Retrieving‎ ‎the‎ ‎Transient‎ ‎Temperature‎ ‎Field and Blood Perfusion‎ ‎Coefficient‎ ‎in‎ ‎the‎ ‎Pennes‎ ‎Bioheat‎ ‎Equation Subject to Nonlocal and Convective Boundary Conditions

Abstract

‎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.