A stable and high-accuracy numerical method for determining the time-dependent coefficient in the bioheat equation

In this paper, we propose a space-time spectral method for time-dependent coefficient identification of the inverse problem with the Ionkin-type nonlocal boundary and integral over- determination conditions. The Legendre-Galerkin method is applied in the spatial direction and the Legendre-tau method is applied in the time direction. And the method is also implemented by the explicit-implicit iterative method. The nonlinear term is collocated at the Chebyshev-Gauss-Lobatto points and computed explicitly by the fast Legendre transform. Tikhonov regularization is applied to employ the blood perfusion coefficient computation with the noisy perturbations. The adopted stabilization scheme presents a good performance in terms of accuracy, effectiveness and robustness on the inverse problem, especially for noisy perturbations. Numerical results are given to show the accuracy and stability of the approach and agree well with theory analysis. Optimal order convergence is also obtained through the estimates in the L 2-norm.