Determine Of Order For The Time-fractional Diffusion Equation

In this paper, we consider an inverse problem to determine the unknown order of Caputo fractional derivative of a multi-term time fractional diffusion equation with a Robin boundary condition. The implicit numerical method is employed to solve the direct problem, the stability and convergence of the proposed method are discussed. For the inverse problem, we first give some important properties of the multi-term Mittag-Leffler function, by using these properties we obtain the formulae of reconstructing the orders of time fractional derivative in the fractional diffusion equation by time history at one fixed spatial point. At last, we show numerical tests for our reconstruction formula.