Some Approximate Algorithms For Partial Differential Equations Of Fractional Order

Fractional calculus equation (FCE) is the natural mathematical promotion of classic calculus equation. FCE has profound physical background and rich theoretical connotation, which has been successfully applied to problems in physical, biological, chemical, and other disciplines. The research of FCE has important academic research value, and has the broad application prospect in engineering. At present, the research of fractional calculus equation has become the hot topic. In this paper, we will study several approximate al-gorithms of fractional partial differential equation (FPDE)(including the ap-proximate analytical algorithm and numerical algorithm).The paper mainly includes three parts:In the first part, we will study the Homotopy analysis method (HAM) in solving FPDE in order to obtain the approximate analytical solution; In the second part, we will study the applications of Local discontinuous Galerkin method (LDG) in solving one-dimensional FPDE; In the third part, we will study the application of Finite element method (FEM) in solving two-dimensional FPDE.In the first part, we mainly study the application of HAM in solving space fractional advection-dispersion equation and time-space fractional diffusion e-quation. The key research of this part is how to structure the scheme and study the validity of it. The numerical experiment shows that the HAM is effective in getting the approximate analytical solution of FPDE. In the sec-ond part, we mainly study the application of LDG in solving time fractional Tricomi-type equation and time fractional Fisher equation. By constructing an implicit, full discrete LDG scheme, we want to get the numerical solu-tion of the above equation. The theoretical analysis shows that the scheme is stable, at the same time we give the detailed error estimation. The numer-ical experiment shows that the full discrete LDG scheme is effective. In the third part, we mainly study the application of FEM in two-dimensional time fractional diffusion equation and time fractional Tricomi-type equation. We will use the finite difference in time direction and the FEM in space direction to structure an unconditional stable and convergent scheme. The theoretical analysis shows that the scheme is stable, at the same time we give the detailed error estimation. The numerical experiment shows that the full discrete finite element scheme is effective.