Fast Numerical Approximation For Nonlocal Diffusion Problem

Nonlocal theory is proposed because the classical theory can not be used to treat the fracture, faults, fractures and so on exotic materials and deformable material.In recent years, nonlocal model has been developed rapidly in the fields of nonlocal diffusion, the continuum theory and image processing, however, especially nonlocal diffusion model has been a period of development, which has been able to highlight its effectiveness in a number of areas, including composite materials fracture and crack instability, fracture and polycrystalline nanofiber networks. Because nonlocal diffusion model itself is global, computation is a very difficult problem, this motivates that we need to find effective numerical algorithms and pre-processing methods.The paper is consists of the following chapters.The first chapter briefly reviews the history and current situation of fractional derivative and existing numerical algorithms, the physical background and development status of nonlocal theory, and summarizes the links and differences between nonlocal theory and the classical theory.In chapter 2, describes the definitions and related properties of fractional order derivative, and the definition and essential properties of Mittag-Leffler function.Chapter 3 gives nonlocal diffusion problems and the definition of nonlocal operator, and spatial semi-discretization to the nonlocal diffusion problem by finite difference, then it leads to a system of differential equations, at the same time discusses the nature of coefficient matrix.In chapter 4, takes into consideration the Mittag-Leffler representation of fractional order differential equations, and analyzes convergence detailly. Applies shift-invert Lanczos pre-processing method to deal with the product of the Mittag-Leffler representation of matrix with a vector, and theoretically proves the effectiveness of their computing time. Finally, convergence and calculated amount are verified by numerical experiments.Fifth chapter exploits the exponential quadrature rule to solve ordinary differential equation systems, and simultaneously the product of matrix exponential with a vector is handled by shift-invert Lanczos pre-processing method. Numerical results illustrate that the computing time by exponential quadrature rule is less than implicit scheme.