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Open Journal of Mathematical Sciences

Vol. 2 (2018), Issue 1, pp. 29 – 38
ISSN: 2523-0212 (Online) 2616-4906 (Print)
DOI: 10.30538/oms2018.0015

Combination of Homotopy Perturbation Method (HPM) and Double Sumudu Transform to Solve Fractional KdV Equations

Hamood ur Rehman\(^{1}\), Muhammad Shoaib Saleem, Ayesha Ahmad
Department of Mathematics, University of Okara, Okara, Pakistan. (H.R & M. S.S & A.A)

\(^{1}\)Corresponding Author: hamood84@gmail.com

Copyright © 2018 Hamood ur Rehman, Muhammad Shoaib Saleem, Ayesha Ahmad. 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.
Received: January 13, 2018 – Accepted: March 9, 2018 – Published: March 11, 2018

Abstract

In this work, we developed homotopy perturbation double Sumudu transform method (HPDSTM) which is obtained by combining homotopy perturbation method, double Sumudu transform and He’s polynomials. The method is applied to find the solution of linear fractional one and two dimensional dispersive KdV and nonlinear fractional KdV equations to illustrate the reliability of the method. It is observed that the solutions obtained by the method converge rapidly to the exact solutions. This method is very powerful, and professional techniques for solving different kinds of linear and nonlinear fractional order differential equations.

Keywords:

Homotopy perturbation method, double Sumudu transform, He’s polynomial, Caputo fractional derivative, fractional KdV equations.

1. Introduction

Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. In recent years, considerable interest in fractional differential equation has been stimulated due to their numerous applications in the areas of physics and engineering. Many important phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are well described by fractional differential equation. Fractional differential equations are increasingly used to model problems in fluid mechanics, acoustics, biology, electromagnetism, diffusion, signal processing, and many other physical processes [1, 2]. In the literature one can find a wide class of methods dealing with the problem of approximate solutions to problems described by nonlinear fractional differential equations, for instance, asymptotic methods and perturbation methods.

A great deal of effort has been expended over the last 10 years or so in attempting to and robust and stable numerical and analytical methods for solving fractional partial differential equations of physical interest. Several numerical methods have been introduced to solve differential equations, such as the homotopy perturbation method (HPM) [3, 4, 5], the Modified homotopy perturbation method (MHPM) [6], the differential transform method (DTM) [7], the variational iteration method (VIM) [8, 9], the homotopy analysis method (HAM) [10, 11], the Sumudu decomposition method [12], the Adomian decomposition method [13, 14] and reproducing kernel method [15, 16, 17]. Among these methods, the HPM is a universal approach which can be used to solve FODEs and FPDEs. On the other hand, various methods are combined with the homotopy perturbation method, such as the variational homotopy perturbation method [18]. Another such combination is the homotopy perturbation transformation method [19]. Another such approach is the use of Homo-Separation of variables for the solution of fractional partial differential equations [20, 21, 22]. There are numerous integral transforms such as the Laplace, Sumudu, Fourier, Mellino solve PDEs. Of these, the Laplace transformation and Sumudu transformation are the most widely used. The Sumudu transformation method is one of the most important transform method. Various methods are combined with the Sumudu transformation method such as the homotopy analysis Sumudu transform method (HASTM) [23]. Another example is the Sumudu decomposition method (SDM) [24].

Singh et al. [25] have made used of studying the solutions of linear and nonlinear partial differential equations by using the homotopy perturbation Sumudu transform method. The nonlinear terms can be easily handled by the use of He's polynomial. The use of He's polynomials in the nonlinear term was first introduced by Ghorbani [26, 27]. There are some important applications and work on fractional differential equation in literature [28, 29, 30].

In this paper, we applied homotopy perturbation double Sumudu transform method to obtain the analytical exact and approximate solutions. The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. The advantage of this method is its capability of combining two powerful methods for obtaining exact solutions for nonlinear equations. The HPDSTM is a combination of double Sumudu transform, HPM, and He's polynomials.

2. Basic definitions and auxiliary theorems related to Sumudu Transform and fractional calculus

The Sumudu transform is an integral transform similar to the Laplace transform, introduced in the early 1990s by Watugala [31] to solve differential equations and control engineering problems. Note that following theorems and definitions will be used in the rest of the paper.

Definition 2.1. The Sumudu transform of a function \(f(t)\) defined for all real number \(t\geq0\) is the function \(F_{s}\left(u\right)\) defined by

\begin{equation} S_{t}\left[f\left(t\right), \left(u\right)\right]=F_{s}\left(u\right)=\int_{0}^{\infty}\frac{1}{u}~e^{\frac{-t}{u}}f\left(t\right)dt \end{equation}
(1)
Many of special properties of the Sumudu transform are mentioned and tabulated in [32, 33]. Some special properties of the Sumudu transform are as follows:
  1. \(S_{t}\left[1\right]=1\)
  2. \(S_{t}\left[\frac{t^{n}}{\Gamma\left(n+1\right)}\right]=u^{n}\)
  3. \(S_{t}\left[f\left(x\right)\mp g\left(x\right)\right]=S\left[f\left(x\right)\right]\mp S\left[g\left(x\right)\right]\)

Definition 2.2. The double Sumudu transform of a function \(f\left(x,t\right)\), defined for all real numbers \(\left(x\geq0, t\geq0\right)\) is defined by

\begin{equation} F\left(u,v\right)=S_{xt}\left[f\left(x,t\right),\left(u,v\right)\right]=\frac{1}{uv}\int\int_{0}^{\infty}~e^{-\left(\frac{t}{v}+\frac{x}{u}\right)}f\left(x,t\right)dxdt \end{equation}
(2)
In the same line of ideas, the double Sumudu transform of second partial derivative with respect to \(x\) is of form [34].
\begin{equation} S_{xt}\left[\frac{\partial^2 f\left(x,t\right)}{\partial x^2};\left(u,v\right)\right]=\frac{1}{u^2}F\left(u,v\right)-\frac{1}{u^2}F\left(0,v\right)-\frac{1}{u}\frac{\partial F\left(0,v\right)}{\partial x} \end{equation}
(3)
Similarly, the double Sumudu transform of second partial derivative with respect to \(t\) is of form [34]
\begin{equation} S_{xt}\left[\frac{\partial^2 f\left(x,t\right)}{\partial t^2};\left(u,v\right)\right]=\frac{1}{v^{2}}F\left(u,v\right)-\frac{1}{v^{2}}F\left(u,0\right)-\frac{1}{v}\frac{\partial F\left(u,0\right)}{\partial t} \end{equation}
(4)

Theorem 2.3. [32] Let \(G\left(u\right)\) be the Sumudu transform of \(f\left(t\right)\), such that \begin{enumerate} \item \(G\left(1/s\right)/s\), is a meromorphic function, with singularities having \(Re\left(s\right) \leq \gamma\) and \item There exists a circular region G with radius R and positive constants, M and k with \(|G\left(1/s\right)/s| < MR^{-k}\): then the function \(f\left(t\right)\) is given by

\begin{equation}S^{-1}\left[G\left(s\right)\right]=\frac{1}{2\Pi\iota}\int^{\gamma+\iota\infty}_{\gamma-\iota\infty} exp\left[st\right]G\left(\frac{1}{s}\right)\frac{ds}{s}=\sum residual\left[e^{st}\frac{G\left(\frac{1}{s}\right)}{s}\right]\end{equation}
(5)
\end{enumerate}

Definition 2.4. The Riemann-Liouville fractional integral operator of order \(\alpha>0\), of a function \(f\left(t\right)\in C_{\mu}\),\(\mu\geq-1\) is defined as

\begin{equation}j^{\alpha}f\left(t\right)=\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{t}\left(t-\tau\right)^{\alpha-1}f\left(\tau\right)d\tau\\ \end{equation}
(6)

Definition 2.5. [10, 27,] The fractional derivative of \(f\left(t\right)\) in the caputo sense is defined as

\begin{eqnarray} D^{\alpha}_{t}f\left(t\right)&=&j^{m-\alpha}D^{n}f\left(t\right)\nonumber\\ &=&\frac{1}{\Gamma\left(n-\alpha\right)}\int_{0}^{t}\left(t-\tau\right)^{m-\alpha-1}f^{m}\tau d\left(\tau\right) \end{eqnarray}
(7)
for \(m-1< \alpha\leq m, m\in N\) and \( t>0\) and \(\Gamma\left(\alpha\right)\) is a Gamma function.

Lemma 2.6. If \(m-1< \alpha\leq m, m\in N\) and \(f\in C_{\mu}^{m}\), and \(\mu \geq -1,\) then

\begin{equation}j^{\alpha}D^{\alpha}_{0}f\left(x\right)=f\left(x\right)-\sum^{m-1}_{k=0}f^{k}\left(0^{+}\right)\frac{x^{k}}{k!}, x>0 \end{equation}
(8)

Definition 2.7. Assume that \(f\left(x\right)\) is a function of \(n\) variables \(x_{i},\ i=1,\cdots,n\) also of class C on \( D\in R_{n}\)

\begin{equation}a\partial_{x}^{\alpha} f=\frac{1}{\Gamma\left(m-\alpha\right)}\int_{a}^{x_{i}} \left(x_{i}-t\right)^{m-\alpha-1}\partial_{x_{i}}^{m} f\left(x_{j}\right)dt \end{equation}
(9)
where \(\partial_{x_{i}}^{m}\) is the usual partial derivative of integer order \(m\).

Theorem 2.8. Let \(\frac{\partial^{i+j}f\left(x,t\right)}{\partial t^{j}\partial x^{i}}\)~\(, i=0,1,\cdots,n,~ j=0,1,\cdots,m\) be of exponential order; i.e \(\left|\frac{\partial^{i+j}f\left(x,t\right)}{\partial t^{j}\partial x^{i}}\right|< \)Me\(^{\frac{x}{\tau_{1}}+ \frac{t}{\tau_{2}}}\) for some \(M,~\tau_{1},~\tau_{2}>0\), then the double Sumudu transform of the Caputo fractional derivative with respect to \(x\) is defined as follows

\begin{equation} S_{xt}\left[\frac{\partial^{n}f\left(x,t\right)}{\partial x^{n}}\right]=u^{-n}S_{xt}\left[f\left(x,t\right)\right]-\sum_{i=1}^{n-1}u^{i-n}S_{xt}\left[\frac{\partial^{i}f\left(0,t\right)}{\partial x^{i}}\right] \end{equation}
(10)
The double Sumudu transform of the Caputo fractional derivative with respect to \(t\) is defined as follows
\begin{equation}S_{xt}\left[\frac{\partial^{m}f\left(x,t\right)}{\partial t^{m}}\right]=u^{-m}S_{xt}\left[f\left(x,t\right)\right]-\sum_{j=1}^{m-1}u^{j-m}S_{xt}\left[\frac{\partial^{j}f\left(x,0\right)}{\partial t^{j}}\right] \end{equation}
(11)

3. Solution by HPDSTM

We illustrate the basic idea of this method, by considering a general fractional nonlinear non- homogeneous partial differential equation with the initial condition of the form of general form
\begin{equation}D_{t}^{\alpha}U\left(x,t\right)=L\left(U\left(x,t\right)\right)+N\left(U\left(x,t\right)\right)+f\left(x,t\right),\\ \alpha>0 \end{equation}
(12)
subject to initial condition
\begin{equation}D_{0}^{k}U\left(x,0\right)=g_{k} , \left(k=0,\cdots,n-1\right) \end{equation}
(13)
\(D_{0}^{n}U\left(x,0\right)=0\), where, \(D_{t}^{\alpha}=\frac{\partial ^{\alpha}}{\partial {t^{\alpha}}}\) denotes without loss of generality the Caputo fraction derivative operator, \(f\) is a known function, \(N\) is the general nonlinear fractional differential operator, and \(L\) represents a linear fractional differential operator. Applying the double Sumudu Transform on both sides of \((12)\), we obtain
\begin{equation}S_{xt}\left[D_{t}^{\alpha}U\left(x,t\right)\right] =S_{xt}\left[L(U\left(x,t\right))\right]+S_{xt}\left[N\left(U\left(x,t\right)\right)\right]+S_{xt}\left[f\left(x,t\right)\right] \end{equation}
(14)
Using the property of the double Sumudu transform, we have
\begin{equation}S_{xt}\left[U\left(x,t\right)\right]=u^{\alpha}S_{xt}\left[L\left(U\left(x,t\right)\right)\right]+u^{\alpha}S_{xt}\left[N\left(U\left(x,t\right)\right)\right]+u^{\alpha} S_{xt}\left[f\left(x,t\right)\right]+g\left(x,t\right) \end{equation}
(15)
Now applying the double Sumudu inverse on both sided of \((15)\), we obtain
\begin{equation}U\left(x,t\right)=S_{xt}^{-1}\left[u^{\alpha}S_{xt}\left[L\left(U\left(x,t\right)\right)\right]+u^{\alpha} S_{xt}\left[N\left(U\left(x,t\right)\right)\right]\right]+G\left(x,t\right) \end{equation}
(16)
where \(G\left(x,t\right)\) represents the term arising from the known function \(f\left(x,t\right)\) and the initial condition [31]. Now apply the HPM
\begin{equation}U\left(x,t\right)=\sum_{n=0}^{\infty}P^{n}U_{n}\left(x,t\right) \end{equation}
(17)
The nonlinear term can be decomposed into
\begin{equation}NU\left(x,t\right)=\sum_{n=0}^{\infty}p^{n}H_{n}\left(U\right)(x,t) \end{equation}
(18)
Using the He's polynomial [32] given as
\begin{equation}H_{n}\left(U_{0},\cdots,U_{n}\right)=\frac{1}{n!}\frac{\partial^{n}}{\partial p^{n}}\left[N\left(\sum_{j=0}^{\infty}p^{j}U_{j}\left(x,t\right)\right)\right] \end{equation}
(19)
Substituting \((17)\) and \((18)\) in \((16)\) \begin{eqnarray} \sum_{n=0}^{\infty}P^{n}U_{n}\left(x,t\right)&=&G\left(x,t\right)+ P\Big[S^{-1}_{xt}\Big[u^{\alpha}S_{xt} \Big[L\Big(\sum_{n=0}^{\infty}P^{n}U_{n}\left(x,t\right)\Big)\Big]\nonumber\\&&+u^{\alpha }S_{xt}\Big[N\Big(\sum_{n=0}^{\infty}P^{n}U_{n}\left(x,t\right)\Big)\Big]\Big]\Big] \end{eqnarray}
(20)
which is the coupling of the double Sumudu transform and the HPM using He's polynomials [34]. Comparing the coefficients of like powers of \(P\), the following approximations are obtained: \begin{eqnarray*} P^{0} :U_{0}\left(x,t\right)&=&G\left(x,t\right)\\ P^{1} :U_{1}\left(x,t\right)&=&S^{-1}_{xt}\left[u^{\alpha}S_{xt}\left[L\left(U_{0}\left(x,t\right)\right)+H_{0}\left(U\right)\right]\right]\\ P^{2} :U_{2}\left(x,t\right)&=&S^{-1}_{xt}\left[u^{\alpha}S_{xt}\left[L\left(U_{1}\left(x,t\right)\right)+H_{1}\left(U\right)\right]\right]\\ P^{3} :U_{3}\left(x,t\right)&=&S^{-1}_{xt}\left[u^{\alpha}S_{xt}\left[L\left(U_{2}\left(x,t\right)\right)+H_{2}\left(U\right)\right]\right]\\ && \vdots\\ P^{n} :U_{n}\left(x,t\right)&=&S^{-1}_{xt}\left[u^{\alpha}S_{xt}\left[L\left(U_{2}\left(x,t\right)\right)+H_{n-1}\left(U\right)\right]\right]\\ \end{eqnarray*} Finally, we approximate the analytical solution \(U\left(x,t\right)\) by truncated series [34]
\begin{equation} U\left(x,t\right)=\lim_{N\rightarrow\infty}\sum_{n=0}^{N}U_{n}\left(x,t\right)\end{equation}
(21)
The above series solutions generally converge very rapidly [34, 35, 36].

4. Applications

In this section, we apply this method for solving solution of one and two dimensional linear fractional dispersive KdV and nonlinear fractional KdV equations to illustrate the reliability of the method.

Example 4.1. Consider the linear one dimensional fractional dispersive KdV equation [37]

\begin{eqnarray} u^{\alpha}_{t}\left(x,t\right)&=&-2u_{x}-u_{xxx} ,t>0\nonumber\\ u\left(x,0\right)&=& \sin x \end{eqnarray}
(22)
Following carefully the steps involved in the HPDSTM, we arrive at the following series solutions \begin{eqnarray*} u_{0}\left(x,t\right)&=& \sin x\\ u_{1}\left(x,t\right)&=& -\frac{t^{\alpha}}{\Gamma\left(\alpha+1\right)} \cos x\\ u_{2}\left(x,t\right)&=& -\frac{t^{2\alpha}}{\Gamma\left(2\alpha+1\right)} \sin x\\ u_{3}\left(x,t\right)&=& \frac{t^{3\alpha}}{\Gamma\left(3\alpha+1\right)} \cos x\\ &&\vdots \end{eqnarray*} we obtained solution in the following form \begin{eqnarray*}u\left(x,t\right)= \sin x-\frac{t^{\alpha}}{\Gamma\left(\alpha+1\right)} \cos x-\frac{t^{2\alpha}}{\Gamma\left(2\alpha+1\right)} \sin x+\frac{t^{3\alpha}}{\Gamma\left(3\alpha+1\right)} \cos x+...\end{eqnarray*} When \(\alpha \rightarrow 1\), we recover the following series approximation \begin{eqnarray*} u\left(x,t\right)&=& \sin x-t \cos x-\frac{t^{2}}{2!} \sin x+\frac{t^{3}}{3!} \cos x+...\\ u\left(x,t\right)&=& \sin x \cos t - \cos x \sin t = \sin \left(x-t\right) \end{eqnarray*} Which is the exact solution of this problem.

Example 4.2. Consider the linear fractional two dimensional dispersive KdV equation [37]

\begin{eqnarray} u^{\alpha}_{t}\left(x,y,t\right)&=&-u_{xxx}-u_{yyy} ,t>0 \nonumber\\u\left(x,y,0\right)&=& \cos \left(x+y\right) \end{eqnarray}
(23)
Following carefully the steps involved in the HPDSTM, we arrive at the following series solutions \begin{eqnarray*} u_{0}\left(x,y,t\right)&=& \cos \left(x+y\right)\\ u_{1}\left(x,y,t\right)&=&-2\frac{t^{\alpha}}{\Gamma\left(\alpha+1\right)} \sin \left(x+y\right)\\ u_{2}\left(x,y,t\right)&=&-4\frac{t^{2\alpha}}{\Gamma\left(2\alpha+1\right)} \cos \left(x+y\right)\\ u_{3}\left(x,y,t\right)&=&8\frac{t^{3\alpha}}{\Gamma\left(3\alpha+1\right)} \sin \left(x+y\right)\\ &&\vdots\\ \end{eqnarray*} We obtained solution in the following form \begin{eqnarray*}u\left(x,y,t\right)&=& \cos \left(x+y\right)-2\frac{t^{\alpha}}{\Gamma\left(\alpha+1\right)} \sin \left(x+y\right)-4\frac{t^{2\alpha}}{\Gamma\left(2\alpha+1\right)} \cos \left(x+y\right) \\&&+8\frac{t^{3\alpha}}{\Gamma\left(3\alpha+1\right)} \sin \left(x+y\right)+...\end{eqnarray*} When \(\alpha\rightarrow 1\) we recover the following series approximation \begin{eqnarray*}u\left(x,y,t\right)&=& \cos \left(x+y\right)-2t \sin \left(x+y\right)-4\frac{t^{2}}{2!} \cos \left(x+y\right)+8\frac{t^{3}}{3!} \sin \left(x+y\right)+... \nonumber\\ &=& \sin \left(x+y+2t\right) \end{eqnarray*} Which is the exact solution of this problem.

Example 4.3. We consider the following nonlinear time fractional KdV equation [38] \begin{eqnarray*}u^{\alpha}_{t}\left(x,t\right)&=&\left(u^{2}\right)_{x}+\left[u\left(u\right)_{xx}\right]_{x},~~t>0 ,x>0,0< \alpha\leq 1\\ u\left(x,0\right)&=& \sinh ^{2}\left(\frac{x}{2}\right) \end{eqnarray*} Applying the double Sumudu transform on both sided, we obtained the following.

\begin{equation} S_{xt}\left[u\left(x,t\right)\right]=S_{xt}\left[ \sinh ^{2}\left(\frac{x}{2}\right)\right]+u^{\alpha}S_{xt}\left[\left(u^{2}\right)_{x}+\left[u\left(u\right)_{xx}\right]_{x}\right]\end{equation}
(24)
Applying the inverse double Sumudu transform, we obtain the following
\begin{equation}u\left(x,t\right)= \sinh ^{2}\left(\frac{x}{2}\right)+S_{xt}^{-1}\left[u^{\alpha}\left[S_{xt}\left[\left(u^{2}\right)_{x}+\left[u\left(u\right)_{xx}\right]_{x}\right]\right]\right] \end{equation}
(25)
Now applying the homotopy perturbation tecnique on the above equation we obtain the following
\begin{eqnarray} \sum_{n=0}^{\infty}P^{n}u_{n}\left(x,t\right)&=& \sinh ^{2}\left(\frac{x}{2}\right)+P\Big[S_{xt}^{-1}\Big[u^{\alpha}\Big[S_{xt}\Big[\sum_{n=0}^ {\infty}P^{n}H_{n}\left(u\right)\nonumber\\&&+ \sum_{n=0}^{\infty}P^{n}H_{n}^{1}\left(u\right)\Big]\Big]\Big]\Big] \end{eqnarray}
(26)
where \( H_{n}\left(u\right)\) and \(H_{n}^{1}\left(u\right)\) are He's polynomials that represent the nonlinear terms.4.1. By comparing the cofficients of like powers \(p\), we have \begin{eqnarray*} P^{0}:u_{0}(x,t)&=& \sinh ^{2}\left(\frac{x}{2}\right)\\ P^{1}:u_{1}\left(x,t\right)&=&S_{xt}^{-1}\left[u^{\alpha}\left[S_{xt}\left[H_{0}\left(u\right)+H_{0}^{1}\left(u\right)\right]\right]\right] =-\frac{t^{\alpha}}{4\Gamma\left(\alpha+1\right)} \sinh \left(x\right)\\ P^{2}:u_{2}\left(x,t\right)&=&S_{xt}^{-1}\left[u^{\alpha}\left[S_{xt}\left[H_{1}\left(u\right)+H_{1}^{1}\left(u\right)\right]\right]\right] =+\frac{t^{2\alpha}}{8\Gamma\left(2\alpha+1\right)} \cosh \left(x\right)\\ &&\vdots\\ \end{eqnarray*} Consequently the third term of the HPDST solution for example 4.3 is given by
\(u\left(x,t\right)= \sinh ^{2}\left(\frac{x}{2}\right)-\frac{t^{\alpha}}{4\Gamma\left(\alpha+1\right)} \sinh \left(x\right)+\frac{t^{2\alpha}}{8\Gamma\left(2\alpha+1\right)} \cosh \left(x\right).\)

5. Conclusion

The aim of this work was to make use of the properties of the double Sumudu transform to solve linear and nonlinear fractional KdV problems. The basic idea of the method combines double Sumudu transform and the HPM using He's polynomial. This combination builds a strong method called the homotopy perturbation double Sumudu transform (HPDSTD). The HPDSTD is an analytical method and runs using the initial conditions only. HPDSTM is a very powerful and efficient method to find approximate solutions as well as numerical solutions.

Competing Interests

The authors do not have any competing interests in the manuscript.

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