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Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

Franklin, Joel N. and Rodemich, Eugene R. (1968) Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation. SIAM Journal on Numerical Analysis, 5 (4). pp. 680-716. ISSN 0036-1429. http://resolver.caltech.edu/CaltechAUTHORS:FRAsiamjna68

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Abstract

G. Fichera [1] and other authors have investigated partial differential equations of the form [Eq. 1.1] in which the matrix (aij(x)) is required to be semidefinite. Equations of this type occur in the theory of random processes. A numerical analysis of some equations of this type has been by Cannon and Hill [9]. In this paper we consider a particular boundary value problem [Eq. 1.2] where we require [Eq. 1.3] and [Eq. 1.4]. A problem of this sort was discussed analytically by W. Fleming [2], but he did not obtain an explicit solution for T(x,0). The solution T(x,y) is related to a randomly-accelerated particle whose position ξ(t) satisfies the stochastic differential equation [Eq. 1.5] where w(t) is white Gaussian noise. If the initial position and velocity are ξ(0) = x and ξ'(0) = y, where |x| < 1, then T(x,y) is the expected value of the first time at which the position ξ(t) equals ±1. We obtain an analytic solution for T(x,y) in terms of hypergeometric functions and confluent hypergeometric functions. We use this analytic solution to test the validity of numerical methods which are applicable to general elliptic-parabolic equations (1.1). We show that, even though the truncation error for the difference equations does not tend to zero, nevertheless the difference methods give convergence of the difference methods. Each difference method requires the solution of a large number of simultaneous linear difference equations. We give iterative methods for solving these equations, and we prove that the iterations converge.


Item Type:Article
Additional Information:©1968 Society for Industrial and Applied Mathematics. Received by the editors May 23, 1968. The work of this author [E.R.R.] was supported by the National Aeronautics and Space Administration.
Record Number:CaltechAUTHORS:FRAsiamjna68
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:FRAsiamjna68
Alternative URL:http://dx.doi.org/10.1137/0705054
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10264
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:21 Apr 2008
Last Modified:26 Dec 2012 09:58

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