Numerical Solution Of Two Inverse Problems For A Time-fractional Diffusion Equation

In recent years,the time fractional diffusion equation has received extensive attention and research,the fractional derivative is a kind of non-local operator,which means that the equation has "memory",the rate of change of state not only depends on the current situation,but also affected by the past,while the integer order diffusion equation,the rate of change of state only depends on the current situation.On this account,compared with the integer diffusion equation,the time-fractional diffusion equation is more suitable to describe the abnormal diffusion in some special materials.Hence,the study on the inverse problem of time fractional diffusion equation has not only theoretical significance,but also has a wide application prospect.In this paper,two kinds of inverse problems of time fractional diffusion equation are studied.The goal is to obtain the numerical solution of time fractional diffusion equation based on two different additional conditions and determine the time-dependent control parameters in the equation.In order to solve these two kinds of inverse problems numerically,the strategy adopted in this paper is to discretize the problem in the spatial direction by using the high-precision Legendre spectral collocation method,and use the formula to approximate the time fractional derivative.For the ill-posedness of the inverse problem,the mollification method is used to regularize it,and finally a stable numerical scheme for solving the inverse problem is obtained.Finally,some numerical examples are given to illustrate the stability and effectiveness of the algorithm.It can be seen from the numerical results that the proposed algorithm has strong stability,and can still get relatively accurate numerical solutions even if the input data contains large noise.