A new modified Adomian decomposition method for nonlinear partial differential equations

Djelloul Ziane1,2, Rachid Belgacem3,∗ and Ahmed Bokhari3 1 Laboratory of mathematics and its applications (LAMAP), University of Oran1 Ahmed Ben Bella, Oran, 31000, Algeria.; djeloulz@yahoo.com 2 Department of Physics, University of Hassiba Benbouali, Ouled Fares, Chlef 02180, Algeria. 3 Department of Mathematics, University of Hassiba Benbouali, Ouled Fares, Chlef 02180, Algeria.; belgacemrachid02@yahoo.fr (R.B); bokhari.ahmed@ymail.com (A.B) * Correspondence: belgacemrachid02@yahoo.fr


Introduction
T he use of integral transforms (Laplace, Sumudu, Natural, Elzaki, Aboodh, Shehu and other transforms) in solving linear differential equations as well as integral equations has developed significantly as a result of the advantages of these transformations. Through these transforms, many problems of engineering and sciences have been solved. However, it was found that these transforms remain limited in solving equations that contain a nonlinear part.
The objective of the present study is to combine two powerful methods, Adomian decomposition method and Shehu transform method to get a better method to solve nonlinear partial differential equations. The modified method is called Shehu transform decomposition method (STDM). We apply our modified method to solve some examples of nonlinear partial differential equations.

Basics of Shehu transform
In this section we define Shehu transform and gave its important properties [21]. Definition 1. The Shehu transform of the function v(t) of exponential order is defined over the set of functions: by the following integralŜ It converges if the limit of the integral exists, and diverges if not. The inverse Shehu transform is given as: Equivalently where s and u are the Shehu transform variables, and α is a real constant and the integral in Equation (4) is taken along s = α in the complex plane s = x + iy.
Theorem 2. (The sufficient condition for the existence of Shehu transform [21]). If the function v(t) is piecewise continues in every finite interval 0 t β and of exponential order α for t > β. Then its Shehu transform V(s; u) exists. [21]). If the function v (n) (t) is the nth derivative of the function v(t) ∈ A with respect to t then its Shehu transform is defined as:

Theorem 3. (Derivative of Shehu transform
Taking n = 1, 2 and 3 in Equation (5), we obtain the following derivatives with respect to t: Now, we summarize some important properties of this transform [21].
Other properties are given in the Table 1.
Proof. By means of integration by parts, we get , then, by using Equation (2) and Equation (9), we get: Proof. We use use mathematical induction to prove (11). By means of Equation (9), the formula (11) is true for n = 1 and supposeŜ Let ∂ n v(x,t) ∂t n = w(x, t) and using (9) and (12), we have:

Shehu transform decomposition method
To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous partial differential equation where is the partial derivative of the function U(x, t) of order m (m = 1, 2, 3), R is the linear differential operator, N represents the general nonlinear differential operator, and g(x, t) is the source term.
Applying the Shehu transform (denoted in this paper byŜ) on both sides of Equation (13), we get Using the properties of Shehu transform, we obtain where m = 1, 2, 3. Thus, we havê Operating the inverse transform on both sides of Equation (16), we get where G(x, t), represents the term arising from the source term and the prescribed initial conditions. The second step in Shehu transform decomposition method, is that we represent the solution as an infinite series given below and the nonlinear term can be decomposed as: where A n are Adomian polynomials [22] of U 0 , U 1 , U 2 , ..., U n and it can be calculated by the formula: Substituting (18) and (19) in (17), we have On comparing both sides of the Equation (21), we get In general, the recursive relation is given as: where m = 1, 2, 3, and n 0. Finally, we approximate the analytical solution U(x, t) by: The above series solutions generally converge very rapidly [23].

Application
Here, we apply Shehu transform decomposition method to solve some nonlinear partial differential equations.

Example 1. Consider the nonlinear KdV equation [24]
: with the initial condition: Applying the Shehu transform on both sides of Equation (25), we get By means of the properties of Shehu transform, we get Taking the inverse Shehu transform on both sides of Equation (28), we obtain By applying the aforesaid decomposition method, we have On comparing both sides of Equation (30), we get The first few components of A n (U) polynomials [22], for example, are given by: Using the iteration formulas (31) and the Adomian polynomials (32), we obtain Based on the formula (24), we get which gives which is an exact solution to the KdV equation as presented in [25]. The graphs of exact solution and approximate solutions of Equation (25) for 3 terms and 4 terms is given in Figure 1.

Example 2.
Consider the nonlinear gas dynamics equation: with the initial condition: Applying the Shehu transform and its inverse on both sides of Equation (36), we get By applying the aforesaid Decomposition Method, we have On comparing both sides of Equation (39), we obtain The first few components of A n (U) polynomials [22] is given by (32), and for B n (U) for example, given as follows: Using the iteration formulas (40) and the Adomian polynomials (32), (41), we get the first terms of the solution series that is given by: So, the approximate series solution of Equation (36) is given as: And in the closed form, is given by: This result is the same as that obtained in [26] using homotopy analysis method. In Figure 2, (a) represents the graph of exact solution, (b) represents the graph of approximate solutions in 5 terms and (c) represents the graph of approximate solutions in 4 terms.
with the initial conditions: Applying the Shehu transform and its inverse on both sides of Equation (45), we get By applying the aforesaid Decomposition Method, we have Comparing both sides of Equation (48), we obtain The first few components of C n (U) and D n (U) Adomian polynomials [22], for example, are given by: and Using the iteration formulas (49) and the Adomian polynomials (50) and (51), we obtain The first terms of the approximate solution of Equation (45), is given by And in the closed form: U(x, t) = x 2 sin(t). (54)

Conclusion
The coupling of Adomian decomposition method (ADM) and Shehu transform method proved very effective to solve nonlinear partial differential equations. We can say that this method is easy to implement and is very effective, as it allows us to know the exact solution after calculate the first three terms only. As a result, the conclusion that comes through this work is that (STDM) can be applied to other nonlinear partial differential equations of higher order, due to the efficiency and flexibility.