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
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.
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.
Keywords:
Asymptotic expansionBoundary layerSingular perturbationSuccessive Complementary Expansion MethodUniformly valid approximationMore Related Videos
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