On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems
TL;DR: In this paper, the authors consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin, and according to the nature of this singularity, they must consider either the two-point boundary-value problem or the onepoint boundary value problem.
Abstract: 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.
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TL;DR: In this paper, the application of different difference schemes (box, trapezoidal, Euler and backward Euler) to numerical solution of boundary value problems for nonlinear first order systems of ordinary differential equations with a singularity of the first kind is examined.
Abstract: 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.
112 citations
TL;DR: 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.
Abstract: 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.
100 citations
TL;DR: In this paper, the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems is discussed, and three possibilities are investigated, their O(h2)-convergence established and illustrated by numerical examples.
Abstract: 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
93 citations
TL;DR: This paper has presented a method based on cubic splines for solving a class of singular two-point boundary value problems and the tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm.
90 citations
TL;DR: In this paper, a fourth order finite difference method for singular two-point boundary value problems was given for all α ≥ 1, where α is the number of points in the boundary value.
Abstract: 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.
85 citations
Cites background from "On the convergence of finite-differ..."
...these papers. Ciarlet et al. [7] and Jamet [ 10 ] had discussed numerical...
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01 Jan 1965-Journal of The Society for Industrial and Applied Mathematics, Series B: Numerical Analysis
TL;DR: A generalized axially symmetric potential (GASP) in n = k + 2 "dimensions" has been studied in this article, where the GASP is defined as a harmonic function in ndimensions having certain symmetry properties.
Abstract: is called a generalized axially symmetric potential (GASP) in n = k + 2 "dimensions". Many authors have studied these functions [9], [10], [17], [27]. When fc is a positive integer, u(x, y) is a harmonic function in n dimen? sions having certain symmetry properties. For example, when k = 1, u(z, r) is an axially symmetric harmonic function in cylindrical coordinates. Several authors [4], [8], [12], [13] have studied numerical methods for this
30 citations
12 citations
TL;DR: In this paper, the authors studied finite-difference methods for elliptic differential equations of the second order whose coefficients are singular on a portion of the boundary; the uniform convergence of the approximations and the existence of a solution of the Dirichlet problem were proved for a class of such equations.
Abstract: I. Introduction. In a previous paper [6], S. V. Parter and the author have studied finite-difference methods for elliptic differential equations of the second order whose coefficients are singular on a portion of the boundary; the uniform convergence of the approximations and the existence of a solution of the Dirichlet problem were proved for a class of such equations. The present work is an extension of those results to parabolic initial boundary-value problems. The class of problems that we consider includes the cases of nonhomogeneous differential equations, of time-dependent coefficients, of time-dependent domains and of over-determined Dirichlet problems.
7 citations