Validated forward integration scheme for parabolic PDEs via Chebyshev series

Abstract

In this paper we introduce a new approach to compute rigorously solutions of Cauchy problems for a class of semi-linear parabolic partial differential equations. Expanding solutions with Chebyshev series in time and Fourier series in space, we introduce a zero finding problem F(a)=0 on a Banach algebra X of Fourier–Chebyshev sequences, whose solution solves the Cauchy problem. The challenge lies in the fact that the linear part L=defDF(0) has an infinite block diagonal structure with blocks becoming less and less diagonal dominant at infinity. We introduce analytic estimates to show that L is an invertible linear operator on X, and we obtain explicit, rigorous and computable bounds for the operator norm ‖L−1‖B(X). These bounds are then used to verify the hypotheses of a Newton–Kantorovich type argument which shows that the (Newton-like) operator T(a)=defa−L−1F(a) is a contraction on a small ball centered at a numerical approximation of the Cauchy problem. The contraction mapping theorem yields a fixed point which corresponds to a classical (strong) solution of the Cauchy problem. The approach is simple to implement, numerically stable and is applicable to a class of PDE models, which include for instance Fisher’s equation and the Swift–Hohenberg equation. We apply our approach to each of these models.