Methodology of Syntheses of Knowledge: Overcoming Incorrectness of the Problems of Mathematical Modeling (revised version, March 2005)

TL;DR

This paper critiques current mathematical modeling practices, emphasizing the importance of well-posed problems, and introduces a new method to transform ill-posed problems into well-posed ones for better numerical solutions.

Contribution

It develops a novel approach to convert ill-posed Fredholm integral equations into well-posed forms, aligning mathematical modeling with Hadamard's principles.

Findings

New method reduces first-kind Fredholm equations to second-kind for numerical stability.

Applicable to boundary-value problems with variable coefficients and complex domains.

Supports the idea that physically meaningful problems should be well-posed.

Abstract

J. Hadamard's ideas about the correct formulation of the problems of mathematical physics have been analyzed. In this connection various interpretations of the directly related Banach theorem about the inverse operator has been touched. The contemporary apparatus of mathematical modeling is shown to be in a drastic contradiction with concepts of J. Hadamard, S. Banach and a number of other outstanding scientists in the sense that the priority is given to the realization of algorithms, which actually imply that ill-posed problems are adequate to real phenomena. A new method is developed for solving problems traditionally associated with the Fredholm integral equation of the first kind that admits of their reduction to Fredholm integral equation of the second kind with properties most favorable for the numerical realization. It is demonstrated that a wide circle of problems can be reduced to two-dimensional Fredholm integral equations of the first kind; these are linear boundary-value and initial-boundary-value problems with variable coefficients, non-canonical domain of definition and others. The elaborated algorithm is shown to be directly applicable to them and may be used for testing their solvability. In discussing the formulation of problems of mathematical physics, considerable attention is paid to methodological aspects. Conclusions about cause-and-effect relations are argued to be essentially illegitimate when the solution of a problem reduces to a primitive renaming of known and unknown functions of a corresponding direct problem. The aim of this work is a constructive realization of J. Hadamard's opinion that physically meaningful problems are always well-posed.