Analytical solutions in solving the parabolic partial differential equation using separation of variables and Fourier series

Abstract

In this paper we present the analytical solution of a one-dimensional (1D) parabolic differential equation (PDE), encountered in many engineering studies under the name of the heat equation.  To solve the equation, we use the method of separation of variables together with Fourier series. After applying the separation of variables, the initial equation decomposes into two ordinary differential equations (ODE). The equations obtained correspond to the spatial (x) and temporal (t) components. The spatial solution, for the first equation, is expressed by sinusoidal functions corresponding to the eigenfunctions, and the temporal component, for the second equation, has a characteristic exponential dependence. The determination of the Fourier series coefficients is based on the initial condition. Thus, we constructed an exact solution in the form of convergent series. The method used in solving the equation, together with the boundary conditions and initial conditions, can be used as a reference for validating numerical methods or practical experiments performed in the laboratory.