(untitled) (ҚазҰУ Хабаршысы. Математика, механика, информатика сериясы, 2022)
The paper develops a method for solving nonlinear equations of heat conduction. Two-layer container complexes have been created, the side faces of which are thermally insulated so that the 1D heat equation can be used. In order not to solve the boundary value problem with a contact discontinuity and lose the accuracy of the solution method, a temperature sensor was placed at the junction of two media, and a mixed boundary value problem is solved in each area (container). To provide initial data for the initial boundary value problem, three temperature sensors were used: two sensors measure air temperatures at the left and right boundaries of the container complex; the third sensor measures the temperature of the soil at the junction of two media. The paper numerically investigates the initial-boundary problem of heat conduction with nonlinear coefficients of heat conduction, heat capacity, heat transfer and density of the material. To solve a nonlinear initial-boundary value problem, the grid method is used. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar sweep method, and the nonlinear difference problem is solved by the Newton method. On the basis of an a priori estimate of the solution of a nonlinear difference problem, we prove the convergence of the second degree of Newton’s method. The numerical calculations carried out show that, for small time intervals, the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%