Simultaneous recovery of time-dependent coeffcients in a one-dimensional heat equation from integral overdetermination data via the Ritz collocation method


  • Prof.Kamal Rashedi Department of Mathematics, University of Science and Technology of Mazandaran



In this work, we study an inverse problem of reconstructing a time-dependent unknown absorption coeffcient and a time-dependent unknown coeffcient in the heat source by means of two integral overdetermination conditions. By applying a pair of transformations, the absorption coeffcient is eliminated and
the problem is converted to an equivalent inverse problem of determining a heat source with initial and
boundary conditions, as well as a nonlocal energy over-specification. We propose a Ritz approximation
as the solution of the unknown temperature distribution and consider a truncated series as the approximation of unknown time-dependent coeffcient in the heat source. Then, the collocation technique is
employed to reduce the solution of the main problem to solve a linear system of algebraic equations. For
the contaminated measurements, we employ the mollification method to derive stable numerical derivatives. Since the problem is ill-posed, numerical discretization of the reformulated problem may produce
ill-conditioned system of equations, therefore, the Tikhonov regularization technique is employed in order to obtain the stable solutions. Numerical simulations while solving two test examples are presented
to show the applicability of the proposed method.

Author Biography

Prof.Kamal Rashedi, Department of Mathematics, University of Science and Technology of Mazandaran

Department of Applied Mathematics