Biological applications and numerical solution based on Monte Carlo method for a two-dimensional parabolic inverse problem Morteza Ebrahimi a,b, * , Rahman Farnoosh b , Somayeh Ebrahimi c a Department of Mathematics, Slamic Azad University of Karaj Branch, P.O. Box 31485-313, Karaj, Iran b School of Mathematics, Iran University of Science and Technology, Narmak, Tehran-16846, Iran c Department of Biology, Faculty of Sciences, Islamic Azad University of Science and Research unit, Tehran, Iran Abstract A numerical algorithm involving the combined use of the finite difference method and Monte Carlo method is proposed as a solution algorithm for a two-dimensional parabolic inverse problem with an unknown boundary condition. The algo- rithm is based on the discretize governing equations by finite difference method. Owing to the application of the finite dif- ference method, some large sparse systems of linear algebraic equations are obtained. An approach of Monte Carlo method is employed to solve the linear systems. The Least squares scheme is proposed to modify unknown boundary con- dition. Furthermore two applications of the present problem in Biological systems are provided. Numerical test is per- formed in order to show the efficiency and accuracy of the present work. Ó 2007 Published by Elsevier Inc. MSC: Primary 35R30, 78A70; Secondary 65M06, 65C05 Keywords: Parabolic inverse problem; Finite difference method; System of linear algebraic equations; Monte Carlo method; Least squares scheme; Complexity; Biological applications 1. Introduction Mathematically, inverse problem belong to the class of ill-posed problems. That is, their solution does not satisfy the general requirement of existence, uniqueness, and stability under small changes to the input data. To date various methods have been developed for the analysis of the parabolic inverse problems involving the estimation of boundary condition from measured temperature inside the material [1–7]. However, most ana- lytical and numerical methods were only employed to deal with one-dimensional inverse problems. Few works were presented for two-dimensional parabolic inverse problems because the difficulty of these problem was more pronounced [2]. The literature reviews showed that Monde [3], Busby and Trujillo [4], Imber [5], Yang 0096-3003/$ - see front matter Ó 2007 Published by Elsevier Inc. doi:10.1016/j.amc.2007.10.048 * Corresponding author. E-mail address: mo_ebrahimi@kiau.ac.ir (M. Ebrahimi). Available online at www.sciencedirect.com Applied Mathematics and Computation 204 (2008) 1–9 www.elsevier.com/locate/amc