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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 $mathcal{L} := DF(0)$ 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 Fishers 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.
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semi-linear par
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs), which are
The tangential condition was introduced in [Hanke et al., 95] as a sufficient condition for convergence of the Landweber iteration for solving ill-posed problems. In this paper we present a series of time dependent benchmark inverse problems for which we can verify this condition.
In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii
In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Fuhrer& Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this