporal-difference learning algo-rithm, as applied to approximating

Documents

Classical Conditioning III: Temporal difference …yael/PSY338/4 Classical Conditioning III...Classical Conditioning III: Temporal difference learning ... Ivan Pavlov recap: animals

Classical Conditioning III: Temporal difference …yael/PSY338/4 Classical Conditioning III...Classical Conditioning III: Temporal difference learning ... Ivan Pavlov recap: animals

Documents

AN EXPLORATION OF COMPACT FINITE DIFFERENCE …AN EXPLORATION OF COMPACT FINITE DIFFERENCE METHODS FOR THE NUMERICAL SOLUTION OF PDE by Mohammad Ozair Ahmed Department of Applied Mathematics

AN EXPLORATION OF COMPACT FINITE DIFFERENCE …AN EXPLORATION OF COMPACT FINITE DIFFERENCE METHODS FOR THE NUMERICAL SOLUTION OF PDE by Mohammad Ozair Ahmed Department of Applied Mathematics

Documents