Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer. (English) Zbl 1093.65075
Singularly perturbed two-point boundary value problems (BVPs) of convection-diffusion type for fourth-order ordinary differential equations (ODEs) with a small positive parameter \(\varepsilon\) multiplying the highest derivative with a discontinuous source term are considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter \(\epsilon\) multiplying the highest derivative, and suitable boundary conditions.
In this paper, a computational method for solving this system is presented. In this method the authors first find a zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by a numerical method which is constructed for this problem and which involves a Shishkin mesh.
In this paper, a computational method for solving this system is presented. In this method the authors first find a zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by a numerical method which is constructed for this problem and which involves a Shishkin mesh.
Reviewer: Fuhua Ling (Milpitas)
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
Keywords:
Fourth-order ordinary differential equation; Singularly perturbed problem; Discontinuous source term; Non-self-adjoint boundary value problem; asymptotic expansion; Boundary layer; Interior layer; Finite difference scheme; singular perturbation; convection-diffusion equation; two-point boundary value problemsReferences:
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