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

      Ford, Neville J.; Pal, Kamal; Yan, Yubin; University of Chester (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
    • Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations

      Pal, Kamal; Liu, Fang; Yan, Yubin; Roberts, Graham; University of Chester (Springer International Publishing, 2015-06-17)
      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.
    • Higher Order Numerical Methods for Fractional Order Differential Equations

      Pal, Kamal (University of Chester, 2015-08)
      This thesis explores higher order numerical methods for solving fractional differential equations.
    • Higher order numerical methods for solving fractional differential equations

      Yan, Yubin; Pal, Kamal; Ford, Neville J.; University of Chester (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.
    • Numerical Solutions of Fractional Differential Equations by Extrapolation

      Pal, Kamal; Liu, Fang; Yan, Yubin; University of Chester (Springer International Publishing, 2015-06-17)
      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.