Research of Thermal Explosion Conditions in Nonlinear Heat Conduction Problems with a Nonlinear Heat Source

Abstract

Using an additional sought function (ASF) and additional boundary conditions (ABCs), an analytical solution of the nonlinear heat conduction problem for a symmetric plate with a nonlinear heat source has been obtained. ASF characterizes the temperature change over time at the center of the plate. Its usage enables the solution reduction of the original partial differential equation to the integration of a temporal ordinary differential equation (ODE). From its solution, exact eigenvalues are found, determined by classical methods from solving the Sturm–Liouville boundary value problem specified in the spatial coordinate domain. Hence, this study considers another direction of their determination, based on solving the temporal ODE with respect to ASF. ABCs are formulated in such a way that their fulfillment by the sought solution is equivalent to satisfying the original differential equation at the boundary points. It leads to its fulfillment within the considered domain, bypassing the direct integration process over the spatial variable and confining it only to the execution of the heat balance integral—the averaged original differential equation.