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# The Regularization Numerical Simulation Method For A Kind of Steady-State Heat Conduction Sideways Problem In Multilayer Media

Directly observing the deep temperature field data of geothermal fields can be a challenging task. In this paper, a mathematical differential equation model of two-dimensional steady-state heat conduction sideways problem in multilayer media is established to calculate the deep structure geothermal field from surface geothermal field observation data. Due to the characteristics of ill-posed problems, a stable regularization numerical solution method is constructed, and various experiments analysis and numerical simulations are carried out. This paper begins by discussing the characteristics of the mathematical model of the steady-state heat conduction sideways problem in the rectangular domain. Next, the non-homogeneous boundary conditions of the model are treated by the homogeneous principle. The non-homogeneous partial differential equation is then transformed into the first Fredholm integral equation using the separation of variables method. Subsequently, the ill-posed integral equation is solved by the regularization method to obtain the geothermal gradient and temperature data on the boundary of the rectangular domain. Finally, the finite difference method is used to calculate the geothermal field in the rectangular domain. For the two-dimensional steady-state heat conduction sideways problem in multilayer media, the temperature field data are obtained layer by layer using the continuity conditions between layers. Numerical examples demonstrate the effectiveness and reliability of the proposed method. This paper has significant reference value for the numerical simulation of geothermal fields in exploring geothermal field thermal resources and studying lithospheric thermal structure.