APPROXIMATE SOLUTION OF STATIONARY HEAT CONDUCTIVITY PROBLEM WITH EXTREME BOUNDARY CONDITION

Abstract

A wide range of applied problems of mathematical physics and astrophysics leads to the search for solutions to problems of minimization of quadratic functionals. In particular, problems of minimization of quadratic functionals of the form (Au)(x) g(x) (x) (Au)(x) g(x) dx inf Rn 2 2 with convolution operators (Au)(x) k(x s)u(s)ds, Rn often found in the theory of mechanisms, linear electric and radio engineering circuits, optimal filters, control systems. Incorrect problems for linear equations also lead to the solution of minimization problems of quadratic functionals. It is known that extremal problems admit solutions in an explicit form only in some individual cases. Therefore, the construction and substantiation of methods for their approximate solution is of significant theoretical and practical interest. The use of numerical methods opens up the possibility of constructing solutions to new extreme problems for the equations of mathematical physics and algorithmizing this process. The paper presents the problem of stationary thermal conductivity with an extreme boundary condition. With the help of the Fourier transform and Sohotsky formulas, it is reduced to the solution of the Riemann matrix problem on the real axis with a non-positive system of partial indices. The equivalence of these problems is proved from the point of view of their solvability and formulas expressing the dependence of the solution of the extremal problem on the solutions of the corresponding Riemann problem. Based on the study of the Riemann problem, the conditions for the normal solvability of the extremal problem have been established. Approximate solutions of the extremal problem are constructed on the basis of approximate solutions of the Riemann problem. A projection method for finding them is proposed and substantiated. Convergence of approximate solutions of the extreme problem to its exact solution was evaluated. The exceptional case of the extremal problem was also studied and the corresponding approximate solutions were constructed. In addition, a numerical experiment was conducted for specific values of the parameters of the problem, the results of which are fully consistent with the theoretical conclusions. This confirms the high efficiency of the proposed method of constructing approximate solutions of extreme problems of mathematical physics. The obtained results can be used in solving problems of the theory of elasticity and thermoelasticity, thermal conductivity and other applied problems.