Background. The interest to obtain analytical solutions to heat conduction problems with variable physical properties and a temperature-dependent internal heat source, is relevant due to the need to study high-temperature processes of thermal ignition and explosion that occur in blow-up regimes. In these regimes heat becomes localized in a bounded region of space and leads to an unbounded increase in temperature (thermal explosion) within a short time interval. The study of such problems is mainly carried out by numerical methods, since their solution using known analytical methods is not feasible. In this regard, the development of analytical methods to get solutions to such problems is of current importance.
Materials and methods. To solve the specified heat conduction problems, an additional function (AF) and additional boundary conditions (ABCs) are introduced. These conditions consist of coordinate relations that ensure the heat conduction equation is satisfied at the boundaries. The AF represents the temporal variation of temperature at a certain point of the spatial variable. Its application makes it possible to reduce the order of the original equation, transforming it from a partial differential equation into an ordinary differential equation. The application of ABCs allows one to eliminate the process of integrating the original differential equation with respect to the spatial variable, as the equation is satisfied at the boundaries.
Results. A proof is presented for a theorem stating that if the equation holds on the boundary, it holds throughout the entire domain of the spatial variable. A solution has been found for the heat conduction problem involving variable physical properties and a temperature-dependent internal heat source. The obtained solution makes it possible to separate stationary regimes from regimes of unbounded temperature growth, which occur when the first eigenvalue is positive. It has been found that the temperature rise occurs within a narrow spatial region over a short time interval, that is, in a blow-up regime. A method is employed that makes it possible to eliminate integration of the equation with respect to the coordinate when a large number of additional boundary conditions are used. This is confirmed by the proof of the theorem. The obtained solution has been used to determine the heat source power at which the temperature gradient of the methane pyrolysis reactor wall does not exceed the permissible (specified) value.
Conclusions. By employing additional boundary conditions and an auxiliary unknown function, an analytical solution has been obtained for a highly complex nonlinear heat conduction problem involving a nonlinear heat source. The method described enables the analysis of combustion and explosion processes characterized by unbounded temperature rise, as well as other thermal processes in structures with internal heat sources, such as the induction heating of metals.