Spectral Methods For Some Fractional Differential Equations

Fractional differential equations(FDEs)are equations containing fractional deriva-tives or fractional integrals,and fractional derivative(or integral)is an extension of the classic integer-order derivative(or integral).In this thesis,the spectral methods for some FDEs are investigated.Firstly,an efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative,which is used to simulate solute transport in watershed catchments and rivers.On establishing an efficient recursive algorithm based on the properties of Jacobi polyno-mials to approximate the Coimbra variable-order fractional derivative operator,spectral collocation method with both temporal and spatial discretisation is used to solve the time variable-order fractional mobile-immobile advection-dispersion model.Numerical ex-amples then illustrate the effectiveness and high order convergence of this approach.The comparisons are made between the present method and the existing ones,which show better performance of the present method.Secondly,two new and efficient fractional spectral collocation methods are pro-posed to solve the multi-term linear and nonlinear fractional differential equations(FDEs)with variable coefficients on the half line,both for the Riemann-Liouville and Caputo type fractional operators.The first method,based on the Lagrange interpola-tion at generalized Laguerre-Gauss type quadrature nodes,has two schemes.The first scheme is based on the three-term recurrence relations and the derivative recurrence re-lations of the generalized Laguerre polynomials.The second scheme is based on the direct fractional integral and derivative relations of the generalized Laguerre polynomi-als.The second method is based on a new fractional Birkhoff interpolation at generalized Laguerre-Gauss-Radau quadrature nodes.This new interpolation leads to new interpola-tion polynomial basis functions which can be used to construct a new fractional colloca-tion scheme.By taking suitable parameter involved in the methods,we can significantly enhance the performance of these methods.Moreover,both methods are easy to extend to solve the variable-order FDEs.The numerical experiments show the efficiency and spectral accuracy of these two methods.Thirdly,the Legendre Galerkin-Chebyshev collocation(LGCC)method for gener-alized fractional Burgers equations is developed.This method is based on the Legendre-Galerkin variational form,but the nonlinear term and the right-hand term are treated by Chebyshev-Gauss interpolation.Error estimates of the fully discrete scheme are giv-en in the L2-norm through introducing some spaces related to the fractional operators.Numerical results indicate that this method is stable and efficient.At last,the LGCC method for space fractional Burgers-like equations with fraction-al nonlinear term and diffusion term is developed.Error estimates of the semi-discrete scheme and the fully discrete scheme are given in the L2-norm through introducing some spaces related to the fractional operators.Numerical results indicate that this method is stable and efficient.