An Efficient Method for Systems of Variable Coefficient Coupled Burgers’ Equation with Time-Fractional Derivative

Abstract

A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. Results indicate that the introduced method is promising for solving other types of systems of nonlinear fractional-order partial differential equations.

1. Introduction

In recent years, the differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, medical sciences, biological research, as well as various chemical, biochemical, and physical fields, viscoelasticity, biology, physics, and engineering. Consequently, considerable attention has been given to the solutions of fractional differential equations and integral equations of physical interest [1–4]. Various powerful methods have been presented so far such as homotopy perturbation method [5, 6], variational iteration method [7], differential transform method [8], homotopy analysis method [9], and Adomian decomposition method [10, 11] for solving different kinds of fractional partial differential equations. In this paper, we construct the solution of a system of variable coefficient coupled Burgers’ equation with time-fractional derivative by extending the idea of [12, 13]. A new version of homotopy perturbation method is proposed, which we called it NHPM and then applied it to the nonlinear systems of variable coefficient coupled Burgers’ equation with time-fractional derivative that can be written as the following basic form: 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝑟₁(𝑡)𝜕²𝑢𝜕𝑥²+𝑠₁(𝑡)𝑢𝜕𝑢𝜕𝑥+𝑝₁(𝑡)𝜕(𝑢𝑣),𝜕𝜕𝑥_𝛼𝑣𝜕𝑡^𝛼=𝑟₂(𝑡)𝜕²𝑣𝜕𝑥²+𝑠₂(𝑡)𝑣𝜕𝑣𝜕𝑥+𝑝₂(𝑡)𝜕(𝑢𝑣),𝜕𝑥 (1) subject to the initial condition 𝑢(𝑥,0)=𝑓(𝑥),𝑣(𝑥,0)=𝑔(𝑥), (2) where the subscripts 𝑟₁(𝑡), 𝑟₂(𝑡), 𝑠₁(𝑡), 𝑠₂(𝑡), 𝑝₁(𝑡), and 𝑝₂(𝑡) are arbitrary smooth functions of 𝑡.

The paper is organized as follows. In Section 2, we begin with an introduction to some necessary definitions of fractional calculus theory. In Section 3, we illustrated a basic idea of the new method. In Section 4, the uses of the new method for solving nonlinear variable coefficient coupled Burgers’ equation are presented. Two examples are solved by the proposed method in this section. Conclusion will appear in Section 5.

2. Fractional Calculus

We give some basic definitions and properties of the fractional calculus theory used in this work. Some of these are Riemann-Liouville, Grunwald-Letnikov, Caputo, and generalized functions approach. The most commonly used definitions are the Riemann-Liouville and Caputo derivatives.

Definition 1. The Riemann-Liouville fractional integral operator 𝐽^𝜇 of order 𝜇 on the usual Lebesgue space 𝐿₁[𝑎,𝑏] is given by 𝐽^𝜇𝑓(𝑥)=1Γ𝜇^𝑥₀(𝑥−𝑡)^𝜇−1𝑓(𝑡)𝐽𝑑𝑡,𝜇>0,₀𝑓(𝑥)=𝑓(𝑥). (3) It has the following properties:(i)𝐽^𝜇 exists for any 𝑥∈[𝑎,𝑏],(ii)𝐽^𝜇𝐽^𝛽=𝐽^𝜇+𝛽,(iii)𝐽^𝜇𝐽^𝛽=𝐽^𝛽𝐽^𝜇,(iv)𝐽^𝜇𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝜇𝑓(𝑥),(v)𝐽^𝜇(𝑥−𝑎)^𝛾=(Γ(𝛾+1)/Γ(𝜇+𝛾+1))(𝑥−𝑎)^𝜇+𝛾,where 𝑓∈𝐿₁[𝑎,𝑏], 𝜇,𝛽≥0, and 𝛾>−1.
The Riemann-Liouville fractional derivative is mostly used by mathematicians, but this approach is not suitable for physical problems of the real world since it requires the definition of fractional order initial conditions which have no physically meaningful explanation yet. Caputo introduced an alternative definition, which has the advantage of defining integer-order initial conditions for fractional order differential equations.

Definition 2. The Caputo definition of fractal derivative operator is given by 𝐷^𝜇𝑓(𝑥)=𝐽^𝑚−𝜇𝐷^𝑛𝑓(𝑥)=1Γ𝑚−𝜇^𝑡₀(𝑥−𝜏)^{𝑚−𝜇−1}𝑓^𝑚(𝜏)𝑑𝜏, (4) where 𝑚−1<𝜇≤𝑚, 𝑚∈𝑁, 𝑥>0.

Lemma 3. If 𝑚−1<𝜇≤𝑚, 𝑚∈ℕ, and 𝑓∈𝐿₁[𝑎,𝑏], then 𝐷^𝜇𝐽^𝜇𝑓(𝑥)(𝑥),𝐽=𝑓_𝜇𝐷^𝜇𝑓(𝑥)=𝑓(𝑥)−^𝑚−1_𝑘=0𝑓^𝑘0⁺(𝑥−𝑎)^𝑘,𝑘!𝑥>0. (5) The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of the problem. In this paper, we have considered some systems of linear and nonlinear FPDEs, where fractional derivatives are taken in Caputo sense as follows.

Definition 4. For 𝑛 to be the smallest integer that exceeds 𝛼, the Caputo time-fractional derivative operator of 𝛼>0 is defined as 𝐷^𝛼_𝑡𝑢()=𝜕𝑥,𝑡_𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=1Γ()×𝑛−𝛼_𝑡0(𝑥−𝜏)^{𝑛−𝛼−1}𝜕^𝑛𝑢(𝑥,𝜏)𝜕𝜏^𝑛𝜕𝑑𝜏,for𝑛−1<𝛼<𝑛_𝑛𝑢(𝑥,𝜏)𝜕𝜏^𝑛,for𝛼=𝑛∈𝑁. (6)

3. Analysis of New Homotopy Perturbation Method

Let us consider the system of nonlinear fractional differential equations 𝐷^𝛼_𝑡𝑢_𝑖(𝑥,𝑡)=𝐴_𝑖𝑢_𝑖+𝑓_𝑖(𝑡,𝑥),𝑥,𝑡∈Ω,𝑖=1,2,…,𝑛, (7) with the following initial conditions: 𝑢_𝑖(𝑥,0)=𝛼_𝑖,𝑖=1,2,…,𝑛, (8) where 𝐴_𝑖 are the operators, 𝑓_𝑖 are known functions, and 𝑢_𝑖 are sought functions. Assume that operators 𝐴_𝑖 can be written as 𝐴_𝑖𝑢_𝑖=𝐿_𝑖𝑢_𝑖+𝑁_𝑖𝑢_𝑖, (9) where 𝐿_𝑖 are the linear operators and 𝑁_𝑖 are the nonlinear operators. Hence, (7) can be rewritten as follows: 𝐷^𝛼_𝑡𝑢_𝑖(𝑥,𝑡)=𝐿_𝑖𝑢_𝑖+𝑁_𝑖𝑢_𝑖+𝑓_𝑖(𝑥,𝑡). (10) For solving system (7) by NHPM, we construct the following homotopy: 𝐻𝑈_𝑖;𝑝=1−𝑝𝐷^𝛼_𝑡𝑈_𝑖(𝑥,𝑡)−𝑢_𝑖,0+𝑝𝐷^𝛼_𝑡𝑢_𝑖(𝑥,𝑡)−𝐿_𝑖𝑈_𝑖−𝑁_𝑖𝑈_𝑖−𝑓_𝑖(𝑡,𝑥)=0, (11) where 𝑝∈[0,1] is an embedding or homotopy parameter, 𝐻(𝑡,𝑥;𝑝)∶Ω×[0,1]→𝑅, and 𝑢_𝑖,0 are the initial approximation of solution of the problem in (10).

Clearly, the homotopy equations 𝐻(𝑈_𝑖∶0)=0 and 𝐻_𝑖(𝑈_𝑖∶1)=0 are equivalent to the equations 𝐷^𝛼_𝑡𝑈_𝑖(𝑥,𝑡)−𝑢_𝑖,0=0 and 𝐷^𝛼_𝑡𝑈_𝑖(𝑥,𝑡)−𝐿_𝑖(𝑈_𝑖)+𝑁_𝑖(𝑈_𝑖)+𝑓_𝑖(𝑡,𝑥)=0, respectively. Thus, a monotonous change of parameter 𝑝 from zero to one corresponds to a continuous change of the trivial problem 𝐷^𝛼_𝑡𝑈_𝑖(𝑥,𝑡)−𝑢_𝑖,0=0 to the original problem. Next, we assume that the solution of equation 𝐻(𝑈_𝑖,𝑝) can be written as a power series in embedding parameter 𝑝 as follows: 𝑈_𝑖=𝑈_𝑖,0+𝑝𝑈_𝑖,1,𝑖=1,2,…,𝑛. (12) Now, let us write (12) in the following form: 𝐷^𝛼_𝑡𝑈_𝑖(𝑥,𝑡)=𝑢_𝑖,0+𝑝𝐿_𝑖𝑈_𝑖+𝑁_𝑖𝑈_𝑖+𝑓_𝑖(𝑡,𝑥). (13) Applying the inverse operator, 𝐽^𝛼_𝑡, which is the Riemann-Liouville fractional integral of order 𝛼≥0, on both sides of (13), we have 𝑈_𝑖(𝑥,𝑡)=𝑈_𝑖(𝑥,0)+𝐽^𝛼_𝑡𝑢_𝑖,0+𝑝𝐽^𝛼_𝑡𝐿_𝑖𝑈_𝑖+𝑁_𝑖𝑈_𝑖+𝑓_𝑖(𝑡,𝑥). (14) Suppose that the initial approximation of (10) has the form 𝑢_𝑖,0(𝑥,𝑡)=^∞_𝑛=0𝑎_𝑖,𝑛(𝑥)𝑝_𝑛(𝑡),𝑖=1,2,…,𝑛, (15) where 𝑎_𝑖,𝑛(𝑥), 𝑛=0,1,2,… are unknown coefficients and 𝑝_𝑛(𝑡), 𝑛=0,1,2,… are specific functions on the problem. By substituting (12) and (15) into (14), we get 𝑈_𝑖,0+𝑝𝑈_𝑖,1=𝑈_𝑖(𝑥,0)+𝐽^𝛼_𝑡^∞_𝑛=0𝑎_𝑖,𝑛(𝑥)𝑝_𝑛(𝑡)+𝑝𝐽^𝛼_𝑡𝐿_𝑖𝑈_𝑖,0+𝑝𝑈_𝑖,1+𝑁_𝑖𝑈_𝑖,0+𝑝𝑈_𝑖,1+𝑓_𝑖(𝑡,𝑥). (16) Equating the coefficients of like powers of 𝑝, we get the following set of equations: coefficientof𝑝⁰∶𝑈_𝑖,0(𝑥,𝑡)=𝑈_𝑖(𝑥,0)+𝐽^𝛼_𝑡^∞_𝑛=0𝑎_𝑖,𝑛(𝑥)𝑝_𝑛(𝑡),coefficientof𝑝¹∶𝑈_𝑖,1(𝑥,𝑡)=𝐽^𝛼_𝑡𝐿_𝑖𝑈_𝑖,0+𝑁_𝑖𝑈_𝑖,0+𝑓_𝑖(𝑡,𝑥). (17) Now, we solve these equations in such a way that 𝑈_𝑖,1(𝑥,𝑡)=0. Therefore, the approximate solution may be obtained as 𝑢_𝑖(𝑥,𝑡)=𝑈_𝑖,0(𝑥,𝑡)=𝑈_𝑖(𝑥,0)+𝐽^𝛼_𝑡^∞_𝑛=0𝑎_𝑖,𝑛(𝑥)𝑝_𝑛(𝑡). (18)

4. Examples

In this section, to illustrate the method and to show the ability of the method, two examples are presented.

Example 1. Consider the following variable coefficient coupled Burgers’ equation: 𝐷^𝛼_𝑡𝑢()=𝑡𝑥,𝑡𝜕1−𝑡²𝑢(𝑥,𝑡)𝜕𝑥²()()−𝑢𝑥,𝑡𝜕𝑢𝑥,𝑡+𝜕𝑥1+𝑡𝜕(𝑢()𝑣(1−𝑡𝑥,𝑡𝑥,𝑡)),𝐷𝜕𝑥_𝛼𝑡𝑣()=𝑡𝑥,𝑡𝜕1+𝑡²𝑣(𝑥,𝑡)𝜕𝑥²()()+𝑣𝑥,𝑡𝜕𝑣𝑥,𝑡−𝜕𝑥1−𝑡𝜕(𝑢()𝑣(1+𝑡𝑥,𝑡𝑥,𝑡)),𝜕𝑥𝑡≠−1,1, (19) subject to the initial condition 𝑢(𝑥,0)=𝑣(𝑥,0)=𝑥. (20) The exact solutions of (19) for the special case 𝛼=1 are 𝑢(𝑥,𝑡)=𝑥/(1−𝑡) and 𝑣(𝑥,𝑡)=𝑥/(1+𝑡).
To obtain the solution of (19) by NHPM, we construct the following homotopy: 1−𝑝𝐷^𝛼_𝑡𝑈(𝑥,𝑡)−𝑢₀(𝐷𝑥,𝑡)+𝑝^𝛼_𝑡𝑈()−𝑡𝑥,𝑡𝜕1−𝑡²𝑈(𝑥,𝑡)𝜕𝑥²()()+𝑈𝑥,𝑡𝜕𝑈𝑥,𝑡−𝜕𝑥1+𝑡𝜕(𝑈()𝑉(1−𝑡𝑥,𝑡𝑥,𝑡))𝐷𝜕𝑥=0,1−𝑝^𝛼_𝑡𝑉(𝑥,𝑡)−𝑣₀()𝐷𝑥,𝑡+𝑝^𝛼_𝑡𝑉()−𝑡𝑥,𝑡𝜕1+𝑡²𝑉(𝑥,𝑡)𝜕𝑥²()()−𝑉𝑥,𝑡𝜕𝑉𝑥,𝑡+𝜕𝑥1−𝑡𝜕(𝑈()𝑉(1+𝑡𝑥,𝑡𝑥,𝑡))𝜕𝑥=0. (21) Applying the inverse operator 𝐽^𝛼_𝑡 of 𝐷^𝛼_𝑡 on both sides of the above equation, we obtain 𝑈(𝑥,𝑡)=𝑈(𝑥,0)+𝐽^𝛼_𝑡𝑢₀(𝑥,𝑡)−𝑝𝐽^𝛼_𝑡𝑢₀()−𝑡𝑥,𝑡𝜕1−𝑡²𝑈(𝑥,𝑡)𝜕𝑥²()()+𝑈𝑥,𝑡𝜕𝑈𝑥,𝑡−𝜕𝑥1+𝑡𝜕(𝑈()𝑉(1−𝑡𝑥,𝑡𝑥,𝑡)),𝑉()()𝜕𝑥𝑥,𝑡=𝑉𝑥,0+𝐽^𝛼_𝑡𝑣₀(𝑥,𝑡)−𝑝𝐽^𝛼_𝑡𝑣₀()−𝑡𝑥,𝑡𝜕1+𝑡²𝑉(𝑥,𝑡)𝜕𝑥²()()−𝑉𝑥,𝑡𝜕𝑉𝑥,𝑡+𝜕𝑥1−𝑡𝜕(𝑈()𝑉(1+𝑡𝑥,𝑡𝑥,𝑡)).𝜕𝑥 (22) For solving system (22), by new homotopy perturbation method, we use the Taylor series of 1=1−𝑡^∞_𝑛=0𝑡^𝑛,1=1+𝑡^∞_𝑛=0(−1)^𝑛𝑡^𝑛. (23) The solution of (19) has the following form: 𝑈(𝑥,𝑡)=𝑈₀(𝑥,𝑡)+𝑝𝑈₁(),𝑉()𝑥,𝑡𝑥,𝑡=𝑉₀(𝑥,𝑡)+𝑝𝑉₁(𝑥,𝑡). (24) Substituting (23) and (24) in (22) and equating the coefficients of like powers of 𝑝, we get the following set of equations: 𝑈₀(𝑥,𝑡)=𝑈(𝑥,0)+𝐽^𝛼_𝑡𝑢₀(),𝑉𝑥,𝑡₀(𝑥,𝑡)=𝑉(𝑥,0)+𝐽^𝛼_𝑡𝑣₀(),𝑈𝑥,𝑡₁(𝑥,𝑡)=𝐽^𝛼_𝑡−𝑢₀(𝑥,𝑡)+𝑡^∞_𝑛=0𝑡^𝑛𝜕²𝑈₀(𝑥,𝑡)𝜕𝑥²−𝑈₀(𝑥,𝑡)𝜕𝑈₀(𝑥,𝑡)+()𝜕𝑥1+𝑡^∞_𝑛=0𝑡^𝑛𝜕𝑈₀(𝑥,𝑡)𝑉₀(𝑥,𝑡),𝑉𝜕𝑥₁(𝑥,𝑡)=𝐽^𝛼_𝑡−𝑣₀(𝑥,𝑡)+𝑡^∞_𝑛=0(−1)^𝑛𝑡^𝑛𝜕²𝑉₀(𝑥,𝑡)𝜕𝑥²+𝑉₀(𝑥,𝑡)𝜕𝑉₀(𝑥,𝑡)−()×𝜕𝑥1−𝑡_∞𝑛=0(−1)^𝑛𝑡^𝑛𝜕𝑈₀(𝑥,𝑡)𝑉₀(𝑥,𝑡).𝜕𝑥 (25) Assuming 𝑢₀(𝑥,𝑡)=∑^∞_𝑛=0𝑎_𝑛(𝑥)𝑝_𝑛(𝑡), 𝑣₀(𝑥,𝑡)=∑^∞_𝑛=0𝑏_𝑛(𝑥)𝑝_𝑛(𝑡), 𝑝_𝑛(𝑡)=𝑡^𝑛𝛼, 𝑈(𝑥,0)=𝑢(𝑥,0), and 𝑉(𝑥,0)=𝑣(𝑥,0) and solving the above equation for 𝑈₁(𝑥,𝑡) and 𝑉₁(𝑥,𝑡) lead to the result 𝑈₁(𝑥,𝑡)=𝑥−𝑎₀(𝑥)𝑡^𝛼Γ()+𝛼+1−𝑎₁(𝑥)+𝑏₀(𝑥)+𝑥𝑑𝑏₀(𝑥)×Γ()𝑡𝑑𝑥+4𝑥𝛼+1^2𝛼Γ()+2𝛼+1−𝑎₂(𝑥)−𝑎₀(𝑥)𝑑𝑎₀(𝑥)+𝑑𝑑𝑥²𝑎₀(𝑥)𝑑𝑥²+12𝑏₁(𝑥)+12𝑥𝑑𝑏₁(𝑥)+𝑑𝑥𝑑𝑎₀(𝑥)𝑏𝑑𝑥⁰(𝑥)+𝑎₀(𝑥)𝑑𝑏₀(𝑥)𝑑𝑥+2𝑎₀(𝑥)+2𝑥𝑑𝑎₀(𝑥)𝑑𝑥+2𝑏₀(𝑥)+2𝑥𝑑𝑏₀(𝑥)×Γ()𝑡𝑑𝑥+4𝑥2𝛼+1^3𝛼Γ()+3𝛼+14𝑥+2𝑎₀(𝑥)+⋯+2𝑥𝑑𝑏₀(𝑥)×Γ()𝑡𝑑𝑥3𝛼+1^4𝛼Γ()𝑉4𝛼+1+⋯,₁(𝑥,𝑡)=−𝑥−𝑏₀(𝑥)𝑡^𝛼Γ()+𝛼+1−𝑏₁(𝑥)−𝑎₀(𝑥)−𝑥𝑑𝑎₀(𝑥)×Γ()𝑡𝑑𝑥+4𝑥𝛼+1^2𝛼Γ()+2𝛼+1−𝑏₂(𝑥)+𝑏₀(𝑥)𝑑𝑏₀(𝑥)−𝑑𝑑𝑥²𝑎₀(𝑥)𝑑𝑥²−12𝑎₁(𝑥)−12𝑥𝑑𝑎₁(𝑥)−𝑑𝑥𝑑𝑎₀(𝑥)𝑏𝑑𝑥⁰(𝑥)−𝑎₀(𝑥)𝑑𝑏₀(𝑥)𝑑𝑥+2𝑎₀(𝑥)+2𝑥𝑑𝑎₀(𝑥)𝑑𝑥+2𝑏₀(𝑥)+2𝑥𝑑𝑏₀(𝑥)×Γ()𝑡𝑑𝑥−4𝑥2𝛼+1^3𝛼Γ()+3𝛼+14𝑥−2𝑎₀(𝑥)+⋯−2𝑥𝑑𝑏₀(𝑥)×Γ()𝑡𝑑𝑥3𝛼+1^4𝛼Γ(4𝛼+1)+⋯. (26) Vanishing 𝑈₁(𝑥,𝑡) and 𝑉₁(𝑥,𝑡) lets the coefficients 𝑎_𝑖,𝑏_𝑖, 𝑖=0,1,2,… have the following values: 𝑎₀(𝑥)=𝑥,𝑎₁(𝑥)=2𝑥,𝑎₂(𝑥)𝑎=3𝑥,₃(𝑥)=4𝑥,𝑎₄(𝑥)=5𝑥,𝑎₅(𝑥)𝑏=6𝑥,…,₀(𝑥)=−𝑥,𝑏₁(𝑥)=2𝑥,𝑏₂(𝑥)𝑏=−3𝑥,₃(𝑥)=4𝑥,𝑏₄(𝑥)=−5𝑥,𝑏₅(𝑥)=6𝑥. (27) Therefore, we obtain the solutions of (19) as 𝑢()𝑡𝑥,𝑡=𝑥+𝑥^𝛼Γ()Γ()𝑡𝛼+1+2𝑥𝛼+1^2𝛼Γ()Γ()𝑡2𝛼+1+3𝑥2𝛼+1^3𝛼Γ()Γ()𝑡3𝛼+1+4𝑥3𝛼+1^4𝛼Γ(4𝛼+1)+⋯=𝑥1+^∞_𝑛=1𝑛Γ((𝑛−1)𝛼+1)𝑡^𝑛𝛼Γ(),𝑣()𝑡𝑛𝛼+1𝑥,𝑡=𝑥−𝑥^𝛼Γ()Γ()𝑡𝛼+1+2𝑥𝛼+1^2𝛼Γ()Γ()𝑡2𝛼+1−3𝑥2𝛼+1^3𝛼Γ()Γ()𝑡3𝛼+1+4𝑥3𝛼+1^4𝛼Γ(4𝛼+1)−⋯=𝑥1+^∞_𝑛=1𝑛(−1)^𝑛Γ((𝑛−1)𝛼+1)𝑡^𝑛𝛼Γ().𝑛𝛼+1 (28) If we put 𝛼→1 in (28) or solve (19) with 𝛼=1, we obtain the exact solution 𝑢(𝑥,𝑡)=𝑥1+𝑡+𝑡²+𝑡³=𝑥+⋯,𝑣()1−𝑡𝑥,𝑡=𝑥1−𝑡+𝑡²−𝑡³=𝑥+⋯.1+𝑡 (29)

Example 2. Consider the following variable coefficient coupled Burgers’ equation: 𝐷^𝛼_𝑡𝑢()𝜕𝑥,𝑡=−²𝑢(𝑥,𝑡)𝜕𝑥²+2𝑒^2𝑡𝑢()()𝑥,𝑡𝜕𝑢𝑥,𝑡()𝜕(𝑢()𝑣(𝜕𝑥−sin2𝑡𝑥,𝑡𝑥,𝑡)),𝐷𝜕𝑥_𝛼𝑡𝑣()=𝜕𝑥,𝑡²𝑣(𝑥,𝑡)𝜕𝑥²−2𝑒^−2𝑡()𝑣()()cos2𝑡𝑥,𝑡𝜕𝑣𝑥,𝑡()𝜕(𝑢()𝑣(𝜕𝑥+cos2𝑡𝑥,𝑡𝑥,𝑡)),𝜕𝑥 (30) subject to the initial condition 𝑢(𝑥,0)=𝑣(𝑥,0)=𝑒^𝑥. (31) The exact solution for 𝛼=1 is 𝑢(𝑥,𝑡)=𝑒^𝑥−𝑡and 𝑣(𝑥,𝑡)=𝑒^𝑥+𝑡.
To obtain the solution of (30) by NHPM, we construct the following homotopy: 1−𝑝𝐷^𝛼_𝑡𝑈(𝑥,𝑡)−𝑢₀(𝐷𝑥,𝑡)+𝑝^𝛼_𝑡𝑈()+𝜕𝑥,𝑡²𝑈(𝑥,𝑡)𝜕𝑥²−2𝑒^2𝑡()𝑈()()sin2𝑡𝑥,𝑡𝜕𝑈𝑥,𝑡()𝜕(𝑈()𝑉(𝜕𝑥+sin2𝑡𝑥,𝑡𝑥,𝑡))𝐷𝜕𝑥=0,1−𝑝^𝛼_𝑡𝑉(𝑥,𝑡)−𝑣₀()𝐷𝑥,𝑡+𝑝^𝛼_𝑡𝑉()−𝜕𝑥,𝑡²𝑉(𝑥,𝑡)𝜕𝑥²+2𝑒^−2𝑡()𝑉()()cos2𝑡𝑥,𝑡𝜕𝑉𝑥,𝑡()𝜕(𝑈()𝑉(𝜕𝑥−cos2𝑡𝑥,𝑡𝑥,𝑡))𝜕𝑥=0. (32) Applying the inverse operator 𝐽^𝛼_𝑡 of 𝐷^𝛼_𝑡 on both sides of the above equation, we obtain 𝑈(𝑥,𝑡)=𝑈(𝑥,0)+𝐽^𝛼_𝑡𝑢₀(𝑥,𝑡)−𝑝𝐽^𝛼_𝑡𝑢₀()+𝜕𝑥,𝑡²𝑈(𝑥,𝑡)𝜕𝑥²−2𝑒^2𝑡()𝑈()()sin2𝑡𝑥,𝑡𝜕𝑈𝑥,𝑡()𝜕(𝑈()𝑉(𝜕𝑥+sin2𝑡𝑥,𝑡𝑥,𝑡)),𝑉()()𝜕𝑥𝑥,𝑡=𝑉𝑥,0+𝐽^𝛼_𝑡𝑣₀(𝑥,𝑡)−𝑝𝐽^𝛼_𝑡𝑣₀()−𝜕𝑥,𝑡²𝑉(𝑥,𝑡)𝜕𝑥²+2𝑒^−2𝑡()𝑉()()cos2𝑡𝑥,𝑡𝜕𝑉𝑥,𝑡()𝜕(𝑈()𝑉(𝜕𝑥−cos2𝑡𝑥,𝑡𝑥,𝑡)).𝜕𝑥 (33) For solving system (33), by new homotopy perturbation method, we use the Taylor series of sin(2𝑡)=^∞_𝑛=0(−1)^𝑛(2𝑡)^2𝑛+1()!,()=2𝑛+1cos2𝑡^∞_𝑛=0(−1)^𝑛(2𝑡)^2𝑛()!,()=2𝑛exp2𝑡^∞_𝑛=0(2𝑡)^𝑛,()=𝑛!exp−2𝑡^∞_𝑛=0(−1)^𝑛(2𝑡)^𝑛.𝑛! (34) The solution of (30) has the following form: 𝑈(𝑥,𝑡)=𝑈₀(𝑥,𝑡)+𝑝𝑈₁(),𝑉()𝑥,𝑡𝑥,𝑡=𝑉₀(𝑥,𝑡)+𝑝𝑉₁(𝑥,𝑡). (35) Substituting (34) and (35) in (33) and equating the coefficients of like powers of 𝑝, we get the following set of equations: 𝑈₀(𝑥,𝑡)=𝑈(𝑥,0)+𝐽^𝛼_𝑡𝑢₀(),𝑉𝑥,𝑡₀(𝑥,𝑡)=𝑉(𝑥,0)+𝐽^𝛼_𝑡𝑣₀(),𝑈𝑥,𝑡₁(𝑥,𝑡)=𝐽^𝛼_𝑡−𝑢₀()−𝜕𝑥,𝑡²𝑈₀(𝑥,𝑡)𝜕𝑥²+2^∞_𝑛=0(2𝑡)^𝑛𝑛!^∞_𝑛=0(−1)^𝑛(2𝑡)^2𝑛+1()!𝑈2𝑛+1⁰(𝑥,𝑡)𝜕𝑈₀(𝑥,𝑡)−𝜕𝑥_∞𝑛=0(−1)^𝑛(2𝑡)^2𝑛+1(2𝑛+1)!𝜕𝑈₀(𝑥,𝑡)𝑉₀(𝑥,𝑡),𝑉𝜕𝑥₁(𝑥,𝑡)=𝐽^𝛼_𝑡−𝑣₀()+𝜕𝑥,𝑡²𝑉₀(𝑥,𝑡)𝜕𝑥²−2^∞_𝑛=0(−1)^𝑛(2𝑡)^𝑛𝑛!^∞_𝑛=0(−1)^𝑛(2𝑡)^2𝑛(2𝑛)!×𝑉₀(𝑥,𝑡)𝜕𝑉₀(𝑥,𝑡)+𝜕𝑥_∞𝑛=0(−1)^𝑛(2𝑡)^2𝑛(2𝑛)!𝜕𝑈₀(𝑥,𝑡)𝑉₀(𝑥,𝑡).𝜕𝑥 (36) Assuming 𝑢₀(𝑥,𝑡)=∑^∞_𝑛=0𝑎_𝑛(𝑥)𝑝_𝑛(𝑡), 𝑣₀(𝑥,𝑡)=∑^∞_𝑛=0𝑏_𝑛(𝑥)𝑝_𝑛(𝑡), 𝑝_𝑛(𝑡)=𝑡^𝑛𝛼, 𝑈(𝑥,0)=𝑢(𝑥,0), and 𝑉(𝑥,0)=𝑣(𝑥,0) and solving the above equation for 𝑈₁(𝑥,𝑡) and 𝑉₁(𝑥,𝑡) lead to the result 𝑈₁(𝑥,𝑡)=−𝑎₀(𝑥)−𝑒^𝑥𝑡^𝛼Γ()+𝛼+1−𝑎₁(𝑥)−𝑑²𝑎₀(𝑥)𝑑𝑥²Γ(𝛼+1)𝑡^2𝛼Γ()+2𝛼+1−𝑎₂(𝑥)−12𝑑²𝑎₁(𝑥)𝑑𝑥²−2𝑏₀(𝑥)𝑒^𝑥+2𝑎₀(𝑥)𝑒^𝑥+2𝑒^𝑥𝑑𝑎₀(𝑥)𝑑𝑥−2𝑒^𝑥𝑑𝑏₀(𝑥)𝑑𝑥+8𝑒^2𝑥Γ(2𝛼+1)𝑡^3𝛼Γ()+3𝛼+1−𝑎₃(𝑥)−2𝑎₀(𝑥)𝑑𝑏₀(𝑥)𝑑𝑥+⋯+8𝑎₀(𝑥)𝑒^𝑥×Γ(3𝛼+1)𝑡^4𝛼Γ()𝑉4𝛼+1+⋯,₁(𝑥,𝑡)=−𝑏₀(𝑥)+𝑒^𝑥𝑡^𝛼Γ()+𝛼+1−𝑏₁(𝑥)−𝑏₀(𝑥)𝑒^𝑥+𝑎₀(𝑥)𝑒^𝑥+𝑒^𝑥𝑑𝑎₀(𝑥)𝑑𝑥−𝑒^𝑥𝑑𝑏₀(𝑥)+𝑑𝑑𝑥₂𝑏₀(𝑥)𝑑𝑥²+4𝑒^2𝑥Γ(𝛼+1)𝑡^2𝛼Γ()+2𝛼+1−𝑏₂(𝑥)−12𝑒^𝑥𝑑𝑏₁(𝑥)−1𝑑𝑥2𝑏₁(𝑥)𝑒^𝑥+12𝑎₁(𝑥)𝑒^𝑥+𝑎₀(𝑥)𝑑𝑏₀(𝑥)𝑑𝑥+4𝑏₀(𝑥)𝑒^𝑥×Γ(2𝛼+1)𝑡^3𝛼Γ()+3𝛼+1−𝑏₁(𝑥)𝑑𝑏₀(𝑥)−1𝑑𝑥3𝑏₂(𝑥)𝑒^𝑥1+⋯+3𝑑²𝑏₂(𝑥)𝑑𝑥²Γ(3𝛼+1)𝑡^4𝛼Γ(4𝛼+1)+⋯. (37) Vanishing 𝑈₁(𝑥,𝑡) and 𝑉₁(𝑥,𝑡) lets the coefficients 𝑎_𝑖,𝑏_𝑖, 𝑖=0,1,2,… have the following values: 𝑎₀(𝑥)=−𝑒^𝑥,𝑎₁(𝑥)=𝑒^𝑥,𝑎₂(𝑥)1=−𝑒2!^𝑥,𝑎₃(𝑥)=1𝑒3!^𝑥,𝑎₄(𝑥)1=−𝑒4!^𝑥𝑏,…,₀(𝑥)=𝑒^𝑥,𝑏₁(𝑥)=𝑒^𝑥,𝑏₂(𝑥)=1𝑒2!^𝑥,𝑏₃(𝑥)=1𝑒3!^𝑥,𝑏₄(𝑥)=1𝑒4!^𝑥,…. (38) Therefore, we obtain the solutions of (30) as 𝑢(𝑥,𝑡)=𝑒^𝑥−𝑒^𝑥𝑡^𝛼Γ(𝛼+1)+𝑒^𝑥Γ(𝛼+1)𝑡^2𝛼Γ()−12𝛼+1𝑒2!^𝑥Γ(2𝛼+1)𝑡^3𝛼Γ()+13𝛼+1𝑒3!^𝑥Γ(3𝛼+1)𝑡^4𝛼Γ(4𝛼+1)−⋯=𝑒^𝑥1+^∞_𝑛=1(−1)^𝑛Γ((𝑛−1)𝛼+1)𝑡^𝑛𝛼()(),𝑣()𝑛−1!Γ𝑛𝛼+1𝑥,𝑡=𝑒^𝑥+𝑒^𝑥𝑡^𝛼Γ(𝛼+1)+𝑒^𝑥Γ(𝛼+1)𝑡^2𝛼Γ()+12𝛼+1𝑒2!^𝑥Γ(2𝛼+1)𝑡^3𝛼Γ()+13𝛼+1𝑒3!^𝑥Γ(3𝛼+1)𝑡^4𝛼Γ(4𝛼+1)+⋯=𝑒^𝑥1+^∞_𝑛=1Γ((𝑛−1)𝛼+1)𝑡^𝑛𝛼()().𝑛−1!Γ𝑛𝛼+1 (39) If we put 𝛼→1 in (39) or solve (30) with 𝛼=1, we obtain the exact solution 𝑢(𝑥,𝑡)=𝑒^𝑥𝑡1−𝑡+²−𝑡2!3³!+⋅⋅⋅=𝑒^𝑥−𝑡,𝑣(𝑥,𝑡)=𝑒^𝑥𝑡1+𝑡+²+𝑡2!3³!+⋅⋅⋅=𝑒^𝑥+𝑡. (40)

5. Concluding Remarks

In this paper, we have used a new homotopy perturbation method for solving a system of two nonlinear time-fractional partial differential equations. The NHPM for solving system of variable coefficient coupled Burgers’ equation with time-fractional derivative is based on two-component procedure and polynomial initial condition. The Computations finally lead to a set of nonlinear equations with one unspecified value in each equation. This set can be readily solved using Maple, and putting these values into the first approximate solution yields the analytical approximate solution. The present study has confirmed that NHPM offers significant advantages in terms of its straightforward applicability, computational efficiency, and accuracy. Thus, we conclude that the new method can be considered as an efficient method for solving linear and nonlinear problems.

Conflict of Interests

The authors of the paper do not have a direct financial relation that might lead to a “conflict of interests” for any of the authors.