# The Numerical Solution of a Non-Characteristic Cauchy Probelm for a Parabolic Equation

• Lars Eldén
Chapter
Part of the Progress in Scientific Computing book series (PSC, volume 2)

## Abstract

We study the problem of solving numerically a parabolic partial differential equation in one space dimension, where boundary values are given on one boundary only. For the analysis and numerical solution the problem is reformulated as a Volterra integral equation of the first kind. The problem is ill-posed and we study the nature of the ill-posedness by computing approximately the singular values and functions of the Volterra operator. The integral equation is discretized giving a system of linear equations. Two methods for solving the discrete problem are briefly discussed. The first is the regularization method, and the other is derived using certain properties of the Volterra equation. It is shown that in the case of time-independent coefficients an approximate solution can be obtained in O(n2) operations, where n is the dimension of the linear system. Numerical examples are given.

## Keywords

Cauchy Problem Singular Value Decomposition Regularization Method Singular Vector Volterra Integral Equation
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