A New Coupled Fractional Reduced Differential Transform Method for the Numerical Solutions of (2+1)-Dimensional Time Fractional Coupled Burger Equations

Abstract

A very new technique, coupled fractional reduced differential transform, has been implemented to obtain the numerical approximate solution of (2 + 1)-dimensional coupled time fractional burger equations. The fractional derivatives are described in the Caputo sense. By using the present method we can solve many linear and nonlinear coupled fractional differential equations. The obtained results are compared with the exact solutions. Numerical solutions are presented graphically to show the reliability and efficiency of the method.

1. Introduction

In the past decades, the fractional differential equations have been widely used in various fields of applied science and engineering [1–10]. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. An immense effort has been expended over the last many years to find robust and efficient numerical and analytical methods for solving such fractional differential equations. In the present analysis, a new approximate numerical technique, coupled fractional reduced differential transform method (CFRDTM), has been proposed which is applicable for coupled fractional differential equations. The proposed method is a very powerful solver for linear and nonlinear coupled fractional differential equations. It is relatively a new approach to provide the solution very efficiently and accurately.

The Burgers model of turbulence is a very important fluid dynamic model and the study of this model and the theory of shock waves have been considered by many authors both to obtain conceptual understanding of a class of physical flows and for testing various numerical methods. The study of coupled Burgers equations is very significant for the system is a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions or colloids, under the effect of gravity [11].

In this paper, (2 + 1)-dimensional coupled time fractional burger equations have been considered. The paper is organized as follows. In Section 2, a brief review of the theory of fractional calculus has been provided for precise purpose of this paper. In Section 3, the coupled fractional reduced differential transform method has been analyzed in detail. In Section 4, CFRDTM has been applied to determine the approximate solutions for the coupled time fractional burger equations. The obtained results show the efficiency and simplicity of the proposed method. Finally, conclusions are presented.

2. Mathematical Preliminaries of Fractional Calculus

The fractional calculus was first anticipated by Leibnitz, one of the founders of standard calculus, in a letter written in 1695. This calculus involves different definitions of the fractional operators as well as the Riemann-Liouville fractional derivative, Caputo derivative, Riesz derivative, and Grunwald-Letnikov fractional derivative [1]. The fractional calculus has gained considerable importance during the past decades mainly due to its applications in diverse fields of science and engineering. For the purpose of this paper Caputo’s definition of fractional derivative will be used, taking the advantage of Caputo’s approach that the initial conditions for fractional differential equations with Caputo’s derivatives take on the traditional form as for integer-order differential equations.

2.1. Definition of Riemann-Liouville Integral

The most frequently encountered definition of an integral of fractional order is the Riemann-Liouville integral [1], in which the fractional integral of order 𝛼 (>0) is defined as 𝐽^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏,𝑡>0,𝛼∈𝐑⁺, (1) where 𝐑⁺ is the set of positive real numbers.

2.2. Definition of Caputo Fractional Derivative

The fractional derivative, introduced by Caputo [12, 13] in the late sixties, is called Caputo Fractional Derivative. The fractional derivative of 𝑓(𝑡) in the Caputo sense is defined by 𝐷^𝛼_𝑡𝑓(𝑡)=𝐽^𝑚−𝛼𝐷^𝑚𝑓(𝑡)=1Γ(𝑚−𝛼)^𝑡₀(𝑡−𝜏)^{(𝑚−𝛼−1)}𝑑^𝑚𝑓(𝜏)𝑑𝜏^𝑚𝑑𝜏,if𝑑𝑚−1<𝛼<𝑚,𝑚∈𝑁_𝑚𝑓(𝑡)𝑑𝑡^𝑚,if𝛼=𝑚,𝑚∈𝑁, (2) where the parameter α is the order of the derivative and is allowed to be real or even complex. In this paper only real and positive α will be considered.

For the Caputo’s derivative we have 𝐷^𝛼𝐶=0,(𝐶isaconstant)𝐷^𝛼𝑡^𝛽=Γ𝑡0,𝛽≤𝛼−1𝛽+1^𝛽−𝛼Γ𝛽−𝛼+1,𝛽>𝛼−1. (3) Being similar to integer-order differentiation, Caputo’s derivative is linear: 𝐷^𝛼𝛾𝑓(𝑡)+𝛿𝑔(𝑡)=𝛾𝐷^𝛼𝑓(𝑡)+𝛿𝐷^𝛼𝑔(𝑡), (4) where 𝛾 and   𝛿 are constants and satisfy so called Leibnitz’s rule: 𝐷^𝛼𝑔(𝑡)𝑓(𝑡)=^∞_𝑘=0𝛼𝑘𝑔^(𝑘)(𝑡)𝐷^𝛼−𝑘𝑓(𝑡), (5) where 𝑓(𝜏) is continuous in [0,𝑡] and 𝑔(𝜏) has 𝑛+1 continuous derivatives in [0,𝑡].

Lemma 1. If 𝑚−1<𝛼<𝑚, 𝑚∈𝑁, then 𝐷^𝛼𝐽^𝛼𝑓(𝑡)=𝑓(𝑡),𝐽^𝛼𝐷^𝛼𝑓(𝑡)=𝑓(𝑡)−^𝑚−1_𝑘=0𝑡^𝑘𝑓𝑘!^(𝑘)(0+),𝑡>0. (6)(7)

Theorem 2 (generalized Taylor’s formula; see [14]). Suppose that 𝐷_𝑎^𝑘𝛼𝑓(𝑡)∈𝐶(𝑎,𝑏] for 𝑘=0,1,…,𝑛+1, where 0<𝛼≤1; we have 𝑓(𝑡)=^𝑛_𝑖=0(𝑡−𝑎)^𝑖𝛼Γ()𝐷𝑖𝛼+1^𝑎𝑖𝛼𝑓(𝑡)_𝑡=𝑎+ℜ^𝛼_𝑛(𝑡;𝑎), (8) with ℜ^𝛼_𝑛(𝑡;𝑎)=((𝑡−𝑎)^(𝑛+1)𝛼/Γ((𝑛+1)𝛼+1))[𝐷_𝑎^(𝑛+1)𝛼𝑓(𝑡)]_𝑡=𝜉, 𝑎≤𝜉≤𝑡,∀𝑡∈(𝑎,𝑏], where 𝐷_𝑎^𝑘𝛼=𝐷^𝛼_𝑎⋅𝐷^𝛼_𝑎⋅𝐷^𝛼_𝑎⋯𝐷^𝛼_𝑎 (k times).

3. Coupled Fractional Reduced Differential Transform Method (CFRDTM)

In order to introduce coupled fractional reduced differential transform, 𝑈(ℎ,𝑘−ℎ) is considered as the coupled fractional reduced differential transform of 𝑢(𝑥,𝑦,𝑡). If function 𝑢(𝑥,𝑦,𝑡) is analytic and differentiated continuously with respect to time 𝑡, then we define the fractional coupled reduced differential transform of 𝑢(𝑥,𝑦,𝑡) as 𝑈()=1ℎ,𝑘−ℎΓ()×𝐷ℎ𝛼+𝑘−ℎ𝛽+1_{𝑡(ℎ𝛼+(𝑘−ℎ)𝛽)}𝑢𝑥,𝑦,𝑡_𝑡=0, (9) whereas the inverse transform of 𝑈(ℎ,𝑘−ℎ) is 𝑢𝑥,𝑦,𝑡=^∞𝑘_𝑘=0ℎ=0𝑈(ℎ,𝑘−ℎ)𝑡^{ℎ𝛼+(𝑘−ℎ)𝛽}, (10) which is one of the solutions of coupled fractional differential equations.

Theorem 3. Suppose that 𝑈(ℎ,𝑘−ℎ), 𝑉(ℎ,𝑘−ℎ), and 𝑊(ℎ,𝑘−ℎ) are the coupled fractional reduced differential transform of the functions 𝑢(𝑥,𝑦,𝑡), 𝑣(𝑥,𝑦,𝑡), and 𝑤(𝑥,𝑦,𝑡), respectively. (i)If 𝑢(𝑥,𝑦,𝑡)=𝑓(𝑥,𝑦,𝑡)±𝑔(𝑥,𝑦,𝑡), then 𝑈(ℎ,𝑘−ℎ)=𝐹(ℎ,𝑘−ℎ)±𝐺(ℎ,𝑘−ℎ).(ii)If 𝑢(𝑥,𝑦,𝑡)=𝑎𝑓(𝑥,𝑦,𝑡), where 𝑎∈𝐑, then 𝑈(ℎ,𝑘−ℎ)=𝑎𝐹(ℎ,𝑘−ℎ).(iii)If 𝑓(𝑥,𝑦,𝑡)=𝑢(𝑥,𝑦,𝑡)𝑣(𝑥,𝑦,𝑡), then 𝐹(ℎ,𝑘−ℎ)=∑^ℎ_𝑙=0∑^𝑘−ℎ_𝑠=0𝑈(ℎ−𝑙,𝑠)𝑉(𝑙,𝑘−ℎ−𝑠).(iv)If 𝑓(𝑥,𝑦,𝑡)=𝐷^𝛼_𝑡𝑢(𝑥,𝑦,𝑡), then 𝐹()=Γ()()ℎ,𝑘−ℎℎ+1𝛼+𝑘−ℎ𝛽+1Γ()𝑈().ℎ𝛼+𝑘−ℎ𝛽+1ℎ+1,𝑘−ℎ (11)(v)If 𝑓(𝑥,𝑦,𝑡)=𝐷^𝛽_𝑡𝑣(𝑥,𝑦,𝑡), then 𝐹()=Γ()ℎ,𝑘−ℎℎ𝛼+𝑘−ℎ+1𝛽+1Γ()𝑉().ℎ𝛼+𝑘−ℎ𝛽+1ℎ,𝑘−ℎ+1 (12)

4. Application of CFRDTM for the Solutions for (2+1)-Dimensional Time Fractional Coupled Burgers Equations

Example 4. Consider the following (2 + 1)-dimensional time fractional coupled Burgers equations [15]: 𝐷^𝛼_𝑡𝑢+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢=1𝜕𝑦𝑅𝜕²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²,𝐷^𝛽_𝑡𝑣+𝑢𝜕𝑣𝜕𝑥+𝑣𝜕𝑣=1𝜕𝑦𝑅𝜕²𝑣𝜕𝑥²+𝜕²𝑣𝜕𝑦², (13)(14) subject to the initial conditions 𝑢=3𝑥,𝑦,04−14(),1+exp𝑅/32−4𝑥+4𝑦𝑣=3𝑥,𝑦,04+14(),1+exp𝑅/32−4𝑥+4𝑦 (15)(16) where 0≤𝑥,𝑦≤1, 𝑡>0, 0<𝛼, 𝛽≤1, and 𝑅 is the Reynolds number.
The exact solutions of (13) and (14), for the special case where 𝛼=𝛽=1, are given by 𝑢=3𝑥,𝑦,𝑡4−14(),𝑣=31+exp𝑅/32−4𝑥+4𝑦−𝑡𝑥,𝑦,𝑡4+14().1+exp𝑅/32−4𝑥+4𝑦−𝑡 (17)
In order to assess the advantages and the accuracy of the CFRDTM, we consider the (2 + 1)-dimensional time fractional coupled burgers equations. Firstly, we derive the recursive formula from (13) and (14). Now, 𝑈(ℎ,𝑘−ℎ) and 𝑉(ℎ,𝑘−ℎ) are considered as the coupled fractional reduced differential transform of 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡), respectively, where 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) are the solutions of coupled fractional differential equations. Here, 𝑈(0,0)=𝑢(𝑥,𝑦,0), 𝑉(0,0)=𝑣(𝑥,𝑦,0) are the given initial conditions. Without loss of generality, the following assumptions have been taken: 𝑈𝑉()0,𝑗=0,𝑗=1,2,3,…,𝑖,0=0,𝑖=1,2,3,…. (18) Applying CFRDTM to (13), we obtain the following recursive formula: Γ(ℎ+1)𝛼+(𝑘−ℎ)𝛽+1Γ()𝑈()=𝜕ℎ𝛼+𝑘−ℎ𝛽+1ℎ+1,𝑘−ℎ₂𝜕𝑥²𝑈()+𝜕ℎ,𝑘−ℎ×𝜕𝑥_{ℎ𝑙=0𝑘−ℎ𝑠=0}𝑈()𝑈()−𝜕ℎ−𝑙,𝑠𝑙,𝑘−ℎ−𝑠𝜕𝑥^ℎ_𝑙=0^𝑘−ℎ_𝑠=0𝑈(ℎ−𝑙,𝑠)𝑉(𝑙,𝑘−ℎ−𝑠). (19)
From the initial condition of (15), we have 𝑈(0,0)=𝑢𝑥,𝑦,0. (20)
In the same manner, we can obtain the following recursive formula from (14): Γℎ𝛼+(𝑘−ℎ+1)𝛽+1Γ()𝑉()=𝜕ℎ𝛼+𝑘−ℎ𝛽+1ℎ,𝑘−ℎ+1₂𝜕𝑥²𝑉()+𝜕ℎ,𝑘−ℎ×𝜕𝑥_{ℎ𝑙=0𝑘−ℎ𝑠=0}𝑉()𝑉()−𝜕𝑙,𝑘−ℎ−𝑠ℎ−𝑙,𝑠𝜕𝑥^ℎ_𝑙=0^𝑘−ℎ_𝑠=0𝑈(𝑙,𝑘−ℎ−𝑠)𝑉(ℎ−𝑙,𝑠). (21)
From the initial condition of (16), we have 𝑉(0,0)=𝑣𝑥,𝑦,0. (22)
According to CFRDTM, using recursive equation (19) with initial condition (20) and also using recursive scheme (21) with initial condition (22) simultaneously, we obtain 𝑈()()𝑅1,0=−exp𝑅/8𝑥−𝑦1281+exp(𝑅/8)𝑥−𝑦²Γ(),𝑉()()𝑅1+𝛼0,1=−exp𝑅/8𝑥−𝑦1281+exp(𝑅/8)𝑥−𝑦²Γ,𝑈()()𝑅1+𝛽1,1=−exp𝑅/4𝑥−𝑦²40961+exp(𝑅/8)𝑥−𝑦⁴Γ,𝑉()𝑅1+𝛼+𝛽0,2=−exp8×𝑅𝑥−𝑦−1+exp8𝑅𝑥−𝑦+exp4𝑥−𝑦𝑅²×𝑅40961+exp8𝑥−𝑦⁴Γ1+2𝛽⁻¹,𝑈()=𝑅2,0exp8×𝑅𝑥−𝑦−1+exp8𝑅𝑥−𝑦+exp4𝑥−𝑦𝑅²×𝑅40961+exp8𝑥−𝑦⁴Γ(1+2𝛼)⁻¹,𝑉()=()𝑅1,1exp𝑅/4𝑥−𝑦²40961+exp(𝑅/8)𝑥−𝑦⁴Γ,𝑈()=𝑅1+𝛼+𝛽1,2exp4×𝑅𝑥−𝑦−1+exp8𝑅𝑥−𝑦+exp4𝑥−𝑦𝑅³×𝑅1310721+exp8𝑥−𝑦⁶Γ1+𝛼+2𝛽⁻¹, (23) and so on.
The approximate solutions, obtained in the series form, are given by 𝑢=𝑥,𝑦,𝑡_{∞𝑘𝑘=0ℎ=0}𝑈(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=𝑈(0,0)+^∞𝑘_𝑘=1ℎ=1𝑈(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=34−14()−()1+exp𝑅/32−4𝑥+4𝑦−𝑡exp𝑅/8𝑥−𝑦𝑅𝑡^𝛼1281+exp(𝑅/8)𝑥−𝑦²Γ()+𝑅1+𝛼exp8×𝑅𝑥−𝑦−1+exp8𝑅𝑥−𝑦+exp4𝑥−𝑦𝑅²𝑡^2𝛼×𝑅40961+exp8𝑥−𝑦⁴Γ(1+2𝛼)⁻¹−𝑅exp4𝑥−𝑦𝑅²𝑡^𝛼+𝛽×𝑅40961+exp8𝑥−𝑦⁴Γ1+𝛼+𝛽⁻¹𝑣=+⋯.𝑥,𝑦,𝑡_{∞𝑘𝑘=0ℎ=0}𝑉(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=𝑉(0,0)+^∞𝑘_𝑘=1ℎ=0𝑉(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=34+14()−()1+exp𝑅/32−4𝑥+4𝑦exp𝑅/8𝑥−𝑦𝑅𝑡^𝛽1281+exp(𝑅/8)𝑥−𝑦²Γ+()𝑅1+𝛽exp𝑅/4𝑥−𝑦²𝑡^𝛼+𝛽40961+exp(𝑅/8)𝑥−𝑦⁴Γ−𝑅1+𝛼+𝛽exp8×𝑅𝑥−𝑦−1+exp8𝑅𝑥−𝑦+exp4𝑥−𝑦𝑅²𝑡^2𝛽×𝑅40961+exp8𝑥−𝑦⁴Γ1+2𝛽⁻¹+⋯. (24)
In special case for 𝛼=1 and 𝛽=1, the solutions in (24) coincide exactly with the Taylor series expansion of the exact solutions in (17) 𝑢=3𝑥,𝑦,𝑡4−14()−()1+exp𝑅/32−4𝑥+4𝑦exp𝑅/8𝑥+𝑦𝑅𝑡128exp(𝑅𝑥/8)+exp𝑅𝑦/8²+exp(𝑅/8)𝑥+𝑦exp(𝑅𝑥/8)−exp𝑅𝑦/8𝑅²𝑡²8192exp(𝑅𝑥/8)+exp𝑅𝑦/8³𝑣=3+⋯,𝑥,𝑦,𝑡4+14()+()1+exp𝑅/32−4𝑥+4𝑦exp𝑅/8𝑥+𝑦𝑅𝑡128exp(𝑅𝑥/8)+exp𝑅𝑦/8²+𝑅exp8×𝑥+𝑦−exp𝑅𝑥8+exp𝑅𝑦8𝑅²𝑡²×8192exp𝑅𝑥8+exp𝑅𝑦8³⁻¹+⋯. (25)
In case of 𝛼=1 and 𝛽=1, Table 1 cites the comparison of results obtained in proposed CRFDTM with variational iteration method (VIM) when y = 1. From these results we can certainly conclude that the proposed method CRFDTM provides remarkable accuracy in comparison to VIM. In the computation of Table 1, the value of 𝑅 has been taken as 100 and the numerical approximate solutions for 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) have been evaluated with four terms in CRFDTM and consequently compared with third approximations in VIM.

Table 1 

The comparison of results obtained in proposed method with VIM when 𝑦 = 1, 𝛼=1, and 𝛽=1.

Example 5. Consider the following (2 + 1)-dimensional time fractional coupled Burgers equations [16]: 𝐷^𝛼_𝑡𝜕𝑢−²𝑢𝜕𝑥²−𝜕²𝑢𝜕𝑦²−2𝑢𝜕𝑢𝜕𝑥−2𝑣𝜕𝑢𝜕𝑦=0,𝐷^𝛽_𝑡𝜕𝑣−²𝑣𝜕𝑥²−𝜕²𝑣𝜕𝑦²−2𝑢𝜕𝑣𝜕𝑥−2𝑣𝜕𝑣𝜕𝑦=0, (26)(27) subject to the initial conditions 𝑢𝑥,𝑦,0=1−tanh−𝑥+2𝑦+1,𝑣𝑥,𝑦,0=1+2tanh−𝑥+2𝑦+1. (28)(29)
First, we derive the recursive formula from (26) and (27). Now, 𝑈(ℎ,𝑘−ℎ) and 𝑉(ℎ,𝑘−ℎ) are considered as the coupled fractional reduced differential transform of 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡), respectively, where 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) are the solutions of coupled fractional differential equations. Here, 𝑈(0,0)=𝑢(𝑥,0), 𝑉(0,0)=𝑣(𝑥,0) are the given initial conditions. Without loss of generality, the following assumptions have been taken:𝑈𝑉()0,𝑗=0,𝑗=1,2,3,…,𝑖,0=0,𝑖=1,2,3,…. (30)
Applying CFRDTM to (26), we obtain the following recursive formula: Γ(ℎ+1)𝛼+(𝑘−ℎ)𝛽+1Γ()𝑈()=𝜕ℎ𝛼+𝑘−ℎ𝛽+1ℎ+1,𝑘−ℎ₂𝜕𝑥²𝑈()+𝜕ℎ,𝑘−ℎ𝜕𝑥^ℎ_𝑙=0^𝑘−ℎ_𝑠=0𝑈()𝑈()−5ℎ−𝑙,𝑠𝑙,𝑘−ℎ−𝑠2𝜕𝜕𝑥^ℎ_𝑙=0^𝑘−ℎ_𝑠=0𝑈(ℎ−𝑙,𝑠)𝑉(𝑙,𝑘−ℎ−𝑠). (31) From the initial condition of (28), we have 𝑈(0,0)=𝑢𝑥,𝑦,0. (32)
In the same manner, we can obtain the following recursive formula from (27): Γℎ𝛼+(𝑘−ℎ+1)𝛽+1Γ()𝑉()=𝜕ℎ𝛼+𝑘−ℎ𝛽+1ℎ,𝑘−ℎ+1₂𝜕𝑥²𝑉()+𝜕ℎ,𝑘−ℎ𝜕𝑥^ℎ_𝑙=0^𝑘−ℎ_𝑠=0𝑉()𝑉()−5𝑙,𝑘−ℎ−𝑠ℎ−𝑙,𝑠2𝜕𝜕𝑥^ℎ_𝑙=0^𝑘−ℎ_𝑠=0𝑈(𝑙,𝑘−ℎ−𝑠)𝑉(ℎ−𝑙,𝑠). (33) From the initial condition of (29), we have 𝑉(0,0)=𝑣𝑥,𝑦,0. (34)
According to CFRDTM, using recursive equation (31) with initial condition (32) and also using recursive scheme (33) with initial condition (34) simultaneously, we obtain successively 𝑈()21,0=−sech²1−𝑥+2𝑦Γ()𝑉()=41+𝛼0,1sech²1−𝑥+2𝑦Γ𝑈()1+𝛽1,1=−16sech⁴1−𝑥+2𝑦Γ𝑉()81+𝛼+𝛽0,2=−sech⁴1−𝑥+2𝑦+2sech²×Γ1−𝑥+2𝑦tanh1−𝑥+2𝑦1+2𝛽⁻¹𝑈(2,0)=8sech²×21−𝑥+2𝑦sech²×(Γ(1−𝑥+2𝑦+tanh1−𝑥+2𝑦1+2𝛼))⁻¹𝑉()=81,1sech⁴1−𝑥+2𝑦Γ,1+𝛼+𝛽 (35) and so on.
The approximate solutions, obtained in the series form, are given by 𝑢𝑥,𝑦,𝑡=^∞𝑘_𝑘=0ℎ=0𝑈(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=𝑈(0,0)+^∞𝑘_𝑘=1ℎ=1𝑈(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=1−tanh−𝑥+2𝑦+1−2𝑡^𝛼sech²1−𝑥+2𝑦Γ()−1+𝛼16𝑡^𝛼+𝛽sech⁴1−𝑥+2𝑦Γ+1+𝛼+𝛽8𝑡^2𝛼sech²×21−𝑥+2𝑦sech²×(Γ(1−𝑥+2𝑦+tanh1−𝑥+2𝑦1+2𝛼))⁻¹+⋯,𝑣𝑥,𝑦,𝑡=^∞𝑘_𝑘=0ℎ=0𝑉(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=𝑉(0,0)+^∞𝑘_𝑘=1ℎ=0𝑉(ℎ,𝑘−ℎ)𝑡^{(ℎ𝛼+(𝑘−ℎ)𝛽)}=1+2tanh1−𝑥+2𝑦+4𝑡^𝛽sech²1−𝑥+2𝑦Γ+1+𝛽8𝑡^𝛼+𝛽sech⁴1−𝑥+2𝑦Γ−1+𝛼+𝛽8𝑡^2𝛽sech⁴1−𝑥+2𝑦+2sech²×Γ1−𝑥+2𝑦×tanh1−𝑥+2𝑦1+2𝛽⁻¹+⋯. (36)(37) When 𝛼=1 and 𝛽=1, the solution in (36) becomes 𝑢𝑥,𝑦,𝑡=1−tanh−𝑥+2𝑦+1−2𝑡sech²1−𝑥+2𝑦+4𝑡²sech²1−𝑥+2𝑦×tanh1−𝑥+2𝑦+⋯. (38)
When 𝛼=1 and 𝛽=1, the solution in (37) becomes 𝑣𝑥,𝑦,𝑡=1+2tanh1−𝑥+2𝑦+4𝑡sech²1−𝑥+2𝑦−8𝑡²sech²1−𝑥+2𝑦tanh1−𝑥+2𝑦+⋯. (39)
The solutions in (38) and (39) are exactly same as the Taylor series expansions of the exact solutions 𝑢𝑥,𝑦,𝑡=1−tanh−𝑥+2𝑦+2𝑡+1=1−tanh−𝑥+2𝑦+1−2𝑡sech²1−𝑥+2𝑦+4𝑡²sech²1−𝑥+2𝑦tanh1−𝑥+2𝑦+⋯𝑣𝑥,𝑦,𝑡=1+2tanh−𝑥+2𝑦+2𝑡+1=1+2tanh1−𝑥+2𝑦+4𝑡sech²1−𝑥+2𝑦−8𝑡²sech²1−𝑥+2𝑦tanh1−𝑥+2𝑦+⋯. (40)(41)
In order to verify whether the proposed methodology lead to higher accuracy, the numerical solutions have been evaluated with four terms for both CRFDTM and homotopy perturbation method (HPM). In case of 𝛼=1and 𝛽=1, Table 2 cites the comparison of results obtained in proposed CRFDTM with HPM when y = 0.3. From these results we can certainly conclude that the proposed method CRFDTM provides remarkable accuracy in comparison to HPM.
Figure 1 explores the comparison of the numerical solutions of 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) for 𝛼=1, 𝛽=1, and 𝑡=0.3 with the exact solutions in (40) and (41) respectively. Similarly, Figure 2 cites the comparison of the numerical solutions of 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) for 𝛼=1, 𝛽=1, and 𝑡=0.1 with the exact solutions in (40) and (41) respectively. It can be observed that the numerical solutions obtained by the proposed method are exactly identical with the exact solution.
Figure 3 cites the numerical solutions of 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) for 𝛼=1/4, 𝛽=1/2, and 𝑡=0.3, respectively, and Figure 4 depicts the numerical solutions of 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) for 𝛼=0.88, 𝛽=0.005, and 𝑡=0.1 respectively. The figures show that when the value of 𝛼 increases solution surface becomes flattened. On the other hand, when the value of 𝛽 decreases solution surface bifurcates into wave.

Table 2 

The comparison of results obtained in proposed method with HPM when 𝑦 = 0.3, 𝛼=1, and 𝛽=1.

(a)

(b)

(c)

(d)

Figure 1 

The surfaces show (a) the numerical approximate solution of 𝑢(𝑥,𝑦,0.3), (b) the exact solution of 𝑢(𝑥,𝑦,0.3), (c) the numerical approximate solution of 𝑣(𝑥,𝑦,0.3), and (d) the exact solution of 𝑣(𝑥,𝑦,0.3) when 𝛼=1 and 𝛽=1.

(a)

(b)

(c)

(d)

Figure 2 

The surfaces show (a) the numerical approximate solution of 𝑢(𝑥,𝑦,0.1), (b) the exact solution of 𝑢(𝑥,𝑦,0.1), (c) the numerical approximate solution of 𝑣(𝑥,𝑦,0.1), and (d) the exact solution of 𝑣(𝑥,𝑦,0.1) when 𝛼=1 and 𝛽=1.

(a)

(b)

Figure 3 

The surfaces show (a) the numerical approximate solution of 𝑢(𝑥,𝑦,0.3) and (b) the numerical approximate solution of 𝑣(𝑥,𝑦,0.3) when 𝛼=1/4 and 𝛽=1/2.

(a)

(b)

Figure 4 

The surfaces show (a) the numerical approximate solution of 𝑢(𝑥,𝑦,0.1) and (b) the numerical approximate solution of 𝑣(𝑥,𝑦,0.1) when 𝛼=0.88 and 𝛽=0.005.

5. Convergence Analysis and Error Estimate

Theorem 6. Suppose that 𝐷_𝑡^𝑘𝛼𝑢(𝑥,𝑡)∈𝐶([0,𝐿]×[0,𝑇]) for 𝑘=0,1,2,…,𝑛+1, where 0<𝛼<1; then 𝑢(𝑥,𝑡)≅^𝑛𝑘_𝑘=0ℎ=0𝑈(ℎ,𝑘−ℎ)𝑡^{ℎ𝛼+(𝑘−ℎ)𝛽}. (42)
Moreover, there exist values 𝜉₁,𝜉₂ where 0≤𝜉₁,𝜉₂≤𝑡 so that the error term 𝐸_𝑛(𝑥,𝑡) has the form 𝐸_𝑛(𝑥,𝑡)=Sup_{0≤𝑥≤𝐿0≤𝑡≤𝑇}𝐷^(𝑛+1)𝛽𝑢(𝑥,0+)Γ()𝑡𝑛+1𝛽+1^(𝑛+1)𝛽,𝑖𝑓𝜉₁,𝜉₂⟶0+. (43)

Proof. From Lemma 1, we have 𝐽^𝛼𝐷^𝛼𝑓(𝑡)=𝑓(𝑡)−^𝑚−1_𝑘=0𝑡^𝑘Γ()𝑓𝑘+1^(𝑘)(0+),𝑚−1<𝛼<𝑚. (44)
The error term is 𝐸_𝑛(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝑢^∗(𝑥,𝑡), (45) where 𝑢(𝑥,𝑡)=^∞𝑘_𝑘=0ℎ=0𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,0)Γ()𝑡ℎ𝛼+𝛽𝑘−ℎ+1^{ℎ𝛼+𝛽(𝑘−ℎ)},𝑢^∗(𝑥,𝑡)=^𝑛𝑘_𝑘=0ℎ=0𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,0)Γ()𝑡ℎ𝛼+𝛽𝑘−ℎ+1^{ℎ𝛼+𝛽(𝑘−ℎ)}. (46) Now, for 0<𝛼<1, 𝐽^{ℎ𝛼+𝛽(𝑘−ℎ)}𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,𝑡)−𝐽^{(ℎ+1)𝛼+𝛽(𝑘−ℎ)}𝐷^{(ℎ+1)𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,𝑡)=𝐽^{ℎ𝛼+𝛽(𝑘−ℎ)}×𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,𝑡)−𝐽^𝛼𝐷^𝛼𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,𝑡)=𝐽^{ℎ𝛼+𝛽(𝑘−ℎ)}𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,0),since0<𝛼<1,using(7)=𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,0)Γ()𝑡ℎ𝛼+𝛽𝑘−ℎ+1^{ℎ𝛼+𝛽(𝑘−ℎ)}. (47)
The 𝑛th order approximation for 𝑢(𝑥,𝑡) is 𝑢^∗(𝑥,𝑡)=^𝑛𝑘_𝑘=0ℎ=0𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,0)Γ()𝑡ℎ𝛼+𝛽𝑘−ℎ+1^{ℎ𝛼+𝛽(𝑘−ℎ)}=^𝑛𝑘_𝑘=0ℎ=0𝐽^{ℎ𝛼+𝛽(𝑘−ℎ)}𝐷^{ℎ𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,𝑡)−𝐽^{(ℎ+1)𝛼+𝛽(𝑘−ℎ)}𝐷^{(ℎ+1)𝛼+𝛽(𝑘−ℎ)}𝑢(𝑥,𝑡),using()=46_𝑛𝑘=0𝐽^𝑘𝛽𝐷^𝑘𝛽𝑢()−𝑥,𝑡_𝑛ℎ=0𝐽^{(ℎ+1)𝛼+𝛽(𝑛−ℎ)}𝐷^{(ℎ+1)𝛼+𝛽(𝑛−ℎ)}𝑢()()+𝑥,𝑡=𝑢𝑥,𝑡^𝑛−1_𝑘=0𝐽^(𝑘+1)𝛽𝐷^(𝑘+1)𝛽𝑢()−𝑥,𝑡_𝑛ℎ=0𝐽^{(ℎ+1)𝛼+𝛽(𝑛−ℎ)}𝐷^{(ℎ+1)𝛼+𝛽(𝑛−ℎ)}𝑢(𝑥,𝑡). (48)
Therefore, from (48), the error term becomes 𝐸_𝑛(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝑢^∗()=𝑥,𝑡_𝑛ℎ=0𝐽^{(ℎ+1)𝛼+𝛽(𝑛−ℎ)}𝐷^{(ℎ+1)𝛼+𝛽(𝑛−ℎ)}𝑢()−𝑥,𝑡_{𝑛−1𝑘=0}𝐽^(𝑘+1)𝛽𝐷^(𝑘+1)𝛽𝑢()=𝑥,𝑡_𝑛𝑖=0𝐽^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}𝐷^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}𝑢()−𝑥,𝑡_{𝑛−1𝑖=0}𝐽^(𝑖+1)𝛽𝐷^(𝑖+1)𝛽𝑢()=𝑥,𝑡_𝑛𝑖=01Γ()()×𝑖+1𝛼+𝛽𝑛−𝑖_𝑡0(𝑡−𝜏)^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)−1}𝐷^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}𝑢()−𝑥,𝜏𝑑𝜏_{𝑛−1𝑖=0}1Γ()𝛽×𝑖+1_𝑡0(𝑡−𝜏)^{(𝑖+1)𝛽−1}𝐷^(𝑖+1)𝛽𝑢()=𝑥,𝜏𝑑𝜏_𝑛𝑖=0𝐷^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}𝑢𝑥,𝜉₁Γ()()𝑡𝑖+1𝛼+𝛽𝑛−𝑖+1^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}−^𝑛−1_𝑖=0𝐷^(𝑖+1)𝛽𝑢𝑥,𝜉₂Γ)𝑡(𝑖+1𝛽+1^(𝑖+1)𝛽,applyingintegralmeanvaluetheorem=^𝑛−1_𝑖=0𝐷^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}𝑢𝑥,𝜉₁Γ()()𝑡𝑖+1𝛼+𝛽𝑛−𝑖+1^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}+𝐷^(𝑛+1)𝛼𝑢𝑥,𝜉₁Γ))𝑡((𝑛+1𝛼+1^(𝑛+1)𝛼−^𝑛−1_𝑖=0𝐷^(𝑖+1)𝛽𝑢𝑥,𝜉₂Γ()𝑡𝑖+1𝛽+1^(𝑖+1)𝛽=^𝑛−1_𝑖=0𝐷^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}𝑢𝑥,𝜉₁Γ()()𝑡𝑖+1𝛼+𝛽𝑛−𝑖+1^{(𝑖+1)𝛼+𝛽(𝑛−𝑖)}−𝐷^(𝑖+1)𝛽𝑢𝑥,𝜉₂Γ()𝑡𝑖+1𝛽+1^(𝑖+1)𝛽+𝐷^(𝑛+1)𝛼𝑢𝑥,𝜉₁Γ))𝑡((𝑛+1𝛼+1^(𝑛+1)𝛼. (49)
Using generalized Taylor’s series formula (8), (49) becomes 𝐸_𝑛()()−𝐷𝑥,𝑡=𝑢𝑥,𝑡^(𝑛+1)𝛼𝑢𝑥,𝜁₁Γ))𝑡((𝑛+1𝛼+1^(𝑛+1)𝛼()+𝐷−𝑢𝑥,𝑡_(𝑛+1)𝛽𝑢𝑥,𝜁₂Γ()𝑡𝑛+1𝛽+1^(𝑛+1)𝛽+𝐷^(𝑛+1)𝛼𝑢𝑥,𝜉₁Γ))𝑡((𝑛+1𝛼+1^(𝑛+1)𝛼, (50) where 0≤𝜁₁,𝜁₂≤max{𝜉₁,𝜉₂}, 𝜉₁,𝜉₂→0+.
This implies 𝐸_𝑛=𝑢(𝑥,𝑡)−𝑢^∗(𝑥,𝑡)=Sup_{0≤𝑥≤𝐿0≤𝑡≤𝑇}𝐷^(𝑛+1)𝛽𝑢𝑥,𝜁₂Γ()𝑡𝑛+1𝛽+1^(𝑛+1)𝛽−𝐷^(𝑛+1)𝛼𝑢𝑥,𝜁₁Γ))𝑡((𝑛+1𝛼+1^(𝑛+1)𝛼+𝐷^(𝑛+1)𝛼𝑢𝑥,𝜉₁Γ))𝑡((𝑛+1𝛼+1^(𝑛+1)𝛼<∞=Sup_{0≤𝑥≤𝐿0≤𝑡≤𝑇}𝐷^(𝑛+1)𝛽𝑢(𝑥,0+)Γ()𝑡𝑛+1𝛽+1^(𝑛+1)𝛽,since𝜉₁,𝜉₂⟶0+. (51) As 𝑛→∞,‖𝐸_𝑛‖→0.
Hence, 𝑢(𝑥,𝑡) can be approximated as 𝑢(𝑥,𝑡)=^∞𝑘_𝑘=0ℎ=0𝑈(ℎ,𝑘−ℎ)𝑡^{ℎ𝛼+(𝑘−ℎ)𝛽}≅^𝑛𝑘_𝑘=0ℎ=0𝑈(ℎ,𝑘−ℎ)𝑡^{ℎ𝛼+(𝑘−ℎ)𝛽}, (52) with the error term given in (51).

6. Conclusion

In this paper, a new approximate numerical technique, coupled fractional reduced differential transform, has been proposed for solving nonlinear fractional partial differential equations. The proposed method is only well suited for coupled fractional linear and nonlinear differential equations. In comparison to other analytical methods, the present method is an efficient tool to determine approximate solution of nonlinear coupled fractional partial differential equations. The obtained results demonstrate the reliability of the proposed algorithm and its applicability to nonlinear coupled fractional evolution equations. It also exhibits that the proposed method is very efficient and powerful technique in finding the solutions of the nonlinear coupled time fractional differential equations. The main advantage of the proposed method is that it requires less amount of computational overhead in comparison to other numerical and analytical approximate methods and consequently introduces a significant improvement in solving coupled fractional nonlinear equations over existing methods available in open literature. The application of the proposed method for the solutions of time fractional coupled (2 + 1) Burger equations satisfactorily justifies its simplicity and efficiency.

Appendix

Proof of Theorem 3 (iii). One has 𝑓𝑣=𝑥,𝑦,𝑡=𝑢𝑥,𝑦,𝑡𝑥,𝑦,𝑡_{∞𝑘𝑘=0ℎ=0}𝑈(ℎ,𝑘−ℎ)𝑡^{ℎ𝜶+(𝑘−ℎ)𝜷}×^∞𝑘_𝑘=0ℎ=0𝑉(ℎ,𝑘−ℎ)𝑡^{ℎ𝜶+(𝑘−ℎ)𝜷}()𝑉()+(𝑈()𝑉()()𝑉(𝑡=𝑈0,00,01,00,0+𝑈0,01,0))^𝜶+(𝑈(0,1)𝑉(0,0)+𝑈(0,0)𝑉(0,1))𝑡^𝜷+(𝑈()𝑉()()𝑉()()𝑉()()𝑉(𝑡1,00,1+𝑈0,11,0+𝑈1,10,0+𝑈0,01,1))^𝜶+𝜷=+⋯_{∞𝑘𝑘=0ℎ=0ℎ𝑙=0𝑘−ℎ𝑠=0}𝑈(ℎ−𝑙,𝑠)𝑉(𝑙,𝑘−ℎ−𝑠)×𝑡^{ℎ𝜶+(𝑘−ℎ)𝜷}. (A.1)
Hence, 𝐹(ℎ,𝑘−ℎ)=∑^ℎ_𝑙=0∑^𝑘−ℎ_𝑠=0𝑈(ℎ−𝑙,𝑠)𝑉(𝑙,𝑘−ℎ−𝑠).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to express his sincere thanks and gratitude to the anonymous learned reviewers for their kind suggestions for the betterment and improvement of the present paper.