The application of certain difference schemes (box, trapezoidal, Euler and backward Euler) to the numerical solution of boundary value problems for nonlinear first order systems of ordinary differential equations with a singularity of the first kind is examined. The solution of the linear eigenvalue problem is also considered.

Difference Methods for Boundary Value Problems with a Singularity of the First Kind

In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. This novel approach possesses main advantages; it can be applied without any limitation on the nature of the problem, the type of singularity, and the number of mesh points. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the presented technique. The results reveal that the algorithm is very effective, straightforward, and simple.

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Solving Singular Two-Point Boundary Value Problems Using Continuous Genetic Algorithm

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<?<1. 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 ??(0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM1 based on just one evaluation off. For uniform mesh we obtain two methodsM2 andM3 each based on three evaluations off. For ?=0,M1 andM2 both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h2)-convergence established and illustrated by numerical examples.

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

Cubic spline for a class of singular two-point boundary value problems

Recently, Chawla et al. described a second order finite difference method for the class of singular two-point boundary value problems:
 
$$y'' + (\alpha /x)y' + f(x,y) = 0, 0< x< 1, y'(0) = 0, y(1) = A, \alpha \geqslant 1.$$

 No higher order finite difference method has been given so far. In the present paper we give a fourth order finite difference method for all α ≥ 1.

A fourth order method for a singular two-point boundary value problem

Consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin; according to the nature of this singularity we must consider either the two-point boundary-value problem or the one-point boundary value problem. Finite-difference schemes are studied; results are given concerning error analysis and monotone convergence.

Pierre Jamet

Papers

On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems