Methodology of Syntheses of Knowledge: Overcoming Incorrectness of the Problems of Mathematical Modeling

TL;DR

This paper critiques current numerical methods for ill-posed problems in mathematical physics, proposing a new approach that ensures correctness by leveraging functional space properties and reducing complex problems to well-posed integral equations.

Contribution

It introduces a novel method for solving ill-posed problems, ensuring correctness through functional space techniques and transforming problems into well-posed integral equations.

Findings

New method ensures correctness of solutions

Reduction of complex problems to Fredholm equations of the second kind

Extension to wide classes of boundary-value problems

Abstract

J. Hadamard's ideas of correct formulation of problems of mathematical physics as well as related Banach's theorem on the inverse operator are analyzed. Modern techniques of numerical simulations are shown to be in drastic contradiction to the 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 inefficient algorithms, based on a belief that ill-posed problems are adequate to real phenomena. A new method of the solution of problems, traditionally associated with Fredholm integral equations of the first kind, is developed. Its key aspect is a constructive use of possibilities of the functional space $l_2$ to ensure the conditions of correctness. A well-known phenomenon of smoothing of information is taken into account by means of a special composition that explicitly involves the sought function and is infinitesimal in the space $L_2$. By relatively simple transformations, the outlined class of problems is reduced to the solution of Fredholm integral equations of the second kind with properties most favorable for the numerical realization. We demonstrate a reduction to Fredholm integral equations of the first kind and, correspondingly, a possibility to extend the suggested approach to wide classes of linear and nonlinear boundary-value and initial-boundary-value problems. We put forward arguments that the determination of causal relationships, based on the formulation restricted to a primitive renaming of known and unknown functions of the corresponding direct problem, is essentially illegitimate.