An effective numerical method to solve a class of nonlinear singular boundary value problems using improved differential transform method

TL;DR

This paper introduces an improved differential transform method (IDTM) for efficiently solving nonlinear singular boundary value problems, providing a convergent series solution with error bounds, validated through physical examples.

Contribution

The paper develops a novel IDTM that uses Adomian polynomials to handle nonlinearities, offering a simple formula for approximate solutions and demonstrating its effectiveness on physical models.

Findings

The method produces accurate approximate solutions.

It provides an explicit error estimation.

Comparisons show it outperforms existing methods.

Abstract

In this work, an effective numerical method is developed to solve a class of singular boundary value problems arising in various physical models by using the improved differential transform method (IDTM). The IDTM applies the Adomian polynomials to handle the differential transforms of the nonlinearities arising in the given differential equation. The relation between the Adomian polynomials of those nonlinear functions and the coefficients of unknown truncated series solution is given by a simple formula, through which one can easily deduce the approximate solution which takes the form of a convergent series. An upper bound for the estimation of approximate error is presented. Several physical problems are discussed as illustrative examples to testify the validity and applicability of the proposed method. Comparisons are made between the present method and the other existing methods.