SOLUTION OF NONLINEAR OPTIMIZATION PROBLEMS OF HEAT TRANSFER THEORY - LEVKIN, DMYTRO; KRAVTSOV, ANDRII; ZAVGORODNIY, OLEXIY; KOTKO, YANA (Herald of Khmelnytskyi National University. Technical sciences, 2025)
The article proposes mathematical models and computational methods for the solution of nonlinear problems of finding local extrema of the objective function. These mathematical models and computational methods create a computational structure for improving the accuracy of applied optimization problems. Computational mathematical models of thermal action on a multilayer material and laser action on an embryo have been developed. It should be noted that the computational mathematical model describing the laser effect on the embryo is a nonlocal boundary value problem with a system of evolutionary, nonstationary heat conduction equations, heat flow boundary conditions, and boundary conditions at the beginning and end of the laser effect on the embryo. Applying the traditional theory of existence and uniqueness of the solution of the boundary value problem, due to the multilayer structure of the embryo and different thermal modes of laser action without averaging the values of the thermophysical characteristics of the object of study, it is impossible to substantiate the correctness of the boundary value problem describing the laser action on the embryo. To substantiate the correctness of this boundary value problem, the article proposes to apply methods from the theory of pseudo-differential operators in the space of slowly increasing distributions that depend smoothly on time variables. This made it possible not only to improve the accuracy of solving boundary value problems, but also to increase the accuracy of solving the general problem of improving the quality of laser action on a multilayer material. Reducing the time required to solve boundary value problems was achieved through the use of functionally oriented blocks that almost instantly implement many computational mathematical models (boundary value problems) on computers. This will increase the efficiency of existing technical means and develop new technical means for automating the design of complex systems containing local sources of thermal action.