The Time-Fractional Coupled-Korteweg-de-Vries Equations

Abstract

We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order 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 integer-order 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 travelling-wave 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 Kuramoto-Sivashinsky 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 time-fractional coupled-KDV 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 one-dimensional Keller-Segel equations, the fractional Jaulent-Miodek and Whitham-Broer-Kaup equations, the fractional Riccati differential equation, fractional nonlinear predator-prey population, and the fractional nonlinear system predator-prey 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 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼+6𝑎𝑢(𝑥,𝑡)𝑢_𝑥(𝑥,𝑡)−2𝑏𝑣(𝑥,𝑡)𝑣_𝑥(𝑥,𝑡)+𝑎𝑢_{𝑥,𝑥,𝑥}()𝜕𝑥,𝑡=0,0<𝛼≤1,_𝛽𝑣(𝑥,𝑡)𝜕𝑡^𝛽+3𝑏𝑢(𝑥,𝑡)𝑣_𝑥(𝑥,𝑡)+𝑏𝑣_{𝑥,𝑥,𝑥}(𝑥,𝑡)=0,0<𝛽≤1. (1) Subject to the initial conditions 𝑢()=𝜆𝑥,0𝑎1sech2𝜆𝑎𝑥²,𝑣()=𝜆𝑥,0√12𝑎sech2𝜆𝑎𝑥². (2) 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 high-order 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 Riemann-Liouville and the Caputo derivatives. For Caputo we have ^𝐶₀𝐷^𝛼_𝑥𝑓(𝑥)=1Γ(𝑛−𝛼)^𝑥₀(𝑥−𝑡)^{𝑛−𝛼−1}𝑑^𝑛𝑓(𝑡)𝑑𝑡^𝑛𝑑𝑡. (3) For the case of Riemann-Liouville we have the following definition: 𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛𝑛𝑥₀(𝑥−𝑡)^{𝑛−𝛼−1}𝑓(𝑡)𝑑𝑡. (4) Each fractional derivative presents some advantages and disadvantages [13, 14]. The Riemann-Liouville 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 first-order derivative might have fractional derivatives of all orders less than one in the Riemann-Liouville sense [15, 16]. Recently, Jumarie (see [17, 18]) proposed a simple alternative definition to the Riemann-Liouville derivative: 𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛𝑛𝑥₀(𝑥−𝑡)^{𝑛−𝛼−1}𝑓(𝑡)−𝑓(0)𝑑𝑡. (5) His modified Riemann-Liouville derivative seems to have advantages of both the standard Riemann-Liouville 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 𝑓(𝑥)−𝑓(0) not for 𝑓(𝑥), this implies that, there is no fractional derivative for some functions that are not defined at the origin, for instance ln(𝑥) [19].

We can point out that Caputo and Riemann-Liouville 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 𝑓(𝑥),𝑥>0 is said to be in the space 𝐶_𝜇,𝜇∈ℝ if there exists a real number 𝑝>μ, such that 𝑓(𝑥)=𝑥^𝑝ℎ(𝑥), where ℎ(𝑥)∈𝐶[0,∞), and it is said to be in space 𝐶^𝑚_𝜇 if  𝑓^(𝑚)∈𝐶_𝜇,𝑚∈ℕ.

Definition 2. The Riemann-Liouville fractional integral operator of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇,𝜇≥−1, is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝐽𝑑𝑡,𝛼>0,𝑥>0,₀𝑓(𝑥)=𝑓(𝑥). (6) Properties of the operator can be found in [15, 16], and one mentions only the following:
for  𝑓∈𝐶_𝜇,𝜇≥−1,𝛼,𝛽≥0, and𝛾>−1: 𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛼+𝛽𝑓(𝑥),𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝛼𝑓(𝑥)𝐽^𝛼𝑥^𝛾=Γ𝛾+1Γ𝑥𝛼+𝛾+1^𝛼+𝛾. (7)

Lemma 3. If 𝑚−1<𝛼≤𝑚,𝑚∈ℕ  and 𝑓∈𝐶^𝑚_𝜇,𝜇≥−1,then 𝐷^𝛼𝐽^𝛼𝑓(𝑥)(𝑥),𝐽=𝑓_𝛼𝐷^𝛼₀𝑓(𝑥)=𝑓(𝑥)−^𝑚−1_𝑘=0𝑓^(𝑘)0⁺𝑥^𝑘𝑘!,𝑥>0. (8)

Definition 4 (partial derivatives of fractional order). Assume now that 𝑓(𝐱) is a function of 𝑛 variables 𝑥_𝑖,𝑖=1,…,𝑛 also of class 𝐶 on 𝐷∈ℝ_𝑛. As an extension of Definition 4, one defines partial derivative of order 𝛼 for 𝑓 with respect to 𝑥_𝑖 the function 𝑎𝜕^𝛼_𝐱1𝑓=Γ(𝑚−𝛼)^𝑥𝑖_𝑎𝑥_𝑖−𝑡^{𝑚−𝛼−1}𝜕^𝑚_𝑥𝑖𝑓𝑥_{𝑗𝑥𝑗=𝑡}𝑑𝑡, (9) if it exists, where  𝜕^𝑚_𝑥𝑖  is the usual partial derivative of integer-order 𝑚.

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: 𝜕^𝛼𝑈(𝑥,𝑡)𝜕𝑡^𝛼=𝐿(𝑈(𝑥,𝑡))+𝑁(𝑈(𝑥,𝑡))+𝑓(𝑥,𝑡),𝛼>0. (10) Subject to the initial condition 𝐷₀^𝛼−𝑘𝑈(𝑥,0)=𝑓_𝑘(𝑥),(),𝐷𝑘=0,…,𝑛−1_0𝛼−𝑛𝑈()[𝛼],𝐷𝑥,0=0,𝑛=_𝑘0𝑈(𝑥,0)=𝑔_𝑘(𝑥),(),𝐷𝑘=0,…,𝑛−1_𝑛0𝑈(𝑥,0)=0,𝑛=[𝛼], (11) where 𝜕^𝛼/𝜕𝑡^𝛼  denotes the Caputo or Riemann-Liouville fraction derivative operator, 𝑓 is a known function, 𝑁 is the general nonlinear fractional differential operator, and 𝐿 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  𝜕^𝛼/𝜕𝑡^𝛼 of both sides of (10) to obtain the following. In the case of Riemann-Liouville fractional derivative   𝑈(𝑥,𝑡)=^𝑛−1_𝑗=1𝑓_𝑗(𝑥)Γ𝑡𝛼−𝑗+1^𝛼−𝑗+1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐿(𝑈((𝑈((𝑥,𝜏))+𝑁𝑥,𝜏))+𝑓𝑥,𝜏)𝑑𝜏. (12) In the case of Caputo fractional derivative 𝑈(𝑥,𝑡)=^𝑛−1_𝑗=1𝑔_𝑗(𝑥)Γ𝑡𝛼−𝑗+1^𝑗+1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐿(𝑈((𝑈((𝑥,𝜏))+𝑁𝑥,𝜏))+𝑓𝑥,𝜏)𝑑𝜏, (13) or in general by putting ^𝑛−1_𝑗=1𝑓_𝑗(𝑥)Γ𝑡𝛼−𝑗+1^𝛼−𝑗=𝑓(𝑥,𝑡)or𝑓(𝑥,𝑡)=^𝑛−1_𝑗=1𝑔_𝑗(𝑥)Γ𝑡𝛼−𝑗+1^𝑗, (14) we obtain the following: 𝑈()()+1𝑥,𝑡=𝑇𝑥,𝑡Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝐿(𝑈((𝑈((𝑥,𝜏))+𝑁𝑥,𝜏))+𝑓𝑥,𝜏)𝑑𝜏. (15) In the homotopy decomposition method, the basic assumption is that the solutions can be written as a power series in 𝑝𝑈𝑥,𝑡,𝑝=^∞_𝑛=0𝑝^𝑛𝑈_𝑛(),𝑈()𝑥,𝑡𝑥,𝑡=lim_𝑝→1𝑈𝑥,𝑡,𝑝, (16a)(16b)

and the nonlinear term can be decomposed as 𝑁𝑈(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑈), (17) where 𝑝∈(0,1] is an embedding parameter. ℋ_𝑛(𝑈) is the He’s polynomials that can be generated by ℋ_𝑛𝑈₀,…,𝑈_𝑛=1𝜕𝑛!𝜕𝑝^𝑛𝑛𝑁^∞_𝑗=0𝑝^𝑗𝑈_𝑗(𝑥,𝑡),𝑛=0,1,2,…. (18) The homotopy decomposition method is obtained by the graceful coupling of homotopy technique with the Abel integral and is given by ^∞_𝑛=0𝑝^𝑛𝑈_𝑛()()=𝑝𝑥,𝑡−𝑇𝑥,𝑡Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝑥,𝜏)+𝐿^∞_𝑛=0𝑝^𝑛𝑈_𝑛(𝑥,𝜏)+𝑁^∞_𝑛=0𝑝^𝑛𝑈_𝑛(𝑥,𝜏)𝑑𝜏. (19) Comparison of the terms of same powers of 𝑝 gives solutions of various orders with the first term: 𝑈₀(𝑥,𝑡)=𝑇(𝑥,𝑡). (20)

3.1. Convergence of the Method and Unicity of the Solution

Theorem 5 (see [19]). Assuming that 𝑋×𝑇⊂ℝ×ℝ⁺ is a Banach space with a well-defined norm ‖⋅‖, over which the series sequence of the approximate solution of (1) is defined, and the operator 𝐺(𝑈_𝑛(𝑥,𝑡))=𝑈_𝑛+1(𝑥,𝑡) defining the series solution of (16b) satisfies the Lipschitzian conditions that is ‖𝐺(𝑈^∗_𝑘)−𝐺(𝑈_𝑘)‖≤𝜀‖𝑈^∗_𝑘(𝑥,𝑡)−𝑈_𝑘(𝑥,𝑡)‖ for all (𝑥,𝑡,𝑘)∈𝑋×𝑇×ℕ, then series solution obtained (16b) is unique.

Proof. Assume that 𝑈(𝑥,𝑡) and 𝑈^∗(𝑥,𝑡) are the series solution satisfying (1), then 𝑈^∗(𝑥,𝑡,𝑝)=∑^∞_𝑛=0𝑝^𝑛𝑈^∗_𝑛(𝑥,𝑡)  with initial guess  𝑇(𝑥,𝑡); 𝑈(𝑥,𝑡,𝑝)=∑^∞_𝑛=0𝑝^𝑛𝑈_𝑛(𝑥,𝑡) also with initial guess 𝑇(𝑥,𝑡); therefore, 𝑈^∗_𝑛(𝑥,𝑡)−𝑈_𝑛(𝑥,𝑡)=0,𝑛=0,1,2,…. (21) By the recurrence for 𝑛=0,𝑈^∗_𝑛(𝑥,𝑡)=𝑈_𝑛(𝑥,𝑡)=𝑇(𝑥,𝑡), assume that for 𝑛>𝑘≥0,  ‖𝑈^∗_𝑘(𝑥,𝑡)−𝑈_𝑘(𝑥,𝑡)‖=0. Then 𝑈^∗_𝑘+1(𝑥,𝑡)−𝑈_𝑘+1(𝑥,𝑡)=𝐺𝑈^∗_𝑘−𝐺𝑈_𝑘≤𝜀𝑈^∗_𝑘(𝑥,𝑡)−𝑈_𝑘(𝑥,𝑡)=0, (22) 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 𝑚←0.

Step 2. Calculating the recursive relation after the comparison of the terms of the same power is done.

Step 3. If ‖𝑈_𝑛+1(𝑥,𝑡)−𝑈_𝑛(𝑥,𝑡)‖<𝑟 with 𝑟 the ratio of the neighbourhood of the exact solution [5] then go to Step 4, else 𝑚←𝑚+1 and go to Step 2

Step 4. Print out 𝑈(𝑥,𝑡)=^∞_𝑛=0𝑈_𝑛(𝑥,𝑡), (23) as the approximate of the exact solution.

Lemma 6. If the exact solution of the fractional partial differential equation (10) exists, then 𝑈_𝑛+1(𝑥,𝑡)−𝑈_𝑛(𝑥,𝑡)<𝑟∀(𝑥,𝑡)∈𝑋×𝑇. (24)

Proof. Let (𝑥,𝑡)∈𝑋×𝑇, then since the exact solution exists, then we have that following: 𝑈_𝑛+1(𝑥,𝑡)−𝑈_𝑛(=𝑈𝑥,𝑡)_𝑛+1(𝑥,𝑡)−𝑈(𝑥,𝑡)+𝑈(𝑥,𝑡)−𝑈_𝑛(≤𝑈𝑥,𝑡)_𝑛+1(𝑥,𝑡)−𝑈(𝑥,𝑡)+−𝑈_𝑛()(≤𝑟𝑥,𝑡+𝑈𝑥,𝑡)2+𝑟2=𝑟. (25) The last inequality follows from [19].

Lemma 7. The complexity of the homotopy decomposition method is of order 𝑂(𝑛).

Proof. The number of computations including product, addition, subtraction, and division are in Step 2 𝑈_𝑜: is 0 because, it is obtained directly form the initial guess 𝑇(𝑥,𝑡) [19]. 𝑈₁: 3 ⋮ 𝑈_𝑛: 3.Now in Step 4, the total number of computations is equal to   ∑^𝑛_𝑗=0𝑈_𝑗(𝑥,𝑡)=3𝑛=𝑂(𝑛).

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: ^∞_𝑛=0𝑝^𝑛𝑢_𝑛()()−𝑝𝑥,𝑡=𝑢𝑥,0Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−16𝑎^∞_𝑛=0𝑝^𝑛𝑢_𝑛^∞_𝑛=0𝑝^𝑛𝑢_𝑛𝑥−2𝑏^∞_𝑛=0𝑝^𝑛𝑣_𝑛^∞_𝑛=0𝑝^𝑛𝑣_𝑛𝑥+^∞_𝑛=0𝑝^𝑛𝑢_{𝑛𝑥,𝑥,𝑥},_∞𝑛=0𝑝^𝑛𝑣_𝑛()()−𝑝𝑥,𝑡=𝑣𝑥,0Γ𝛽^𝑡₀(𝑡−𝜏)^𝛽−16𝑎^∞_𝑛=0𝑝^𝑛𝑢_𝑛^∞_𝑛=0𝑝^𝑛𝑢_𝑛𝑥+3𝑏^∞_𝑛=0𝑝^𝑛𝑢_𝑛^∞_𝑛=0𝑝^𝑛𝑣_𝑛𝑥×𝑏^∞_𝑛=0𝑝^𝑛𝑢_{𝑛𝑥,𝑥,𝑥}. (26) If we compare the terms of the same power of 𝑝  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 ℎ, 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 𝑝⁰∶𝑢₀(𝑥,𝑡)=𝑢(𝑥,0),𝑢₀()(),𝑝𝑥,0=𝑢𝑥,0₀∶𝑢₀(𝑥,𝑡)=𝑢(𝑥,0),𝑢₀()(),𝑝𝑥,0=𝑢𝑥,0₁∶𝑢₁()1𝑥,𝑡=−Γ(𝛼)×^𝑡₀(𝑡−𝜏)^𝛼−16𝑎𝑢₀𝑢_0𝑥−2𝑏𝑣₀𝑣_0𝑥+𝑎𝑢_{0𝑥𝑥𝑥}𝑢𝑑𝜏,₁()𝑝𝑥,0=0,₁∶𝑣₁()1𝑥,𝑡=−Γ𝛽^𝑡₀(𝑡−𝜏)^𝛽−1×3𝑏𝑢₀𝑣_0𝑥+𝑏𝑢_{0𝑥𝑥𝑥}𝑣𝑑𝜏,₁()⋮𝑝𝑥,0=0,_𝑛∶𝑢_𝑛()1𝑥,𝑡=−Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1×6𝑎^𝑛−1_𝑖=0𝑢_𝑖𝑢_{𝑛−𝑖−1𝑥}−2𝑏^𝑛−1_𝑖=0𝑣_𝑖𝑣_{𝑛−𝑖−1𝑥}+𝑎𝑢_{𝑛−1𝑥𝑥𝑥}𝑢𝑑𝜏,_𝑛()𝑝𝑥,𝑡=0,_𝑛∶𝑣_𝑛()1𝑥,𝑡=−Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1×3𝑏^𝑛−1_𝑖=0𝑢_𝑖𝑣_{𝑛−𝑖−1𝑥}+𝑏𝑣_{𝑛−1𝑥𝑥𝑥}𝑣𝑑𝜏,_𝑛(𝑥,𝑡)=0. (27) Integrating the above, we obtain the following series solutions: 𝑢₀()=𝜆𝑥,𝑡𝑎1sech2𝜆𝑎𝑥²,𝑣₀()=𝜆𝑥,𝑡√12𝑎sech2𝜆𝑎𝑥². (28) For the sake of simplicity we put the following: 𝜆𝑑=𝑎,𝑑₁=𝜆√12𝑎,𝑚=2𝜆𝑎,𝑢₁()=𝑥,𝑡4𝑚𝑡^𝛼Γ()×1+𝛼−𝑏𝑑₁²+𝑎𝑑3𝑑−5𝑚²+𝑎𝑑𝑚²()×((cosh2𝑚𝑥sech𝑚𝑥))⁴(),𝑣tanh𝑚𝑥₁(𝑥,𝑡)=2𝑏𝑑₁𝑚𝑡^𝛽Γ1+𝛽3𝑑−10𝑚²+2𝑚²()×((cosh2𝑚𝑥sech𝑚𝑥))⁴(),𝑢tanh𝑚𝑥₂()=1𝑥,𝑡Γ()ΓΓ()Γ×21+𝛼1+𝛽0.5+𝛼1+𝛼+𝛽_1−2𝛼𝑚²√𝜋𝑡^𝛼Γ×1+𝛽−2𝑏²𝑑²₁𝑡^𝛽×−12𝑑+44𝑚²+9𝑑−38𝑚²×cosh(2𝑚𝑥)+2𝑚²()()cosh4𝑚𝑥×Γ1+2𝛼+𝑎𝑡^𝛼×−82𝑏𝑑²₁−3𝑑+13𝑚²+𝑎𝑑18𝑑²−111𝑑𝑚²+151𝑚⁴+4𝑏𝑑²₁−9𝑑+49𝑚²+3𝑎𝑑36𝑑²−272𝑑𝑚²+397𝑚⁴×cosh(2𝑚𝑥)−4𝑚²4𝑏𝑑²₁−15𝑎𝑑𝑑−2𝑚²×cosh(4𝑚𝑥)+𝑎𝑑𝑚⁴((cosh6𝑥𝑚)×Γ1+𝛼+𝛽(sech𝑚𝑥))⁸,𝑣₂()=1𝑥,𝑡Γ()ΓΓ()Γ×21+𝛼1+𝛽0.5+𝛼1+𝛼+𝛽_1−2𝛽𝑚²√𝜋𝑡^𝛽Γ(1+𝛼)(sech(𝑚𝑥))⁸×𝑏𝑡^𝛽−27𝑑²+411𝑑𝑚²−1208𝑚⁴+36𝑑²−124𝑑𝑚²+397𝑚⁴×cosh(2𝑚𝑥)+3𝑚²9𝑑−40𝑚²×cosh(4𝑚𝑥)+𝑚⁴cosh(6𝑚𝑥)×Γ1+𝛼+𝛽+12𝑡^𝛼−𝑏𝑑²₁+𝑎𝑑3𝑑−5𝑚²+𝑎𝑑𝑚²()Γ×((cosh2𝑚𝑥1+2𝛽sinh𝑚𝑥))². (29) 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: 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃()𝑣()𝑥,𝑡+⋯,𝑥,𝑡=𝑣₀(𝑥,𝑡)+𝑣₁(𝑥,𝑡)+𝑣₂(𝑥,𝑡)+𝑣₃(𝑥,𝑡)+⋯. (30)

4.1. Numerical Solutions

The following figures show the graphical representation of the approximated solution of the system of the time-fractional coupled-Korteweg-de-Vries equations for 𝜆=1,𝑎=𝑏=1.

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 𝛼 and 𝛽. The approximate solution of main problem has been depicted in Figures 1, 2, 3, and 4 which is plotted in Mathematica according to different 𝛼  and 𝛽 values.

Figure 1 

Approximate solution for 𝛼=0.75 and 𝛽=0.45.

Figure 2 

Approximate solution for 𝛼=0.75 and 𝛽=0.45.

Figure 3 

Approximate solution for  𝛼=1 and 𝛽=0.9.

Figure 4 

Approximate solution for 𝛼=1 and 𝛽=1.

It is important to note that if 𝛼=𝛽,𝑎=1, and  𝑏=3, the exact solution of the coupled-KDV equations is given as 𝑢()=𝜆𝑥,0𝑎1sech2𝜆𝑎𝑥−𝜆𝑡²,𝑣()=𝜆𝑥,0√12𝑎sech2𝜆𝑎𝑥−𝜆𝑡². (31) 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 

Numerical values of the approximate, exact solutions and the results obtained in [20, 21].

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 fractional-coupled 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 well-organized analytical method for solving exact and approximate solutions of nonlinear fractional partial differential equations.