Identification Of Source Term In A Multi-term Time-fractional Diffusion Equation

The fractional diffusion equations have more practical significance in describing some complex transport diffusion mechanisms,such as the anomalous diffusion behavior in highly heterogeneous media and the behavior of complex viscoelastic materials.Inverse problem based on fractional diffusion equation is a popular topic,At present,the theory of dealing with the ill-posedness of inverse source problem of the fractional diffusion equation has become mature.As a special case of the timefractional diffusion equation,the multi-term time-fractional diffusion equations have the characteristics of higher precision and wider application background in describing anomalous diffusion phenomena.However,due to the complexity of the analytic solution and the ill-posedness of the inverse source problem,there are a few studies on using of regularization methods to solve the inverse source problem of the multi-term time-fractional diffusion equation.In this paper,the source term identification problem of multi-term time-fractional diffusion equation based on terminal measurement data is discussed,that is to determine a space-dependent source term in the multi-term time-fractional diffusion equation from a noisy final data.The uniqueness and a conditional stability of the inverse source problem are derived based on expression of forward problem solution and some properties of the multinomial Mittag-Leffler function.Furthermore,the quasi-boundary regularization method and the landweber regularization method are provided to deal with the inverse source problem,and two convergence estimates are obtained by using an a priori and an a posteriori regularization parameter choice rule.In addition,in terms of numerical reconstruction,with the help of finite difference method,finite element method,Galerkin spectral method to solve the numerical solution of multi-term time-fractional diffusion equation,Galerkin spectral method is extended as a regularization method to deal with the inverse source problem of multi-term time-fractional diffusion equation.This method is also known as the projection regularization method.Finally,the effectiveness and feasibility of each regularization method are verified by numerical examples.The research of this paper will provide theoretical basis for pollution source detection in heterogeneous media.