Abstract and Applied Analysis

Volume 2013 (2013), Article ID 746401, 12 pages

http://dx.doi.org/10.1155/2013/746401

Research Article

Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order

Veyis Turut¹ and Nuran Güzel²

¹Department of Mathematics, Faculty of Arts and Sciences, Batman University, Batman, Turkey

²Department of Mathematics, Faculty of Arts and Sciences, Yıldız Technical University, İstanbul, Turkey

Received 27 November 2012; Revised 14 January 2013; Accepted 14 January 2013

Academic Editor: Hassan Eltayeb

Copyright © 2013 Veyis Turut and Nuran Güzel. This 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.

Abstract.

Two tecHniques were implemented, the Adomian decomposition method (ADM) and multivariate Padé approximation (MPA), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM), then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.

1. Introduction

Recently, differential equations of fractional order have gained much interest in engineering, physics, chemistry, and other sciences. It can be said that the fractional derivative has drawn much attention due to its wide application in engineering physics [1–9]. Some approximations and numerical techniques have been used to provide an analytical approximation to linear and nonlinear differential equations and fractional differential equations. Among them, the variational iteration method, homotopy perturbation method [10, 11], and the Adomian decomposition method are relatively new approaches [5–9, 12, 13].

The decomposition method has been used to obtain approximate solutions of a large class of linear or nonlinear differential equations [12, 13]. Recently, the application of the method is extended for fractional differential equations [6–9, 14].

Many definitions and theorems have been developed for multivariate Padé approximations MPA (see [15] for a survey on multivariate Padé approximation). The multivariate Padé Approximation has been used to obtain approximate solutions of linear or nonlinear differential equations [16–19]. Recently, the application of the unvariate Padé approximation is extended for fractional differential equations [20, 21].

The objective of the present paper is to provide approximate solutions for initial value problems of nonlinear partial differential equations of fractional order by using multivariate Padé approximation.

2. Definitions

For the concept of fractional derivative, we will adopt Caputo’s definition, which is a modification of the Riemann-Liouville definition and has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order, which is the case in most physical processes. The definitions can be seen in [3, 4, 22, 23].

3. Decomposition Method [24]

Consider

The decomposition method requires that a nonlinear fractional differential equation (1) is expressed in terms of operator form as

where is a linear operator which might include other fractional derivatives of order less than , is a nonlinear operator which also might include other fractional derivatives of order less than is the Caputo fractional derivative of order ,  and is the source function [24].

Applying the operator [3, 4, 22, 23], the inverse of the operator , to both sides of (5) Odibat and Momani [24] obtained

From this, the iterates are determined in [24] by the following recursive way:

4. Multivariate Padé Aproximation [25]

Consider the bivariate function with Taylor series development

around the origin. We know that a solution of unvariate Padé approximation problem for

is given by

Let us now multiply th row in and by and afterwards divide th column in and by . This results in a multiplication of numerator and denominator by . Having done so, we get

if .

This quotent of determinants can also immediately be written down for a bivariate function . The sum shall be replaced by th partial sum of the Taylor series development of and the expression by an expression that contains all the terms of degree in . Here a bivariate term is said to be of degree . If we define

Then it is easy to see that and are of the form

We know that and are called Padé equations [25]. So the multivariate Padé approximant of order for is defined as

5. Numerical Experiments

In this section, two methods, ADM and MPA, shall be illustrated by two examples. All the results are calculated by using the software Maple12. The full ADM solutions of examples can be seen from Odibat and Momani [24].

Example 1. Consider the nonlinear time-fractional advection partial differential equation [24]

subject to the initial condition

Odibat and Momani [24] solved the problem using the decomposition method, and they obtained the following recurrence relation [24]:

where are the Adomian polynomials for the nonlinear function . In view of (15), the first few components of the decomposition series are derived in [24] as follows:

and so on; in this manner, the rest of components of the decomposition series can be obtained [24].

The first three terms of the decomposition series are given by [24]

For (16) is

Now, let us calculate the approximate solution of (18) for and by using Multivariate Padé approximation. To obtain multivariate Padé equations of (18) for and , we use (10). By using (10), we obtain

So, the multivariate Padé approximation of order for (17), that is,

For (17) is

For simplicity, let ; then

Using (10) to calculate the multivariate Padé equations for (22) we get

recalling that , we get multivariate Padé approximation of order for (21), that is,

For (17) is

For simplicity, let ; then

Using (10) to calculate the multivariate Padé equations and then recalling that , we get multivariate Padé approximation of order for (25), that is,

Table 1, Figures 1(a), 1(b), 1(c), 2(a), 2(b), 2(c), and 2(d) shows the approximate solutions for (13) obtained for different values of using the decomposition method (ADM) and the multivariate Padé approximation (MPA). The value of is for the exact solution [24].

Table 1: Numerical values when , , and for (13).

0.01
0.01
0.001118860666
0.001118859998

0.00009999999987
0.0001000000000
0.0001

0.02
0.02
0.003138022015
0.003138007574

0.0003999999915
0.0004000000000
0.0004

0.03
0.03
0.005716640278
0.005716554826

0.0008999999028
0.0009000000000
0.0009

0.04
0.04
0.008727882362
0.008727584792

0.001599999454
0.001600000000
0.0016

0.05
0.05
0.01209607907
0.01209530419

0.002499997917
0.002500000000
0.0025

0.06
0.06
0.01576873408
0.01576705530

0.003599993779
0.003600000000
0.0036

0.07
0.07
0.01970633078
0.01970312778

0.004899984313
0.004900000000
0.0049

0.08
0.08
0.02387754051
0.02387197301

0.006399965047
0.006400000000
0.0064

0.09
0.09
0.02825661342
0.02824760190

0.008099929141
0.008100000000
0.0081

0.1
0.1
0.03282181204
0.03280802591

0.009999866667
0.010000000000
0.01

(a)

(b)

(c)

Figure 1: (a) Exact solution of Example 1 for (b) ADM solution of Example 1 for (c) Multivariate Padé approximation of ADM solution for in Example 1.

(a)

(b)

(c)

(d)

Figure 2: (a) ADM solution of Example 1 for (b) Multivariate Padé approximation of ADM solution for in Example 1 (c) ADM solution of Example 1 for (d) Multivariate Padé approximation of ADM solution for in Example 1.

Example 2. Consider the nonlinear time-fractional hyperbolic equation [24]

subject to the initial condition

Odibat and Momani [24] solved the problem using the decomposition method, and they obtained the following recurrence relation in [24]:

where are the Adomian polynomials for the nonlinear function . In view of (30), the first few components of the decomposition series are derived in [24] as follows:

and so on; in this manner the rest of components of the decomposition series can be obtained.

The first three terms of the decomposition series (7) are given [24] by

For (43) is

Now, let us calculate the approximate solution of (33) for and by using multivariate Padé approximation. To obtain multivariate Padé equations of (33) for and , we use (10). By using (10), we obtain

So, the multivariate Padé approximation of order for (33), that is,

For (32) is

For simplicity, let ; then

Using (10) to calculate multivariate Padé equations of (37) for and , we use (10). By using (10), we obtain

recalling that , we get multivariate Padé approximation of order for (36), that is,

For (32) is

For simplicity, let ; then

Using (10) to calculate multivariate Padé equations of (41) for and , we use (10). By using (10), we obtain

recalling that , we get multivariate Padé approximation of order for (40), that is,

Table 2, Figures 3(a), 3(b), 3(c), 4(a), 4(b), 4(c), and 4(d) show the approximate solutions for (28) obtained for different values of using the decomposition method (ADM) and the multivariate Padé approximation (MPA). The value of is for the exact solution [24].

Table 2: Numerical values when , , and for (28).

0.01
0.01
0.00009844537131
0.00009844533110
0.00009811632420
0.00009811625316
0.00009802960500
0.00009802960486
0.00009802960494

0.02
0.02
0.0003889833219
0.0003889809228
0.0003855443171
0.0003855410232
0.0003844675200
0.0003844675123
0.0003844675125

0.03
0.03
0.0008664034309
0.0008663775704
0.0008529749012
0.0008529437792
0.0008483364450
0.0008483363176
0.0008483363182

0.04
0.04
0.001527388854
0.001527250241
0.001492265504
0.001492112228
0.001479290880
0.001479289940
0.001479289941

0.05
0.05
0.002370102454
0.002369595331
0.002296248891
0.002295720729
0.002267578126
0.002267573695
0.002267573696

0.06
0.06
0.003393997434
0.003392539471
0.003258613123
0.003257161294
0.003204002880
0.003203987182
0.003203987184

0.07
0.07
0.004599726730
0.004596176215
0.004373819216
0.004370404791
0.004279895445
0.004279849765
0.004279849769

0.08
0.08
0.005989109024
0.005981449991
0.005637043845
0.005629880400
0.005487083520
0.005486968446
0.005486968450

0.09
0.09
0.007565131844
0.007550068937
0.007044140677
0.007030367394
0.006817867605
0.006817607941
0.006817607945

0.1
0.1
0.009331981000
0.009304436794
0.008591616573
0.008566895894
0.008265000000
0.008264462804
0.008264462810

(a)

(b)

(c)

Figure 3: (a) Exact solution of Example 2 for (b) ADM solution of Example 2 for (c) Multivariate Padé approximation of ADM solution for in Example 2.

(a)

(b)

(c)

(d)

Figure 4: (a) ADM solution of Example 2 for (b) Multivariate Padé approximation of ADM solution for in Example 2 (c) ADM solution of Example 2 for (d) Multivariate Padé approximation of ADM solution for in Example 2.

6. Concluding Remarks

The fundamental goal of this paper has been to construct an approximate solution of nonlinear partial differential equations of fractional order by using multivariate Padé approximation. The goal has been achieved by using the multivariate Padé approximation and comparing with the Adomian decomposition method. The present work shows the validity and great potential of the multivariate Padé approximation for solving nonlinear partial differential equations of fractional order from the numerical results. Numerical results obtained using the multivariate Padé approximation and the Adomian decomposition method are in agreement with exact solutions.

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