Stability Estimate And Reconstruction For Inverse Problems Of Time-fractional Diffusion Equation

The study of fractional diffusion equation arises from modeling of a class of anomalous diffusion phenomena with long-range correlation and historical memory effect, which has a wide range of practical applications in many applied sciences, such as porous system model, viscous fluid, turbulent diffusion, quantum optics, etc. Hence, it is significant and has broad prospects in theory and practical applications to study the related forward and inverse problems. However, due to the fractional differential usually does not satisfy the basic properties of the classical differential, such as Lebnizi formula, integration by parts, the chain rule, semigroup nature, etc. This causes great difficulties for the study of fractional differential equations. In recent years, there exists a wealth of results on the study of the forward problem. However, the inverse problem is in its infancy, concentrated in certain regularization theory of the inverse problem and numerical methods. For the conditional stability of the inverse problems which is significant in theory and numerical implementation of inverse problems, there only exist the conditional stability results for the Cauchy problem and inversion of the coefficient for zero-order derivative term.In this thesis, we mainly discuss the conditional stability for identifying the diffusion coefficient, ie, the coefficient for second-order derivative term via some additional conditions, and the regularization by projection method for a backward problem of the time-fractional diffusion equation. Based on the iterative regulariza-tion method, we derive stable inversion of the diffusion coefficient. Under appropri-ate regularity assumptions of the exact solution, a uniform error estimate with an optimal convergence rate between the reconstructed solution and the exact one is obtained for a priori parameter choice strategy. Based on the balancing principal, we also obtain the optimal convergence rate of the regularized solution for a posteri-ori parameter choice strategy which does not involve the knowledge of regularity for the exact solution. Finally, numerical examples are presented to illustrate a poste-riori parameter choice strategy shows better results than a priori parameter choice strategy and gives a stable numerical inversion of the exact solution.First of all, a brief introduction of research background on fractional diffusion equations is given in Chapter 1, including the history and development of fractional calculus, the fractional diffusion equation based upon the continuous-time random walk model, and the research status of the forward and inverse problems in fractional diffusion equation and the contents of the thesis.Some preliminary knowledge which will be involved is given in Chapter 2, in-cluding the introduction of inverse problems and regularization theory, Carleman estimate and its application, and some definitions, lemmas regarding to fractional calculus.In Chapter 3, by introducing a new scalar product into H4(Ω), we improve the regularity of the solution for the fractional diffusion equation based on the regu-larity for elliptic equation and the analytical solution from eigenfunction expansion up to H1(δ0, T; H2(Ω) ∩ H01(Ω)) ∩ L2(δ0, T;H4(Ωf2) n H01(Ω)). Under this regularity, by transforming the fractional diffusion equation to a parabolic equation of order 4 in the space variable, we prove a Carleman estimate which includes the forth-order derivative with respect to space variable for general 1-dimensional fractional diffusion equation of half order with variable coefficient. Finally, we give a brief introduction of some basic numerical methods for forward problem of fractional diffusion equations.To identify the diffusion rate, i.e., the heterogeneity of medium, we consider an inverse coefficient for second-order derivative term problem utilizing finite mea-surements. Based on the Carleman estimate we established in Chapter 3, we prove a local Holder type conditional stability via two Carleman estimates for the corre-sponding differential equations of integer orders. By using iterative regularization method and the finite difference scheme for initial boundary value problem of time fractional diffusion equation in Chapter 3, we give the stable inversion of the diffu- sion coefficient.In Chapter 5, we investigate a backward time fractional diffusion equation. Ap-plying the regularization by projection method, for higher regularity assumptions of the exact solution, a uniform Holder type error estimate with an optimal con-vergence order is obtained for a priori a parameter choice rule. Under a posteriori parameter choice rule which is based on the balancing principal, we give the same order optimal error estimate between the reconstructed solution and the exact one. Although a posteriori parameter choice rule does not involve the regularity proper-ties of the exact solution, however, by the comparison of the numerical results, we find that a posteriori parameter choice strategy shows better results than a priori parameter choice strategy and gives almost optimal inversion results. Even in some cases of failure of a priori parameter choice strategy, a posteriori parameter choice strategy can give relatively accurate results.