High Accuracy Algorithms For Fractional Differential Equations

fractional differential equations have been proved to be a valuable tool in the model-ing of many phenomena, especially in physics, mechanics, biology, engineering, finance, hydrology, and fractional-order controllers. In recent years, due to its profound physical background, the subject of the fractional differential equations is gaining much importance and attention. Generally, most fractional differential equations can not be resolved analyti-cally. Therefore, the research of numerical method of the fractional differential equation is of important theory significance and practical value. In this thesis, we consider the high order approximation method for fractional ordinary differential equations and fractional par-tial differential equations.In Chapter1, we mainly review the development history and research background of fractional calculus and fractional differential equations, and the main work of this paper is summed up. For the sake of convenience, the basic definitions and properties of fractional derivatives are presented at the same time.In Chapter2, we concerns with the numerical solution of nonlinear fractional integro-differential equation (FIDE). A hybrid collocation method is used which combines a non-polynomial collocation used on the first subinterval and graded piecewise polynomial col-location used on the rest of the interval. A theoretical analysis for the convergence order of the method is presented.In Chapter3, we propose and analyze a spectral Jacobi-collocation method for the numerical solution of general linear fractional integro-differential equations. First, we use some function and variable transformations to change the equation into a Volterra integral equation defined on the standard interval [-1,1]. Then the Jacobi-Gauss points are used as collocation nodes and the Jacobi-Gauss quadrature formula is used to approximate the integral equation. Later, the convergence order of the proposed method is investigated in the L∞norm.In Chapter4, singular initial value problems of the Lane-Emden type in the fractional-order ordinary differential equations has been studied. We have applied a spectral Jacobi-collocation to solve the Lane-Emden type equations. The method changes solving the equa-tion to solving a Volterra integral equations. The Jacobi-Gauss points are used as collocation nodes and the Jacobi-Gauss quadrature formula is employed in Volterra terms. For the spec-tral Jacobi-collocation method, a rigorous error analysis in both the L∞and weighted L2norms is given.In Chapter5, we consider a numerical approximation scheme for solving the nonlinear Stokes’first problem for a heated generalized second grade fluid with fractional derivative. A high order finite difference/spectral scheme combining the finite difference method in time and Legendre spectral method in space is produced and analyzed. The stability and L2norm convergence of the method are proved rigorously.