On a method for approximate solution nonlinear heat conduction equation

V. B. Bednova

Čebyševskij sbornik, Tome 17 (2016) no. 4, pp. 11-22

Résumé

In this paper a one-dimensional non-stationary heat conduction problem, modeling the process of rapid local heating of the sample beam type on the lateral surface is considered. The character of heating is such that it is possible to allocate only a certain directionof the heat propagation. Temperature fields are determined by an approximate method based on the idea of the thermal front. The solution is sought in the form of a power series in the coordinate with coefficients depending on time. The boundaries of the front heat distribution as a function of time are determined by the condition of the integral satisfaction of the heat conduction equation. Considered temperature fields arise in many industrial processes, such as laser material processing, when due to large temperature gradients can arise thermal stresses, leading to microcracking inner layers or the destruction of structural elements. Analytical view of the heat conduction problem’s solution allows to obtain analytical expressions for the thermal stresses and further facilitates the results analysis. The paper presents the solution of problems with boundary conditions of the first and second kinds for two monotonic and one non-monotonic dependencies of the thermal conductivity coefficient on temperature. The approximate solution and the exact solution of the non-stationary linear problem are compared and shows the suitability of the method for future use. Bibliography: 16 titles.

Détail

Keywords: Physical non-linearity of the heat conduction equation, approximate solution of the heat conduction equation, thermal front.