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A method based on the tau approach for the identification of a time-dependent coefficient in the heat equation subject to an extra measurement

In this work, the problem of parameter identification in a parabolic partial differential equation is discussed. The shifted Legendre-tau method is presented for finding the unknown function and also the unknown coefficient in the one-dimensional diffusion model. The method consists of expanding the required approximate solution as the elements of a shifted Legendre polynomial and by using operational matrices we reduce the problem to a set of algebraic equations. Several examples are given to demonstrate the validity and applicability of the technique and a comparison is made with existing results. The method is easy to implement and yields very accurate results.