Numerical methods for singular boundary value problems involving the p-laplacian

Abstract

In this work, we are concerned about a singidar boundary value problem for a nonlinear ordinary differential equation involving the N-dimensional p-laplacian. This equation may be considered as a generalized Emden-Fowler equation and arises in models of fliud mechanics, elasticity theory and other fields of physics. The main feature of the considered boundary value problem is that it has two singidarities at the endpoints of the considered interval.We analyze the asymptotic behavior of the solutions near these singidarities and propose computational methods that take this behavior into consideration. Numerical examples are presented and dicussed.

Finite difference solution of a singular boundary value problem for the p-Laplace operator

“…As it was shown in [19], in this case the function h, defined by (7), is a lower solution of (3) and (4) if…”

“…In [19] we have extended the results obtained in [17] to the multidimensional case (N > 1). Upper and lower solutions were obtained for this case and stable shooting methods were applied to its solution.…”

“…These solutions expansions where used to implement a shooting method that provided very accurate results and also to determine upper and lower solutions. Let us resume the main results of [19] concerning the singular behavior of the solution in the neighborhood of the endpoints. Theorem 1 If p > 1, σ > −1, a < 0, n < −1 and N ≥ 1, the following singular initial value problem:…”

“…In [19] we have determined one-parameter families of solutions of suitable initial value problems, describing the solution in the neighborhood of the singular points. These solutions expansions where used to implement a shooting method that provided very accurate results and also to determine upper and lower solutions.…”

“…The different problems considered in [17][18][19] and [20] have one common feature: the equation (3) has at least one singularity at one of the endpoints r = 0 or r = 1. In all the cited papers, the asymptotic behavior of the solutions of (3) was analyzed near each singularity and this information was used to reduce the boundary condition at the singular point to an equivalent boundary condition in a neighbor point.…”