Numerov's Method for a Class of Nonlinear Multipoint Boundary Value Problems

The purpose of this paper is to give a numerical treatment for a class of nonlinear multipoint boundary value problems. The multipoint boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition. The problems are discretized by the fourth-order Numerov's method. The existence and uniqueness of the numerical solution are investigated by the method of upper and lower solutions. The convergence and the fourth-order accuracy of the method are proved. An accelerated monotone iterative algorithm with the quadratic rate of convergence is developed for solving the resulting nonlinear discrete problems. Some applications and numerical results are given to demonstrate the high efficiency of the approach.

Finite Difference Method for the Reverse Parabolic Problem

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example.

SUCCESSIVE ITERATION AND POSITIVE SOLUTIONS FOR A p-LAPLACIAN MULTIPOINT BOUNDARY VALUE PROBLEM

In this paper, we study the existence of positive solutions for the one-dimensional p-Laplacian differential equation, subject to the multipoint boundary condition by applying a monotone iterative method.