The Adomian Decomposition Method with Discretization for Second Order Initial Value Problems

In this paper, Adomian Decomposition Method with Discretization (ADMD) is applied to solve both linear and nonlinear initial value problems (IVP). Comparison with Adomian Decomposition Method (ADM) is presented. To illustrate the efficiency and accuracy of the method, five examples are considered. The result shows that ADMD is more efficient and accurate than ADM.


The Adomian Decomposition Method with Discretization for Second Order Initial Value Problems
Dagnachew Mengstie Tefera and Awoke Andargie Abstract-In this paper, Adomian Decomposition Method with Discretization (ADMD) is applied to solve both linear and nonlinear initial value problems (IVP). Comparison with Adomian Decomposition Method (ADM) is presented. To illustrate the efficiency and accuracy of the method, five examples are considered. The result shows that ADMD is more efficient and accurate than ADM.
Index Terms-Decomposition method, Adomian polynomial, initial value problems, infinite series.

I. INTRODUCTION
A DOMIAN decomposition method (ADM) [1], [2], which was first introduced by the American Physicist George Adomian, has been used to solve effectively and easily a large class of linear and nonlinear ordinary and partial differential equations. This method generates a solution in the form of a series whose terms are determined by a recursive relationship using the Adomian polynomials [3], [4].
The non-linear problems are solved easily and elegantly without linearizing the problem by using ADM. It also avoids the linearization, perturbation and discretization unlike other classical techniques [5], [6]. The main advantage of this method is that it can be applied directly to all types of differential and integral equations, either linear or non-linear, homogeneous or inhomogeneous, with constant or variable coefficients. Another important advantage is that, the method is capable of greatly reducing the size of computational works while still maintaining high accuracy of the numerical solution [7]. The decomposition method produced reliable results with less iteration, than the Taylor series method and the Runge-Kutta methods [8], [9]. The convergence of the ADM has been investigated by a number of authors [10], [11].
In this study, we applied the ADMD to solve the initial value problem of linear and non-linear second order ODE. ADMD differs from ADM, i.e. it divides the interval into a finite number of subintervals and for each subinterval it generates a solution in the form of a series whose terms are determined by a recursive relationship using the Adomian polynomials. The results show that the ADMD is more accurate and suitable solution than the ADM. Also, if the order of the differential equation increases, the solution using ADMD is better than ADM.

II. ADOMIAN DECOMPOSITION METHOD
Consider the following second order ordinary differential equation Ly where L = d 2 dx 2 is a linear operator, R is the remaining linear lower order derivative, N is a nonlinear operator and g is any function. Integrating (1) yields for the initial value problem, where L −1 (·) = x 0 x 0 (·)dtdt. The Adomian Decomposition Method assumes that the unknown function y can be expressed by infinite series of the form y = sum ∞ n=0 y n .
The ADM assumes that the nonlinear operator N(y) can be decomposed by an infinite series of polynomial given by where A n are the Adomian's polynomials defined as A n = A n (y 0 , y 1 , y 2 , . . . , y n ). Substituting (3) and (4) into equation (2) and using the fact that R is a linear operator, we obtain ∞ ∑ n=0 y n = y(0)+xy (0)+L −1 g−L −1 (5) Therefore, the formal recurrence algorithm could be defined by The Adomian polynomial A n was first introduced by Adomian himself. It was defined via the following general formula The first few Adomian polynomials are III. ADOMIAN DECOMPOSITION METHOD WITH DISCRETIZATION Consider the following second-order ordinary differential equation Ly + Ry + Ny = g(x) (9) where L = d 2 dx 2 is a linear operator, R is the remaining linear lower order derivative, N is a nonlinear operator and g is any function.
ADMD divides the working interval ADMD assumes that the unknown function can be expressed by infinite series and the ADMD assumes that the nonlinear operator N(y i ) can be decomposed by an infinite series of polynomial given by where A n,i are the Adomian's polynomials defined as A n,i = A n,i (y 0,i , y 1,i , y 2,i , . . . , y n,i ). Substituting (3) and (4) into equation (2) and using the fact that R is a linear operator, we obtain for i = 0, 1, 2, . . . , m − 1. Therefore, the formal recurrence algorithm could be defined by The Adomian polynomial A n,i was first introduced by Adomian himself. It was defined via the following general formula The exact solution is y = sin x. By using linear operator L, (17) can be written as Ly − y = 0 with y(0) = 0 and y (0) = 1 By using ADM, we apply L −1 to both sides of (18) and using initial condition, we obtain y = y(0) + xy (0) + L −1 (y) The first five terms of the series are , y 4 = x 9 362880 More components in the decomposition series can be calculated to enhance the accuracy of the approximation. By computing five terms of the solution series, we obtain y = 4 ∑ n=0 y n = x + x 3 6 + x 5 120 + x 7 5040 + x 9 362880 for 0 ≤ x ≤ 1.

Upon using the decomposition series for the solution is
This leads to the following recursive equation Thus on the first interval, we select α = 0 to obtain y 0,0 = x y n+1,0 = The result of ADM and ADMD are illustrated in Table I  By applying L −1 to both sides in (18) and by using the initial condition, we obtain y = λ L −1 (e y ) By using the ADM, we assume y and nonlinear term as infinite series given by where A n is Adomian polynomial representation for the nonlinear term e y gives By using ADMD, We divide the interval [0, 1] into m subintervals h = x i+1 − x i for i = 0, 1, 2, . . . , m − 1 where x 0 = 0 and x m = 1. We apply L −1 to both sides in (20) and by using the initial condition, we obtain Using the decomposition method assumes y and nonlinear term as infinite series, given by A n for i = 0, 1, 2, . . . , m − 1. This leads to the following recursive relation where A n,i is the Adomian polynomial representation for the nonlinear term e y , gives  Table 2 below with five terms of the series for λ = 1 and λ = 2 at h = 0.25. Clearly, ADMD is more accurate than ADM and the accuracy of the ADM and ADMD increased for Bratu's equation if λ is small and choosing a small step size or by adding more terms of the solution series.  Table III below with five terms of the series at h = 0.1 and h = 0.25. Clearly, ADMD is more accurate than ADM. Figure 3 shows the comparison between ADM and ADMD at h = 0.1 and h = 0.25.      Table IV below with five terms of the series at h = 0.25. Clearly, ADMD is more accurate than ADM. Figure 4 shows the comparison between ADM and ADMD at h = 0.25. e) Example 5: Consider the following non-homogeneous linear differential equation y − y = 2 with y(0) = 0 and y (0) = 0 The exact solution is y(x) = e x +e −x −2. The result of ADM and ADMD are illustrated in Table V below with five terms of the series at h = 0.1 and h = 0.25. Clearly, ADMD is more accurate than ADM. Figure 5 shows the comparison between ADM and ADMD at h = 0.1 and h = 0.25.

IV. CONCLUSIONS
In this paper, the comparison ADM and ADMD methods for solving second order linear and nonlinear initial value problems is presented. The numerical efficiency of the ADM and ADMD methods is tested by considering five examples. The results show that the ADMD is more effective and accurate than ADM. The accuracy of the ADMD can be improved by taking small step size or by adding more terms of the solution series.