scholarly journals A Spectral Collocation Method for Systems of Singularly Perturbed Boundary Value Problems

2017 ◽  
Vol 108 ◽  
pp. 725-734
Author(s):  
N. Sharp ◽  
Manfred Trummer
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Spectral Collocation Method ◽  
Spectral Collocation

scholarly journals Rational Spectral Collocation Combined with the Singularity Separated Method for a System of Singularly Perturbed Boundary Value Problems

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Lufeng Yang
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Numerical Experiments ◽  
Original Problem ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Barycentric Form ◽  
Theoretical Results

A novel rational spectral collocation method is presented combined with the singularity-separated technique for a system of singularly perturbed boundary value problems. The solution is expressed as u=w+v, where w is the solution of the corresponding auxiliary boundary value problem and v is a singular correction with explicit expressions. The rational spectral collocation method in barycentric form with the sinh transformation is applied to solve the auxiliary third boundary problem. The parameters of the singular correction can be determined by the boundary conditions of the original problem. Numerical experiments are carried out to support theoretical results and provide a favorable comparison with research results of other work.

scholarly journals The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Lufeng Yang
Keyword(s):  
Boundary Value Problem ◽  
Collocation Method ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Spectral Collocation Method ◽  
The Novel ◽  
Spectral Collocation ◽  
Nonsmooth Data

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. A novel rational spectral collocation combined with a singularity-separated method for this problem is presented. The solution is expressed as u = w + v , where w is the solution of corresponding auxiliary boundary value problem and v is a singular correction with direct expressions. The rational spectral collocation method combined with a sinh transformation is applied to solve the weakened singularly boundary value problem. According to the asymptotic analysis, the sinh transformation parameters can be determined by the width and position of the boundary layers. The parameters in the singular correction can be determined by the boundary conditions of the original problem. Numerical experiment supports theoretical results and shows that compared with previous research results, the novel method has the advantages of a high computational accuracy in singularly perturbed reaction-diffusion problems with nonsmooth data.

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