Rational spectral collocation and differential evolution algorithms for singularly perturbed problems with an interior layer. (English) Zbl 1460.65094
Summary: We are concerned with using high accuracy numerical methods for solving singularly perturbed problems whose solutions exhibit an interior layer. A rational spectral collocation method in barycentric form with sinh transformation is firstly applied to find the approximate solution to the given problem. The sinh transformation consists of two parameters: the location and the width of the interior layer. With the tool of the asymptotic analysis, the interior layer can be easily located. Then, in determining the width of the interior layer, we design a nonlinear unconstrained optimization problem which is solved by using the basic differential evolution (DE) algorithm. The feasibility and effectiveness of the proposed method is verified by numerical results. The proposed method can also be applied to other singularly perturbed problems.
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L11 | Numerical solution of singularly perturbed problems involving ordinary differential equations |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
Keywords:
singularly perturbed problems; rational spectral collocation method; optimization problem; differential evolution algorithmSoftware:
CMA-ESReferences:
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