Approximations for a Cauchy problem for the heat equation. (English) Zbl 0645.35094

Eldén, Lars

Inverse Probl. 3, 263-273 (1987).

A Cauchy problem for the heat equation in the quarter plane is considered. Data are given along the line \(x=1\) and the solution at \(x=0\) is sought. The problem is ill-posed: the solution does not depend continuously on the data. In order to solve the problem numerically it is necessary to modify the equation so that a bound on the solution is imposed (explicitly or implicitly). We study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the solution of the Cauchy problem for the heat equation.

MSC:

35R25	Ill-posed problems for PDEs
35K15	Initial value problems for second-order parabolic equations
65M30	Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs

Keywords:

Cauchy problem; heat equation; quarter plane; ill-posed; Error estimates; modified equation