G.V.Druzhinin, N.M.Bodunov

Kazan State Technical University

K.Marx Str., 10, Kazan, 420111, Russia

e-mail: root@kaiadm.kazan.ru

I.M.Zakirov

KNIAT-Gosinprom

Dement'ev Str, 2в, Kazan, 420036, Russia

e-mail: nurzak@mail.ru

We suggest a unified approach to the construction of basic functions (eigenfunctions) in handling the problems of continuum, cubature and quadrature problems, and also the problems on (hyper)surfaces approximation. We present the description of numerical analytical methods for obtaining the approximate solutions of inner and outer boundary-value problems of continuum (both linear and nonlinear). The method is based on the expansion of invariant solutions of partial differential equations in series of the basic functions. Also presented is the linearization of partial differential equations and the reduction of nonlinear boundary-value problems to the systems of linear algebraic equations (SLAE) solvable for the coefficients unknown; in the stage of basic function selection, no use is made of the conventional auxiliary techniques of linearization.

The group method is at present the most effective tool in searching for particular (invariant and partially invariant) solutions of mathematical physics equations; it is used also when solving the problems on linearization of partial differential equations. We must note, however, that a significant drawback of this method is its local character of use: the boundary conditions for the factor systems are determined only in accordance with the transforms obtained, i.e., it is impossible to meet the arbitrary boundary conditions for the so-obtained solution of the equation (or the system of equations). The herein-presented paper deals with the improvement of methods for solving linear and nonlinear boundary-value problems of continuum based on the global or local approximation of sought solutions of equations and boundary conditions by the functions with use of invariant solutions expansion is series of the basic functions.

We would like to note the distinguishing features of herein-presented basic functions:

the basic functions obtained in the form of invariant solutions of equations of mathematical physics possess the good structure, and proper analytical and computational properties (e.g., the solid spherical harmonics for the Laplace's equation have a scatter of coefficients and while the scatter of the herein-presented functions have and ;

the space dimension reduces in many cases by unity; the solutions are represented in the analytical form and the very formulation and solution of problems of parametric identification and inverse problems become more simple;

the mathematical models (both linear and nonlinear) are reduced to the linear algebraic equations; having disposed of the analytical solution and the boundary conditions, we can eliminate all difficulties of SLAE solution by using the internal parameters ;

on substituting the basic functions (i.e., the equation solution that corresponds to the Poisson bracket) into a definite class of nonlinear differential equations, the latters are reduced to SLAE without resort to conventional auxiliary methods of linearization (quasilinearization method, method of disturbances, and others), and also without use of the conventional Newtonian method of linearization as applied to the systems of nonlinear algebraic equations; the linearization occurs at the stage of basic functions selection. All what has been said above can be represented in the form of the following sequence:

nonlinear differential system of algebraic

equation equations.