Spline finite difference methods for singular two point boundary value problems

Abstract

Summary. In this paper we discuss the construction of a spline function for a class of singular two-point boundary value problem x-'(x'u')'=f (x,u), u(0)=A, u(1)=B, 0<c~< 1 or ~= 1, 2. The boundary conditions may also be of the form u'(0)=0, u(1)=B. Three point finite difference methods, using the above splines, are obtained for the solution of the boundary value problem. These methods are of second order and are illustrated by four numerical examples.

Computational technique for singular boundary value problems by using Numerical Quadrature

“…The maximum absolute errors in the results obtained are given in Tables 1 and 2. The results obtained by our method are superior as compared to the results of [3], [6], and [7]. The results also show the second order convergence of the method.…”

Non polynomial splines and weakly singular two-point boundary value problems

“…In the present paper we shall consider an application of simple nonpolynomial splines to a numerical solution of a weakly singular two-point boundary value problem: (1) x-~(x~y') ' ./(x,y); 0 < x 5 1 with boundary conditions The spline s satisfies the homogeneous differential equation: (4) {x-=(x=s')'}'(x) = O; xj < x < xj,~ i.e., x-=(x's'y(x) is a step function defined on a uniform partition of [0,1]: 0 = x0 <xl <"" < x. = 1.…”

“…The condition 8ffl3y > 0 on [(3, 1) x (-o0, oo) in [I]- [4] required to insure the existence and the convergence of the approximate solution could be replaced with a weaker on the existence of the isolated solution, while the condition 8f/Oy > 0 is sufficient for the existence of the isolated solution. …”

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

“…The associated boundary conditions are given by (4). For = 0, 1, or 2, the differential equation (18) shows planar, cylindrical, or spherical geometries (see, [25,26]). …”