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

1 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. A finite error in the determination of the data by measurements can also be represented in the same manner. 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 boundary-value and initial-boundary-value problems characterized by variable coefficients, non-canonical domains, as well as by other factors complicating their solution. Basically similar techniques are developed for the solution of initial-boundary-value problems for non-linear differential equations of the evolutionary type. Also considered are boundary-value problems for substantially nonlinear equations and equations of a mixed type, the cases of nonlinearities in the boundary conditions, the presence of a small factor by the highest-order derivative, the inverse problem of the restoration of the variable coefficient of the differential operator , and the problem of the Stefan type. 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. The suggested approach may be considered as a constructive realization of J. Hadamard's ideas of the existence of correct formulations of physically meaningful problems.