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Difference Methods for Boundary Value Problems in Ordinary Differential Equations

Keller, H. B. and White, A. B., Jr. (1975) Difference Methods for Boundary Value Problems in Ordinary Differential Equations. SIAM Journal on Numerical Analysis, 12 (5). pp. 791-802. ISSN 0036-1429. http://resolver.caltech.edu/CaltechAUTHORS:20120809-094752016

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Abstract

A general theory of difference methods for problems of the form Ny ≡ y' - f(t,y) = O, a ≦ t ≦ b, g(y(a),y(b))= 0, is developed. On nonuniform nets, t_0 = a, t_j = t_(j-1) + h_j, 1 ≦ j ≦ J, t_J = b, schemes of the form N_(h)u_j = G_j(u_0,•••,u_J) = 0, 1 ≦ j ≦ J, g(u_0,u_J) = 0 are considered. For linear problems with unique solutions, it is shown that the difference scheme is stable and consistent for the boundary value problem if and only if, upon replacing the boundary conditions by an initial condition, the resulting scheme is stable and consistent for the initial value problem. For isolated solutions of the nonlinear problem, it is shown that the difference scheme has a unique solution converging to the exact solution if (i) the linearized difference equations are stable and consistent for the linearized initial value problem, (ii) the linearized difference operator is Lipschitz continuous, (iii) the nonlinear difference equations are consistent with the nonlinear differential equation. Newton’s method is shown to be valid, with quadratic convergence, for computing the numerical solution.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1137/0712059DOIUNSPECIFIED
http://epubs.siam.org/doi/abs/10.1137/0712059PublisherUNSPECIFIED
Additional Information:© 1975 Society for Industrial and Applied Mathematics. Received by the editors July 24, 1974. This work was supported by the U.S. Atomic Energy Commission under Contract AT(04-3-767), Project Agreement 12.
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Funding AgencyGrant Number
Atomic Energy CommissionAT(04-3-767)
Record Number:CaltechAUTHORS:20120809-094752016
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120809-094752016
Official Citation:Difference Methods for Boundary Value Problems in Ordinary Differential Equations Keller, H. and White, Jr., A. SIAM Journal on Numerical Analysis 1975 12:5, 791-802
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:33044
Collection:CaltechAUTHORS
Deposited By: Aucoeur Ngo
Deposited On:09 Aug 2012 17:43
Last Modified:26 Dec 2012 15:54

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