Research On Regularization Methods And Algorithms For Inverse Problems Of Three Kinds Of Fractional Partial Differential Equations

Fractional derivatives and integrals provide a great tool for describing phenomena of non-locality and memory features.Fractional derivatives and fractional differential equations have also been widely applied in many scientific fields such as biology,physics,chemistry,and so on So far,there are many forms of fractional integrals and derivatives,such as Riemann-Liouville,Gr¨unwald-Letnikov,Riesz,Caputo,Hadamard and Caputo-Fabrizio et.al.This article considers the inverse problem of three types of fractional derivative partial differential equations.In Chapter 2,the inverse problem for identifying the unknown source on time fractional diffusion equation with Caputo-Hadamard derivative is considered.This problem is ill-posed and its solution stability is restored by regularization method.Firstly,we prove that the problem is ill-posed,and give the conditional stability result and its the optimal error bound.Secondly,the fractional Landweber iterative regularization method and fractional Tikhonov regularization method are used to solve the problem,and the convergence error estimates under the priori and posteriori regularization parameter selection rules are given respectively.Finally,we use numerical examples to show the effectiveness of the two regularization methods and compare them.In Chapter 3,the inverse problem for identifying the unknown initial value on time fractional diffusion equation with Caputo-Hadamard derivative is considered.Analyze the ill-posedness of the problem,and give the conditional stability result and optimal error bound.The Quasi-boundary regularization method and fractional Landweber iterative regularization method are used to solve the ill-posed problem,and the error estimates of the exact solution and its regularization solution under the priori and posteriori regularization parameter selection rules are given respectively.Finally,we use numerical examples to show the effectiveness of the two regularization methods.In Chapter 4,the inverse problem of initial values identification for diffusionwave equation with space-time fractional derivatives is studied.Analyze the illposedness of the inverse problem and give the conditional stability result.The Tikhonov regularization method is used to restore the stability of the solution,and the error estimates between the regularization solution and the exact solution are given under the priori and posteriori regularization parameter selection rules,respectively.Finally,numerical examples show that the Tikhonov regularization method is effective for solving such inverse problems.In Chapter 5,we study the inverse problem of identifying source term and initial value simultaneously for the time-fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator.Firstly,we prove that the problem is ill-posed and give the conditional stability result.Then,the Tikhonov regularization method is used to solve the ill-posed problem,and the error estimates between the exact solution and the regularization solution are given under the priori and posteriori regularization parameter selection rules.Finally,we discretize the fractional derivative and give numerical examples to show that the Tikhonov regularization method is effective for solving such inverse problems.In Chapter 2 and Chapter 3,we use two regularization methods to restore the stability of the solution,and compare them.In Chapter 3 and Chapter 4,the Tikhonov regularization method is used to solve ill-posed problems.The inverse problem of identifying two unknown terms simultaneously studied in Chapter 4 is a new topic,which is worth in-depth study.