The Finite Element Methods For Caputo-type Evolution Equation With Fractional Laplacian

A physical concept underlying the notion of the fractional Laplacian is the fractional diffusive flux,describes an anomalous diffusions due to long-range interactions executed by Levy flights.It has become an important and frontier research topic in fractional partial differential equations.At present,two challenging tasks usually involved in its numerical treatment:the handling of highly singular kernels and the need to cope with an unbounded region of integration.Thus,this dissertation mainly focuses on the following problems:1.We will solve the extended local problem instead of dealing with the nonlocal operator by adopting Caffarelli-Silvestre lifted technique,and construct finite element scheme.The cor-responding of convergence and error estimates will be obtained.2.The finite-part integral in the sense of Hadamard of principal-value integral will be constructed,and the finite element method is used to solve the equations.This dissertation aims at the study on the finite element methods for Caputo-type evolu-tion parabolic equation with fractional Laplacian.The main five contents of this dissertation are given as follows:(1)In the first chapter,consider a one-dimensional random walk dispersion model un-der variable intuitive reflect microscopic particle motion process,the physical meaning of fractional Laplacian is given,and mathematical definition of the principal value fractional Laplacian is presented.(2)Chapter 2 is devoted to introduce Caffarelli-Silvestre extension and the property of the modified Bessel function of the second kind.Finally,the decay of solution by extension technique are considered.(3)Chapter 3 adopts the extension technique to construct the finite element fully dis-cretization scheme,the time discretization is based on the Diethelm method.The weighted elliptic projector and corresponding error estimates are considered.Some numerical exam-ples are given for verification of our theoretical analysis.(4)In chapter 4,we prove the equivalence between time fractional derivative and its finite-part integral,and adopt the corresponding finite-part integral to deal with the spatial principal value of fractional Laplacian.The finite element scheme is established to solve the equation,numerical results illustrate the effectiveness of the scheme.(5)In chapter 5,the properties of long-term time fractional derivative in weighted space are considered.The finite element error estimations for long-term integration are studied.