  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.
