%0 Journal Article
%T A Point Source Identification Problem for a Time Fractional Diffusion Equation
%A Xiao-Mei Yang
%A Zhi-Liang Deng
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/485273
%X An inverse source identification problem for a time fractional diffusion equation is discussed. The unknown heat source is supposed to be space dependent only. Based on the use of Green¡¯s function, an effective numerical algorithm is developed to recover both the intensities and locations of unknown point sources from final measurements. Numerical results indicate that the proposed method is efficient and accurate. 1. Introduction Let be a bounded domain in and let be the boundary of . Consider the following time fractional diffusion process: where is the uniformly elliptic operator, is the outward normal at the boundary , and , are known constants which are not simultaneously zero. Here, stands for the Caputo fractional derivative operator of order defined by where is the standard -function and the prime denotes the general derivative. From the last few decades, fractional calculus grabbed great attention of not only mathematicians and engineers but also many scientists from all fields (e.g., see [1¨C4]). Fractional diffusion equations describe anomalous diffusions on fractals (physical objects of fractional dimension, like some amorphous semiconductors or strongly porous materials; see [5, 6] and references therein). Indeed, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main superiority of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. For the detailed theory and application of fractional calculus, one can refer to [1¨C4] and references therein. Not only have differential equations of fractional order attracted people¡¯s attention, but also theories and applications related to physics and geometry of fractal dimension have been well studied (e.g., [7¨C11]). If the initial condition is nonhomogeneous, that is, , we are always able to simplify the system (1) into two components; that is, , where solves the homogeneous equation with nonhomogeneous initial condition and satisfies the nonhomogeneous equation with homogeneous initial condition. As we know, the initial value/boundary value problem associated with is well-posed and there exist many works on such forward problem, for example, [12, 13]. In the following, instead of nonhomogeneous initial condition, we only focus on the system (1) with homogeneous initial condition. Ordinarily, when is a known function, we are asked to determine the solution function so as to satisfy (1). So posed, this is a direct problem. However, the
%U http://www.hindawi.com/journals/amp/2013/485273/