A Numerical Method for an Ill-posed Problem

The noncharacterisitic initial value problem for the one-dimensional heat equation (the solution and its rst-order spatial derivative speciied on an interval of the time axis) is well known to be ill-posed. Nevertheless, the author has proved in 4] that nonnegative solutions of this problem depend continuously on the initial data. However, this result does not solve the problem of constructing approximate solutions from given approximate initial data. The problem is that given approximate initial data is unlikely to be genuine initial data for a nonnegative solution of the Cauchy problem, and it is easy to show that arbitrarily near any given initial data, there are analytic initial data for which the solutions grow arbitrarily fast. What is needed, and what is provided here, is a method for selecting, as near as possible to given approximate initial data, genuine initial data for a Cauchy problem with a nonnegative solution. The solution for this data will be well-behaved, and the result on continuous dependence implies that small diierences in the genuine intial data that is selected lead to small diierences in the solutions. problem for the one-dimensional heat equation; u t (x; t) = u xx (x; t); 0 < x < a; 0 t T; u(0+; t) = f(t); 0 t T; u x (0+; t) = g(t); 0 t T; where the solution u and its derivative u x are required to be continuous on 0; a) 0; T]. The theorem below follows routinely from Lemma 2 of 4]. Theorem 1. Let a; T > 0. Then for 0 < b < a, and 0 < T 1 < T 2 < T, there are constants A and B, depending only on a; T; b; T 1 ; T 2 such that if f; g 2 C0; T], and P(a; T; f; g) has a nonnegative solution u, then ju(x; t)j Akfk + Bkgk for all (x; t) 2 0; b] T 1 ; T 2 ] where k k denotes the supremum norm on 0; T]. In 4], the constants A and B are given explicitly in terms of the geometric parameters; for a more general result without the explicit constants, see 7]. The next theorem now follows from Theorem 1 and from 1] or 2, Thm. 11.4.1]. for all (x; t) 2 0; b] T 1 ; T 2 ]. Theorem 2 also follows from Theorem 1 and …