In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques for self-validating numerics for initial and boundary value problems in ordinary and partial differential equations including eigenvalue problems will be presented. A fairly detailed introductions are given for the author's own method related to second-order elliptic boundary value problems. Many references which seem to be useful for readers are supplied at the end of the article.

Numerical verification methods for solutions of ordinary and partial differential equations

This article describes a survey on numerical verification methods for second-order semilinear elliptic boundary value problems introduced by authors and their colleagues. Here “numerical verification” means a computer-assisted numerical method for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. Three kinds of methods based on the infinite dimensional fixed-point theorems using Newton-like operator will be presented. In each verification method, a projection into a finite dimensional subspace and constructive error estimates of the projection play an important and essential role. It is shown that these methods are really useful for actual problems by illustrating numerical examples.

Numerical verification methods for solutions of semilinear elliptic boundary value problems

In this paper we show some structural properties of a p -quasihyponormal operator via Lowner-Heinz inequality and Hansen inequality. As important applications, it is shown that the normal parts of quasisimilar injective p -quasihyponormal operators are unitarily equivalent and quasisimilar injective p -quasihyponormal operators have same spectra and essential spectra. Mathematics subject classification (2000): 47B20.

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On p-quasihyponormal operators and quasisimilarity

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Numerical Verification Methods for Solutions of Ordinary and Partial Differential Equations (Relevance and Feasibility of Mathematical Analysis on the Computer)

In this paper, we consider numerical techniques which enable us to verify the existence of solutions for the free boundary problems governed by two kinds of elliptic variational inequalities(EVIs). Based upon the finite element approximations and the explicit a priori error estimates for some simple EVIs, we present effective verification procedures that, through numerical computation, generate a set which includes exact solutions. We describe a survey of the previous works as well as show some newly obtained results up to now.

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Numerical verification methods for solutions of the free boundary problem

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for the elasto-plastic torsion problems governed by the variational inequality. Based upon the finite element approximations and the explicit a priori error estimates for a simple problem, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. This paper is an extension of the previous paper [1] in which we mainly dealt with the obstacle problems, but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem. A numerical example is illustrated.

Numerical verification of solutions for elasto-plastic torsion problems

The equation AX = BX implies when A and B are normal (Fuglede-Putnam theorem). In this paper, the hypotheses on A and B can be relaxed by usin a Hilbert-Schmidt operator X: Let A be p-quasihyponormal and let be invertible p-quasihyponormal such that AX = XB for a Hilbert-Schmidt operator X and .Then .

Sang-Hun Lee

Papers

Numerical verification of solutions for elasto-plastic torsion problems

AN EXTENSION OF THE FUGLEDE-PUTNAM THEOREM TO p-QUASITHYPONORMAL OPERATORS

Some Remarks on The Structure of $k^*$-Paranormal Operators

Some properties of certain nonhyponormal operators

A computational verification method of existence of solution for elastoplastic torsion problems with uniqueness