retrieving‎ ‎the‎ ‎transient‎ ‎temperature‎ ‎field and blood perfusion‎ ‎coefficient‎ ‎in‎ ‎the‎ ‎pennes‎ ‎bioheat‎ ‎equation subject to nonlocal and convective boundary conditions

‎in this paper‎, ‎we delve into a coefficient inverse problem linked to the bioheat equation‎, ‎a pivotal component in medical research concerning phenomena such as temperature response and blood perfusion during surface heating‎. ‎by considering factors like heat transfer between tissue and blood in capillaries and incorporating the geometric intricacies of the skin‎, ‎we confine our analysis to a one-dimensional domain‎. ‎our approach involves transforming the original problem into one concerning the reconstruction of a multiplicative source term within a parabolic equation‎. ‎subsequently‎, ‎we utilize integral conditions to derive a specific integro-differential equation‎, ‎accompanied by the requisite initial and boundary conditions‎. ‎leveraging a spectral method‎, ‎we streamline the modified problem into a linear system of algebraic equations‎. ‎to accomplish this‎, ‎we employ appropriate regularization algorithms to obtain stable approximations for the derivatives of perturbed boundary data and to effectively solve the resultant system of equations‎.