Fourier method for one dimensional parabolic inverse problem with Dirichlet boundary conditions

Abstract

The finite difference method, spectral method, and double shifted Lagrange’s polynomials have been discussed for the one-dimensional inverse problem of the heat equation with control parameters and the source term in literature. Here, we present, Fourier method for the one- dimensional parabolic inverse problem with Dirichlet boundary conditions. In this study, after analyzing the control parameters, the initial condition and the source term are used to track a temperature distribution at a point in the interval. We validated that desired temperature distribution and measured temperature distribution (or the point evaluation) at an internal point overlap each other for the derived values of control parameters (source term and initial distribution) using the Fourier method. Moreover, we validated the temperature distribution at a point in the domain and tracked the desired harmonic and linear temperature distributions using numerical simulations. Finally, we simplified the above numerical simulations using the COMSOL software and illustrated some figures to the given point.