High-order Numerical Methods For Fractional Differential Equations

Fractional calculus is as old as the classical calculus, which can be dated back to17th century. In recent years, fractional differential equations have been found useful and applicable in many fields of science and engineering. In this thesis, we investigate the spectral approximations to the fractional integral and the Caputo derivative, and explore high-order numerical algorithms for time-fractional subdiffusion equations.Firstly, we propose efficient formulas for calculating the fractional integral of the Ja-cobi orthogonal polynomials based on the three-term recursive formula of the Jacobi poly-nomials and their properties. The derived method is unconditionally stable. The formula for calculating the Caputo derivative is obtained from the formula for the fractional inte-gral of the Jacobi polynomials. The fractional differential matrix is also obtained with its spectral radius being numerically studied. The spectral collocation methods based on the fractional differential matrix are presented to solve the initial and boundary value problems. Sufficient numerical examples are presented to verify the derived algorithms and the com-parison between other methods are made, which show good performances of the present methods.Secondly, we propose two fully discrete finite element algorithms for the βth-order Caputo type time-fractional subdiffusion equation with the time being discretized by the fractional linear multistep methods, where β∈(0,1). Theoretical analysis shows that the two methods are unconditionally stable. The global convergence order and the average convergence order in time are (1-β) and (1.5-β), respectively. The optimal error estimates in the sense of L2norm in space are also obtained. The two methods are reduced to the classical Crank-Nicolson (CN) method when βâ†'1. Then, four improved algorithms with unconditional stability are proposed such that the global convergence order in time are up to2. Numerical examples are presented to verify the theoretical analysis, which shows that the numerical results are somewhat better than the theoretical results. The comparisons are made between the present methods and the existing ones, which show better performances of the present methods.Thirdly, we propose a new type CN finite element scheme for the (1-β)th-order Riemann-Liouville type time-fractional subdiffusion equation, where β∈(0,1). The new time discretization based on the nonuniform time grids is used to discretize the time-fractional Riemann-Liouville derivative, which is similar to that of the classical L1method. The new CN finite element scheme is unconditionally stable and convergent of order (1+β) in time. The new method is reduced to the classical CN method when βâ†'1. The new time discretization is also used to discretize the fractional cable equation, which leads to the unconditionally stable numerical algorithms.At last, the alternating direction implicit (ADI) algorithms for the two-dimensional βth-order Caputo type time-fractional subdiffusion equation are studied, where β∈(0,1). Based on the time discretization introduced in Chapter3, six fully discrete ADI algorithms are presented. All the algorithms are unconditionally stable and convergent of order from min{1-β,1+β} to (1+β), which is better than the existing ones. The optimal error estimates in the L2norm in space are obtained. Numerical experiments are presented to verify the theoretical analysis, and the comparisons between the present methods and the existing ones are made to show better performances of the present methods.