Source Term Identification Of Time-Fractional Diffusion/Wave Equations With Robin Boundary Condition

In recent years,the inverse problem of mathematical physical equation has attracted the attention of many scholars at home and abroad.To the best of our knowledge,there are abundant results under Dirichlet or Neumann boundary conditions,while few results under Robin boundary condition.Inspired by that,in this paper,source term identification of time-fractional diffusion/wave equations with Robin boundary condition is considered.This kind of problem is ill-posed,so regularization methods are needed to obtain regularized approximate solutions.In Chapter 3,we first study the source term identification of time-fractional diffusion equation under Robin boundary condition(0<α<1).That is,for given r(t)and Z(x,t),one shall identify the source term f(x)in the variable separation form of F(x,t)=r(t)f(x)+Z(x,t).For this problem,we analyze the ill-posedness and give the conditional stability result of the problem.In addition,we use four regularization methods to obtain regularized approximate solutions,which are classical Landweber iterative regularization method,fractional Landweber iterative regularization method,TSVD method,combining TSVD method and fractional Landweber iterative regularization method.For these four methods,we obtain the error estimates of the exact source term and the regularized approximate solutions under the a-priori and a-posteriori parameter selection rules.In the numerical part,we verify the effectiveness of the four regularization methods through several examples of different types.On the basis of Chapter 3,we extend the diffusion equation to the diffusion wave equation in Chapter 4,then the source term identification of time-fractional diffusion wave equation under Robin boundary condition is considered(1<α<2).Similar to the previous chapter,we adopt four regularization methods to deal with this problem,all of which obtains their regularized approximate solutions.Due to the change of value range of α,the solution of the time-fractional diffusion wave equation no longer has good properties.We have difficulty in solving the solution in view of redefining the regularized approximate solutions.Holder type error estimates of exact source term and regularized approximate solutions are also obtained.Through several numerical examples,we can find that the four methods are effective in solving the source term identification of time-fractional diffusion wave equation.