Fractional collocation methods for multi-term linear and nonlinear fractional differential equations with variable coefficients on the half line

Abstract

ABSTRACTIn this paper, two new and efficient fractional spectral collocation schemes are proposed 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 fractional operators. Our main task is to find corresponding fractional pseudospectral differentiation matrices (F-PSDMs) which are computed by two different methods. The first one is based on the three-term recurrence relations and the derivative recurrence relations of the generalized Laguerre polynomials. The second one is based on the direct fractional integral and derivative relations of the generalized Laguerre polynomials. Based on the F-PSDMs, the corresponding spectral collocation schemes are developed. By taking suitable parameter involved in the methods, we can significantly enhance the performance of these schemes. Moreover, both schemes are easy to extend to solve the variable-order FDEs. Numerical examples are presented, which show the efficiency and spectral accuracy of our methods.