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Finite difference method (4)

Error estimates (2)Caputo derivative (1)Caputo fractional derivative (1)error estimates (1)Fractional differential equation (1)fractional diffusion wave equation (1)Riemann-Liouville fractional derivative (1)Space-fractional partial differential equations (1)space-fractional partial differential equations (1)View MoreJournalBIT Numerical Mathematics (1)Computational Methods in Applied Mathematics (1)Journal of Computational Physics (1)Authors
Yan, Yubin (4)

Pal, Kamal (3)Ford, Neville J. (2)Du, Ruilian (1)Liang, Zongqi (1)Liu, Fang (1)Roberts, Graham (1)TypesArticle (3)Book chapter (1)

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Higher order numerical methods for solving fractional differential equations

Yan, Yubin; Pal, Kamal; Ford, Neville J. (Springer, 2013-10-05)

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 < α < 1. The order of convergence of the numerical method is O(h^(3−α)). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α >0. The order of convergence of the numerical method is O(h^3) for α ≥ 1 and O(h^(1+2α)) for 0 < α ≤ 1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations

Pal, Kamal; Liu, Fang; Yan, Yubin; Roberts, Graham (Springer International Publishing, 2015-06)

Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order O(Δx^(3−α )),10 , where Δt,Δx denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.

An algorithm for the numerical solution of two-sided space-fractional partial differential equations.

Ford, Neville J.; Pal, Kamal; Yan, Yubin (de Gruyter, 2015-08-20)

We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order

A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation

Du, Ruilian; Yan, Yubin; Liang, Zongqi (Elsevier, 2018-10-05)

A new high-order finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k-1}) \big ), k=1, 2, \dots, N, $ with the convergence order $O(\Delta t^{4-\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4-\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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