Studies For Simultaneous Determination Of Source Term And Initial Value In The Diffusion Equation By Two Regularization Methods

For integer order diffusion equation Source term and initial value depends on the spatial variable is an important ill-posed problem in the inverse problem.As a result of the classical regularization method can not deal with the inverse problem of determining source term and initial date simultaneously in a parabolic equation Hence,some special skills are used to solve this problem.In this paper,we investi-gate the inverse problem of determining source term and initial date simultaneously in a parabolic equation where data is given at a fixed time.We can get the exact solution of the problem by the measured data of two fixed time for a heat conduc-tion equation.For ill-posedness of problem,using the two kinds of regularization method to solve it,i.e.,the Mollification regularization method and the Modified quasi-reversibility regularization method.At the same time,the fractional diffusion equation is studied by using the Modified Quasi-reversibility regularization method,and the optimal error estimation is given.Next,the simplified method is used to split the original equation.For ill-posedness of problem,using the two kinds of regularization method solve it.Finally,We give the selection of regularization pa-rameters for the proposed methods and some numerical examples including smooth and non-smooth functions to show that this method is effective and feasible.