Numerical Verifications of Solutions for Nonlinear Hyperbolic Equations

In this paper, we consider a numerical technique to enclose the solutions with guaranteed error bounds for nonlinear hyperbolic initial boundary value problems as well as to verify the existence of solutions. Using a finite element approximation and explicit error estimates for a certain simple linear hyperbolic problem, we construct, by computer, a set of functions which satisfies the condition of Schauder’s fixed point theorem in some appropriate function space. In order to obtain such a numerical set, we use a kind of multivalued iterative procedure with efficient use of an initial approximate solution. A numerical example is provided.