Finite difference methods and their convergence for a class of singular two point boundary value problems

Abstract

Summary. We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x~y')'=f(x,y), y(0)=A, y(1)=B, 0<c~<l. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all c~e(0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over 1-0, 1], we obtain a method M1 based on just one evaluation off For uniform mesh we obtain two methods M 2 and M 3 each based on three evaluations off For c~=0, M 1 and M 2 both reduce to the classical second-order method based on one evaluation of f. These three methods are investigated, their O(hZ)-convergence established and illustrated by numerical examples.

Spline finite difference methods for singular two point boundary value problems

“…The results obtained by our method are superior as compared to the results of [1]. The results also show the second order convergence of the method (5).…”

“…Recently [1] have proposed three point difference methods of second order under appropriate conditions using the boundedness of f', f" and f'". For ~= 1, Russell and Shampine [9] wrote the differential equation in the form (x u')' + xf (x, u) = 0 and considered the discretization…”

An application of Chawla's identity to a different scheme for singular problems

“…We consider an application of our difference scheme for a numerical solution of the following examples in [2]. EXAMPLE 1.…”

Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method

“…By using FDM, equations with variable coefficients and even nonlinear problem can be easily solved. There are many studies about FDM applications for different engineering problems in literature (Forsythe and Wasow, 1960;Chapel and Smith, 1968;Chawla and Katti, 1982;Strikwerda 1990;Cocchi and Cappello, 1990;Thomee, 1990; In FDM approach, derivatives in the partial differential equation are approximated by linear combinations of function values at the grid points and are expressed as difference functions. The general difference representations of differential equations are as follows:…”