A second order spline finite difference method for singular two-point boundary value problems. (English) Zbl 1034.65061
A second order spline finite difference method on a non-uniform mesh for problems of the form
\[
x^{-\alpha}(x^{\alpha}u')'=f(x,u), 0<x\leq 1, u(0)=A, u(1)=B
\]
with \(0<\alpha<1\) is presented. Assumptions on \(f\) are continuity on \([0,1]\times \mathbb{R}\), \(\partial f/ \partial u\) exists, is continuous and \(\geq 0\). A spline with the three-point finite difference method is constructed for a non-uniform mesh. \(O(h^2)\) convergence under appropriate conditions is shown. Two numerical examples are presented.
Reviewer: Eugene L. Allgower (Fort Collins)
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
Keywords:
singular two-point boundary value problems; spline solution; finite difference method; non-uniform mesh; convergence; numerical examplesReferences:
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