method of boundary states, method of boundary states with the perturbation, the perturbation method, full parametric solution, analytical solution.

Classical solutions of problems of mathematical physics before introduction of electronic computing machines into computing practice had analytical form and used for bodies of simple geometry with the typical boundary conditions (including the method of Schwarz). The reorientation to numerical computer technology has created powerful computational methods (finite-difference, boundary integral equations (boundary element method), the Ritz energy methods (finite element method), Galerkin’s methos, least squares, Kantorovich’s method and their modifications. This has allowed them to find solutions to quite complex tasks. Their disadvantage was the need of the recalculation of the solutions when changing the parameters of the problem. The use of interpolation procedures can be successfully increased their level of flexibility, although the procedures of Schwartz become more accessible. The creation of modern computing systems based on "computer algebra allowed us to obtain the solution in numerical form. New variational method of boundary states allows us to construct full parametric analytical solution (solutions that contains all task parameters) for an arbitrary geometric configuration of bodies, different types of boundary conditions and all the constants of the physical medium. For including medium parameters and geometrical parameters of the body in an analytical solution it is possible to apply interpolation (resource-intensive approach). The perturbation method in combination with the method of boundary states were fundamentally reduces the cost of computing resources. The relevance and purpose of the work as defined by the organization of this approach and suggest a number of tasks: 1) development of procedures for the use of the perturbation method to include medium parameters in the solution; 2) development of algorithm for constructing full parametric solution of the problem by the method of boundary states with perturbation; 3) full parametric solution of the second basic problem for a single finite body with several types of boundary conditions. These results have been achieved and illustrated in an analytical form and graphically for a single finite body with three different generic parts of the border.

Novikova Olga Sergeevna e-mail: _o_l_g_a_@bk.ru, Postgraduate student, Lipetsk State Technical University, Lipetsk, Russia, Penkov Viktor Borisovich e-mail: vbpenkov@mail.ru, Dr. Sci. Phys. & Math., Professor, Lipetsk State Technical University, Lipetsk, Russia, Levina Lyubov Vladimirovna e-mail: satalkina_lyubov@mail.ru, Ph.D., Associate Professor, Lipetsk State Technical University, Lipetsk, Russia.