Explicit solutions for a non-classical heat conduction problem for a semi-infinite strip with a non-uniform heat source

Abstract

A non-classical initial and boundary value problem for a non-homogeneous one-dimensional heat equation for a semi-infinite material with a zero temperature boundary condition is studied. It is not a standard heat conduction problem because a non-uniform heat source dependent on the heat flux at the boundary is considered. The purpose of this article is to find explicit solutions and analyze how to control their asymptotic temporal behavior through the source term. Explicit solutions independent of the space or temporal variables, solutions with separated variables and solutions by an integral representation depending on the heat flux at the boundary are given. The controlling effects of the source term are analyzed by comparing the asymptotic temporal behavior of solutions corresponding to the same problem with and without source term. Finally, a relationship between the problem considered here with another non-classical problem for the heat equation is established, and explicit solutions for this second problem are also obtained. In this article, we give explicit solutions and analyze how to control them through the source term for several non-classical heat equation problems. In addition, our results enable us to compute the asymptotic temporal behavior of the heat flux at the boundary for each explicit solution obtained. As a consequence of our study, several solved non-classical problems for the heat equation that can be used for testing new numerical methods are given.