INPE – National Institute for Space Research São José dos Campos – SP – Brazil – July 26-30, 2010 Solution for anomalous diffusion equation with source term Marcelo T. Araujo 1 , Elso Drigo Filho 2 1Unesp, São José do Rio Preto, Brazil, marcelot-araujo@hotmail.com 2Unesp, São José do Rio Preto, Brazil, edrigof@gmail.com Abstract: In literature the phenomenon of diffusion has been widely studied, however for non-extensive systems which are governed by a stochastic nonlinear dynamics, there are few soluble models. The purpose of this paper is to present the solution of the non-linear Fokker-Planck equation for a model of potential with barrier considering a term of absorption. Systems of this nature can be observed in various chemical or biological process and its solution enriches the studies of non-extensive systems exist. keywords: Stochastic Dynamics, Nonextensive systems, Applications of Nonlinear Sciences. 1. BASIC INFORMATION A large number of phenomena can be related to kinetic processes, this is the case, for example, electrochemical reactions in metals or in membranes. In these cases, the atoms or particles that are interacting at the interface between the environments, bulk and the interior of the material. This difference between the environments in many cases leads to a diffusion process in which particles of the medium are absorbed into the structure. This absorption can occur at a constant rate or in some cases, has time dependence [1]. The Fokker-Planck equation is widely used in literature to study stochastic processes of diffusion, where one can consider the presence of forces acting on the interaction between the bodies [2]. In the problem addressed, we will make a brief review of nonlinear diffusion in a simple system consisting of two regions separated by a barrier of potential. For the study, we adopt the non-linear Fokker-Planck equation (NLFP) with an additional term time, which is interpreted as a source term or absorption that favors the passage of particles through the potential barrier. 2. PURPOSE The non-linear Fokker-Planck equation can be write as follows [3]: (1) In this equation, f(x) represents a force of interaction among the particles of the external environment with the internal and P(x,t) can be considered the density of particles of a medium. An additional term, proportional to the function μ(t) is included in equation (1). This term is related to the a source of the model. Thus, we obtain: (2) The nonlinearity of the Fokker-Planck equation is due to the exponent ν present in second derivative. Equation (1) is employed in the study of non-extensive systems, where there are long-range interactions and the thermodynamic relations serve the proposal introduced by Tsallis [4-5]. Figure 1: Representation of the potential considered with . In this paper, the function f(x) is given by derivative of potential: , (3) whose characteristic graph is shown in Figure 1. The solution for this system is of the form: (4) where D(t), β(t) and ς(t) are dependent functions of the time to be defined later. Initially, for a time equal to zero, the Ansatz above gives the distribution of the stationary Fokker-Planck equation. As we are working in non I II