Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in the braking systems of vehicles, which are geometrically depicted as isotropic spatial heat-active media that are subject to internal local thermal heating, have been developed. With the use of classical methods, it is not possible to obtain analytical solutions of linear and nonlinear boundary value problems of mathematical physics in a closed form. This is especially the case when the right-hand sides of differential equations with partial derivatives and boundary conditions are discontinuous functions. The given approach is based on the application of the apparatus of generalized functions to describe the local concentration of thermal influence. This made it possible to apply the integral transformation and, on this basis, to obtain analytical solutions of both linear and nonlinear boundary value problems. In the case of a nonlinear boundary value problem, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a linearized second-order differential equation with partial derivatives and boundary conditions with a discontinuous right-hand side were obtained. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. On the basis of the developed mathematical models, a computational algorithm was created and on this basis, software tools were created, using which the heat exchange processes in the middle of the brake structures for the selected materials of the brake pads were analyzed in terms of their effectiveness, as well as the determination of the optimal temperature values for the effective operation of the braking system of vehicles. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active media with internal heating make it possible to analyze their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and individual elements, but also the entire structure.
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