Modified Differential Transform Method for Two Singular Boundary Values Problems

Abstract

This paper deals with the two singular boundary values problems of second order. Two singular points are both boundary values points of the differential equation. The numerical solutions are developed by modified differential transform method (DTM) for expanded point. Linear and nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, we give the convergence of this new method.

Large Interval Solution of the Emden–fowler Equation Using a Modified Adomian Decomposition Method With An integrating factor

“…We recommend a new reliable and efficient numerical method, the ADM with an integrating factor, for solving Emden-Fowler equation. Because of the singularity, this m ||R|| in [0, 6] m ||R|| in [6,10] m ||R|| in [10,12] type of equation cannot be solved by the traditional ADM. Our numerical results show that only a few terms of a series solution are enough to achieve very high accuracy.…”

“…Again (2.15) is not used due to its long computing time. We choose a = 0.1 in this case, and first solve the solution in [0, 6], then extend it to the intervals [6,10] and [10,12]. The maximal errors of residual R(x) in these three intervals are listed in Table 5.…”

“…For example, Guo et al [6] used a fixed point theorem to obtain such existence and uniqueness. Recently, much attention has been paid to the computation of their numerical solutions [2,12,[17][18][19]. Chowdhury and Hashim [2], for example, used a homotopy asymptotic method to find their approximate solutions.…”

A new implicit-explicit local differential method for boundary value problems

“…Thus, DTM based methods for BVPs can be considered to be shooting methods [25][26][27][28][29][30][31][32]. Various kinds of BVPs were solved by DTMs in literature [25][26][27][28][29][30][31][32]. Generally, those semi-analytic approaches give good agreement with exact solution only about one of the boundaries due to the structure of the Taylor series expansion.…”