Fourier collocation algorithm for identifying the spacewise-dependent source in the advection-diffusion equation from boundary data measurements

In this study, we investigate the inverse problem of identifying an unknown spacewise-dependent source F (x) in the one-dimensional advection-diffusion equation u(t) = Du(xx) - vu(x) + F(x)H(t), (x, t) is an element of (0, 1) x (0, T], based on boundary concentration measurements g(t) := u(l, t). Most studies have attempted to reconstruct an unknown spacewise-dependent source F(x) from the final observation u(T)(x) := u(x, T), but from an engineering viewpoint, the above boundary data measurements are feasible. Thus, we propose a new algorithm for reconstructing the spacewise-dependent source F(x). This algorithm is based on Fourier expansion of the direct problem solution followed by minimization of the cost functional by taking a partial K-sum of the Fourier expansion. Tikhonov regularization is then applied to the ill-posed problem that is obtained. The proposed approach also allows us to estimate the degree of ill-posedness for the inverse problem considered in this study. We then establish the relationship between the noise level gamma > 0, the parameter of regularization alpha > 0, and the truncation (or cut-off) parameter K. A new numerical filtering algorithm is proposed for smoothing the noisy output data. Our numerical results demonstrated that the results obtained for random noisy data up to noise levels of 7% had sufficiently high accuracy for all reconstructions. (C) 2015 IMACS. Published by Elsevier B.V. All rights reserved.