Modeling of temperature processes in orthotropic boards of radio-electronic devices

. The design of electronic equipment that is resistant to external influences is a topical problem. The arrangement of individual circuit parts is modeled in the form of an orthotropic plate. The paper describes the solutions of homogeneous and inhomogeneous thermal conductivity equation for an orthotropic rectangular plate, obtained by the recurrence-operator method. Numerical results are given for two cases of temperature distribution, satisfying zero boundary conditions, during cooling and heating and given initial conditions


Introduction
Development of mathematical methods and development of programs for computer simulation of physical processes open up broad prospects for building systems of computeraided design of electronic equipment, resistant to external influences.
In electronic equipment, blocks and units made in the form of monolithic structures by pouring the units with various resins, foams and other fillers are widely used. The monograph [1] builds spatial, planar and rod models of blocks and units on the basis of artificial discretization of the corresponding differential equations.
In [2] the technique of solving problems on the basis of the finite difference method is outlined, an estimate of errors in numerical solution of problems is given, examples of solutions of some of them on the computer are given, the problems of automation of designing equipment, stable to mechanical and thermal influences are covered.
When designing electronic devices, it is desirable to arrange individual circuit parts so as to create a certain structure, which can be modeled as an orthotropic plate or an orthotropic parallelepiped.
In this paper, without resorting to artificial discretization, we present solutions of the heat conduction equation for an orthotropic medium.
t y x f external influence of the temperature field. To solve (1) we apply recurrence-operator method [3,4].
We first look for the solution of homogeneous equation (1) are constant coefficients determined from equation (1). Substituting (2) into (1), we obtain, , and combining all sums into one, we obtain For this equality to hold for all values of the common factor, the expression in square brackets must be equated to zero. As a result, the problem is reduced to the solution of the following numerical recurrence relation Here are the first few coefficients j i R , : 20  ,  10  ,  4  ,   ,  10  ,  6  ,  3  ,   ,  4  ,  3  ,  2  ,   ,  ,  ,  ,  1   3  1   3  0  3  ,  3   2  1   3  0  2  ,  3  1   3  0  1  ,  3   3  0  0  ,  3   3  1   2  0  3  ,  2   2  1   2  0  2  ,  2  1   2  0  1  ,  2   2  0  0  ,  2   3  1  0  3  ,  1   2  1  0  2  ,  1  1  0  1  ,  1  0  0  ,  1   3  1  3  ,  0   2  1  2  ,  0  1 By the method of total induction it can be shown that the solution of the recurrence equation (3) under condition (4) has the form Changing the order of summation, we present (2) in the form: Substituting (6) into this expression and minimizing the internal sum, we obtain the solution of equation (1) in the following operator form: where then series (7) will collapse, and we obtain exact final expressions. The solutions of equation (8) are [3]: where q i Q -constant coefficients determined from the recurrence relation under initial conditions The solution of (10), (11) will be ! )! 1 ( Thus, for example, assuming ! ) ( n y y g n  , we obtain polynomial solutions of equation (8) p q r i n For isotropic media the solutions of polycaloric equations are obtained in [5]. It can be shown that for function (2) the following differentiation formulas are true: using which we can make sure that (2) satisfies equation (1). The solution of (2) satisfies the following initial condition If in (2) where a n b a m a n m    (Fig. 1a). For the case of concentrated heating at an arbitrary point of the plate, the formula of the plate (Fig. 1a). For the case of concentrated heating at an arbitrary point of the plate, the formula

Conclusion
The last internal sum in (25) appears when the integration limits are substituted at each integration step. In case other boundary conditions on the plate contour are given, solutions of homogeneous equation (2) or (7) should be added to partial solution (25) and the function from boundary conditions should be determined.