Numerical analysis, verified numerical computation, computer-assisted proofs, partial differential equations, delay differential equations, ordinary differential equations, differential equations in the complex domain, blow-up solution of differetial equations, infinite dimensional dynamical systems, fixed point methods, spectral methods, finite element method, semigroup theory
Verified numerical computation is a numerical method based on interval arithmetic, which takes all errors of numerical computations into account. It yields mathematically rigorous result using numerical computations. My research interest is rigorous numerics of solutions to differential equations (partial differential equations, delay differential equations, and ordinary differential equations). Rigorous numerics reveal all errors that may occur in an approximation of solutions to differential equations. We derive a sufficient condition of a fixed-point theorem, which can be validated by verified numearical computations. Then, by using rigorous numerics, we can prove existence and local uniqueness of a solution to differential equations in the neighborhood of a numeriically computed approximate solution.
A number of verified numerical methods for, in particular, various elliptic equations have been developed and improved in the recent three decades by many researchers. In this field, the finite element method (FEM) is a powerful approach of obtaining approximate solutions to PDEs on general domains. We have developed a verified numerical method based on the FEM for semilinear elliptic equations defined on arbitrary polygonal domains. Our method can easily deal with a variety of problems, which depends on the shape of domain.
We have also developed a verified numerical method for the initial-boundary value problems of semilinear parabolic equations. This method is based on the semigroup theory (analytic semigroups and evolution operators). Developing rigorous integrator for time-dependent PDEs is one of main themes in my works.
Furthermore, we have constructed an approach for validating the blow-up solutions of ODEs by combining dynamical systems with theories of resolution of singularity in the field of algebraic geometry. It is worth mentioning that the study of blowups and global existence of solutions in differential equations is beginning to be studied with the tools of rigorous numerics.
My current interest is to develop a rigorous integrator for DDEs, general (parabolic/hyperbolic/dispersive) PDEs on the basis of spectral methods.