Applied Numerical Mathematics 78 (2014) 49–67 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum A unified approach to identifying an unknown spacewise dependent source in a variable coefficient parabolic equation from final and integral overdeterminations Alemdar Hasanov ∗ , Burhan Pekta ¸ s Department of Mathematics and Computer Science, Izmir University, Üçkuyular 35350, Izmir, Turkey article info abstract Article history: Received 13 May 2013 Received in revised form 10 October 2013 Accepted 27 November 2013 Available online 14 December 2013 Keywords: Inverse source problem Parabolic equation Final and time-average temperature overdeterminations Integral representation formula Filter function Conjugate gradient algorithm An adjoint problem approach with subsequent conjugate gradient algorithm (CGA) for a class of problems of identification of an unknown spacewise dependent source in a variable coefficient parabolic equation u t = (k(x)u x ) x + F (x) H(t ), (x, t ) ∈ (0, l) × (0, T ] is proposed. The cases of final time and time-average, i.e. integral type, temperature observations are considered. We use well-known Tikhonov regularization method and show that the adjoint problems, corresponding to inverse problems ISPF1 and ISPF2 can uniquely be derived by the Lagrange multiplier method. This result allows us to obtain representation formula for the unique solutions of each regularized inverse problem. Using standard Fourier analysis, we show that series solutions for the case in which the governing parabolic equation has constant coefficient, coincide with the Picard’s singular value decomposition. It is shown that use of these series solutions in CGA as an initial guess substantially reduces the number of iterations. A comparative numerical analysis between the proposed version of CGA and the Fourier method is performed using typical classes of sources, including oscillating and discontinuous functions. Numerical experiments for variable coefficient parabolic equation with different smoothness properties show the effectiveness of the proposed version of CGA.  2013 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction Heat source identification problems are the most commonly encountered inverse problems in heat conduction. These problems have been studied over several decades due to their significance not only in a variety of scientific and engineering applications, but also to their significance in the theory of inverse source problems for PDEs (see [1–6,8–11,13,15,17,16, 18–21,28–34,37] and references therein). The final overdetermination for one-dimensional heat equation has first been con- sidered by Tikhonov [35] in study of geophysical problems. In this work the heat equation with prescribed lateral and final data is studied in half-plane and the uniqueness of the bounded solution is proved. For parabolic equations in a bounded domain, when in addition to usual initial and boundary condition, a solution is given at the final time, well-posedness of inverse source problem has been proved by Isakov [19,20]. Certain existence, uniqueness and conditional stability questions for various inverse source problems have been analyzed in [3,8–11,28,31,34,38]. An adjoint problem approach for the inverse source problems related to linear and nonlinear parabolic equations has been proposed in [4–6]. This approach is then devel- oped in [10,15,17], where, in particular, Fréchet differentiability of the cost functional and Lipschitz continuity of its gradient is proved. Some uniqueness theorems for inverse spacewise dependent source problems related to nonlinear parabolic and * Corresponding author. E-mail address: alemdar.hasanoglu@izmir.edu.tr (A. Hasanov). 0168-9274/$36.00  2013 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apnum.2013.11.006