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A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer

Süleyman Cengizci1, Mehmet Tarık Atay2, Aytekin Eryılmaz3

  • 1Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey.

Springerplus
|March 24, 2016
PubMed
Summary

This study presents an efficient Successive Complementary Expansion Method (SCEM) for singularly perturbed nonlinear ordinary differential equations with a single boundary layer. The SCEM provides accurate, uniformly valid approximations, outperforming existing techniques.

Keywords:
Asymptotic expansionBoundary layerSingular perturbationSuccessive Complementary Expansion MethodUniformly valid approximation

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Area of Science:

  • Applied Mathematics
  • Numerical Analysis
  • Differential Equations

Background:

  • Singularly perturbed nonlinear ordinary differential equations present challenges in numerical solutions.
  • Solutions often exhibit boundary layers, complicating approximation accuracy.
  • Existing methods may lack uniform validity or ease of application.

Purpose of the Study:

  • To introduce and validate the Successive Complementary Expansion Method (SCEM) for a specific class of problems.
  • To obtain uniformly valid approximate solutions for singularly perturbed nonlinear ODEs with a single boundary layer.
  • To demonstrate the efficiency and accuracy of SCEM compared to existing methods.

Main Methods:

  • Application of the Successive Complementary Expansion Method (SCEM).
  • Analysis of two-point boundary value problems for singularly perturbed nonlinear ODEs.
  • Numerical validation using four distinct test problems.

Main Results:

  • SCEM successfully generates uniformly valid approximate solutions.
  • Numerical results show excellent agreement with exact and literature solutions.
  • The method is effective for problems with a single boundary layer.

Conclusions:

  • SCEM is a highly effective and accurate method for the studied differential equations.
  • The proposed method offers advantages in ease of application and overall effectiveness.
  • SCEM provides a superior alternative to existing numerical techniques for these problems.