A new aftertreatment for improving adomian modified decomposition method for solving nonlinear differential equations

The theory of the Adomian modified decomposition method (AMDM) for solving linear and nonlinear differential equations is well established. However, the solutions obtained by using the current AMDM are valid only for a very small region. In this paper, a new aftertreatment technique is proposed to improve the accuracy of the AMDM during a wide region. Based on the proposed aftertreatment technique, the truncated series solution obtained by the AMDM can be expressed as another series in terms of the independent sine and cosine trigonometric functions. Two numerical examples are presented and compared to those obtained from the numerical 4th-order Runge-Kutta algorithm. It is shown that the AMDM with the proposed aftertreatment technique offers an accurate and effective method for solving nonlinear differential equations in a wide applicable region.


Introduction
The Adomian decomposition method (ADM) [1-3] is a useful and powerful method for solving linear and nonlinear differential equations. The goal of ADM is to find the solution of linear and nonlinear, ordinary or partial differential equation without depending on any small parameter, such as the case with the perturbation method. The main advantages of ADM are computational simplicity and do not involve any linearization or smallness assumptions [1]. However, the result by using ADM is usually considered as a truncated series solution which only gives a good approximation to the accurate solution in a small region and has very small convergence radius. To overcome this drawback, two aftertreatment techniques have been proposed. One method is called as multistage ADM [4][5][6], which divide the solution space into M small regions. The solution is found an approximate solution in a closed and recursive form in a sequence of intervals [0, t 1 ], [t 1 , t 2 ], …, [t M ; T] such that the initial condition in sub-division interval [t i , t i +1] is taken to be the condition at t i . The accuracy of this aftertreatment technique is strongly dependence on the subdivision interval length. Another aftertreatment technique is based on Padé approximants, Laplace transform and its inverse to deal with the truncated series solution obtained by ADM. The combining Laplace ADM with Padé approximants have been investigated by many researchers and obtained plentiful achievement. No attempt will be made here to present a bibliographical account of previous work in this area. A few selective recent papers are quoted [7][8][9][10] which provide further references on the subject. In this paper, a modified type of ADM, termed as the Adomian modified decomposition method (AMDM), proposed by Wazwaz and EI-Sayed [11] is used to solve nonlinear differential equations. And a new aftertreatment technique for improving the accuracy of the AMDM is proposed. The truncated series solution obtained by the AMDM can be expressed as another series in terms of the independent sine and cosine trigonometric functions based on the proposed aftertreatment technique. Finally, two numerical examples are given to demonstrate the feasibility of the proposed aftertreatment technique.

A brief review of Adomian modified decomposition method (AMDM)
In this section, the concept of Adomian modified decomposition method (AMDM) is briefly introduced. Consider the general nonlinear functional equation: where L is a linear invertible operator of highest-order derivative with respect to x. R is the remainder linear operators. N is the nonlinear operator, and g(x) is the source term.
As L is invertible, and applying the inverse operator L -1 to both sides of equation (1), we obtain where L -1 can be an integral operator defined from 0 to x. And  0 is the kernel of the inverse operator L -1 and satisfies L 0 = 0.
According to the standard Adomian modified decomposition method [11][12][13], it defines the solution u(x) by an infinite series of the form where the unknown coefficients C m will be determined recurrently. And for the non-linear term Nu(x), it defines [11] (5) where the A m are the classical Adomian's polynomials [1,14]. Notice that g(x) in equation (1) is also required to decompose as an infinite series Substituting equations (4), (5) and (6) into equation (3), we obtain The recurrence relation with respect to initial condition is applied to determine the coefficient C k in equation (7). However, in practice, all series in equation (7) cannot be obtained exactly. The solution of u(x) is expressed by a truncated series and will be rewritten as

Aftertreatment technique
It is well-known that the main disadvantage of the AMDM is that solution's series may only have very small convergence radius and the truncated series solution may be inaccurate in some regions [12].

ICPMS2018
IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1053 (2018) 012019 doi :10.1088/1742-6596/1053/1/012019 The main goal of this paper is to extend the convergence radius of the AMDM with a new proposed aftertreatment technique.
Assume that the truncated series solution in equation (8) can be expressed as another series in terms of the independent sine and cosine trigonometric functions: sin cos (9) where S k , W k , p k , and q k are the unknown values. Notice that the both side of equation (9) can be expanded as a power series, yields From equation (10), it can be found that the unknown values S k , W k , p k , and q k in equation (9) can be determined by the following nonlinear algebraic equations, such as By using the MATLAB functions sovle or MATHEMATICA function Nsolve, equation (11) can be directly solved. Then the solution u(x) in equation (9) can be obtained.

Numerical calculations
In order to verify the proposed aftertreatment technique for the AMDM, two numerical examples will be discussed in this section. 4.1. Example 1: Consider a nonlinear differential equation (12) with the initial conditions (13) According to AMDM presented in Section 2, the solution y(t) and nonlinear term [y(t)] 3  The Adomian's polynomials for nonlinear term [y(t)] 3 can be found in Ref. [14], and the first several terms are listed below for the convenience of the reader.  3 To determine the unknown coefficients C m , a linear operator Finally, using equations (13), (15) and (17), the coefficients C m in equation (11) can be determined by using the following recurrence relations Substituting the solved C m into equation (11) and solving the system of nonlinear algebraic equations, the values S k , W k , p k , and q k in equation (11) can be determined. In this case, we set the series summation limit M = 7 and K = 2 in equations (8) and (9), respectively. Based on proposed aftertreatment technique, the solution for equation (12) can be written as In order to check the accuracy of the proposed aftertreatment technique, the numerical result based on the four order Runge-Kutta solution (RK4) is also calculated. Figure 1 shows the comparisons between the AMDM solution, the AMDM with aftertreatment solution and the RK4 solution of equation (12). From figure 1, it can be found that the AMDM solution diverges rapidly when t > 0.8. However, the AMDM with aftertreatment solution shows a good convergence in comparison with the RK4 solution for t < 10. If more terms of C m are used for aftertreatment, the accuracy of the solution can be improved. Figure 2 shows the solution when M = 11 and K = 3 in equations (8) and (9) are used. It illustrates that the AMDM with aftertreatment solution coincides with RK4 solution for t <30.
For the sake of simplicity, the series summation limit M and K in equations (8) and (9) will be simply truncated to M = 7 and K = 2 in all the subsequent calculations.