Solvability and factorization of mixed problem for the heat conduction equation

The aim of this paper is to propose and prove some properties and theorems of common fixed point in the function weighted b- metric spaces and its application, its space is a generalization of the function weighted metric spacesThe study aims to analyse the solvability of mixed problems for the heat conduction equation. To achieve the objectives, analytical methods, methods of mathematical analysis of mixed problems and heat conduction equations were applied. Functional analysis and differential equation theory methods were also used to study the solvability. The study determined that the equation under consideration has a wide range of applications and plays an important role in describing heat transfer processes in various fields of science and technology. The study revealed that analysing mixed problems involving a variety of initial and boundary conditions provides a key tool for gaining a deeper understanding of heat transfer processes. Furthermore, properly chosen initial and boundary conditions have a significant impact on the long-term system operation. The study results have important implications for understanding and modelling heat conduction processes in a variety of physical systems. The results provide a deeper understanding of heat conduction processes and their mathematical description in a variety of systems, which opens new perspectives for improving the prediction and control of heat conduction phenomena. This study represents a significant contribution to the field of thermal conduction, not only expanding the theoretical understanding of this physical phenomenon but also providing a basis for practical applications.