A Modification Regularization Method For A Backward-Problem Of The Time-fractional Parabolic Equation

We always study the classic time-fractional parabolic equation and the research of this kind of equation is more perfect. However,not many, on the study of this kind of problem for the inverse problem in many engineering problems are often used to this kind of inverse problem. For example, that the inside of the physical problems, such as temperature, density, for such a problem only through one point measurements to inversion. This paper mainly studies the 0<γ<1 time fractional inverse problem: We define the fractional derivative of order 7 in the sense of Caputo:It is our aim to determine u(x,t) for t €[O, T) from the measurement data of u(·,T). The problem is ill-posed:if the solution exists, it is discontinuous rely on the known data. Therefore, we need to deal with the regularization of the problem.First of all, we assume that the initial value of u(x,0) meet the prior condition:‖u(x,O)‖Hop(O,π)< E, p> O.In addition, suppose that gδ(x) is the final measured value g(x) disturbance data satisfying ‖gδ(x)-g(x)‖L2(0,π)≤ δ.So we need to consider the following questions: To the best of our knowledge, there are few results on the backward problem of the time-fractional one up to now. An initialization of such a problem can be found in [23] where a regularization scheme inspired by the quasi-reversibility method is considered to solve the problem.Another discussion about the prob-lem can be found in [24], it proposes a regularization by projection method.In the following section, we will propose another different regularization method about the backward problem:a mollification method to deal with the ill-posed problem. And given in ‖·‖Hop estimate form under a priori information, and the way of the selection of regularization parameter is given. If we select then we obtain the following error estimate: where C=C(γ,T)·C2 and C2 depends on the fractional order γ0,γ1.Finally, we will pass the corresponding numerical example to verify the feasibility and effectiveness of the regularization format.