On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems

Abstract

Multi-order fractional differential equations are motivated by their flexibility to describe complex multi-rate physical processes. This paper is concerned with the convergence behavior of a spectral collocation method when used to approximate solutions of nonlinear multi-order fractional initial value problems. The collocation scheme and its convergence analysis are developed based on the novel spectral collocation method which has been recently presented by Wang et al. (J Sci Comput 76(1):166-188, 2018) for single fractional-order boundary value problems. The proposed method is an indirect approach since we act on the equivalent Volterra integral equation of the second kind. More precisely, the spectral rate of convergence for the proposed method is established in the L 2 -and L ∞ -norms. The method enjoys high accuracy for problems with smooth solutions. Exponentially rapid convergence is observed with a small number of degree of freedoms and for all samples of fractional orders. Numerical examples are presented to support the theoretical finding.

A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions

“…For b = 3, () becomes the same example investigated in the literature 24,25 . The corresponding , which is smooth on the interval [− 1,1] for .…”

“…, which is smooth on the interval [− 1,1] for ∈ N. From Table 1, we can see the proposed method is applicable and gives high accurate results comparing with the results given in previous studies. 24,25 For b = 2.5, we get Y (x) = 3.75…”

“…For b = 3, (21) becomes the same example investigated in the literature. 24,25 The corresponding Y (x) = 6…”

Applications of Distributed-Order Fractional Operators: A Review

“…Similar to the Galerkin spectral method, various collocation methods have been developed starting from (1) Legendre basis (LCM) [ 100 , 134 ] and (2) Jacobi basis (JCM) [ 105 , 152 ]. Zaky constructed a LCM to solve both linear and nonlinear boundary value problems [ 100 ], and later extended this method to simulate initial value DODEs [ 99 , 153 ]. Results indicated that the convergence error decays exponentially with an increasing number of Gauss–Legendre points.…”

Jacobi Spectral Galerkin Method for Distributed-Order Fractional Rayleigh–Stokes Problem for a Generalized Second Grade Fluid

“…In this paper, we use a non-local representation of the solution of the distributed-order time-fractional Rayleigh-Stokes problem to introduce spectral solutions. The spectral and pseudospectral methods are well-known for their high accuracy and have been used extensively in scientific computation, see [33][34][35][36][37][38][39][40][41] and the references therein. The main contribution of this paper is to develop Jacobi-Galerkin algorithms for solving the multidimensional distributed-order time-fractional Rayleigh-Stokes problem (1).…”