High order algorithms for numerical solution of fractional differential equations

From Springer Nature via Jisc Publications RouterHistory: received 2020-09-23, accepted 2021-02-02, registration 2021-02-03, online 2021-02-17, pub-electronic 2021-02-17, collection 2021-12Publication status: PublishedAbstract: In this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms

Numerical Solutions of Fractional Differential Equations by Extrapolation

An extrapolation algorithm is considered for solving linear fractional differential equations in this paper, which is based on the direct discretization of the fractional differential operator. Numerical results show that the approximate solutions of this numerical method has the expected asymptotic expansions