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The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law...
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An effective numerical method to solve a class of nonlinear singular boundary value problems using improved

Lie-Jun Xie1, Cai-Lian Zhou1, Song Xu1

  • 1Department of Mathematics, Ningbo University, Fenghua Road 818, Jiangbei District, Ningbo City, 315211 Zhejiang Province People's Republic of China.

Springerplus
|July 28, 2016
PubMed
Summary
This summary is machine-generated.

An improved differential transform method (IDTM) effectively solves singular boundary value problems in physics. This method uses Adomian polynomials for nonlinearities, providing a convergent series solution with error estimation.

Keywords:
Adomian polynomialsApproximate series solutionsDifferential transform methodImproved differential transform methodSingular boundary value problem

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Area of Science:

  • Numerical Analysis
  • Applied Mathematics
  • Mathematical Physics

Background:

  • Singular boundary value problems (SBVPs) are prevalent in diverse physical phenomena.
  • Existing numerical methods often struggle with the singularities and nonlinearities inherent in SBVPs.
  • The differential transform method (DTM) offers a framework for solving differential equations, but requires enhancements for complex problems.

Purpose of the Study:

  • To develop an effective numerical technique for solving a class of singular boundary value problems.
  • To introduce an enhancement to the differential transform method (DTM) for improved accuracy and applicability.
  • To provide a reliable method for analyzing physical models governed by SBVPs.

Main Methods:

  • The improved differential transform method (IDTM) is proposed, building upon the standard DTM.
  • Adomian polynomials are integrated into the IDTM to accurately handle nonlinear terms within the differential equations.
  • A straightforward formula is established to relate Adomian polynomials to the coefficients of the series solution, simplifying the derivation process.

Main Results:

  • The IDTM successfully generates approximate solutions in the form of convergent series for SBVPs.
  • An analytical upper bound for estimating the approximate error of the obtained solutions is presented.
  • The method's validity and applicability are demonstrated through several illustrative examples from various physical domains.

Conclusions:

  • The IDTM is a robust and efficient numerical method for tackling singular boundary value problems.
  • The integration of Adomian polynomials significantly enhances the DTM's capability in handling nonlinearities.
  • The proposed method offers a valuable alternative to existing techniques, with demonstrated accuracy and applicability in physical modeling.