A note on time-fractional Navier–Stokes equation and multi-Laplace transform decomposition method

*Correspondence: hgadain@ksu.edu.sa 1College of Science, Mathematics Department, King Saud University, P.O. Box 2455, 11451 Riyadh, Saudi Arabia Full list of author information is available at the end of the article Abstract In this study, the double Laplace Adomian decomposition method and the triple Laplace Adomian decomposition method are employed to solve oneand two-dimensional time-fractional Navier–Stokes problems, respectively. In order to examine the applicability of these methods some examples are provided. The presented results confirm that the proposed methods are very effective in the search of exact and approximate solutions for the problems. Numerical simulation is used to sketch the exact and approximate solution.


Introduction
Fractional partial differential equations as generalizations of classical partial differential equations, and they have been proposed and applied to many applications in various fields of physical sciences and engineering such as electromagnetic, acoustics, visco-elasticity and electro-chemistry. Recently, the solution of fractional partial differential equations has been obtained through a double Laplace decomposition method by the authors [1][2][3]. The natural transform decomposition method has been successfully used to handle linear and nonlinear problems appearing in physical and engineering disciplines [4,5]. The Navier-Stokes equations are the fluid dynamics identical to Newton's second law, force equals mass times acceleration, and they are of crucial significance in fluid dynamics. Also the Navier-Stokes equations are vector equations. Recently, many powerful methods have been used to obtain different type solution of time-fractional Navier-Stokes equation such as the Adomian decomposition method [6], the q-homotopy analysis transform scheme [7], the modified Laplace decomposition method [8,9], the natural homotopy perturbation method [10] and a reliable algorithm based on the new homotopy perturbation transform method [11]. The one-dimensional Navier-Stokes equation with timefractional derivative has been given in operator form [12]. The main objective of this work is to find the exact and approximate solution of time-fractional Navier-Stokes equations by using the double and triple Laplace Adomian decomposition methods, respectively.

Basic definitions and preliminaries concepts
In this section, we give some essential definitions, properties and theorems of fractional calculus and double Laplace transform, which should be used in the present study.

Definition 1
In [13] 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 complex variables, and further triple Laplace transforms of the partial derivatives are shown by Likewise, the triple Laplace transform for the second partial derivative with respect to x, y and t are defined by The inverse triple Laplace transform L -1 In the next theorem, one can introduce the triple Laplace transform of the partial fractional Caputo derivatives. Theorem 1 ([17]) 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 the triple Laplace transforms 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 defined by

2)
and In the following part, the relations between Mittag-Leffler function and Laplace transform are considered, which are helpful in the in the current study. The Mittag-Leffler function is defined by the following series: , z ∈ C, (β) > 0, (2.4) the Mittag-Leffler function with two parameters is defined by , z ∈ C, (α) > 0, (2.5) see [18,19]. If we put β = 1 in Eq. (2.5) we obtain Eq. (2.4). It follows from Eq. (2.5) that and in general Triple Laplace transforms of some Mittag-Leffler functions are given by , ,

Analysis of the double Laplace decomposition method
In this secction, we give the essential idea of the double Laplace Adomian decomposition method (DLADM) for the time-fractional Navier-Stokes equations. With a view to showing the fundamental scheme of the double Laplace Adomian decomposition method, we consider the following time-fractional Navier-Stokes equations: subject to the condition where D α t = ∂ α ∂t α is the fractional Caputo derivative, D 2 x = ∂ 2 ∂x 2 , D x = ∂ ∂x and the right-handside function f (x, t) is the source term. In order to apply the double Laplace Adomian decomposition method, we multiply first Eq. (3.1) by x, we obtain implementing the double Laplace transform on both sides of Eq. (3.2), we have by using Theorem 1, we get Immediately, implementing the differentiation property of the Laplace transform, we get d dp after an algebraic manipulation, we obtain By taking the integral for both sides of Eq. (3.5) from 0 to p with respect to p, we get the double Laplace Adomian decomposition solution u(x, t) is defined by the following infinite series: by using DLADM, we introduce the iterative relations (3.9) and the remaining components can be written as Hence, u 0 (x, t) and u m (x, t) can be obtained by applying the inverse double Laplace transform to Eqs. (3.9) and (3.10), respectively, and we have and where L x L t is the double Laplace transform with respect to x, t and the double inverse Laplace transform denoted by L -1 p L -1 s is with respect to p, s. We supposed that the double inverse Laplace transform exists for Eqs. (3.11) and (3.12).

Analysis of the triple Laplace decomposition method
In this part of the paper, we give the fundamental idea of the triple Laplace Adomian decomposition method (TLADM) for the two-dimensional time-fractional Navier-Stokes equations. In order to show the fundamental plan of the triple Laplace Adomian decomposition method, we consider the following system of two-dimensional time-fractional Navier-Stokes equations: On using the differentiation property of the Laplace transform, we get By applying the triple inverse Laplace transformation for Eq. (4.3), we get the solutions u(x, y, t) and v(x, y, t) are defined by the following series: moreover, the nonlinear terms uu x , vu y , uv x and vv y are determined by and Taking the inverse Laplace transformation to Eqs. (4.7) and (4.8) we have by using DLADM, we introduce the recursive relations 11) and the remaining components u n+1 and v n+1 , n ≥ 0 are given by and where L x L y L t is the triple Laplace transform with respect to x, y, t and triple inverse Laplace transform denoted by L -1 p L -1 q L -1 s is with respect to p, q, s. We assume that the triple inverse Laplace transform with respect to p, q and s exist for Eqs. (4.11), (4.12) and (4.13).

Numerical examples
In this part of paper, we discuss the achievement of our present methods and examine its accuracy by using the decomposition method with connection of the Laplace transform. Three problems are given.

Problem 1
Consider the homogeneous one-dimensional motion of a viscous fluid in a tube given by subject to the initial condition One can write Eq. (5.1) in the form where K = -∂r ρ∂z , multiplying the above equation with x, we have Implementing the double Laplace transform on both sides of Eq. (5.4), we get using the differentiation property of the Laplace transform and Theorem 1, we obtain d dp substituting the initial condition and arranging Eq. (5.6), we have dU(p, s) dp = 1 s d dp by integrating both sides of Eq. (5.7) from 0 to p with respect to p, we have The inverse double Laplace transform of Eq. (5.8) is denoted by we assume an infinite series solution of the unknown function u(x, t) is given by substituting Eq. (5.10) into Eq. (5.9), we get The zeroth component u 0 is proposed by Adomian method, is constantly contains initial condition and the nonhomogeneous term, both of which are assumed to be known. Accordingly, we put The remaining components u m+1 , m ≥ 0 are given by using the relation by substituting m = 0, into Eq. (5.12), we get similarly at m = 1, [0] dp = 0, (5.14) at m = 2, we have Hence, the solution of Eq. (5.1) can be can be found to be The result is the same as given by [6,10].

Problem 2
The nonhomogeneous time-fractional Navier-Stokes equation x, t > 0, (5.15) subject to the initial condition Applying the double Laplace transform on both sides of Eq. (5.15), subject to the initial condition Eq. (5.16), we have (xD x u) dp. (5.17) Working with the double Laplace inverse on both sides of Eq. (5.17) gives (xD x u) dp . (5.18) By using the above-mentioned method the solution of Eq. (3.7), is given by u m (x, t) dp , (5.19) the first few terms of the double Laplace decomposition series are given by and Hence, at m = 0, we get , In the same manner, The series solution is therefore given by where E denotes the Mittag-Leffler function. On setting α = 1 in Eq. (5.15), we get the exact solution of the non-time-fractional Navier-Stokes equation under the same condition u(x, 0) = x 2 . The solution is given by In the following problem, the suggested method is applied to the two-dimensional timefractional model of the Navier-Stokes equation, in Eq. (5.20). We let h 1 = 1 ρ ∂r ∂x = -h 2 = -1 ρ ∂r ∂y = h as follows.

Problem 3
Consider the time-fractional order two-dimensional Navier-Stokes equation [9,13] D α t u + uu x + vu y = ρ 0 (u xx + u yy ) + h, x, y, t > 0, on using the differentiation property of the Laplace transform, we have substituting the initial condition and arranging Eq. (5.21), we have L x L y L t u(x, y, t) = -p + q s(p 2 + 1)(q 2 + 1) -1 s α L x L y L t (uu x + vu y ) Now, implementing the inverse triple Laplace transform for Eq. (5.22) The zeroth components u 0 and v 0 are found by the Adomian method. Always it contains initial condition and the source term, both of which are assumed to be known. Accordingly, we set The remaining components u n+1 , u n+1 , n ≥ 0 are given by using the relations the first few terms of the Adomian polynomials A n , B n , C n and D n are given by in the same way, we have similarly at n = 1, and v 2 = -L -1 p L -1 q L -1 at n = 2, we have and In the same manner, we have The solution of Eq. (5.20) is given by (-2ρ 0 ) n t nα (nα + 1) (-2ρ 0 ) n t nα (nα + 1) -ht α (α + 1) , at α = 1 and h = 0, we obtain the exact solution of the classical Navier-Stokes equation for the velocity: v(x, y, t) = sin(x + y)e -2ρ 0 t .

Numerical result
In  The three-dimensional surface in Fig. 2(a) shows the solution of Eqs. (5.1) at (α = 0.5) and Fig. 2(b) shows the exact solution of the time-fractional Navier-Stokes equation with α = 1 in normal form.
In the same manner, the exact solution and approximate solution of Eq. The three-dimensional surface in Fig. 4(a) shows the solution of Eqs. (5.15) at (α = 0.5) and Fig. 4(b) shows the exact solution of the time-fractional Navier-Stokes equation at α = 1 in standard form equal to x 2 e t .
It is clear from the solutions of Eqs. (5.1) and (5.15) that the double Laplace transform decomposition method shows good agreement with the exact solutions of the problems. Figure 5 consists of two graphs, namely Fig. 5(a) and Fig. 5(b). Figures 5(a) and 5(b) represent the functions u(x, y, t) and u(x, y, t) of the Navier-Stokes equation, respectively, of Eq. (5.20) when ρ = 0.5, α = 0.5 and t = 0.5. Figure 6 consists of two graphs, namely Fig. 6(a) and Fig. 6(b). Figures 6(a) and 6(b) represent the functions u(x, y, t) and u(x, y, t) of the Navier-Stokes equation, respectively, of Eq. (5.20) when ρ = 0.5, α = 0.5 and t = 0.05.

Conclusion 1
In this work, double and triple Laplace Adomian decomposition methods are suggested for solving one-and two-dimensional time-fractional Navier-Stokes equations. These methods have been proved to be a powerful tool which enable us to manage fractional order differential equations and allow one to reach the desired accuracy. All we