Local Discontinuous Finite Element Methods For The Variable-order Mobile-immobile Advection-Dispersion Equation And The Caputo-type Variable-order Reaction Diffusion Equation

Fractional calculus is the development and extension of integral calculus,and its practical significance has attracted the wide attention of many scholars.Fractional calculus is widely used in many fields such as fluid mechanics,optical and thermal systems,signal and image processing,industrial production and technology,fractal and dispersion in porous media,etc.The fractional integral operator is nonlocal and weakly singular,which makes it more beneficial for modeling anomalous diffusion and transport dynamics processes in various complex systems.In this paper,the fully discrete local discontinuous Galerkin method for Coimbra variable fractional-order mobile-immobile advection-dispersion equation and Caputo-type variable-order reaction diffusion equation is studied,including the construction of numerical schemes,stability,error estimation and specific numerical experiments.The structure of the paper is as follows:In Chapter 1,fractional calculus and local discontinuous Galerkin method are introduced briefly.Secondly,the research status of variable-fractional differential equations is given.Finally,some preparatory knowledge is introduced.In Chapter 2,we present a fully discrete local discontinuous Galerkin method for solving the variable-order mobile-immobile advection-dispersion equation with Coimbra fractional order derivative operator.The method proposed in this chapter is based on the finite difference method in time and the local discontinuous Galerkin method in space.We prove that the scheme is unconditionally stable and convergent by selecting appropriate numerical flux,and the convergence order is O(Δt+hk+1).Finally,the results of the theoretical analysis and the convergence of the scheme are verified by numerical experiments.In Chapter 3,we construct and analyze a numerical method for solving the reaction diffusion equation with Caputo-type variable-order.Based on the generalized numerical fluxes,space and time discretization,we obtain a fully discrete local discontinuous Galerkin scheme.After detailed theoretical analysis,we prove that the scheme is unconditionally stable and converget with O(hk+1+(Δt)2-α).Finally,numerical experiment is given to verify the validity of the method.