Mathematical Modeling and Analysis of Temperature Regime in Digital Devices Due to Surface Heating

Linear and nonlinear mathematical models for determining the temperature field and, subsequently, for analyzing temperature regimes in an isotropic plate due to near-surface thermal loading have been developed. For the case of a thermosensitive plate (the thermophysical parameters of the structural material depend on temperature), the Kirchhoff transformation has been applied, using which the nonlinear heat conduction equation and nonlinear boundary conditions have been linearized, and as a result, a linear second-order differential equation with partial derivatives and a discontinuous right-hand side and partially linearized boundary conditions have been obtained. For the final linearization of the boundary conditions, the temperature has been approximated by the spatial coordinate on the boundary surface of the thermosensitive plate by a segment-constant function, which made it possible to obtain a linear boundary problem with respect to the Kirchhoff variable. To solve the obtained boundary value problems, the integral Fourier transform was used and, as a result, analytical and analytical-numerical solutions in the form of improper convergent integrals were obtained. For a thermally sensitive medium, as an example, the linear dependence of the thermal conductivity coefficient of the structural material of the structure on temperature was chosen, which is often used to solve many practical problems. Software tools have been developed, using which a numerical analysis of the behavior of temperature as a function of spatial coordinates for given values of geometric and thermophysical parameters has been performed, and on this basis the behavior of the temperature field has been geometrically depicted. The developed linear and nonlinear mathematical models for determining the temperature field in spatial environments with near-surface heating make it possible to analyze their thermal stability and on this basis it is possible to prevent overheating, which can cause failure not only of structural units and their individual elements, but also of the electronic device as a whole.