Solution of nonlinear problems of heat conduction of thermosensitive solids by the method of stage-by-stage linearization (Q2759497)

The temperature field in a material with a simple nonlinearity (the coefficient of heat conductivity \(\lambda_t(t)\) and the volume heat capacity \(c_v(t)\) depend on the temperature and their ratio, the thermal diffusivity \(a=\lambda_t(t)/c_v(t)\) is constant) is studied. A technique for the construction of analytic-numerical solutions of non-stationary problems for one-dimensional solids in Cartesian, cylindrical and spherical coordinates under convective heat exchange at their ends is proposed. This technique is based on the introduction of the Kirchhoff variable and thus reducing the problem to a linear one. After a specification of the dependence \(\lambda_t(t)\), spline approximations of the temperature are constructed. Their coefficients are determined from conditions of equality of splines at nodes of approximation to boundary values of the temperature.