A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method

Abstract

a b s t r a c tBased on the Adomian decomposition method, a new analytical and numerical treatment is introduced in this research to investigate linear and non-linear singular two-point BVPs. The effectiveness of the proposed approach is verified by several linear and non-linear examples.

An approach for solving singular two point boundary value problems: analytical and numerical treatment

“…Table 1 compares the absolute errors obtained by Ebaid, Cui and Geng [13,22], and the modified decomposition method proposed in this contribution at s = 1 and s = 3 , respectively. Importantly, this table shows the accuracy increases with increasing s, especially at s = 3 in which a very small absolute error is obtained, where the error is assumed to be zero for an exponent less than −16.…”

“…The present work focuses on creating a canonical form of recursive relation which contains all the boundary conditions so that the zeroth component is explicitly determined without additional computations, as are all other components. The method proposed in this contribution may be considered as a further extension of the method by Ebaid [13] and provides a more accurate approximation than previous methods [8,13,22]. Several numerical examples illustrate the accuracy of the proposed method, and the numerical results obtained by it and other efficient methods are presented and compared.…”

“…The convergence of the decomposition series has been investigated by several researchers [11,12]. In this work, we obtain E23 approximate solutions of the singular two-point boundary value problems [13] u (x) + p(x)u (x) + q(x)f(u(x)) = r(x), x ∈ (a, b),…”

“…Various efficient numerical methods have been proposed to solve this type of problem [10,13,16]. These methods include the homotopy analysis method [17], differential transformation method [16], modified adm [18,19,20,21], and an improved adm [13].…”

“…Accordingly, it is desirable to have a modification of the decomposition method which explicitly determines the zeroth component of the solution. One such method was suggested by Ebaid [13], which is based on the adm and a modification of Lesnic's work [4] to create a canonical form of recursive relation with an E24 explicitly determined zeroth component. The efficiency and reliability of this method has been well demonstrated in various singular problems.…”