1. Australian Journal of Basic and Applied Sciences, 5(12): 3323-3330, 2011 ISSN 1991-8178 Estimation of Diffusion Coefficient in Gas Exchange Process with in Human Respiration Via an Inverse Problem M. Ebrahimi Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. http://www.kiau.ac.ir Abstract: This paper is intended to provide a stochastic approach involving the combined use of the Feynman-Kac formula and Monte Carlo method as a solution algorithm for estimating the time- dependent effective diffusivity in a one-dimensional parabolic inverse problem. The inverse problem is purposed to design a mathematical model for the gas-diffusion process within the alveolar region of the human's lung. The model depends on a representative physical property of the alveolar region termed the effective diffusivity. In the present study, the functional form of the effective diffusivity is unknown a priori. The unknown effective diffusivity is approximated by the polynomial form. To modify the coefficients of the polynomial form of the unknown effective diffusivity, we introduce a deterministic optimization method based on least squares minimization. A numerical test is performed in order to show the efficiency and accuracy of the present work. Key words: Gas-exchange process, Human respiration, Inverse problem, Feynman-Kac formula, Monte Carlo method. INTRODUCTION In the present work a specific stochastic combined algorithm is used for obtaining the solution of inverse diffusion equation as part of a parabolic inverse problem that arise in gas-diffusion process within the alveolar region of the lung during human respiration. This algorithm uses Feynman-Kac formula to represent the solution of the parabolic inverse problem at a point as the expected value of functionals of Brownian motion trajectories started at the point of interest. We are interested to have solution of a parabolic inverse problem without any need to discretize the problem domain. For many problems described by partial differential equations (PDEs) such solutions are delivered by the so-called Feynman-Kac formulas (Csaki, E., 1993; Modeste, N., 2006; Budaev, B.V. and D.B. Bogy, 2003). The literature reviews showed that E. Csaki (1993) have applied Feynman- Kac formula to solve initial-boundary value problems. In (Csaki, E., 1993) a discrete Feynman-Kac formula has employed for linear parabolic PDEs with zero boundary conditions. Modeste et al. (2006) have used a kind of nonlinear Feynman-Kac formula to give a probabilistic interpretation of the solutions of parabolic quasi-linear PDEs. Budaev and Bogy (2003) presented a probabilistic approach to systems of PDEs on the basis of the well- known Feynman-Kac formula. To date various methods have been developed for the analysis of the parabolic inverse problems involving the estimation of boundary condition or diffusion coefficient from measurement inside the material (Shidfar, A., 2009; Wang, J. and N. Zabaras, 2004; Shidfar, A., 2007; Shidfar, A., 2006; Shidfar, A., 2006; Farnoosh, R. and M. Ebrahimi, 2010; Dehghan, M., 2005). Shidfar et al. (2009) have applied an algorithm based on conjugate gradient method to estimate the unknown time dependent melt depth during laser material processing in liquid phase. In this article the determination of the melt depth is treated as a one- dimensional, transient, inverse heat conduction problem. Numerical procedure shows a good agreement with experimental and analytical results. In (Wang, J. and N. Zabaras, 2004), Wang and Zabaras have used a Bayesian inference approach to the inverse heat conduction problem. Their work is based on Monte Carlo method and experiment results show a good estimation on the linear inverse problems in two dimensions. Shidfar et al. (2007) have used an accurate and stable numerical algorithm based on finite difference method to solve an inverse parabolic problem in one dimension. To our best knowledge the problem of gas-exchange in human respiratory system with unknown time dependent diffusion coefficient has not been studied. Furthermore, according to latest information from the research works it is believed that the solution of inverse problem based on stochastic algorithm included the Feynman-Kac formula has been investigated for the first time in the present study. Description of the Problem: Formulation of the direct and inverse problem is given as follows: I) Direct Problem: The mathematical formulation of a one dimensional linear parabolic problem is given as follows: Corresponding Author: M. Ebrahimi, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. E-mail: mo.ebrahimi@kiau.ac.ir.ebrahimi@kiau.ac.ir 3323