An Efficient Algorithm for Some Highly Nonlinear Fractional PDEs in Mathematical Physics

Abstract

In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and subsequently Reduced Differential Transform Method (RDTM) is applied on the transformed system of linear and nonlinear time-fractional PDEs. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs and hence can be extended to other complex problems of diversified nonlinear nature.

Introduction

Fractional differential equations arise in almost all areas of physics, applied and engineering sciences [1]–[8]. In order to better understand these physical phenomena as well as further apply these physical phenomena in practical scientific research, it is important to find their exact solutions. The investigation of exact solution of these equations is interesting and important. In the past several decades, many authors mainly had paid attention to study the solution of such equations by using various developed methods. Recently, the variational iteration method (VIM) [1]–[3] has been applied to handle various kinds of nonlinear problems, for example, fractional differential equations [4], nonlinear differential equations [5], nonlinear thermo elasticity [6], nonlinear wave equations [7]. In Refs. [8]–[13] Adomian's decomposition method (ADM), homotopy perturbation method (HPM), homotopy analysis method (HAM) and variation of parameter method (VPM) are successfully applied to obtain the exact solution of differential equations. In the present article, we used reduced differential transform method (RDTM) [14]–[18], to construct an appropriate solution of some highly nonlinear time-fractional partial differential equations of mathematical physics.

Preliminaries

In this section, we give some basic formula and results about fractional calculus, and then we discuss the analysis reduced differential transform method (RDTM) to fractional partial differential equations.

1 Jumarie's Fractional Derivative

Some useful results and properties of Jumarie's fractional derivative were summarized [20].

(1)

(2)

(3)

(4)

(5)

2 Fractional Complex Transform

The fractional complex transform was first proposed [19] and is defined as

(6)

where p, q, k, and l are unknown constants,

3 Reduced Differential Transform Method (RDTM)

To demonstrate the basic idea of the DTM, differential transform of derivative of a function which is analytic and differentiated continuously in the domain of interest, is defined as

(7)

The differential inverse transform of is defined as follow

(8)

Eq. (8) is known as the Taylor series expansion of around. Combining Eq. (7) and (8)

(9)

when above equation reduces to

(10)

and Eq. (2) reduces to

(11)

Theorem 1: If the original function is then the transformed function is

Theorem 2: If then

Theorem 3: If then

Theorem 4: If then

Theorem 5: If then

Theorem 6: If then

Theorem 7: If then

Theorem 8: If then where

Theorem 9: If then

4 Numerical Applications of RDTM

In this section, we shall apply the reduced differential transform method (RDTM) to construct approximate solutions for some nonlinear fractional PDEs in mathematical physics and then compare approximate solutions to the exact solutions as follows.

4.1 Fornberg-Whitham (FW) Equation [21]

(12)

with the initial conditions

(13)

Applying the transformation [19], we get the following partial differential equation

(14)

Applying the differential transform to Eq. (14) and Eq. (13), we obtain the following recursive formula

(15)

using the initial condition, we have

(16)

Substituting Eq. (16) into (15), we obtain the following values of successively,

The series solution is given by

The inverse transformation will yields

(17)

This solution is convergent to the exact solution [22]

(18)

Fig. 1 (a–d) : Surface plot of approximate and exact solutions of (12) for different values of using only 3^th order of RDTM solution are:

Figure 1. Surface plot of approximate and exact solutions of (12) for different values of α, using only 3rd order of RDTM solution.

4.2 Modified Fornberg-Whitham (MFW) Equation [23]

(19)

with the initial conditions

(20)

where

Applying the transformation [19], we get the following partial differential equation

(21)

Applying the differential transform to Eq. (21) and Eq. (20), we obtain the following recursive formula

(22)

using the initial condition, we have

(23)

Now, substituting Eq. (21) into (20), we obtain the following values successively,

Finally, after applying the inverse transformation the approximate solution is

(24)

The exact solution [23] of this problem is

(25)

Fig. 2 (a–d) : Surface plot of approximate and exact solutions of (19) for different values of using only 3^th order of RDTM solution are:

Figure 2. Surface plot of approximate and exact solutions of (19) for different values of α, using only 3rd order of RDTM solution.

4.3 Sharma-Tasso-Olver (STO) Equation [24]

(26)

with the initial conditions

(27)

Applying the transformation [19], we get the following partial differential equation

(28)

Applying the differential transform to Eq. (28) and (27), we obtain the following recursive formula

(29)

using the initial condition, we have

(30)

Now, substituting Eq. (30) into (29), we obtain the following values successively,

The series solution is given by

Finally, the inverse transformation will yields the solution

(31)

Where the exact solution is

(32)

Fig. 3 (a–d) : Surface plot of approximate and exact solutions of (26) for different values of using only 3^th order of RDTM solution are:

Figure 3. Surface plot of approximate and exact solutions of (26) for different values of α, using only 3rd order of RDTM solution.

4.4 Gardner Equation [25]

(33)

with the initial condition

(34)

Applying the transformation [19], we get the following partial differential equation

(35)

Applying the RDTM to (35) and (34), we obtain the recursive relation

(36)

using the initial condition, we have

(37)

Substituting Eq. (37) into Eq. (36), we obtain the following values successively,

The series solution is given by

Finally, the inverse transformation will yields the solution

(38)

Where the exact solution is

(39)

Fig. 4 (a–d) : Surface plot of approximate and exact solutions of (33) for different values of using only 3^th order of RDTM solution are:

Figure 4. Surface plot of approximate and exact solutions of (33) for different values of α, using only 3rd order of RDTM solution.

4.5 Variant Water Wave (VWW) equation [26]

(40)

with initial condition

(41)

Applying the transformation [19], we get the following partial differential equation

(42)

Applying the RDTM to (42) and (41), we obtain the recursive relation

(43)

using the initial condition, we have

(44)

Substituting Eq. (44) into (43), we obtain the following values successively,

The series solution is given by

Finally, the inverse transformation will yields the solution

(45)

The exact solution [26] is given by

(46)

Fig. 5 (a–d) : Surface plot of approximate and exact solutions of (32) for different values of using only 3^th order of RDTM solution are:

Figure 5. Surface plot of approximate and exact solutions of (40) for different values of α, using only 3rd order of RDTM solution.

Conclusions

Applied fractional complex transform (FCT) proved very effective to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and the same is true for its subsequent effect in Reduced Differential Transform Method (RDTM) which was implemented on the transformed system of linear and nonlinear time-fractional PDEs. The solution obtained by Reduced Differential Transform Method (RDTM) is an infinite power series for appropriate initial condition, which can in turn express the exact solutions in a closed form. The results show that the Reduced Differential Transform Method (RDTM) is a powerful mathematical tool for solving partial differential equations with variable coefficients. Computational work fully re-confirms the reliability and efficacy of the proposed algorithm and hence it may be concluded that presented scheme may be applied to a wide range of physical and engineering problems.

Funding Statement

The authors have no support or funding to report.