Mathematics and Computers in Simulation 54 (2000) 159–167 Numerical procedures for determining unknown source parameter in parabolic equations Afet Fatullayev a,∗ , Emine Can b a Baskent University, Applied Science High School, Ankara, Turkey b University of Kocaeli, Department of Mathematics, 41100 Anitpark, Izmit, Kocaeli, Turkey Received 1 August 2000; accepted 11 August 2000 Abstract Numerical procedures for the solution of an inverse problem of determining unknown source parameter in a parabolic equation are considered. Two different numerical procedures are studied and their comparison analysis is presented. One of these procedures obtained by introducing transformation of unknown function, while the other based on trace type functional formulation of the problem. © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Parabolic equations; Inverse problems; Unknown source parameter; Finite difference method; Additional specification 1. Introduction In this paper, we study numerical procedures for the following inverse source problem for a parabolic equation: Find u = u(x,t) and p = p(t) which satisfy u t = u xx + p(t)u(x,t) + f(x,t), (x,t) ∈ Q 0 = (0, 1) × (0,T ], (1) u(x, 0) = ϕ(x), x ∈ [0, 1], (2) u(0,t) = µ 1 (t), u(1,t) = µ 2 (t), t ∈ (0,T ], (3) subject to the additional specification u(x ∗ ,t) = E(t), 0 ≤ t ≤ T. (4) where f(x,t), ϕ(x), µ 1 (t), µ 2 (t) and E(t) are known functions, and x ∗ is a fixed prescribed interior point in (0, 1). If u is a temperature then (1)–(4) can be regarded as a control problem finding the control p(t) ∗ Corresponding author. Fax: +90-262-3393106. E-mail address: afet@baskent.edu.tr (A. Fatullayev). 0378-4754/00/$20.00 © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0378-4754(00)00221-4