Applied Mathematics and Computation 224 (2013) 166–177

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Numerical solution of Burgers’ equation with modified cubicB-spline differential quadrature method

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.08.071

⇑ Corresponding author.E-mail addresses: geetadma@gmail.com (G. Arora), bksingh0584@gmail.com (B.K. Singh).

Geeta Arora, Brajesh Kumar Singh ⇑Department of Mathematics, School of Allied Sciences, Graphic Era Hill University, Dehradun 248002, Uttarakhand, India

a r t i c l e i n f o

Keywords:Burgers’ equationCubic B-splineModified cubic B-splineDifferential quadrature method (DQM)Thomas algorithmSSP-RK43 scheme

a b s t r a c t

In this paper, a new numerical method, ‘‘modified cubic-B-spline differential quadraturemethod (MCB-DQM)’’ is proposed to find the approximate solution of the Burgers’ equation.The modified cubic-B-spline basis functions are used in differential quadrature to determinethe weighting coefficients. The MCB-DQM is used in space, and the optimal four-stage,order three strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme isused in time for solving the resulting system of ordinary differential equations. To checkthe efficiency and accuracy of the method, four examples of Burgers’ equation are includedwith their numerical solutions, L2 and L1 errors and comparisons are done with the resultsgiven in the literature. The proposed method produces better results as compared to theresults obtained by almost all the schemes available in the literature, and approaching tothe exact solutions. The presented method is seen to be easy, powerful, efficient and eco-nomical to implement as compared to the existing techniques for finding the numericalsolutions for various kinds of linear/nonlinear physical models.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

We consider the well known one dimensional nonlinear Burgers’ equation

@u@tþ au

@u@x� m

@2u@x2 ¼ 0; ðx; tÞ 2 X� ½0; T�; ð1:1Þ

where X ¼ ða; bÞ, with the initial condition

uðx;0Þ ¼ f ðxÞ; x 2 ½a; b�; ð1:2Þ

and the boundary conditions

uða; tÞ ¼ 0 and uðb; tÞ ¼ 0; t 2 ½0; T�; ð1:3Þ

where m > 0 is a small parameter known as the coefficient of kinematic viscosity and a is some positive constant. Such typeof equations was first introduced by Bateman [5]. Also, he proposed the steady-state solution of the problem. Burgers [6,7]has introduced this equation to capture some features of turbulent fluid in a channel caused by the interaction of the oppo-site effects of convection and diffusion, and hence Eq. (1.1) is referred to as ‘‘Burgers’ equation’’. The structure of Burgers’equation is similar to the one dimensional Navier–Stoke’s equation without the stress term. It is the simplest model of non-linear partial differential equation for diffusive waves in fluid dynamics. This model arises in many physical problems

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 167

including one-dimensional sound/shock waves in a viscous medium, waves in fluid filled viscous elastic tubes, magneto-hydrodynamic waves in a medium with finite electrical conductivity, mathematical modeling of turbulent fluid, and in con-tinuous stochastic processes.

In the last years, a great deal of effort to compute efficiently the numerical solutions of the Burgers’ equation for small andlarge both values of the kinematic viscosity has been expanded. The Burgers’ equation is solved for both the infinite and thefinite domain [9]. The various numerical techniques to compute the numerical solutions of the Burgers’ equation are: auto-matic differentiation method [1], finite elements method [32,34], Galerkin finite element method [13], cubic B-splines col-location method [12], cubic B-spline quasi-interpolation [15], modified cubic B-spline collocation method [31], spectralcollocation method [27], sinc differential quadrature method [22], polynomial based differential quadrature method [23],cubic B-spline differential quadrature method [20], quartic B-splines differential quadrature method [24], quartic B-splinescollocation method [38], quadratic B-splines finite difference element method [4,33], fourth-order finite difference method[14], factorized diagonal Padé approximation [2], non-polynomial spline approach [37], a novel numerical scheme [45], ex-plicit and exact-explicit finite difference methods [25], Hopf–Cole transformation [11,25], least-squares quadratic B-splinesfinite element method [26], reproducing kernel function method [41], implicit fourth-order compact finite difference [29],weighted average differential quadrature method [16], variational method [3], parameter-uniform implicit differencescheme [17], one dimensional fourier expansion [30], etc.

The differential quadrature method (DQM), is an efficient technique to solve partial differential equations (PDEs), was firstintroduced by Bellman et al. [8]. It was further improved by Quan and Chang [35,36] to solve the weighting coefficients. Var-ious kinds of test functions such as spline functions, Lagrange interpolation polynomials, sinc function [35,36,22], etc. are used todetermine the weighting coefficients, for further details on DQM we refer to [42,43,40,36].

B-splines are a set of special spline functions that can be used to construct piece-wise polynomial by computing theappropriate linear combination. These functions have their computational advantage from the fact that any B-spline basisfunction of order m is nonzero over at most m adjacent intervals and zero otherwise. Due to smoothness and capabilityto handle local phenomena, B-spline basis functions offer distinct advantages in comparison to other basis functions. CubicB-spline functions have already been used as basis functions to solve many physical models. Recently, Korkmaz and Dag [21]used cubic B-spline functions with DQM to solve advection–diffusion equation. Mittal and Jain [31] solved Burgers’ equationby modified cubic B-spline collocation method.

In this paper, a new numerical method, ‘‘modified cubic-B-spline differential quadrature method (MCB-DQM)’’ is pro-posed to find the approximate solution of the Burgers’ equation. In this method, the modified cubic-B-spline basis func-tions are used in DQM to determine the weighting coefficients (i.e., spatial derivatives) which produces the system offirst order ordinary differential equations (ODEs). The resulting system of ODEs is solved by the optimal four-stage, orderthree strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme [44]. The SSP-RK43 scheme needs lessstorage space that causes less accumulation of numerical errors. This is why we preferred SSP-RK43 scheme. TheMCB-DQM solutions to the Burgers’ equation have been computed without transforming the equation and without usingthe linearization. The comparison of the MCB-DQM numerical solutions with analytical solutions are presented to illus-trate the efficiency and adaptability of the method. The L2 and L1 errors are also evaluated and compared with resultsgiven in the literature.

This paper is organized as follows. In Section 2, the description of the modified cubic B-spline differential quadraturemethod is given. In Sections 3, procedure for implementation of method is described. Numerical examples are given to estab-lish the applicability and accuracy of the proposed method in Section 4. The conclusion is given in Section 5 that briefly sum-marizes the numerical outcomes.

2. Description of the method

The differential quadrature method (DQM) is an approximation to derivatives of a function using the weightedsum of the functional values at certain discrete points. Since the weighting coefficients are dependent on the spatialgrid spacing only, one can assume uniformly distributed N nodes/knots: a ¼ x1 < x2; . . . ; xN�1 < xN ¼ b such thatxiþ1 � xi ¼ h on the real axis. The first and second order spatial derivatives of the uðx; tÞ at any time on the knotxi for i ¼ 1; . . . ;N are given by

uxðxi; tÞ ¼XN

j¼1

aijuðxj; tÞ; for j ¼ 1; . . . ;N; ð2:1Þ

uxxðxi; tÞ ¼XN

j¼1

bijuðxj; tÞ; for j ¼ 1; . . . ;N; ð2:2Þ

where aij and bij are weighting coefficients of the first and second order derivatives with respect to x, respectively [8].The cubic B-spline basis functions at the knots are defined as follows

168 G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

ujðxÞ ¼1

h3

ðx� xj�2Þ3 x 2 ðxj�2; xj�1Þðx� xj�2Þ3 � 4ðx� xj�1Þ3 x 2 ðxj�1; xjÞðxjþ2 � xÞ3 � 4ðxjþ1 � xÞ3 x 2 ðxj; xjþ1Þðxjþ2 � xÞ3 x 2 ðxjþ1; xjþ2Þ0 otherwise;

8>>>>>>><>>>>>>>:

ð2:3Þ

where fu0;u1; . . . ;uN;uNþ1g forms a basis over the region ½a; b�. The values of cubic B-splines and its derivatives at the nodalpoints are tabulated in Table 0.1.

The cubic B-spline basis functions are modified in such way that the resulting matrix system of equations is diagonallydominant. The modified cubic B-spline basis functions at the knots are defined as follows [31]

/1ðxÞ ¼ u1ðxÞ þ 2u0ðxÞ/2ðxÞ ¼ u2ðxÞ �u0ðxÞ/jðxÞ ¼ ujðxÞ for j ¼ 3; . . . ;N � 2/N�1ðxÞ ¼ uN�1ðxÞ �uNþ1ðxÞ/NðxÞ ¼ uNðxÞ þ 2uNþ1ðxÞ

9>>>>>>=>>>>>>;

ð2:4Þ

where f/1;/2; . . . ;/Ng forms a basis over the region ½a; b�.

2.1. To determine the weighting coefficients

The first order derivative approximation is given by

/0kðxiÞ ¼XN

j¼1

aij/kðxjÞ; for i ¼ 1; . . . ;N; k ¼ 1; . . . ;N ð2:5Þ

For the first knot point x1, the approximation can be given as

/0kðx1Þ ¼XN

j¼1

a1j/kðxjÞ; for k ¼ 1; . . . ;N ð2:6Þ

which gives a tridiagonal system of equation as

6 10 4 1

1 4 1. .

. . .. . .

.

1 4 11 4 0

1 6

2666666666664

3777777777775

a11

a12

..

.

a1N�1

a1N

266666664

377777775¼

�6=h

6=h

0...

00

26666666664

37777777775

We apply well known ‘‘Thomas algorithm’’ to solve the resulting tridiagonal system of equations whose solution providesthe coefficients a11; a12; . . . ; a1N . Similarly, for the second knot point x2, the approximation can be given as

/0kðx2Þ ¼XN

j¼1

a2j/kðxjÞ; for k ¼ 1; . . . ;N ð2:7Þ

which again gives a tridiagonal system of equations as

Table 0.1The coefficients of cubic B-splines and its derivatives at knots xj .

xj�2 xj�1 xj xjþ1 xjþ2

ujðxÞ 0 1 4 1 0u0jðxÞ 0 3=h 0 �3=h 0

u00j ðxÞ 0 6=h2 �12=h2 �6=h2 0

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 169

6 10 4 1

1 4 1. .

. . .. . .

.

1 4 11 4 0

1 6

2666666666664

3777777777775

a21

a22

..

.

a2N�1

a2N

266666664

377777775¼

�3=h

03=h

0...

0

26666666664

37777777775

whose solution provides the coefficients a21; a22; . . . ; a2N . Proceeding in the similar manner, up to the second last knot pointxN�1 the coefficients ak1; ak2; . . . ; akN for k ¼ 3 . . . N � 1 are determined. At the knot point xN�1, the tridiagonal system of equa-tions is given as

6 10 4 1

1 4 1. .

. . .. . .

.

1 4 11 4 0

1 6

2666666666664

3777777777775

aN�11

aN�12

..

.

aN�1N�1

aN�1N

266666664

377777775¼

0...

0�3=h

03=h

26666666664

37777777775

Now, the matrix system of equations corresponding to the last knot point xN is given as

6 10 4 1

1 4 1. .

. . .. . .

.

1 4 11 4 0

1 6

2666666666664

3777777777775

aN1

aN2

..

.

aNN�1

aNN

266666664

377777775¼

0...

0�6=h

6=h

26666664

37777775;

which provides the coefficients aN1; aN2; . . . ; aNN . Thus, we have evaluated the weighting coefficient aij fori ¼ 1;2; . . . ;N; j ¼ 1;2; . . . ;N.

Using these coefficients, the weighting coefficient bij for i ¼ 1;2; . . . ;N; j ¼ 1;2; . . . ;N is evaluated as follows [40]

bij ¼ 2aij aij �1

xi � xj

� �; for i – j; and bii ¼ �

XN

i¼1;i – j

bij:

3. Implementation of method

On substituting the first and second order approximation of the spatial derivatives, obtained by using MCB-DQM, the Bur-gers’ Eq. (1.1) can be rewritten as

@ui

@t¼ m

XN

j¼1

bijuðxjÞ � auðxiÞXN

j¼1

aijuðxjÞ; i ¼ 1; . . . ;N: ð3:1Þ

Thus, Eq. (3.1) is reduced into a set of ordinary differential equations in time, that is, for i ¼ 1; . . . ;N, we have

dui

dt¼ LðuiÞ; ð3:2Þ

where L denotes a spatial nonlinear differential operator. There are various methods to solve this system of ODE. We pre-ferred the optimal four-stage, order three strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme [44]to solve the system of ODE. In this scheme the Eq. (3.2) is integrated from time t0 to t0 þ Mt through the following operations

uð1Þ ¼ um þ Mt2

LðumÞ

uð2Þ ¼ uð1Þ þ Mt2

Lðuð1ÞÞ

uð3Þ ¼ 23

um þ uð2Þ

3þ Mt

6Lðuð2ÞÞ

umþ1 ¼ uð3Þ þ Mt2

Lðuð3ÞÞ;

170 G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

and consequently the solution uðx; tÞ at a particular time level is completely known.

4. Numerical experiments and discussion

In this section, the numerical solutions by the proposed method (MCB-DQM) are evaluated for some examples of Burgers’equation. Existence of analytical solutions help to measure the accuracy of numerical methods. In the present study, theaccuracy and the efficiency of this method is measured for various numerical examples by evaluating the discrete L2 andL1 error norms which are defined as follows

Table 1Compar

x

0.20

0.40

0.6

0.8

L2 ¼ hXN

j¼1

uexactj � u�j

h i2 !1=2

; and L1 ¼maxN

j¼1uexact

j � u�j��� ���;

where u�j represent the numerical solution at node j.

Example 1. The Burgers’ Eq. (1.1) with a ¼ 1 is solved over the region ½0;1:2� and the initial and boundary conditions are asgiven in [1,4,23]

uðx;1Þ ¼ x1þ exp 1

4m ðx2 � 14Þ

� � ; and uð0; tÞ ¼ 0; uð1:2; tÞ ¼ 0; for t > 1:

In this problem the initial condition is taken at t ¼ 1. The exact solution of the problem is

uðx; tÞ ¼xt

1þ tt0

� �1=2exp x2

4mt

� � ; for t P 1; where t0 ¼ exp1

8m

� �:

In this example, the parameter values are taken as m ¼ 0:005;h ¼ 0:01;Mt ¼ 0:01. The comparison of the numerical solutionsobtained by MCB-DQM, at different time levels are presented with the solutions obtained by Mittal and Jain [31], Shu et al.[41] and the exact solutions in Table 1.1. In Table 1.2, L2 and L1 errors at different time levels t 6 3:50 are compared with theerrors obtained by several earlier schemes. Further, the L2 and L1 errors at t ¼ 3:60 are compared with errors obtained by thethree methods recently proposed by Korkmaz–Dag [20] and are reported in Table 1.3. It is found that our results are muchbetter than all the three methods. It is evident from Table 1.1, 1.2 and 1.3 that our method produces better approximate nu-meric solutions than almost all the earlier schemes and approaching towards the exact solutions.

The absolute errors at t ¼ 3:5 are plotted in Fig. 1. It is evident from Fig. 1 that the absolute errors are very small ascompared to that given by Mittal and Jain [[31] Fig. 10]. The absolute errors for different time levels are also plotted in Fig. 2.It is found that the errors are decreasing with increasing time and the maximum error is shifting towards the boundaryx ¼ 1:2 only. The physical behavior of the numerical solutions at m ¼ 0:005 for different time levels t 6 3:5 is depicted inFig. 3.

Example 2. In this example, we take the particular solution of the Burgers’ Eq. (1.1), for a ¼ 1, over the region ½0;2� as con-sidered in [32]:

.1ison of the MCB-DQM numerical solutions of Example 1 with exact solutions, for m ¼ 0:005.

t Shu et al. [41] with h ¼ 10�4;Mt ¼ :01 Mittal & Jain [31] MCB-DQM Exact value

b ¼ 1 b ¼ 0:5 h ¼ 0:005;Mt ¼ 10�3 h ¼ 0:01;Mt ¼ 0:01

1.7 0.1176565 0.1174841 0.1176452 0.1176450 0.11764522.5 0.0800527 0.0798389 0.0799990 0.0799989 0.07999903.0 0.0667147 0.0665176 0.0666658 0.0666658 0.06666583.5 0.0571820 0.0570060 0.0571422 0.0571422 0.05714221.7 0.2332111 0.2348504 0.2351690 0.2351680 0.23516772.5 0.1591735 0.1596608 0.1599771 0.1599770 0.15997693.0 0.1328314 0.1330273 0.1333211 0.1333210 0.13332093.5 0.1139606 0.1140077 0.1142780 0.1142780 0.11427791.7 0.2940048 0.2961269 0.2958570 0.2959160 0.29590972.5 0.2347876 0.2376699 0.2381299 0.2381200 0.23812073.0 0.1973222 0.1990478 0.1994839 0.1994800 0.19948053.5 0.1697753 0.1708231 0.1712257 0.1712240 0.17122421.7 0.0008917 0.0006640 0.0006381 0.0006464 0.00064652.5 0.1103866 0.1036067 0.1021325 0.1020930 0.10209573.0 0.2088346 0.2093735 0.2088032 0.2088380 0.20883593.5 0.2119293 0.2143409 0.2145938 0.2145870 0.2145869

Table 1.2Comparison of L2 and L1 errors in the MCB-DQM solutions of Example 1 for m ¼ 0:005 at different time levels t 6 3:50 with the errors obtained by earlierschemes.

Methods N Mt t ¼ 1:7 t ¼ 2:4 t ¼ 3:1 t ¼ 3:25

L2 � 103 L1 � 103 L2 � 103 L1 � 103 L2 � 103 L1 � 103 L2 � 103 L1 � 103

MCB-DQM 121 0.01 0.00191 0.00777 0.00086 0.00308 0.00065 0.00331 0.001341 0.00918QRTDQ [24] 101 0.001 0.109 0.434 0.100 0.339 0.091 0.266BS.FEM [10] 50 0.100 0.857 2.576 0.423 1.242 0.230 0.680C.S.C.[39] 50 0.100 0.857 2.576 0.423 1.242 0.235 0.688Galerkin [46] 200 0.010 0.857 2.576 0.423 1.242 0.235 0.688QBCM1[28] 200 0.010 0.017 0.061 0.012 0.058 0.601 4.434QBCM2 [28] 200 0.010 0.358 1.211 0.251 0.807 0.630 4.790PDQ [18] 200 0.010 0.015 0.056 0.011 0.064 0.584 4.301CBCDQ [19] 101 0.001 0.210 0.680 0.190 0.530

t ¼ 2:5

QBCM [12] 200 0.010 0.072 0.311 0.051 0.189 1.129 8.983CBCM [12] 200 0.010 2.466 27.577 2.111 25.15 1.925 21.084QRKM [12] 200 0.010 0.026 0.091 0.031 0.115 1.111 8.000

t ¼ 3:00 t ¼ 3:50

MCB-DQM 121 0.010 0.00191 0.00777 0.00778 0.00275 0.00056 0.0017 0.006177 0.04335MCB-CM [31] 241 0.010 0.0252 0.0994 0.0151 0.0549 0.0118 0.0414 0.0117 0.0486[41] (b ¼ 0:5) 12001 0.010 0.38421 1.34728 0.49135 1.55470 0.51508 1.5529 0.525855 1.52196[41](b ¼ 1) 12001 0.010 3.08966 10.4040 2.72048 8.29747 2.39922 6.9880 2.12110 5.94321MCB-DQM 121 0.010 0.00191 0.00777 0.00778 0.00275 0.00056 0.0017 0.006177 0.04335

Table 1.3Comparison of L2 and L1 errors in the MCB-DQM solutions of Example 1 for m ¼ 0:005 at t ¼ 3:6 with the errors obtained in [20].

MCB-DQM Korkmaz & Dag [20]

Method I Method II Method III

L2 � 103 0:01 0:18 0:16 0:14

L1 � 103 0:07 0:46 0:52 0:54

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

4

4.5 x 10−5

x

Abso

lute

Erro

r

Fig. 1. Absolute errors in the MCB-DQM numeric solutions of Example 1 for m ¼ 0:005 at t ¼ 3:5 with h ¼ 0:01;Mt ¼ 0:01.

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 171

uðx; tÞ ¼ 2pmsinðpxÞexpð�p2m2tÞ þ 4 sinð2pxÞexpð�4p2m2tÞ

4þ cosðpxÞexpð�p2m2tÞ þ 2 cosð2pxÞexpð�4p2m2tÞ ; for x 2 ð0;2Þ and t P 0; ð4:1Þ

where the initial condition is evaluated from (4.1), and the boundary conditions are taken to be uð0; tÞ ¼ 0 and uð2; tÞ ¼ 0.In this example, we have computed L1 and L2 errors at t ¼ 0:1;1:0 with the parameter h ¼ 0:01;Mt ¼ 0:01, and at the

different values of m. The comparison of the computed errors with the errors obtained by Mittal and Jain [[31] Table 5.1] arereported in Table 2.1. It is evident that as the value of m decreases the absolute error decreases rapidly. Also, for a given valueof m, the computed errors are less than that obtained by Mittal and Jain [31], and hence, the numerical solutions produced by

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

4

4.5 x 10−5

x

Abso

lute

Erro

r

t=1.7t=2.5t=3.0t=3.5

Fig. 2. Absolute errors in the MCB-DQM numeric solutions of Example 1 for m ¼ 0:005 at t 6 3:5 with h ¼ 0:01;Mt ¼ 0:01.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

u(x,

t)

t=0.0t=1.7t=2.5t=3.0t=3.5

Fig. 3. Physical behavior of the MCB-DQM numeric solutions of Example 1 for m ¼ 0:005 at t 6 3:5 with h ¼ 0:01;Mt ¼ 0:01.

Table 2.1Comparison of L1 and L2 in the MCB-DQM numeric solutions of Example 2 with the errors obtained in [31].

m t ¼ 0:1 t ¼ 1:0

Mittal & Jain [31] MCB-DQM Mittal & Jain [31] MCB-DQM

h ¼ 0:025;Mt ¼ 10�3 h ¼ 0:1;Mt ¼ 0:01 h ¼ 0:025;Mt ¼ 10�3 h ¼ 0:1;Mt ¼ 0:01

L1 L2 L1 L2 L1 L2 L1 L2

10�2 4:41E� 03 3:55E� 03 3:89E� 03 3:41E� 03 3:13E� 02 2:66E� 02 2:92E� 02 2:63E� 02

10�3 4:60E� 05 3:72E� 05 4:09E� 05 3:55E� 05 4:45E� 04 3:59E� 04 3:93E� 04 3:45E� 04

10�4 4:62E� 07 3:74E� 07 4:11E� 07 3:56E� 07 4:61E� 06 3:72E� 06 4:09E� 06 3:55E� 06

10�5 4:62E� 09 3:74E� 09 4:11E� 09 3:56E� 09 4:62E� 08 3:74E� 08 4:11E� 08 3:56E� 08

10�6 4:62E� 11 3:74E� 11 4:11E� 11 3:56E� 11 4:62E� 10 3:74E� 10 4:11E� 10 3:56E� 10

172 G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

our method are accurate than [31], for the current problem. Also, the absolute errors at t ¼ 1 taking v ¼ 10�4;10�5;10�6

have been shown in Fig. 4. The physical behavior of the numerical solutions at m ¼ 0:01 for different time levels are depictedin Fig. 5.

Example 3. The initial and the boundary conditions of the Burgers’ Eq. (1.1) over the region ½0;1� with a ¼ 1, are consideredas in [26,34,16]:

uðx;0Þ ¼ 4xð1� xÞ and uð0; tÞ ¼ uð1; tÞ ¼ 0:

The description of the numerical solutions of this example for m ¼ 0:1 and 0:01 is given below:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−8

−6

−4

−2

0

2

4

6

8 x 10−4

x

Erro

rs

v=1.0 E−06

v=1.0 E−05

v=1.0 E−04

Fig. 4. Errors in the MCB-DQM numeric solutions of Example 2 at t ¼ 1 for m ¼ 10�4;10�5 and 10�6 taking h ¼ 0:01;Mt ¼ 0:01.

Table 3.1Comparison of the computed results form Example 3 to Jiwari et al. [16], Kutluay et al. [26] for m ¼ 0:1, and the exact solutions.

x T [26] [16] MCB-DQM ExactMt ¼ 0:0001 Mt ¼ 0:0001 Mt ¼ 0:001h ¼ 0:0125 h ¼ 0:04 h ¼ 0:025

0.25 0.4 0.32091 0.31744 0.317526 0.317520.8 0.20211 0.19952 0.199558 0.199561.0 0.16782 0.16557 0.165601 0.165603.0 0.02828 0.02775 0.027761 0.02775

0.50 0.4 0.58788 0.58443 0.584541 0.584540.8 0.37111 0.36733 0.367406 0.367401.0 0.30183 0.29830 0.298352 0.298343.0 0.04185 0.04106 0.041069 0.04106

0.75 0.4 0.65054 0.64556 0.645641 0.645620.8 0.39068 0.38526 0.385369 0.385341.0 0.30057 0.29582 0.295885 0.295863.0 0.03106 0.03043 0.030443 0.03044

Table 3.2Comparison of the MCB-DQM numerical solutions of Example 3 with the numeric solutions due to Mittal & Jain [31] for m ¼ 0:01.

x t [31] MCB-DQM Exacth ¼ 0:025 h ¼ 0:025Mt ¼ 0:001 Mt ¼ 0:001

0.25 0.4 0.36225 0.36226 0.362260.6 0.28202 0.28204 0.282040.8 0.23044 0.23045 0.230451.0 0.19468 0.19469 0.194693.0 0.07613 0.07613 0.07613

0.50 0.4 0.68368 0.68368 0.683680.6 0.54832 0.54832 0.548320.8 0.45371 0.45371 0.453711.0 0.38567 0.38568 0.385683.0 0.15218 0.15218 0.15218

0.75 0.4 0.92052 0.92049 0.920500.6 0.78300 0.78297 0.782990.8 0.66272 0.66271 0.662721.0 0.56932 0.56932 0.569323.0 0.22782 0.22775 0.22774

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 173

(a) In Table 3.1, we have computed the numerical solutions with parameter values m ¼ 0:1;h ¼ 0:025;Mt ¼ 0:001 at dif-ferent time levels. The comparison of our results with the results obtained in [16,26] are repored in Table 3.1. Also, thenumerical solutions obtained by Jiwari et al. [[16]Table 3] are better than the solutions obtained in [26,34]. Thus, wefound that our solutions are more accurate than the solutions obtained in [16,26,34] and approaching to exactsolutions.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 00.2

0.40.6

0.81

−0.1

−0.05

0

0.05

0.1

tx

u(x,

t)

Fig. 5. Physical behavior of the MCB-DQM numeric solutions of Example 2 for m ¼ 0:01 at t 6 1 with m ¼ 0:01; h ¼ 0:01;Mt ¼ 0:01.

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

tx

u(x,

t)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

tx

u(x,

t)

Fig. 6. Physical behavior of the MCB-DQM numeric solutions of Example 3 at m ¼ 0:01 (left) and at m ¼ 0:1 (right) for t 6 1 with h ¼ 0:025;Mt ¼ 0:001.

Table 4.1Comparison of the MCB-DQM numeric solutions of Example 4 for m ¼ 1:0 with the numercal solutions obtained by Dag et al. [12], Korkmaz [22], Mittal & Jain[31], and exact solutions.

x t [12] [31] [22] MCB-DQM Exacth ¼ 0:0125 h ¼ 0:025 h ¼ 0:025 h ¼ 0:025

Mt ¼ 10�4 Mt ¼ 0:00025 Mt ¼ 0:000125 Mt ¼ 0:00025

0.25 0.4 0.01357 0.01354 0.01363 0.0135710 0.013570.6 0.00189 0.00188 0.00190 0.0018888 0.001890.8 0.00026 0.00026 0.00026 0.0002624 0.000261.0 0.00004 0.00004 0.00003 0.0000365 0.000043.0 0.00000 0.00000 0.00000 0.0000000 0.00000

0.50 0.4 0.01923 0.01920 0.01932 0.0192336 0.019230.6 0.00267 0.00266 0.00269 0.0026719 0.002670.8 0.00037 0.00037 0.00037 0.0003712 0.000371.0 0.00005 0.00005 0.00005 0.0000516 0.000053.0 0.00000 0.00000 0.00000 0.0000000 0.00000

0.75 0.4 0.01362 0.01360 0.01369 0.0136298 0.013630.6 0.00189 0.00188 0.00190 0.0018899 0.001890.8 0.00026 0.00026 0.00026 0.0002625 0.000261.0 0.00004 0.00004 0.00003 0.0000365 0.000043.0 0.00000 0.00000 0.00000 0.0000000 0.00000

174 G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

(b) In Table 3.2, we have computed the numerical solutions with parameter values m ¼ 0:01; h ¼ 0:025;Mt ¼ 0:001. Sincethe numerical solutions obtained by Mittal and Jain [[31] Table 5.1] are better than the solutions obtained in [1,2,45],hence the comparison of the obtained solutions with the exact solutions as well as with the solutions obtained in [31]are reported in Table 3.2. It is observed that our solutions are accurate than the results obtained in [1,2,31,45].

Table 4.2Comparison of the MCB-DQM numeric solutions of Example 4 for m ¼ 1:0 with the solutions obtained by Dag et al. [12].

x [12] MCB-DQM [12] MCB-DQM Exacth ¼ 0:0125 h ¼ 0:025 h ¼ 0:00625 h ¼ 0:0125

Mt ¼ 10�5Mt ¼ 10�4

Mt ¼ 10�5Mt ¼ 10�4

0.1 0.10952 0.109530 0.10953 0.109526 0.109540.2 0.20975 0.209771 0.20977 0.209766 0.209790.3 0.29184 0.291860 0.29186 0.291855 0.291900.4 0.34785 0.347874 0.34788 0.347869 0.347920.5 0.37149 0.371517 0.37153 0.371512 0.371580.6 0.35896 0.358981 0.35900 0.358975 0.359050.7 0.30983 0.309845 0.30986 0.309839 0.309910.8 0.22776 0.227773 0.22778 0.227766 0.227820.9 0.12065 0.120666 0.12067 0.120659 0.12069

Table 4.3Comparison of the MCB-DQM numeric solutions of Example 4 for m ¼ 0:1 with the numercal solutions obtained by Dag et al. [12],Korkmaz [22], Mittal & Jain [31], and exact solutions.

x t [12] [31] [22] MCB-DQM Exacth ¼ 0:0125 h ¼ 0:025 h ¼ 0:025 h ¼ 0:025

Mt ¼ 10�4 Mt ¼ 0:0025 Mt ¼ 0:00125 Mt ¼ 0:004

0.25 0.4 0.30890 0.30892 0.30910 0.3089280 0.308890.6 0.24075 0.24077 0.24093 0.2407550 0.240740.8 0.19569 0.19572 0.19586 0.1956840 0.195681.0 0.16258 0.16261 0.16274 0.1625700 0.162563.0 0.02720 0.02718 0.02720 0.0272047 0.02720

0.50 0.4 0.56965 0.56970 0.56973 0.5696530 0.569630.6 0.44723 0.44729 0.44736 0.4472170 0.447210.8 0.35925 0.35930 0.35943 0.3592450 0.359241.0 0.29192 0.29195 0.29213 0.2919250 0.291923.0 0.04019 0.04016 0.04032 0.0402085 0.04021

0.75 0.4 0.62538 0.62520 0.62573 0.6253490 0.625440.6 0.48715 0.48694 0.48760 0.4872040 0.487210.8 0.37385 0.37365 0.37434 0.3739350 0.373921.0 0.28741 0.28724 0.28788 0.2874930 0.287473.0 0.02976 0.02974 0.29881 0.0297753 0.02977

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

tx

u(x,

t)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

tx

u(x,

t)

Fig. 7. Physical behavior of MCB-DQM numeric solutions at m ¼ 0:1 (left) and at m ¼ 1:0 (right) of Example 4 for t 6 1 with h ¼ 0:025;Mt ¼ 0:001.

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 175

The physical behavior of the current problem for different time levels t 6 1, is depicted in Fig. 6. The similar figures are alsodepicted in [1,31].

Example 4. In this example, the initial condition for the Burgers’ Eq. (1.1), for a ¼ 1, over the region ½0;1� is considered asgiven in [22]:

176 G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

uðx;0Þ ¼ sinðpxÞ; ð4:2Þ

and the boundary conditions

uð0; tÞ ¼ uð1; tÞ ¼ 0: ð4:3Þ

The analytical solution of this problem is given by Cole [11] in terms of an infinite series as

uðx; tÞ ¼4pm

P1j¼1jIjð 1

2pmÞ sinðjpxÞexpð�j2p2mtÞI0ð 1

2pmÞ þ 2P1

j¼1Ijð 12pmÞ cosðjpxÞexpð�j2p2mtÞ

; ð4:4Þ

where Ij are the modified Bessel’s functions.The numerical solutions of this example for different values of m are given below:

(a) In Table 4.1, we have computed the numerical solutions of this example at different time levels with parameter valuesm ¼ 1:0;h ¼ 0:025 and Mt ¼ 0:00025. The comparison of our results with the exact solutions as well as the solutionsobtained in [12,31,22] are reported in Table 4.1. It is found that our method produces comparable results as obtainedin [12,31,22].Further, the numerical solutions are computed at t ¼ 0:1 with parameter values h ¼ 0:025;0:0125m ¼ 1:0;Mt ¼ 0:0001, and compared with the solutions obtained in Dag et al. [[12] Table 1]. The results are reportedin Table 4.2. It is found that we require half of the grid points to produces the results similar to [12].

(b) Also, the numerical solutions of this example are computed at different time levels with parameter valuesm ¼ 0:1;h ¼ 0:025 and Mt ¼ 0:004. The comparison of our solutions with the exact solutions as well as the solutionsobtained in [12,31,22] are reported in Table 4.3. It is evident that the MCB-DQM numerical solutions are better as com-pared to the results obtained in [12,31,22].

The physical behavior of this example for m ¼ 0:1 and m ¼ 1:0 are depicted in Fig. 7.

5. Conclusion

In this paper, we have developed a method (MCB-DQM) to solve nonlinear partial differential equations. In this method,the modified cubic B-splines are used in differential quadrature method as basis function to evaluate the weighting coeffi-cients, and hence the derivatives. In this way, we find a system of ordinary differential equations (ODEs) which is solved bySSP-RK43 scheme. To check the efficiency and accuracy of the method, four examples of Burgers’ equation are included withtheir numerical solutions, L2 and L1 errors and done the comparisons with the results given in the literature. It is evident thatour method produces better results as compared to the results obtained by almost all the schemes available in the literature,and approaching to the exact solutions.

The cubic B-spline basis functions are modified in such a way that it reduces matrix size and complexity when appliedwith differential quadrature method. In this method we require less number of grid points as compared to the earlier givenmethods. This method is hence easy to implement and economical in terms of data complexity, which results in less errorsand so, the easiness of the implementation of MCB-DQM, and low memory storage can be counted as advantages of thismethod. Also, this method can be easily implemented to solve two-dimensional nonlinear partial differential equations.

Acknowledgement

The authors thank the anonymous referees for their time, effort, and extensive comments which improve the quality ofthe presentation of the paper.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.amc.2013.08.071.

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