The Finite Element Algorithms For Three Types Of Fractional Partial Differential Equations

In this dissertation, the theory analysis for the fractional calculus and the finite element algorithms for three types of fractional partial differential equations are stud-ied. The first part focuses on the fractional integrability and differentiability of a given function, the second part deals with the finite element algorithms for the generalized nonlinear space fractional Fokker-Planck equation, the nonlinear time-space fractional differential equations with subdiffusion and superdiffusion, and the time-space frac-tional telegraph eqaution.In details, chapter II is devoted to discussing the fractional integrability and dif-ferentiability of the considered function, in the senses of Riemann-Liouville integral, Riemann-Liouville derivative and Caputo derivative, respectively. Important issues on these fractional integral and derivatives are also included.Chapter III is to formulate a fully discrete scheme to numerically solve the gener-alized nonlinear space fractional Fokker-Planck equation, which can be used to describe the Levy flights. The error estimates for the fully discrete scheme are derived in details. The numerical examples are also included which agree with the theoretical analysis.Chapter IV is to propose a new fractional finite element method for the nonlinear time-space fractional differential equations with subdiffusion and superdiffusion. The semi-discrete and fully discrete numerical approximations are both analyzed. In spatial direction, we use the fractional finite element method, and in temporal direction, we use the fractional finite difference methods. For the subdiffusion problem, we use the fractional Euler backword difference method. For the superdiffusion problem, we use the fractional center difference method. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are also included to confirm the theoretical analysis. During our simulations, an interesting fractional diffusion phenomenon of particles is also observed.Chapter V is to numerically study the time-space fractional telegraph equation, which has a multi-fractional order characteristically describing the random walks of the individual particles in the suspension flow, and the anomalous diffusion during the transmission of the voltage wave or the current wave. In temporal direction, we use the fractional difference method, including the fractional Euler backword difference scheme and the fractional center difference scheme. In spatial direction, we use the fractional finite element method. We derive the semi-discrete scheme and the fully discrete scheme separately for the considered equation. We prove the existence and uniqueness of the discrete solution and then derive the error estimates. The numerical results are inline with the theoretical results.