Polynomial Spectral Collocation Method For Space Fractional Advection Diffusion Equations

Fractional calculus and fractional differential equations have been widely applied to describing phenomenons in various practical fields, including physics, chemistry, biology, mechanical engineering, signal processing, systems identification, control theory, finance and etc. By now it is still difficult to solve most of fractional differential equations analyti-cally due to the properties of fractional differential operators that they are global operators and the adjoint of a fractional differential operator is not the negative of itself, although several analytical methodologies, such as, Laplace transform, Fourier transform and Mellin transform, are restored to obtain the analytical solutions of the fractional equations by many authors. Hence the study of numerical approximation becomes very important. It is well known that the challenge of solving fractional differential equations essentially comes from the nonlocal properties of fractional derivatives. As a’global’numerical method, spectral method seems to be a natural choice for obtaining high order numerical schemes of solving fractional differential equations, and spectral collocation method has its special advantage for solving nonlinear problems.This paper discusses the spectral collocation method for numerically solving nonlo-cal problems:steady state fractional advection dispersion equations; one dimensional s-pace fractional advection-diffusion equation; and two dimensional linear/nonlinear space fractional advection-diffusion equation. In Chapter2, the differentiation matrixes of the fractional operators: Riemann-Liouville fractional integrals, Riemann-Liouville fractional derivatives and Caputo fractional derivatives, are derived for any collocation points within the given interval [a, b], which brings the applicability of the spectral collocation method for the fractional integral or differential equations. Then the spectral collocation schemes are designed for numerically solving the steady state fractional advection dispersion equa-tion in one and two dimensions in Chapter3, several numerical examples are computed to testify the efficiency of the numerical schemes and confirm the exponential convergence. In Chapter4, we have established the polynomial spectral collocation schemes for space frac-tional advection-diffusion equations, Several numerical examples with different boundary conditions are illustrated to testify the efficiency of our numerical schemes. Appendix A discusses the stabilities in the time direction of the semi-discrete and full-discrete schemes of the one dimensional space fractional advection-diffusion equation mentioned in Chapter4.