Further developments to the decomposition method for solving singular initial-value problems

Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods

“…For Examples 4.2 and 4.3, it is difficult to solve them directly using the existing method such as the modified decomposition method, the variational iteration method and the homotopy perturbation method [6][7][8][9][10][11]. However, the present method can solve them successfully.…”

“…The main idea of this paper is to present an algorithm for computing the solutions of singular initial value problems including Lane-Emden-type equations of the form (1.1) where α, β, k 1 , k 2 are real constants, a(x), b(x) are continuous and maybe a(0) = 0, b(0) = 0, f (x, y) is a continuous real valued function, and g(x) ∈ C (0, 1], i.e., the case when the function g(x) may be undefined at the origin. Such problems have attracted much attention and have been studied by many authors [3][4][5][6][7][8][9][10][11][12]. There is considerable interest in numerical methods on singular initial value problems.…”

“…In these references, the authors selected g(x) as a well-defined function at the origin. In [11], Aslanov and Abu-Alshaikh introduced a new Adomian decomposition method for solving (1.1) with k 1 = 2, a(x) = x, k 2 = 0 and g(x) ∈ C (0, 1].…”

The Homotopy Perturbation Method for Solving Singular Initial Value Problems

“…Now, we consider the following singular initial value problem [5], By applying the procedures of Problem 1, and assuming the solution as in Eq. (6), we have Λ (*)=!> Some numerical results related to the problem under consideration are shown in Table The homotopy perturbation method (HPM) was applied to singular initial value problems in the second-order ordinary differential equations.…”