An accelerated solution for some classes of nonlinear partial differential equations

Background Many physics and engineering problems are modeled by partial differential equations (PDEs). In many instances, these equations are nonlinear and the exact solutions are difficult to be obtained. Several methods were developed over some time to find approximate solutions to these nonlinear equations, such as homotopy analysis method (HAM) [1–4], homotopy perturbation method (HPM) [1, 5, 6], and Adomian decomposition method (ADM) [7–15]. In this paper, we introduce an accelerated version of the ADM for solving some classes of NPDEs. In ADM, the nonlinear term is replaced by a series of what are called Adomian polynomials which were introduced by Adomian and his colleagues have so far. Some other authors have suggested different formulas for computing Adomian polynomials [16–23]. This work aims to apply the accelerated formula proposed by El-Kalla in [21] for solving some classes of nonlinear partial differential equations. The main advantages of this accelerated version of Adomian polynomials can be summarized in the following main three points:

The paper is organized as follows. In "The method" section, the standard ADM and the accelerated version of ADM are introduced. In "Convergence analysis" section, the convergence analysis of the accelerated version is introduced, while in "Numerical examples" section, some examples are solved to illustrate the effectiveness of this version.

The method
Consider the nonlinear partial differential equation given in the operator form: where L t (.) = ∂ k (.) ∂t k , R is the linear remainder operator that could include partial derivatives with respect to x , N is the nonlinear operator, and g is the nonhomogeneous term.
Put (1) in the following form Applying L −1 t to both sides of (2), we obtain where �(x, t) is the solution of L t u(x, t) = 0 satisfied by the given initial conditions and ADM assumes that the solution u can be decomposed into infinite series and the nonlinear term Nu by: The components u n , n ≥ 0 of the solution u can be determined by using the recursive relation: where A n = A n (u 0 , u 1 , . . . , u n ) is Adomian polynomials that can be determined by the traditional polynomials formula, (1) L t u(x, t) + R(u(x, t)) + N (u(x, t)) = g(x, t), (2) L t u(x, t) = g(x, t) − R(u(x, t)) − N (u(x, t)). or by El-Kalla formula [21], where the partial sum S n = n i=0 u i (x, t). For example, Table 1 shows the first four polynomials of the nonlinear term u 2 generated by both the traditional polynomials formula (7) and El-Kalla polynomials formula (8).
Clearly, the first four polynomials generated by El-Kalla formula (8) include the first four polynomials generated by the traditional formula (7) in addition to other terms that should appear in A 4 , A 5 , . . . using formula (7). Thus, the solution obtained using El-Kalla polynomials converges faster than the solution obtained using the traditional polynomials.

Convergence analysis
) and let u, u * ∈ E . Then, Table 1 The first four Adomian polynomials and the first four El-Kalla polynomials of the nonlinear term u 2

Adomian polynomials of u 2
El-Kalla polynomials of u 2 Under the condition 0 < α < 1 , the mapping F is contraction; therefore, by the Banach fixed-point theorem for contraction, there exists a unique solution to problem (1).
Proof Let, S n and S m be arbitrary partial sums with n > m . We are going to prove that {S n } is a Cauchy sequence in Banach space E . From Theorem 1, we write Using the triangle inequality, we have

Theorem 3 (Error estimate) An estimate for the truncation error of the series solution (4) to problem (1) is given by:
Proof From (9) in Theorem 2, we have

Numerical examples
In this section, we present some numerical examples to illustrate the effectiveness of the proposed version of ADM. All the results are calculated using Mathematica 11.
Example 1 Consider the following nonlinear partial differential equation: with initial condition which has exact solution u(x, t) = xt.
Solution Equation (10) is rewritten in the form: where Nu = uu x and L t = ∂ ∂t . (12), we get Based on the recurrence relation (6) and substituting the initial value, we get using the traditional polynomials formula (7), Then, from (14) and (15) we get and using El-Kalla polynomials formula (8), Then, from (14) and (17) we get Table 2 shows the absolute relative error (ARE) for the sixth-order approximate solution using the proposed version of ADM and the seventh-order approximate solution using the standard ADM at t = 1 for some values of x in Example 1.
Example 2 Consider a nonlinear partial differential equation: (16) A 0 = u 0 u 0x , u + u 2 = x 2 sin 2 πt 2 , with initial condition This problem was solved in [24] by using the standard Adomian decomposition method. Now, we will apply the proposed accelerated version of ADM and compare the results in Table 3.

Conclusion
An accelerated technique based on ADM is proposed. In this proposed technique, there is no need for differentiation in calculations of the Adomian polynomials. Consequently, it makes programming easier and saves much time on the same processor compared with the calculations using traditional Adomian polynomials. Convergence analysis of this version is discussed, and the error analysis of the series solution is estimated. Results of numerical examples show the effectiveness of the proposed technique. Accordingly, in the future, this accelerated version is recommended for solving nonlinear equations with different complicated piece-wise differentiable nonlinearity terms.