Inverse Problems For The Variable-Order Time Fractional Diffusion Equation With Variable Coefficient

In recent decades,more and more scholars have devoted themselves to the practical application of fractional calculus.Many scholars have done a lot of research on the anomalous diffusion problem in complex media.They found that the fractional order of the equations of some important dynamic processes will change with time or space,which means that variable fractional differential equations will likely provide an effective mathematical idea for these complex dynamic problems.The objective of this thesis is to deal with numerical solution and the related inverse problems for the variable-order time fractional diffusion equation with variable coefficient.Chapter 1 introduces significance of the topic,and briefly describes the current research and development trend of the research field,and gives the research motivation and main work.Chapter 2 mainly introduces the concept and relationship of variable fractional derivatives,and gives the Legendre polynomials and their properties which are utilized in the subsequent chapters.Chapter 3 deals with the 1D variable-order time fractional diffusion equation with variable coefficient.The variable fractional derivative is discretized to create a difference scheme and numerical solutions of the forward problem are obtained.Assuming that the diffusion coefficient and the variable fractional order are unknown,the observations at one inner point of the region are used as additional data,and the homotopy regularization algorithm is applied to solve the inverse problem with the orthogonal basis of Legendre polynomials.The inversion results with exact and noisy data are presented,and the impacts of the additional data and the noise levels on the inversion algorithm are discussed.In chapter 4,the 2D variable-order time fractional diffusion equation with variable coefficient is considered.The finite difference scheme for solving the forward problem is proposed similar to the 1D case,and then numerical inversions for the fractional order and the diffusion coefficients are performed respectively also by the homotopy regularization algorithm.Numerical examples are presented with noisy data under several cases and numerical stability of the inversion algorithm is testified.