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)F(x) in the one-dimensional advection–diffusion equation ut=Duxx−vux+F(x)H(t)ut=Duxx−vux+F(x)H(t), (x,t)∈(0,1)×(0,T](x,t)∈(0,1)×(0,T], based on boundary concentration measurements g(t):=u(ℓ,t)g(t):=u(ℓ,t). Most studies have attempted to reconstruct an unknown spacewise-dependent source F(x)F(x) from the final observation uT(x):=u(x,T)uT(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)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 γ>0γ>0, the parameter of regularization α>0α>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.