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

Mizuguchi, Makoto  ,  Takayasu, Akitoshi  ,  Kubo, Takayuki  ,  Oishi, Shin'ichi

55 ( 2 )  , pp.980 - 1001 , 2017-04 , SIAM journal on numerical analysis
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.

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