Analytical Solution for an Inhomogeneous Heat Equation with Dirichlet Boundary Conditions

Abstract

In the subject of temperature distribution in a rod, the problem that is dealt with in the literature is one in which heat is transferred only through the cross section of a rod, without heat sources and with homogeneous boundary conditions. In this article, the one-dimensional heat equation is studied, considering heat transfer by convection on the lateral surface of a rod, Dirichlet boundary conditions and an initial arbitrary temperature distribution. This heat equation is derived from Fourier's law of conduction, the conservation of thermal energy, and Newton's law of cooling. A variant of an analytical method was used to find a solution, which meets the established initial condition and the boundary conditions. In the case of the initial condition, the convergence is not good at the ends of the rod due to the Gibbs phenomenon in the Fourier series. It is proved that such a solution can model the solution of less general problems. The solution found, contains a discontinuity when the convective thermal conductance is equal to zero, but taking the limit at this value, the function converges to the analytical solution where the convection factor is equal to zero.