  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 $\mathcal{L} := DF(0)$ has an
infinite block diagonal structure with blocks becoming less and less diagonal
dominant at infinity. We introduce analytic estimates to show that
$\mathcal{L}$ is a boundedly invertible linear operator on $X$, and we obtain
explicit, rigorous and computable bounds for the operator norm $\|
\mathcal{L}^{-1}\|_{B(X)}$. These bounds are then used to verify the hypotheses
of a Newton-Kantorovich type argument which shows that the (Newton-like)
operator $\mathcal{T}(a) := a - \mathcal{L}^{-1} F(a)$ 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.
