A note on singular two-dimensional fractional coupled Burgers’ equation and triple Laplace Adomian decomposition method

The present article focuses on how to find the exact solutions of the time-fractional regular and singular coupled Burgers’ equations by applying a new method that is called triple Laplace and Adomian decomposition method. Furthermore, the proposed method is a strong tool for solving many problems. The accuracy of the method is considered through the use of some examples, and the results obtained are compared with those of the existing methods in the literature.


Introduction
Fractional Burgers' equation has received significant attention. The solution of this equation becomes very important for mathematical and physical phenomena. This equation has been discovered to explain different kinds of events, for example, it is a mathematical model of turbulence and approximate theory of a flow over a shock wave traveling in a viscous fluid [1,2]. Local fractional homotopy analysis method has been explained in [3,4]. The authors in [5] applied the new semianalytical method which is called the homotopy analysis Shehu transform method to solve multidimensional fractional diffusion equations. The researchers in [6,7] examined the numerical solutions of three-dimensional Burger's equation and Riccati differential equations by applying Laplace decomposition methods. From a recent couple of years, important dedication has been given to Laplace decomposition method and its modifications for studying mathematical model [8,9]. In the literature several authors have suggested different types of approximation and exact methods for solving fractional Burger's equation [10][11][12]. The authors in [13] applied the variational iteration method to obtain Burger's equation. In [14] the Laplace decomposition method (LDM) is suggested to solve the two-dimensional nonlinear Burgers' equations. The authors in [15] used the double Laplace decomposition methods to solve sin-gular Burgers' equation and coupled Burgers' equations. In this study, we propose a new hybrid triple Laplace Adomian decomposition method to obtain exact solutions of the time-fractional regular and singular coupled Burgers' equations. Finally, two examples are provided to illustrate the suggested approach.
Here we recall some definitions, notation of Laplace transform, and fractional calculus facts which are useful in this article. Definition 1 ([16]) Let f be a function of three variables x, y, and t, where x, y, t > 0. The triple Laplace transform of f is defined by where p, q, s are Laplace variables. Further, the triple Laplace transform of the partial derivatives is denoted by Similarly, the triple Laplace transform for the second partial derivatives with respect to x, y, and t are defined as follows: The inverse triple Laplace transform L -1 In the following theorem, we present the triple Laplace transforms of the partial fractional Caputo derivatives. Theorem 1 ([20]) Let α, β, γ > 0, n -1 < α ≤ n, m -1 < β ≤ m, r -1 < γ ≤ r, and n, m, p ∈ N, so that f ∈ C l (R + × R + × R + ), l = max{n, m, p}, f (l) ∈ L 1 [(0, a) × (0, b) × (0, c)] for any a, b, c > 0, |f (x, y, t)| ≤ we xτ 1 +yτ 2 +tτ 3 , x > a > 0, y > b > 0, and t > c > 0. Then the triple Laplace transform of Caputo's fractional derivatives D α t u(x, y, t), D α t u(x, y, t) and D α t u(x, y, t) are given by Below, we establish the relation between Mittag-Leffler function and Laplace transform, which will be useful in this paper. The Mittag-Leffler function is defined by the following series representation: , z ∈ C, (β) > 0, (1.4) the Mittag-Leffler function with two parameters is defined by , z ∈ C, (α) > 0, (1.5) see [21]. If we put β = 1 in Eq. (1.5), we obtain Eq. (1.4). It follows from Eq. (1.5) that and in general, In the following, we introduce the Laplace transforms of some Mittag-Leffler functions which are useful in this work: , ,

Analysis of the triple Laplace decomposition method
The main objective of this section is to address the use of triple Laplace Adomian decomposition method (TLADM) for solving two-dimensional time-fractional coupled Burger's equation. We consider two-dimensional time-fractional coupled Burger's equation in the form: with the initial condition where D α t = ∂ α ∂t α is the fractional Caputo derivative, is the Reynolds number, and the velocity components are given by u(x, y, t) and v(x, y, t) in the x and y direction, respectively. For the purpose of finding the solution of Eq. (2.1), we apply triple Laplace Adomian decomposition method as follows: Step 1. Taking the triple Laplace transform for Eq. (2.1), we obtain Step 2. Now, employing the differentiation property of the Laplace transform, we have Step 3. By implementing the triple inverse Laplace transformation of Eq. (2.3), we obtain Step 4. The Laplace Adomian decomposition solution functions u(x, y, t) and v(x, y, t) are given by the infinite series further, the nonlinear terms uu x , v ∂u ∂y , u ∂v ∂x , and v ∂v ∂y are given by: Step 5. Using Laplace Adomian decomposition method, we introduce the recursive relations and get: 9) and the remainder components u n+1 and v n+1 , for n ≥ 0, are given by where a few first terms of the Adomian polynomials A n , B n , C n , and D n are given by (2.14) and where L x L y L t is the triple Laplace transform with respect to x, y, t and the triple inverse Laplace transform with respect to p, q, s is denoted by L -1 p L -1 q L -1 s . We supposed that the triple inverse Laplace transform with respect to p, q, and s exists in Eqs. (2.9), (2.10), and (2.11).

Triple Laplace Adomian decomposition method and singular two-dimensional fractional coupled Burgers' equation
The principal algorithm of the triple Laplace decomposition method will be applied to singular two-dimensional fractional coupled Burgers' equation of the form x, y, t > 0, (3.1) associated with the initial condition u(x, y, 0) = f 1 (x, y), v(x, y, 0) = g 1 (x, y), (3.2) where D α t = ∂ α ∂t α is the fractional Caputo derivative and 1 x (xu x ) x , 1 y (yu y ) y are called Bessel operators, u(x, y, t) and v(x, y, t) are the velocity components to be determined, f (x, y, t); g(x, y, t); f 1 (x, y) and g 1 (x, y) are known functions. In order to obtain the solution of Eq. (3.1), we use the following steps: Step 1. Multiplying both sides of Eq. (3.1) by xy, we have x, y, t > 0, Step 2. Applying the triple Laplace transform to both sides of Eq. (3.3), we obtain and Now, using the differentiation property of the Laplace transform yields (3.6) Step 3. Integrating both sides of Eq. (3.6) from 0 to p and from 0 to q with respect to p and q, respectively, we have U(p, q, s) = 1 s p 0 q 0 ∂ 2 ∂p∂q F 1 (p, q) dq dp and V (p, q, s) = 1 s p 0 q 0 ∂ 2 ∂p∂q G 1 (p, q) dq dp x + x(yv y ) y dq dp -1 s α p 0 q 0 L x L y L t (yuv x + xvv y ) dq dp 0 L x L y L t xyg(x, y, t) dq dp. (3.8) Step 4. By taking the triple inverse Laplace transformation of Eqs. (3.7) and (3.8), we obtain and v(x, y, t) = L -1 Step 5. Substituting Eqs. F 1 (p, q) dq dp u ny y dq dp B n dq dp G 1 (p, q) dq dp D n dq dp x, y, t) dq dp .
Step 6. Using Laplace Adomian decomposition method, we introduce the recursive relations and get x, y, t) dq dp (3.11) and the other components u n+1 and v n+1 , for n ≥ 0, are given by u ny y dq dp B n dq dp (3.13) and D n dq dp , (3.14) where L x L y L t is the triple Laplace transform with respect to x, y, t and the triple inverse Laplace transform with respect to p, q, s is denoted by L -1 p L -1 q L -1 s . We assumed that the triple inverse Laplace transform with respect to p, q, and s exists for Eqs. (3.11), (3.12), (3.13), and (3.14). In the following example we apply the triple Laplace Adomian decomposition method to solve singular two-dimensional time-fractional coupled Burgers' equations.
Example 2 Consider singular two-dimensional time-fractional coupled Burgers' equations given by x, y, t > 0, (3.15) with the initial condition As stated by the above steps, we have and v(x, y, t) = x 2y 2 + 1 s α p 0 q 0 L x L y L t y(xv x ) x + x(yv y ) y dq dp By applying Eqs. (3.11), (3.12), (3.13), and (3.14), we obtain 18) and the other components u n+1 and v n+1 , for n ≥ 0, are given by q 0 L x L y L t (yA n + xB n ) dq dp (3.19) and v n+1 (x, y, t) = L -1 p L -1 q L -1 s 1 s α p 0 q 0 L x L y L t y(xv nx ) x + x(yv ny ) y dq dp where a few first terms of the Adomian polynomials A n , B n , C n , and D n are given by Eqs. (2.12), (2.13), (2.14), and (2.15), respectively. By substituting n = 0 into Eqs. (3.19) and (3.20), we get In the same manner for n = 1, we obtain that The solution of Eq. (3.15) is given by u(x, y, t) = u 0 + u 1 + u 2 + · · · + u n , v(x, y, t) = v 0 + v 1 + v 2 + · · · + v n .

Conclusion
In this study, the triple Laplace transform and Adomian decomposition have been successfully combined to obtain a new powerful method named a hybrid triple Laplace Adomian decomposition method (TLADM). This method has been used to solve regular and singular coupled Burgers' equations. By applying this method on some examples, we have obtained new efficient relations to solve our problems. It allows more realistic series solution that converges very quickly to the true solution.