INTEGRAL FORMULAS OF SOLUTIONS OF THE BASIC LINEAR DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS WITH VARIABLE FACTORS

In paper initial-regional problems for linear differential equations are
considered The mathematical physics (elliptic, hyperbolic and parabolic) with variables In the factors depending on coordinates and time. Such equations together with input datas we will be To name  initial. The equations with variable factors describe processes in the  composite Materials at which mechanical performances change or a  saltus or it is continuous in Boundary region between phases. Many  problems from various sections linear and nonlinear Mechanics are  reduced to a solution of simple equations with variable factors. In  case of periodic factors on coordinates one of popular modes of a solution of the equations The method of average of Bahvalova– Pobedri (MBP), based on representation of a solution is initial  Problems in the form of an asimptotical series on degrees of the  small geometrical parametre equal to the ratio Characteristic size of a mesh of periodicity to a characteristic size of a skew field. In this  method the initial The boundary value problem is reduced to two  recurrent sequences of problems. The first recurrent The sequence  consists in determination of periodic solutions of auxiliary problems  in a mesh Periodicity. The second sequence consists in a solution of  initial-regional problems for the equation with In constant effective factors. These factors are after a solution on a mesh Periodicity of  auxiliary problems. As base of a recursion in the second sequence in MBP serves Solution of a initial-regional problem for the equation  with effective factors in definition range, Having the same form and  it is exact with the same input datas, as an initial problem. Input  datas in each of recurrent sequences on what or a pitch are only after that as the previous recurrent problems are solved all. In the  present paper the new integral formulas are received, allowing to  express a solution of the initial Problems for the equation with the variable factors depending on co-ordinates and time, through a  solution The same problem for the equation with constant factors. The equation with constant factors Is called as the accompanying  equations, and the problem according to accompanying a problem.  In the kernel The integral formula the Green function and a  difference of factors initial and accompanying enters The equations.  By means of expansion of an accompanying solution in a many  dimensional Taylor series from the integral Formulas equivalent representation of a solution of an initial problem in the form of a  series on the various is received Derivative of a solution of an accompanying problem. Factors at derivatives are called as structural Functions. They are continuous functions of coordinates and time,  converted in zero at Coincidence of initial and accompanying factors. For definition of structural functions it is constructed System of the recurrent equations. Through structural functions factors of the  accompanying are defined The equations, coinciding in a periodic  case with effective factors in MBP. Unlike Method of Bahvalova–Pobedri in the new approach it is necessary to solve one recurrent  sequence of problems For determination of structural functions and  once to solve a problem for a homogeneous skew field with the  effective In performances.

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