Validated forward integration scheme for parabolic PDEs via Chebyshev series
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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 := DF(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 a boundedly 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) := a - ℒ^-1 F(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, the Kuramoto-Sivashinsky equation, the Swift-Hohenberg equation and the phase-field crystal (PFC) equation. We apply our approach to each of these models and report plausible experimental results, which motivate further research on the method.
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