Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 543848, 10 pageshttp://dx.doi.org/10.1155/2013/543848

Research ArticleFractional Variational Iteration Method and Its Application toFractional Partial Differential Equation

Asma Ali Elbeleze,1 Adem KJlJçman,2 and Bachok M. Taib1

1 Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800 Nilai, Malaysia2 Department of Mathematics, Faculty of Science, University Putra Malaysia, 43400 UPM, Serdang, Selangor Darul Ehsan, Malaysia

Correspondence should be addressed to Adem Kılıcman; akilicman@putra.upm.edu.my

Received 13 March 2013; Accepted 5 June 2013

Academic Editor: Mufid Abudiab

Copyright © 2013 Asma Ali Elbeleze et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations influidmechanics and in financial models.The fractional derivatives are described in Riemann-Liouville sense. To show the efficiencyof the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, andfractional Black-Scholes equation are investigated.

1. Introduction

The topic of fractional calculus (theory of integration and dif-ferentiation of an arbitrary order) was started over 300 yearsago. Recently, fractional differential equations have attractedmany scientists and researchers due to the tremendous use influidmechanics, mathematical biology, electrochemistry, andphysics. For example, differential equations with fractionalorder have recently proved to be suitable tools to modelingofmany physical phenomena [1] and the fluid-dynamic trafficmodelwith fractional derivative [2], andnonlinear oscillationof earthquake can be modeled with fractional derivatives [3].

There are several types of time fractional differentialequations.

(1) Fractional Klein-Gordon equations

𝜕𝛼

𝑢 (𝑥, 𝑡)

𝜕𝑡𝛼−𝜕2

𝑢 (𝑥, 𝑡)

𝜕𝑥2+ 𝑎𝑢 (𝑥, 𝑡) + 𝑏𝑢

2

+ 𝑐𝑢3

= 𝑓 (𝑥, 𝑡) , 𝑥 ∈ 𝑅.

(1)

This model is obtained by replacing the order timederivative with the fractional derivative of order 𝛼.The linear and nonlinear Klein-Gordon equations areused to modeling many problems in classical andquantum mechanics and condensed matter physics.

For example, nonlinear sine Klein-Gordon equationmodels a Josephson junction [4, 5].

(2) Fractional Burger’s equation

𝜕𝛼

𝑢 (𝑥, 𝑡)

𝜕𝑡𝛼=𝜕2

𝑢 (𝑥, 𝑡)

𝜕𝑥2+𝜕𝑢 (𝑥, 𝑡)

𝜕𝑥+ 𝑓 (𝑥, 𝑡) , 𝑥 ∈ 𝑅. (2)

In general, fractional Burger’s model is derived fromwell-known Burger’s equation model by replacing theordinary time derivatives to fractional order timederivatives. Reference [6] has investigated unsteadyflows of viscoelastic fluids with fractional Burger’smodel and fractional generalized Burger’s modelthrough channel (annulus) tube and solutions forvelocity field.

(3) Fractional Black-Scholes European option pricingequationsIn financial model the fractional Black-Scholes equa-tion is obtained by replacing the order of derivativewith a fractional derivative order [10].

𝜕𝛼V

𝜕𝑡𝛼+𝜎𝑥2

2

𝜕2V

𝜕𝑥2+ 𝑟 (𝑡) 𝑥

𝜕V

𝜕𝑥− 𝑟 (𝑡) V = 0,

(𝑥, 𝑡) ∈ 𝑅+

× (0, 𝑇) , 0 < 𝛼 ≤ 1,

(3)

2 Mathematical Problems in Engineering

where V(𝑥, 𝑡) is the European call option price at assetprice 𝑥 and at time 𝑡, 𝑇 is the maturity, 𝑟(𝑡) is the risk-free interest rate, and 𝜎(𝑥, 𝑡) represents the volatilityfunction of underlying asset.The payoff functions are

V𝑐(𝑥, 𝑡) = max (𝑥 − 𝐸, 0) ,

V𝑝(𝑥, 𝑡) = max (𝐸 − 𝑥, 0) ,

(4)

where V𝑐(𝑥, 𝑡) and V

𝑝(𝑥, 𝑡) are the value of the Euro-

pean call and put options, respectively, 𝐸 denotesthe expiration price for the option, and the functionmax(𝑥, 0) gives the large value between 𝑥 and 0. TheBlack-Scholes equation is one of the most significantmathematical models for a financial market. Thisequation is used to submit a reasonable price for callor put options based on factors such as underlyingstock volatility and days to expiration.

Formerly, [7] investigated approximate analytical solu-tion of fractional nonlinear Klein-Gordon equation (1) when0 < 𝛼 ≤ 1 by using HPM, while [8] solved this equationby using HAM also when 1 ≤ 𝛼 < 2. Reference [9] solvedthe coupled Klein-Gordon equation with time fractionalderivative by ADM. References [10, 11] solved fractionalBlack-Scholes equations by using HPM using Sumudu andLaplace transforms, respectively. Reference [12] gave the exactsolution of fractional Burgers equation, while [13] used DTMto find the approximate and exact solution of space- andtime fractional Burgers equations. Reference [14] solved thisequation by using VIM.

The variational iteration method [15–29] is one ofapproaches to provide an analytical approximation solutionsto linear and nonlinear problems. The fractional variationaliterationmethod with Riemann-Liouville derivative was pro-posed byWu and Lee [30] and applied to solve time fractionaland space fractional diffusion equations. Furthermore Wu[31] explained a possible use of the fractional variationaliteration method as a fractal multiscale method. Recentlyfractional variational iteration method has been used toobtain approximate solutions of fractional Riccati differentialequation [32].

The objective of this paper is to extend the applicationof the fractional variational iteration method to obtain ana-lytical approximate solution for some fractional partial dif-ferential equations.These equations include fractional Klein-Gordon equation (1), Burgers equation (2), and fractionalBlack-Scholes equations (3).

Motivated and inspired by the ongoing research in thisfield, we will consider the following time fractional differen-tial equation:

𝜕𝛼

𝑢 (𝑥, 𝑡)

𝜕𝑡𝛼= 𝑅 [𝑥] 𝑢 (𝑥, 𝑡) + 𝑞 (𝑥, 𝑡) ,

0 < 𝛼 ≤ 1, 𝑥 ∈ R, 𝑡 > 0,(5)

with initial condition

𝑢 (𝑥, 0) = 𝑓 (𝑥) , (6)

where 𝜕𝛼/𝜕𝑡𝛼 is modified Riemann-Liouville derivative [33–35] of order 𝛼 defined in Section 2, 𝑓(𝑥) and 𝑞(𝑥, 𝑡) arecontinuous functions, 𝑅[𝑥]𝑢(𝑥, 𝑡) are linear and nonlinearoperators, and 𝑢(𝑥, 𝑡) is unknown function.

To solve the problem (1)-(2), we consider the FVIM inthis work. This method is based on variational iterationmethod [19, 36] and modified Riemann-Liouville derivativesproposed by Jumarie.

This paper is organized as follows. in Section 2 somebasic definitions of fractional calculus theory are given. InSection 3, the solution procedure of the fractional iterationmethod is given; we present the application of the FVIMfor some fractional partial differential equations in Section 4.The conclusions are drawn in Section 5.

2. Fractional Calculus

2.1. Fractional Derivative via Fractional Difference

Definition 1. The left-sides Riemann-Liouville fractional inte-gral operator of order 𝛼 ≥ 0, of a function 𝑓 ∈ 𝐶

𝜇, 𝜇 ≥ −1, is

a defined as

𝐽𝛼

𝑓 (𝑥) =1

Γ (𝛼)∫

𝑥

0

(𝑥 − 𝑡)𝛼−1

𝑓 (𝑡) 𝑑𝑡, 𝛼 > 0,

𝑥 > 0, 𝐽0

𝑓 (𝑥) = 𝑓 (𝑥) .

(7)

Definition 2. Themodified Riemann-Liouville derivative [34,35] is defined as

𝐷𝑥

𝛼𝑓 (𝑥) =

1

Γ (𝑛 − 𝛼)

𝑑𝑛

𝑑𝑥𝑛∫

𝑥

0

(𝑥 − 𝑡)𝑛−𝛼

(𝑓 (𝑡) − 𝑓 (0)) 𝑑𝑡,

(8)

where 𝑥 ∈ [0, 1], 𝑛 − 1 ≤ 𝛼 < 𝑛, and 𝑛 ≥ 1.

Definition 3. Let𝑓 : 𝑅 → 𝑅, 𝑥 → 𝑓(𝑥) denote a continuous(but not necessarily differentiable) function, and let ℎ > 0

denote a constant discretization span. Define the forwardoperator FW(ℎ) by the equality

FW (ℎ) 𝑓 (𝑥) := 𝑓 (𝑥 + ℎ) . (9)

Then the fractional difference of order 𝛼, 0 < 𝛼 < 1, of 𝑓(𝑥)is defined by the expression

Δ(𝛼)

𝑓 (𝑥) := (FW − 1)𝛼

𝑓 (𝑥)

=

∞

∑

𝑘=0

(−1)𝑘

(𝛼

𝑘)𝑓 [𝑥 + (𝑎 − 𝑘) ℎ] ,

(10)

and its fractional derivative of order 𝛼 is defined by the limit

𝑓𝛼

(𝑥) = lim𝑥→0

Δ(𝛼)

[𝑓 (𝑥) − 𝑓 (0)]

ℎ𝛼. (11)

Equation (11) is defined as Jumarie fractional derivative oforder 𝛼 which is equivalent to (8). For more details we referthe reader to [35].

Mathematical Problems in Engineering 3

For 0 < 𝛼 ≤ 1, some properties of the fractional modifiedRiemann-Liouville derivative.

Fractional Leibnitz product law:

0𝐷𝛼

𝑥(𝑢V) = 𝑢(𝛼)V + 𝑢V(𝛼), (12)

fractional Leibnitz formulation:

0𝐼𝛼

𝑥𝐷𝛼

𝑥(𝑢V) = 𝑓 (𝑥) − 𝑓 (0) , (13)

The fractional integration by parts formula:

𝑎𝐼𝛼

𝑏(𝑢(𝛼)V) = (𝑢V)|𝑏

𝑎−𝑎𝐼𝛼

𝑏(𝑢V(𝛼)) . (14)

Definition 4. Fractional derivative of compounded function[34, 35] is defined as

𝑑𝛼

𝑓 ≅ Γ (1 + 𝛼) 𝑑𝑓, 0 < 𝛼 < 1. (15)

Definition 5 (see [34, 35]). The integral with respect to (𝑑𝑡)𝛼 isdefined as the solution of the fractional differential equation

𝑑𝑥 ≅ 𝑓 (𝑥) (𝑑𝑡)𝛼

, 𝑡 ≥ 0, 𝑥 (0) = 0, 0 < 𝛼 < 1. (16)

Lemma 6 (see [34, 35]). Let 𝑓(𝑥) denote a continuousfunction; then the solution of (2) is defined as

𝑦 = ∫

𝑥

0

𝑓 (𝜏) (𝑑𝜏)𝛼

= 𝛼∫

𝑥

0

(𝑥 − 𝜏)𝛼−1

𝑓 (𝜏) 𝑑𝜏, 0 < 𝛼 < 1,

(17)

that is,

𝐽𝛼

𝑓 (𝑥) = (1

Γ (𝛼))∫

𝑥

0

(𝑥 − 𝜏)𝛼−1

𝑓 (𝜏) 𝑑𝜏

=1

(Γ (𝛼 + 1))∫

𝑥

0

𝑓 (𝜏) (𝑑𝜏)𝛼

.

(18)

For example, with 𝑓(𝑥) = 𝑥𝛽 in (7), one obtains

∫

𝑥

0

𝑡𝛽

(𝑑𝑡)𝛼

=Γ (𝛽 + 1) Γ (𝛼 + 1)

Γ (𝛼 + 𝛽 + 1)𝑥𝛼+𝛽

, 0 < 𝛼 < 1. (19)

Definition 7. The Mittag-Leffler function 𝐸𝛼(𝑧) with 𝛼 > 0

is defined by the following series representation, valid in thewhole complex plane [37]:

𝐸𝛼(𝑧) =

∞

∑

0

𝑧𝑛

Γ (𝛼𝑛 + 1). (20)

3. Fractional Variational Iteration Method

To describe the solution procedure of fractional variationaliterationmethod, we consider the time-fractional differentialequations (1)–(3).

According to variational iteration method we constructthe following correction function:

𝑢𝑛+1

(𝑥, 𝑡)

= 𝑢𝑛(𝑥, 𝑡) + 𝐽

𝛼

𝑡[𝜇(

𝜕𝛼

𝑢 (𝑥, 𝑠)

𝜕𝑠𝛼− 𝑅 [𝑥] �� (𝑥, 𝑠) − 𝑞 (𝑥, 𝑠))]

= 𝑢𝑛(𝑥, 𝑡) +

1

Γ (𝛼)

× ∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

{𝜇 (𝑠) (𝜕𝛼

𝑢 (𝑥, 𝑠)

𝜕𝑠𝛼

−𝑅 [𝑥] �� (𝑥, 𝑠) − 𝑞 (𝑥, 𝑠))} 𝑑𝑠,

(21)

where 𝜇 is the general Lagrange multiplier which can bedefined optimally via variational theory [22] and ��(𝑥, 𝑡) is therestricted variation, that is, 𝛿��(𝑥, 𝑡) = 0.

By using (7), we obtain a new correction functional

𝑢𝑛+1

(𝑥, 𝑡) = 𝑢𝑛(𝑥, 𝑡) +

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜇 (𝑠) (𝜕𝛼

𝑢 (𝑥, 𝑠)

𝜕𝑠𝛼− 𝑅 [𝑥] �� (𝑥, 𝑠)

−𝑞 (𝑥, 𝑠))} (𝑑𝑠𝛼

) .

(22)

Making the above functional stationary the following condi-tions can be obtained:

𝛿𝑢𝑛+1

(𝑥, 𝑡) = 𝛿𝑢𝑛(𝑥, 𝑡) +

𝛿

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜇 (𝑠) (𝜕𝛼

𝑢 (𝑥, 𝑠)

𝜕𝑠𝛼− 𝑅 [𝑥] �� (𝑥, 𝑠)

−𝑞 (𝑥, 𝑠))} (𝑑𝑠𝛼

) .

(23)

Now, we can get the coefficients of 𝛿𝑢 to zero:

1 + 𝜇 (𝑠) = 0,𝜕𝛼

𝜇 (𝑠)

𝜕𝑠𝛼= 0. (24)

So, the generalized Lagrange multiplier can be identified as

𝜇 = −1. (25)

4 Mathematical Problems in Engineering

Then we obtain the following iteration formula by substitut-ing (25) in (23):

𝛿𝑢𝑛+1

(𝑥, 𝑡) = 𝛿𝑢𝑛(𝑥, 𝑡) −

𝛿

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜇 (𝑠) (𝜕𝛼

𝑢 (𝑥, 𝑠)

𝜕𝑠𝛼− 𝑅 [𝑥] �� (𝑥, 𝑠)

−𝑞 (𝑥, 𝑠))} (𝑑𝑠𝛼

) ,

(26)

where 0 < 𝛼 ≤ 1 and 𝑢0(𝑥, 𝑡) is an initial approximation

which can be freely chosen if it satisfies the initial andboundary conditions of the problem.

4. Applications

In this section, we have applied fractional variational iterationmethod (FVIM) to fractional partial differential equations.

Example 8. In this example we consider the following frac-tional nonlinear Klein-Gordon differential equation:

𝜕𝛼

𝑢

𝜕𝑡𝛼−𝜕2

𝑢

𝜕𝑥2+ 𝑢2

= 0, 𝑡 ≥ 0, 0 < 𝛼 ≤ 1, (27)

subject to initial condition

𝑦 (𝑥, 0) = 1 + sin (𝑥) . (28)

Substituting (𝑎 = 0, 𝑏 = 0 and 𝑐 = 1) in (1). Construction theiteration formula as follows:

𝑢𝑛+1

(𝑥, 𝑡) = 𝑢𝑛(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢𝑛

𝜕𝑠𝛼−𝜕2

𝑢𝑛

𝜕𝑥2+ 𝑢2

𝑛} (𝑑𝑠)

𝛼

.

(29)

Taking the initial value 𝑢0(𝑥, 𝑡) = 1+ sin(𝑥)we can derive the

first approximate 𝑢1(𝑥, 𝑡) as follows:

𝑢1(𝑥, 𝑡) = 𝑢

0(𝑥, 𝑡) −

1

Γ(𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢0

𝜕𝑠𝛼−𝜕2

𝑢0

𝜕𝑥2+ 𝑢2

0} (𝑑𝑠)

𝛼

= 1 + sin (𝑥) − 𝑡𝛼+1

Γ (𝛼 + 1)

× (1 + 3 sin (𝑥) + sin2 (𝑥)) ,

𝑢2(𝑥, 𝑡) = 𝑢

1(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢1

𝜕𝑠𝛼−𝜕2

𝑢1

𝜕𝑥2+ 𝑢2

1} (𝑑𝑠)

𝛼

= 1 + sin (𝑥) − 𝑡𝛼+1

Γ (𝛼 + 1)(1 + 3 sin (𝑥) + sin2 (𝑥))

+𝑡2𝛼+1

Γ (2𝛼 + 1)(11 sin (𝑥) + 12sin2 (𝑥) + 2sin3 (𝑥)) ,

𝑢3(𝑥, 𝑡) = 𝑢

2(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢2

𝜕𝑠𝛼−𝜕2

𝑢2

𝜕𝑥2+ 𝑢2

2} (𝑑𝑠)

𝛼

= 1 + sin (𝑥) − 𝑡𝛼

Γ (𝛼 + 1)(1 + 3 sin (𝑥) + sin2 (𝑥))

+𝑡2𝛼

Γ (2𝛼 + 1)(11 sin (𝑥) + 12sin2 (𝑥) + 2sin3 (𝑥))

+𝑡3𝛼

Γ (3𝛼 + 1)(18 − 57 sin (𝑥) − 160sin2 (𝑥)

−82sin3 (𝑥) − 10sin4 (𝑥)) .(30)

Thus, the approximate solution is

𝑢 (𝑥, 𝑡) = 1 + sin (𝑥) − 𝑡𝛼

Γ (𝛼 + 1)

× (1 + 3 sin (𝑥) + sin2 (𝑥)) + 𝑡2𝛼

Γ (2𝛼 + 1)

× (11 sin (𝑥) + 12sin2 (𝑥) + 2sin3 (𝑥))

+𝑡3𝛼

Γ (3𝛼 + 1)(18 − 57 sin (𝑥) − 160sin2 (𝑥)

− 82sin3 (𝑥) − 10sin4 (𝑥)) + ⋅ ⋅ ⋅ .(31)

In Figures 1 and 2 we have shown the surface of 𝑢(𝑥, 𝑡)corresponding to the values 𝛼 = 0.01, 0.5, 1 for FVIM andHPM; the two figures indicate that the differences amongVIM and HPM, and the exact solution in Example 8 arenegligible when 𝛼 = 0.5, 1 while when 𝛼 = 0.01 the resultsof VIM and HPM somewhat diverge from the exact solution.

Mathematical Problems in Engineering 5

0

0.002

0.004

0.006

0.008

−2

−1

1

0

0−50

−100

−150

−200

x

t

u(x,t)

(a)

0

0.002

0.004

0.006

0.008

−2

−1

0

1

0−50

−100

−150

−200

x

t

u(x,t)

(b)

0

0.002

0.004

0.006

0.008

0.01

−2

−1

0

1

x

t

u(x,t)

1.81.41

0.60.2

(c)

0

0.002

0.004

0.006

0.008

0.01

−2

−1

0

1

x

t

u(x,t)

1.81.41

0.60.2

(d)

Figure 1:The surface shows the solution 𝑢(𝑥, 𝑡) for (27) with initial condition (28): FVIM results are, respectively, (a) 𝛼 = 0.01 and (c) 𝛼 = 0.5;HPM [7] results are, respectively, (b) 𝛼 = 0.01 and (d) 𝛼 = 0.5.

Example 9. Weconsider the one-dimensional linear inhomo-geneous fractional Burger equation

𝜕𝛼

𝑢

𝜕𝑡𝛼+𝜕𝑢

𝜕𝑥−𝜕2

𝑢

𝜕𝑥2=

2𝑡2−𝛼

Γ (3 − 𝛼)+ 2𝑥 − 2,

𝑡 > 0, 𝑥 ∈ 𝑅, 0 < 𝛼 ≤ 1,

(32)

subject to initial condition

𝑢 (𝑥, 0) = 𝑥2

. (33)

By construction the iteration formula as follows:

𝑢𝑛+1

(𝑥, 𝑡) = 𝑢𝑛(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢𝑛

𝜕𝑡𝛼+𝜕𝑢𝑛

𝜕𝑥−𝜕2

𝑢𝑛

𝜕𝑥2

−2𝑡2−𝛼

Γ (3 − 𝛼)− 2𝑥 + 2} (𝑑𝑠)

𝛼

.

(34)

6 Mathematical Problems in Engineering

0

0.002

0.004

0.006

0.008

−2

−1

0

1

x

t

u(x,t)

1.81.41

0.60.2

(a)

0

0.002

0.004

0.006

0.008

−2

−1

1

0x t

u(x,t)

1.8

1.20.80.4

(b)

0

0.002

0.004

0.006

0.008

0.01

−2

−1

0

1

2

x

t

u(x,t)

1.81.20.80.4

(c)

Figure 2: The surface shows the solution 𝑢(𝑥, 𝑡) for (27) with initial condition (28): (a) FVIM when 𝛼 = 1, (b) HPM [7] when 𝛼 = 1, and (c)exact solution.

Taking the initial value 𝑢0(𝑥, 𝑡) = 0 we can derive the first

approximate 𝑢1(𝑥, 𝑡) as follows:

𝑢1(𝑥, 𝑡) = 𝑢

0(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢0

𝜕𝑡𝛼+𝜕𝑢0

𝜕𝑥−𝜕2

𝑢0

𝜕𝑥2

−𝑡2−𝛼

Γ (3 − 𝛼)− 2𝑥 + 2} (𝑑𝑠)

𝛼

= 𝑥2

+ 𝑡2

,

𝑢2(𝑥, 𝑡) = 𝑢

1(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢1

𝜕𝑡𝛼+𝜕𝑢1

𝜕𝑥−𝜕2

𝑢0

𝜕𝑥2

−𝑡2−𝛼

Γ (3 − 𝛼)− 2𝑥 + 2} (𝑑𝑠)

𝛼

= 𝑥2

+ 𝑡2

...𝑢𝑛(𝑥, 𝑡) = 𝑥

2

+ 𝑡2

. (35)

Mathematical Problems in Engineering 7

1012

864200

1

2

3 0.01

0.02

0.03

0.04

0.05

t

x

u(x,t)

(a)

10864200

1

2

3 0.01

0.02

0.03

0.04

0.05

t

x

u(x,t)

(b)

10864200

1

2

3 0.01

0.02

0.03

0.04

0.05

t

x

u(x,t)

(c)

Figure 3: The surface shows the solution 𝑢(𝑥, 𝑡) for (36) with initial condition (37): (a) FVIM (𝛼 = 1), (b) HPM [10] (𝛼 = 1), and (c) FVIM(𝛼 = 0.01).

So, the exact solution 𝑢(𝑥, 𝑡) = 𝑥2

+ 𝑡2 follows immediately.

The exact solution is obtained by using two iterations and thisis dependent on proper selection of initial guess 𝑢

0(𝑥, 𝑡).

Example 10. We consider the following fractional Black-Scholes option pricing equation [38] as follows:

𝜕𝛼

𝑢

𝜕𝑡𝛼=𝜕2

𝑢

𝜕𝑥2+ (𝑘 − 1)

𝜕𝑢

𝜕𝑥− 𝑘𝑢, 0 < 𝛼 ≤ 1, (36)

where 𝑘 is the risk-free interest rate subject to initial condition

𝑢 (𝑥, 0) = max (𝑒𝑥 − 1, 0) . (37)

The exact solution for special case 𝛼 = 1 is given by

𝑢 (𝑥, 𝑡) = max (𝑒𝑥 − 1, 0) 𝑒−𝑘𝑡 +max (𝑒𝑥, 0) (1 − 𝑒−𝑘𝑡) . (38)

8 Mathematical Problems in Engineering

By construction the iteration formula as follows:

𝑢𝑛+1

(𝑥, 𝑡) = 𝑢𝑛(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢𝑛

𝜕𝑠𝛼−𝜕2

𝑢𝑛

𝜕𝑥2

+ (𝑘 − 1)𝜕𝑢𝑛

𝜕𝑥− 𝑘𝑢𝑛} (𝑑𝑠)

𝛼

.

(39)

Taking the initial value 𝑢0(𝑥, 𝑡) = max(𝑒𝑥−1, 0)we can derive

the first approximate 𝑢1(𝑥, 𝑡) as follows:

𝑢1(𝑥, 𝑡) = 𝑢

0(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢0

𝜕𝑠𝛼−𝜕2

𝑢0

𝜕𝑥2+ (𝑘 − 1)

𝜕𝑢0

𝜕𝑥− 𝑘𝑢0

} (𝑑𝑠)𝛼

= max (𝑒𝑥 − 1, 0) −max (𝑒𝑥, 0)(−𝑘𝑡𝛼

)

Γ (𝛼 + 1)

+max (𝑒𝑥 − 1, 0)(−𝑘𝑡𝛼

)

Γ (𝛼 + 1),

𝑢2(𝑥, 𝑡) = 𝑢

1(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢1

𝜕𝑠𝛼−𝜕2

𝑢1

𝜕𝑥2

+ (𝑘 − 1)𝜕𝑢1

𝜕𝑥− 𝑘𝑢1} (𝑑𝑠)

𝛼

= max (𝑒𝑥 − 1, 0) −max (𝑒𝑥, 0)

× ((−𝑘𝑡𝛼

)

Γ (𝛼 + 1)+

(−𝑘𝑡𝛼

)2

Γ (2𝛼 + 1))

+max (𝑒𝑥 − 1, 0)((−𝑘𝑡𝛼

)

Γ (𝛼 + 1)+

(−𝑘𝑡𝛼

)2

Γ (2𝛼 + 1))

...

𝑢3(𝑥, 𝑡) = 𝑢

2(𝑥, 𝑡) −

1

Γ (𝛼 + 1)

× ∫

𝑡

0

{𝜕𝛼

𝑢2

𝜕𝑠𝛼−𝜕2

𝑢2

𝜕𝑥2

+ (𝑘 − 1)𝜕𝑢2

𝜕𝑥− 𝑘𝑢2} (𝑑𝑠)

𝛼

= max (𝑒𝑥 − 1, 0) −max (𝑒𝑥, 0)

× ((−𝑘𝑡𝛼

)

Γ (𝛼 + 1)+

(−𝑘𝑡𝛼

)2

Γ (2𝛼 + 1)+

(−𝑘𝑡𝛼

)3

Γ (3𝛼 + 1))

+max (𝑒𝑥 − 1, 0)

× ((−𝑘𝑡𝛼

)

Γ (𝛼 + 1)+

(−𝑘𝑡𝛼

)2

Γ (2𝛼 + 1)+

(−𝑘𝑡𝛼

)3

Γ (3𝛼 + 1))

...

𝑢𝑛(𝑥, 𝑡) = max (𝑒𝑥 − 1, 0) 𝐸

𝛼(−𝑘𝑡𝛼

)

+max (𝑒𝑥, 0) (1 − 𝐸𝛼(−𝑘𝑡𝛼

)) ,

(40)

so that the solution 𝑢(𝑥; 𝑡) of the problem is given by

𝑢𝑛(𝑥, 𝑡) = max (𝑒𝑥 − 1, 0) 𝐸

𝛼(−𝑘𝑡𝛼

)

+max (𝑒𝑥, 0) (1 − 𝐸𝛼(−𝑘𝑡𝛼

)) ,

(41)

where 𝐸𝛼(𝑧) is Mittag-Leffler function in one parameter.

Equation (41) represents the closed form solution of thefractional Black-Scholes equation (36). Now for the standardcase 𝛼 = 1, this series has the closed form of the solution𝑢(𝑥; 𝑡) = max(𝑒𝑥 − 1, 0)𝑒−𝑘𝑡 + max(𝑒𝑥, 0)(1 − 𝑒−𝑘𝑡), which isan exact solution of the given Black-Scholes equation (36) for𝛼 = 1.

In Figure 3 we have shown the surface of 𝑢(𝑥, 𝑡) corre-sponding to the value (𝛼 = 1 for FVIM&HPM and for FVIM𝛼 = 0.01).

5. Conclusion

Variational iteration method has been known as a powerfulmethod for solving many fractional equations such as partialdifferential equations, integrodifferential equations, and somany other equations. In this paper, based on the variationaliteration method and modified Riemann-Liouville deriva-tive, we have presented a general framework of fractionalvariational iteration method for analytical and numericaltreatment of fractional partial differential equations in fluidmechanics and in financial models. All of the examplesconcluded that the fractional variational iteration methodis powerful and efficient in finding analytical approximatesolutions as well as numerical solutions. For example, theresults of Examples 8 and 10 illustrate that the presentmethodis in excellent agreement with those of HPM and exactsolution, where the obtained solution is shown graphically.Further, in Example 9 we got the exact solution in twoiterations. The basic idea described in this paper is expectedto be further employed to solve other similar linear andnonlinear problems in fractional calculus. Maple has beenused for presenting graph of solution in the present paper.

Mathematical Problems in Engineering 9

Acknowledgments

The second author gratefully acknowledges that this researchpartially supported by Ministry of Higher Education(MOHE), Malaysia under the ERGS Grant 5527068.

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