A CLASS OF INVERSE PROBLEMS FOR THE HEAT EQUATION WITH INVOLUTIVE PERTURBATION - Mussirepova Э.; Urmatova А.; Altynbekov Ш. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2024)
Inverse problems for the equation of deflected thermal conductivity with involution are one of the most relevant research topics in the field of mathematical physics. This study is devoted to the study of solutions of deviating equations characterizing the process of thermal conductivity, and the development of methods for solving inverse problems, taking into account their involutional properties. Such tasks are widely used in practical applications such as studies of thermal properties of materials, problems of reverse distribution, and engineering tasks for managing thermal processes. The equations of deflected thermal conductivity with involution are a general modified version of the problem of thermal conductivity, which allows us to more accurately describe various physical processes. In such equations, higher sequences of time derivatives or additional involutive terms are introduced, which complicates the model, but brings it closer to real processes. The theory of inverse problems includes important questions in the search for solutions to the equations of thermal conductivity. By determining unknown coefficients, initial or boundary conditions based on actual data, these tasks allow a deeper understanding of thermal processes. The specificity of deviating equations is due to the need to preserve the stability and uniqueness of their solutions. The purpose of this work is to study inverse problems for equations of deflected thermal conductivity with involution, and to develop analytical methods for their solution. The paper considers the issues of setting inverse problems, studying the conditions for their correct formulation, proving loneliness and stability of solutions. In addition, effective methods for solving problems are proposed. The novelty of the work lies in the presentation of a new formulation of inverse problems for equations with involution and the study of their analytical solutions. These models allow us to describe specific physical phenomena, such as the thermal conductivity of complex materials or changes caused by external factors. In addition, the results of the study contribute to improving the accuracy of the model in solving many engineering and scientific problems. The results of the work make a significant contribution to the development of the theory of inverse problems, as well as to the construction of new mathematical models of thermal conductivity processes. The results of the research can be used in scientific research, engineering reports and optimization of technological processes. A class of inverse problems for the equation of deflected thermal conductivity with involution is considered using four different boundary conditions. The solutions were obtained in the form of series classification using sets orthogonal to each report. The completeness of the solutions received was also discussed.