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

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 𝐷^𝛼_∗𝑡𝑢(𝑥,𝑡)=𝑓𝑢,𝑢_𝑥,𝑢_𝑥𝑥+𝑔(𝑥,𝑡),𝑚−1<𝛼≤𝑚. (1) The decomposition method requires that a nonlinear fractional differential equation (1) is expressed in terms of operator form as 𝐷^𝛼_∗𝑡𝑢(𝑥,𝑡)+𝐿𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)=𝑔(𝑥,𝑡),𝑥>0, (2) where 𝐿 is a linear operator which might include other fractional derivatives of order less than 𝛼, 𝑁 is a nonlinear operator which also might include other fractional derivatives of order less than 𝛼,𝐷^𝛼_∗𝑡=𝜕^𝛼/𝜕𝑡^𝛼 is the Caputo fractional derivative of order 𝛼,  and 𝑔(𝑥,𝑡) is the source function [24].

Applying the operator 𝐽^𝛼 [3, 4, 22, 23], the inverse of the operator 𝐷^𝛼_∗𝑡, to both sides of (5) Odibat and Momani [24] obtained 𝑢(𝑥,𝑡)=^𝑚−1_𝑘=0𝜕^𝑘𝑢𝜕𝑡^𝑘𝑥,0⁺𝑡^𝑘𝑘!+𝐽^𝛼𝑔(𝑥,𝑡)−𝐽^𝛼[𝐿𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)]_∞𝑛=0,𝑢_𝑛(𝑥,𝑡)=^𝑚−1_𝑘=0𝜕^𝑘𝑢𝜕𝑡^𝑘𝑥,0⁺𝑡^𝑘𝑘!+𝐽^𝛼𝑔(𝑥,𝑡)−𝐽^𝛼𝐿^∞_𝑛=0𝑢_𝑛(𝑥,𝑡)+^∞_𝑛=0𝐴_𝑛. (3) From this, the iterates are determined in [24] by the following recursive way: 𝑢₀(𝑥,𝑡)=^𝑚−1_𝑘=0𝜕^𝑘𝑢𝜕𝑡^𝑘𝑥,0⁺𝑡^𝑘𝑘!+𝐽^𝛼𝑔(),𝑢𝑥,𝑡₁(𝑥,𝑡)=−𝐽^𝛼𝐿𝑢₀+𝐴₀,𝑢₂(𝑥,𝑡)=−𝐽^𝛼𝐿𝑢₁+𝐴₁,⋮𝑢_𝑛+1(𝑥,𝑡)=−𝐽^𝛼𝐿𝑢_𝑛+𝐴_𝑛. (4)

4. Multivariate Padé Aproximation [25]

Consider the bivariate function 𝑓(𝑥,𝑦) with Taylor series development 𝑓𝑥,𝑦=^∞_𝑖,𝑗=0𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗 (5) around the origin. We know that a solution of unvariate Padé approximation problem for 𝑓(𝑥)=^∞_𝑖=0𝑐_𝑖𝑥^𝑖 (6) is given by 𝑝(𝑥)=^𝑚_𝑖∑₌₀𝑐_𝑖𝑥^𝑖𝑥∑^𝑚−1_𝑖=0𝑐_𝑖𝑥^𝑖…𝑥∑^{𝑛𝑚−𝑛}_𝑖=0𝑐_𝑖𝑥^𝑖𝑐_𝑚+1𝑐_𝑚…𝑐_{𝑚+1−𝑛}𝑐⋮⋮⋱⋮_𝑚+𝑛𝑐_{𝑚+𝑛−1}…𝑐_𝑚𝑞(𝑥)=,1𝑥…𝑥^𝑛𝑐_𝑚+1𝑐_𝑚…𝑐_{𝑚+1−𝑛}𝑐⋮⋮⋱⋮_𝑚+𝑛𝑐_{𝑚+𝑛−1}…𝑐_𝑚. (7)(8) Let us now multiply 𝑗th row in 𝑝(𝑥) and 𝑞(𝑥) by 𝑥^{𝑗+𝑚−1}(𝑗=2,…,𝑛+1) and afterwards divide 𝑗th column in 𝑝(𝑥) and 𝑞(𝑥) by 𝑥^𝑗−1(𝑗=2,…,𝑛+1). This results in a multiplication of numerator and denominator by 𝑥^𝑚𝑛. Having done so, we get 𝑝(𝑥)𝑞(𝑥)=^{∑𝑚𝑖=0𝑐𝑖𝑥𝑖∑𝑚−1𝑖=0𝑐𝑖𝑥𝑖⋯∑𝑚−𝑛𝑖=0𝑐𝑖𝑥𝑖𝑐𝑚+1𝑥𝑚+1𝑐𝑚𝑥𝑚⋯𝑐𝑚+1−𝑛𝑥𝑚+1−𝑛𝑐⋮⋮⋱⋮𝑚+𝑛𝑥𝑚+𝑛𝑐𝑚+𝑛−1𝑥𝑚+𝑛−1⋯𝑐𝑚𝑥𝑚}_{𝑐11⋯1𝑚+1𝑥𝑚+1𝑐𝑚𝑥𝑚⋯𝑐𝑚+1−𝑛𝑥𝑚+1−𝑛𝑐⋮⋮⋱⋮𝑚+𝑛𝑥𝑚+𝑛𝑐𝑚+𝑛−1𝑥𝑚+𝑛−1⋯𝑐𝑚𝑥𝑚} (9) if (𝐷=det𝐷_𝑚,𝑛≠0).

This quotent of determinants can also immediately be written down for a bivariate function 𝑓(𝑥,𝑦). The sum ∑^𝑘_𝑖=0𝑐_𝑖𝑥^𝑖 shall be replaced by 𝑘th partial sum of the Taylor series development of 𝑓(𝑥,𝑦) and the expression 𝑐_𝑘𝑥^𝑘 by an expression that contains all the terms of degree 𝑘 in 𝑓(𝑥,𝑦). Here a bivariate term 𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗 is said to be of degree 𝑖+𝑗. If we define 𝑝=𝑥,𝑦_𝑚𝑖+∑_𝑗=0𝑐_𝑖𝑗𝑥^𝑖𝑦∑^𝑗𝑚−1_𝑖+𝑗=0𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗⋯∑^𝑚−𝑛_𝑖+𝑗=0𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗_𝑖+𝑗=∑_𝑚+1𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗∑_{𝑖+𝑗=𝑚}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗⋯∑_{𝑖+𝑗=𝑚+1−𝑛}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗∑⋮⋮⋱⋮_{𝑖+𝑗=𝑚+𝑛}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗∑_{𝑖+𝑗=𝑚+𝑛−1}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗⋯∑_{𝑖+𝑗=𝑚}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗,𝑞=∑𝑥,𝑦11⋯1_{𝑖+𝑗=𝑚+1}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗∑_{𝑖+𝑗=𝑚}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗⋯∑_{𝑖+𝑗=𝑚+1−𝑛}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗∑⋮⋮⋱⋮_{𝑖+𝑗=𝑚+𝑛}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗∑_{𝑖+𝑗=𝑚+𝑛−1}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗⋯∑_{𝑖+𝑗=𝑚}𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗. (10) Then it is easy to see that 𝑝(𝑥,𝑦) and 𝑞(𝑥,𝑦) are of the form 𝑝𝑥,𝑦=^{𝑚𝑛+𝑚}_{𝑖+𝑗=𝑚𝑛}𝑎_𝑖𝑗𝑥^𝑖𝑦^𝑗,𝑞𝑥,𝑦=^{𝑚𝑛+𝑛}_{𝑖+𝑗=𝑚𝑛}𝑏_𝑖𝑗𝑥^𝑖𝑦^𝑗. (11) We know that 𝑝(𝑥,𝑦) and 𝑞(𝑥,𝑦) are called Padé equations [25]. So the multivariate Padé approximant of order (𝑚,𝑛) for 𝑓(𝑥,𝑦) is defined as 𝑟_𝑚,𝑛=𝑝𝑥,𝑦𝑥,𝑦𝑞𝑥,𝑦. (12)

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] 𝐷^𝛼_∗𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)𝑢_𝑥(𝑥,𝑡)=𝑥+𝑥𝑡²,𝑡>0,𝑥∈𝑅,0<𝛼≤1, (13) subject to the initial condition 𝑢(𝑥,0)=0. (14) Odibat and Momani [24] solved the problem using the decomposition method, and they obtained the following recurrence relation [24]: 𝑢₀(𝑥,𝑡)=𝑢(𝑥,0)+𝐽^𝛼𝑥+𝑥𝑡²𝑡=𝑥^𝛼Γ()+𝛼+12𝑡^𝛼+2Γ(),𝑢𝛼+3_𝑗+1(𝑥,𝑡)=−𝐽^𝛼𝐴_𝑗,𝑗≥0, (15) where 𝐴_𝑗 are the Adomian polynomials for the nonlinear function 𝑁=𝑢𝑢_𝑥. In view of (15), the first few components of the decomposition series are derived in [24] as follows: 𝑢₀()𝑡𝑥,𝑡=𝑥^𝛼Γ()+𝛼+12𝑡^𝛼+2Γ(),𝑢𝛼+3₁()Γ()𝑡𝑥,𝑡=−𝑥2𝛼+1^3𝛼Γ(𝛼+1)²Γ()+()𝑡3𝛼+14Γ2𝛼+3^3𝛼+2Γ()Γ()Γ()+()𝑡𝛼+1𝛼+33𝛼+34Γ2𝛼+5^3𝛼+4Γ(𝛼+3)²Γ(),𝑢3𝛼+5₂()Γ()Γ()𝑡𝑥,𝑡=2𝑥2𝛼+14𝛼+1^5𝛼Γ(𝛼+1)³Γ()Γ()+()Γ()𝑡3𝛼+15𝛼+18Γ2𝛼+54𝛼+7^5𝛼+6Γ(𝛼+1)³Γ()Γ(),3𝛼+55𝛼+7+⋯ (16) 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] 𝑢()𝑡𝑥,𝑡=𝑥^𝛼Γ()+𝛼+12𝑡^𝛼+2Γ()−Γ()𝑡𝛼+32𝛼+1^3𝛼Γ(𝛼+1)²Γ()−()𝑡3𝛼+14Γ2𝛼+3^3𝛼+2Γ()Γ()Γ().𝛼+1𝛼+33𝛼+3+⋯ (17) For 𝛼=1 (16) is 𝑢(𝑥,𝑡)=𝑥𝑡+0.1×10⁻⁹𝑡³−0.1333333333𝑥𝑡⁵. (18) Now, let us calculate the approximate solution of (18) for 𝑚=4 and 𝑛=2 by using Multivariate Padé approximation. To obtain multivariate Padé equations of (18) for 𝑚=4 and 𝑛=2, we use (10). By using (10), we obtain 𝑝()=𝑥,𝑡𝑥𝑡+0.1×10⁻⁹𝑡³𝑥𝑡𝑥𝑡00.1×10⁻⁹𝑡³0−0.1333333333𝑥𝑡⁵00.1×10⁻⁹𝑡³=0.1333333333×10⁻¹⁰×𝑡²+0.7500000002×10⁻⁹𝑥³𝑡⁷,𝑞()=𝑥,𝑡11100.1×10⁻⁹𝑡³0−0.1333333333𝑥𝑡⁵00.1×10⁻⁹𝑡³=0.1333333333×10⁻¹⁰×𝑡²+0.7500000002×10⁻⁹𝑥²𝑡⁶. (19) So, the multivariate Padé approximation of order (4,2) for (17), that is, [4,2]_(𝑥,𝑡)=𝑡²+0.7500000002×10⁻⁹𝑥𝑡𝑡²+0.7500000002×10⁻⁹. (20) For 𝛼=0.5 (17) is 𝑢(𝑥,𝑡)=1.128379167𝑥𝑡^0.5−0.9577979850𝑥𝑡^1.5+0.6018022226𝑥𝑡^2.5−0.7005608116𝑥𝑡^3.5. (21) For simplicity, let 𝑡^1/2=𝑎; then 𝑢(𝑥,𝑎)=1.128379167𝑥𝑎−0.9577979850𝑥𝑎³+0.6018022226𝑥𝑎⁵−0.7005608116𝑥𝑎⁷. (22)Using (10) to calculate the multivariate Padé equations for (22) we get

𝑝()=𝑥,𝑎1.128379167𝑥𝑎−0.9577979850𝑥𝑎³+0.6018022226𝑥𝑎⁵1.128379167𝑥𝑎−0.9577979850𝑥𝑎³1.128379167𝑥𝑎−0.9577979850𝑥𝑎³00.6018022226𝑥𝑎⁵0−0.7005608116𝑥𝑎⁷00.6018022226𝑥𝑎⁵=−0.49078545071.201464294𝑎⁴−1.231832347𝑎²−0.8326662354𝑥³𝑎¹¹,𝑞(𝑥,𝑎)=11100.6018022226𝑥𝑎⁵0−0.7005608116𝑥𝑎⁷00.6018022226𝑥𝑎⁵=0.49078545070.7379312378+1.718058483𝑎²𝑥³𝑎¹⁰ (23)

recalling that 𝑡^1/2=𝑎, we get multivariate Padé approximation of order (6,2) for (21), that is, [6,2]_(𝑥,𝑡)=−1.201464294𝑡²𝑥√−1.231832347𝑡−0.8326662354𝑡×(0.7379312378+1.718058483𝑡)⁻¹. (24) For 𝛼=0.75 (17) is 𝑢(𝑥,𝑡)=0.00007125345441𝑥𝑡^7.5+0.1764791440×10⁻⁵𝑥𝑡^9.5−0.1238343301×10⁻¹⁷𝑥𝑡^22.5−0.2897967272×10⁻¹⁹𝑥𝑡^24.5. (25) For simplicity, let 𝑡^1/2=𝑎; then 𝑢(𝑥,𝑎)=0.00007125345441𝑥𝑎¹⁵+0.1764791440×10⁻⁵𝑥𝑎¹⁹−0.1238343301×10⁻¹⁷𝑥𝑎⁴⁵−0.2897967272×10⁻¹⁹𝑥𝑎⁴⁹. (26) Using (10) to calculate the multivariate Padé equations and then recalling that 𝑡^1/2=𝑎, we get multivariate Padé approximation of order (49,2) for (25), that is, [49,2]_(𝑥,𝑡)=−0.8398214310×10⁻³⁹𝑥³𝑡^113/2×−0.00007125345441−0.1764791440×10⁻⁵𝑡²+0.1238343301×10⁻¹⁷𝑡¹⁵×0.8398214310×10⁻³⁹𝑥²𝑡⁴⁹⁻¹. (27) 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 𝛼 using the decomposition method (ADM) and the multivariate Padé approximation (MPA). The value of 𝛼=1 is for the exact solution 𝑢(𝑥,𝑡)=𝑥𝑡 [24].

Table 1 

Numerical values when 𝛼=0.5, 𝛼=0.75, and 𝛼=1.0 for (13).

(a)

(b)

(c)

Figure 1 

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

(a)

(b)

(c)

(d)

Figure 2 

(a) ADM solution of Example 1 for 𝛼=0.5 (b) Multivariate Padé approximation of ADM solution for 𝛼=0.5 in Example 1 (c) ADM solution of Example 1 for 𝛼=0.75 (d) Multivariate Padé approximation of ADM solution for 𝛼=0.75 in Example 1.

Example 2. Consider the nonlinear time-fractional hyperbolic equation [24] 𝐷^𝛼_∗𝑡𝑢()=𝜕𝑥,𝑡𝑢()()𝜕𝑥𝑥,𝑡𝜕𝑢𝑥,𝑡,𝜕𝑥𝑡>0,𝑥∈𝑅,1<𝛼≤2, (28) subject to the initial condition 𝑢(𝑥,0)=𝑥²,𝑢_𝑡(𝑥,0)=−2𝑥². (29) Odibat and Momani [24] solved the problem using the decomposition method, and they obtained the following recurrence relation in [24]: 𝑢₀(𝑥,𝑡)=𝑢(𝑥,0)+𝑡𝑢_𝑥(𝑥,0)=𝑥²(),𝑢1−2𝑡_𝑗+1(𝑥,𝑡)=𝐽^𝛼𝐴_𝑗𝑥,𝑗≥0, (30) where 𝐴_𝑗 are the Adomian polynomials for the nonlinear function 𝑁=𝑢𝑢_𝑥. In view of (30), the first few components of the decomposition series are derived in [24] as follows: 𝑢₀(𝑥,𝑡)=𝑥²(),𝑢1−2𝑡₁(𝑥,𝑡)=6𝑥²𝑡^𝛼Γ()−𝛼+14𝑡^𝛼+1Γ()+𝛼+28𝑡^𝛼+2Γ(),𝑢𝛼+3₂(𝑥,𝑡)=72𝑥²𝑡^2𝛼Γ()−2𝛼+14𝑡^2𝛼+1Γ()+2𝛼+28𝑡^2𝛼+2Γ()−()𝑡2𝛼+32Γ𝛼+2^2𝛼+1Γ(𝛼+1)Γ(2𝛼+2)+72𝑥²8Γ(𝛼+3)𝑡^2𝛼+2Γ()Γ()−()𝑡𝛼+22𝛼+316Γ𝛼+4^2𝛼+3Γ()Γ(),⋮𝛼+32𝛼+4 (31) 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 𝑢(𝑥,𝑡)=𝑥²(1−2𝑡)+6𝑥²×𝑡^𝛼Γ()−𝛼+14𝑡^𝛼+1Γ()+𝛼+28𝑡^𝛼+2Γ(𝛼+3)+72𝑥²𝑡^2𝛼Γ().2𝛼+1+⋯ (32) For 𝛼=2 (43) is 𝑢(𝑥,𝑡)=𝑥²(1−2𝑡)+6𝑥²0.5000000000𝑡²−0.6666666668𝑡³+0.3333333334𝑡⁴+3.000000000𝑥²𝑡⁴. (33) Now, let us calculate the approximate solution of (33) for 𝑚=4 and 𝑛=2 by using multivariate Padé approximation. To obtain multivariate Padé equations of (33) for 𝑚=4 and 𝑛=2, we use (10). By using (10), we obtain 𝑝()=𝑥𝑥,𝑡₂(1−2𝑡)+3.000000000𝑥²𝑡²𝑥²(1−2𝑡)𝑥²−4.000000001𝑥²𝑡³3.000000000𝑥²𝑡²−2𝑥²𝑡5.000000000𝑥²𝑡⁴−4.000000001𝑥²𝑡³3.000000000𝑥²𝑡²=−20.00000000𝑡⁴0.28×10⁻⁹𝑡²−0.34×10⁻⁹𝑡(−0.04999999986)𝑥⁶,𝑞()=𝑥,𝑡111−4.000000001𝑥²𝑡³3.000000000𝑥²𝑡²−2𝑥²𝑡5.000000000𝑥²𝑡⁴−4.000000001𝑥²𝑡³3.000000000𝑥²𝑡²=20.00000000𝑡⁴0.0499999999+0.1000000001𝑡+0.0500000004𝑡²𝑥⁴. (34) So, the multivariate Padé approximation of order (4,2) for (33), that is, [4,2]_(𝑥,𝑡)=−1.0000000000.28×10⁻⁹𝑡²−0.34×10⁻⁹𝑡−0.04999999986)𝑥²×0.0499999999+0.1000000001𝑡+0.0500000004𝑡²⁻¹. (35) For 𝛼=1.5 (32) is 𝑢(𝑥,𝑡)=𝑥²(1−2𝑡)+6𝑥²×0.7522527782𝑡^1.5−1.203604445𝑡^2.5+0.6877739683𝑡^3.5+12.00000000𝑥²𝑡^3.0. (36) For simplicity, let 𝑡^1/2=𝑎; then 𝑢(𝑥,𝑎)=𝑥²1−2𝑎²+6𝑥²×0.7522527782𝑎³−1.203604445𝑎⁵+0.6877739683𝑡⁷+12.00000000𝑥²𝑎⁶.=𝑥²−𝑥²2𝑎²+4.513516669𝑥²𝑎³−7.221626670𝑥²𝑎⁵+4.126643810𝑥²𝑎⁷+12.00000000𝑥²𝑎⁶. (37) Using (10) to calculate multivariate Padé equations of (37) for 𝑚=7 and 𝑛=2, we use (10). By using (10), we obtain 𝑝()=𝑥𝑥,𝑎²−𝑥²2𝑎²+4.513516669𝑥²𝑎³−7.221626670𝑥²𝑎⁵𝑥²−𝑥²2𝑎²+4.513516669𝑥²𝑎³𝑥²−𝑥²2𝑎²+4.513516669𝑥²𝑎³12.00000000𝑥²𝑡⁶−7.221626670𝑥²𝑎⁵04.126643810𝑥²𝑎⁷12.00000000𝑥²𝑎⁶−7.221626670𝑥²𝑎=49.519725728.235760151𝑎⁵+0.879185531𝑎⁴+1.253427636𝑎³+1.403426542𝑎²+1.750000001𝑎+1.053153890𝑥⁶𝑎¹⁰,𝑞(𝑥,𝑎)=11112.00000000𝑥²𝑎⁶−7.221626670𝑥²𝑎⁵04.126643810𝑥²𝑎⁷12.00000000𝑥²𝑎⁶−7.221626670𝑥²𝑎=49.519725721.053153890+1.750000001𝑎+3.509734322𝑎²𝑥⁴𝑎¹⁰, (38)

recalling that 𝑡^1/2=𝑎, we get multivariate Padé approximation of order (7,2) for (36), that is, [7,2]_(𝑥,𝑡)=8.235760151𝑡^5/2+0.879185531𝑡²+1.253427636𝑡^3/2√+1.403426542𝑡+1.750000001𝑡+1.053153890𝑥²×1.053153890+1.750000001√𝑡+3.509734322𝑡)⁻¹. (39) For 𝛼=1.75 (32) is 𝑢(𝑥,𝑡)=𝑥²(1−2𝑡)+6𝑥²0.6217515726𝑡^1.75−0.9043659240𝑡^2.75+0.4823284927𝑡^3.75+6.189965715𝑥²𝑡^3.5. (40) For simplicity, let 𝑡^1/4=𝑎; then 𝑢(𝑥,𝑎)=𝑥²1−2𝑎⁴+6𝑥²0.6217515726𝑎⁷−0.9043659240𝑎¹¹+0.4823284927𝑎¹⁵+6.189965715𝑥²𝑎¹⁴=𝑥²−2𝑥²𝑎⁴+3.730509436𝑥²𝑎⁷−5.426195544𝑥²𝑎¹¹+2.893970956𝑥²𝑎¹⁵+6.189965715𝑥²𝑎¹⁴. (41) Using (10) to calculate multivariate Padé equations of (41) for 𝑚=15 and 𝑛=2, we use (10). By using (10), we obtain 𝑝()=𝑥𝑥,𝑎₂−2𝑥²𝑎⁴+3.730509436𝑥²𝑎⁷−5.426195544𝑥²𝑎¹¹𝑥²−2𝑥²𝑎⁴+3.730509436𝑥²𝑎⁷−5.426195544𝑥²𝑎¹¹𝑥²−2𝑥²𝑎⁴+3.730509436𝑥²𝑎⁷−5.426195544𝑥²𝑎¹¹6.189965715𝑥²𝑎¹⁴002.893970956𝑥²𝑎¹⁵6.189965715𝑥²𝑎¹⁴0=−38.31567556𝑥⁶𝑎²⁸−1+2𝑎⁴−3.730509436𝑎⁷+5.426195544𝑎¹¹,𝑞(𝑥,𝑎)=1116.189965715𝑥²𝑎¹⁴002.893970956𝑥²𝑎¹⁵6.189965715𝑥²𝑎¹⁴0=38.31567556𝑥⁴𝑎²⁸, (42)

recalling that 𝑡^1/4=𝑎, we get multivariate Padé approximation of order (15,2) for (40), that is, [15,2]_(𝑥,𝑡)=−38.31567556𝑥⁶𝑡⁷×−1+2𝑡−3.730509436𝑡^7/4+5.426195544𝑡^11/4×38.31567556𝑥⁴𝑡⁷⁻¹. (43) 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 𝛼 using the decomposition method (ADM) and the multivariate Padé approximation (MPA). The value of 𝛼=2 is for the exact solution 𝑢(𝑥,𝑡)=(𝑥/𝑡+1)² [24].

Table 2 

Numerical values when 𝛼=1.5, 𝛼=1.75, and 𝛼=2.0 for (28).

(a)

(b)

(c)

Figure 3 

(a) Exact solution of Example 2 for 𝛼=2.0 (b) ADM solution of Example 2 for 𝛼=2.0 (c) Multivariate Padé approximation of ADM solution for 𝛼=2.0 in Example 2.

(a)

(b)

(c)

(d)

Figure 4 

(a) ADM solution of Example 2 for 𝛼=1.5 (b) Multivariate Padé approximation of ADM solution for 𝛼=1.5 in Example 2 (c) ADM solution of Example 2 for 𝛼=1.75 (d) Multivariate Padé approximation of ADM solution for 𝛼=1.75 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.