Recent Developments in Integral Transforms, Special Functions, and Their Extensions to Distributions Theory
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The TimeFractional CoupledKortewegdeVries Equations
Abstract
We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupledKortewegdeVries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractionalorder derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).
1. Introduction
Fractional calculus has been used to model physical and engineering processes, which are found to be best described by fractional differential equations. It is worth nothing that the standard mathematical models of integerorder derivatives, including nonlinear models, do not work adequately in many cases. In the recent years, fractional calculus has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, notably control theory, signal image processing, and groundwater problems. In the past several decades, the investigation of travellingwave solutions for nonlinear equations has played an important role in the study of nonlinear physical phenomena. In [1], homotopy analysis method is applied to obtain approximate analytical solution of the modified KuramotoSivashinsky equation. In addition to that an excellent literature of this can be found in [2–11]. Analytical solutions of these equations are usually not available. Since only limited classes of equations are solved by analytical means, numerical solution of these nonlinear partial differential equations is of practical importance.
In this paper, we extend the application of the homotopy decomposition method (HDM) in order to derive analytical approximate solutions to nonlinear timefractional coupledKDV equations. This coupled system is used to describe iterations of water waves proposed by Hirota and Satsuma [12]. The HDM was recently applied to solve the fractional modified Kawahara equation, fractional model of HIV infection of CD4+T cells, the attractor fractional onedimensional KellerSegel equations, the fractional JaulentMiodek and WhithamBroerKaup equations, the fractional Riccati differential equation, fractional nonlinear predatorprey population, and the fractional nonlinear system predatorprey population. The relatively new technique that approached the HDM is a promising analytical technique to solve nonlinear fractional partial and ordinary differential equations. The fractional systems of partial differential equations under investigation here are given as Subject to the initial conditions The remaining of this paper is structured as follows: in Section 2 we present a brief history of the fractional derivative order and their properties. We present the basic ideal of the homotopy decomposition method for solving highorder nonlinear fractional partial differential equations. We present the application of the HDM for system fractional nonlinear differential equations (1) and numerical results in Section 4. The conclusions are then given in Section 5.
2. Fractional Derivative Order
2.1. Brief History
In the literature, one can find several definitions of fractional derivatives. The most common used are the RiemannLiouville and the Caputo derivatives. For Caputo we have For the case of RiemannLiouville we have the following definition: Each fractional derivative presents some advantages and disadvantages [13, 14]. The RiemannLiouville derivative of a constant is not zero while Caputo’s derivative of a constant is zero but demands higher conditions of regularity for differentiability: to compute the fractional derivative of a function in the Caputo sense, we must first calculate its derivative. Caputo derivatives are defined only for differentiable functions while functions that have no firstorder derivative might have fractional derivatives of all orders less than one in the RiemannLiouville sense [15, 16]. Recently, Jumarie (see [17, 18]) proposed a simple alternative definition to the RiemannLiouville derivative: His modified RiemannLiouville derivative seems to have advantages of both the standard RiemannLiouville and Caputo fractional derivatives: it is defined for arbitrary continuous (nondifferentiable) functions and the fractional derivative of a constant is equal to zero. However, the Jumarie fractional derivative gives the fractional derivative of not for , this implies that, there is no fractional derivative for some functions that are not defined at the origin, for instance [19].
We can point out that Caputo and RiemannLiouville may have their disadvantages but they still remain the best definitions of the fractional derivative. Every definition must be used accordingly [19].
2.2. Properties and Definitions
Definition 1. A real function is said to be in the space if there exists a real number , such that , where , and it is said to be in space if .
Definition 2. The RiemannLiouville fractional integral operator of order , of a function , is defined as
Properties of the operator can be found in [15, 16], and one mentions only the following:
for , and:
Lemma 3. If and then
Definition 4 (partial derivatives of fractional order). Assume now that is a function of variables also of class on . As an extension of Definition 4, one defines partial derivative of order for with respect to the function if it exists, where is the usual partial derivative of integerorder .
3. Basic Idea of the HDM
To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous fractional partial differential equation with initial conditions of the following form: Subject to the initial condition where denotes the Caputo or RiemannLiouville fraction derivative operator, is a known function, is the general nonlinear fractional differential operator, and represents a linear fractional differential operator. The method first step here is to transform the fractional partial differential equation to the fractional partial integral equation by applying the inverse operator of both sides of (10) to obtain the following. In the case of RiemannLiouville fractional derivative In the case of Caputo fractional derivative or in general by putting we obtain the following: In the homotopy decomposition method, the basic assumption is that the solutions can be written as a power series in
and the nonlinear term can be decomposed as where is an embedding parameter. is the He’s polynomials that can be generated by The homotopy decomposition method is obtained by the graceful coupling of homotopy technique with the Abel integral and is given by Comparison of the terms of same powers of gives solutions of various orders with the first term:
3.1. Convergence of the Method and Unicity of the Solution
Theorem 5 (see [19]). Assuming that is a Banach space with a welldefined norm , over which the series sequence of the approximate solution of (1) is defined, and the operator defining the series solution of (16b) satisfies the Lipschitzian conditions that is for all , then series solution obtained (16b) is unique.
Proof. Assume that and are the series solution satisfying (1), then with initial guess ; also with initial guess ; therefore, By the recurrence for , assume that for , . Then which completes the proof.
3.2. Complexity of the Homotopy Decomposition Method
It is very important to test the computational complexity of a method or algorithm. Complexity of an algorithm is the study of how long a program will take to run, depending on the size of its input and long of loops made inside the code. We compute a numerical example which is solved by the homotopy decomposition method. The code has been presented with Mathematica 8 according to the following code [19].
Step 1. Set .
Step 2. Calculating the recursive relation after the comparison of the terms of the same power is done.
Step 3. If with the ratio of the neighbourhood of the exact solution [5] then go to Step 4, else and go to Step 2
Step 4. Print out as the approximate of the exact solution.
Lemma 6. If the exact solution of the fractional partial differential equation (10) exists, then
Proof. Let , then since the exact solution exists, then we have that following: The last inequality follows from [19].
Lemma 7. The complexity of the homotopy decomposition method is of order .
Proof. The number of computations including product, addition, subtraction, and division are in Step 2 : is 0 because, it is obtained directly form the initial guess [19]. : 3 : 3.Now in Step 4, the total number of computations is equal to .
4. Application
In learning science, examples are useful than rules (Isaac Newton). In this section, we apply this method for solving system of fractional differential equation. Following carefully the steps involved in the HDM, we arrive at the following equations: If we compare the terms of the same power of we obtain the following integral equations. Note that when comparing this approach with the methodology of the homotopy perturbation method, one will obtain in this step a set of ordinary differential equations something which needs to be also solved with care, because one will need to choose an appropriate initial guest. But with the current approach, the initial guess is straightforwardly obtained as the Taylor series of the exact solution of the problem under investigation; this is one of the advantages that the approach has over the HPM [22]. On the other hand, when comparing this approach with the variational iteration method [23], one will find out that we do need the Lagrange multiplier here or the correctional function. Also this approach provides us with a convenient way to control the convergence of approximation series without adapting , as in the case of [24] which is a fundamental qualitative difference in analysis between HDM and other methods. Therefore, comparing the terms of the same power we obtain Integrating the above, we obtain the following series solutions: For the sake of simplicity we put the following: And so on, using the package Mathematica, in the same manner, one can obtain the rest of the components. But, here, few terms were computed and the asymptotic solution is given by the following:
4.1. Numerical Solutions
The following figures show the graphical representation of the approximated solution of the system of the timefractional coupledKortewegdeVries equations for .
Note that the below figure show that the coupled solution of KDV equation is not only the function of time and space but also an increasing function of the fractional order derivative, which are and . The approximate solution of main problem has been depicted in Figures 1, 2, 3, and 4 which is plotted in Mathematica according to different and values.
It is important to note that if , and , the exact solution of the coupledKDV equations is given as Thus, to test the accuracy of the relatively new analytical technique, we represent in Table 1 the numerical values of the approximate and the exact solutions and the results obtained in [20].

Table 1 comparison shows that the solutions obtained in this paper are more accurate than those obtained in [20].
5. Conclusions
We derived approximated solutions of nonlinear fractionalcoupled KDV equations using the relatively new analytical technique, the HDM. We presented the brief history and some properties of fractional derivative concept. It is demonstrated that HDM is a powerful and efficient tool of FPDEs. In addition, the calculations involved in HDM are very simple and straightforward. Comparing the methodology HDM to HPM, ADM [25], VIM, and HAM have the advantages. Disparate the ADM, the HDM is free from the need to use the Adomian polynomials. In this method, we do not need the Lagrange multiplier, correction functional, stationary conditions, or calculating heavy integrals, as the solutions obtained are noise free [26], which eliminate the complications that exist in the VIM. In contrast to the HAM, this method is not required to solve the functional equations in iteration since the efficiency of HAM is very much dependant on choosing auxiliary parameter. In contract to HPM, we do not need to continuously deform a difficult problem to another that is easier to solve. We can easily conclude that the homotopy decomposition method is a wellorganized analytical method for solving exact and approximate solutions of nonlinear fractional partial differential equations.
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Copyright © 2013 Abdon Atangana and Aydin Secer. 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.