A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

Abstract

Abstract. This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem is transformed into a fixed-point form using an analytic semigroup. The sufficient condition is derived from Banach's fixed-point theorem. This paper also introduces a recursive scheme to extend a time interval in which the validity of the solution can be verified. As an application of this method, the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation.

Validated forward integration scheme for parabolic PDEs via Chebyshev series

“…It is worth mentioning that the development of rigorous computational methods to study the flow of dissipative PDEs has received its fair share of attention in the last fifteen years. Let us mention the topological method based on covering relations and self-consistent bounds [1,2,3,4,5,6,7], the C 1 rigorous integrator of [8], the semi-group approach of [9,10,11,12], and the finite element discretization based approach of [13,14,15]. This interest is perhaps not surprising as dissipative PDEs naturally lead to the notion of infinitedimensional dynamical systems in the form of semi-flows, and understanding the asymptotic and bounded dynamics of these models is strongly facilitated by a rigorous investigation of the flow.…”

Numerical verification for positive solutions of Allen–Cahn equation using sub- and super-solution method

“…Nakao, Kinoshita, and Kimura succeeded in verifying the existence of solutions to parabolic problems by estimating the norm of an inverse operator related to parabolic operators [12,13,14]. Subsequently, Mizuguchi, Takayasu, Kubo, and the third author of this paper proposed another method based on semigroup theories [17,18,19]. These two approaches were summarized in detail in a recent survey [15].…”

Rigorous FEM for 1D Burgers equation

“…The method has been used successfully to verify the solutions for elliptic (using FEM or Fourier basis) [10,11,13,19,20,29] and parabolic [21,[23][24][25][26]35] PDEs (but this mainly for periodic boundary conditions). The up to date information about these techniques can be found in the recent monograph [27].…”