Accurate method of verified computing for solutions of semilinear heat equations
For researchers in numerical analysis and PDEs, it offers an improved verification technique for semilinear heat equations, though the improvement is incremental.
This paper presents a verification method for semilinear heat equations that numerically proves existence and local uniqueness of exact solutions near approximate ones, achieving enclosure for longer time intervals than previous analytic semigroup methods.
We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact solution in a neighborhood of a numerically computed approximate solution. Our method is based on a fixed-point formulation using the evolution operator, an iterative numerical verification scheme to extend a time interval in which the validity of the solution can be verified, and rearranged error estimates for avoiding the propagation of an overestimate. As a result, compared with the previous verification method using the analytic semigroup, our method can enclose the solution for a longer time. Some numerical examples are presented to illustrate the efficiency of our verification method.