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Error estimates (8)

Caputo fractional derivative (3)Laplace transform (3)Finite difference method (2)Fractional differential equation (2)Riemann-Liouville fractional derivative (2)Caputo derivative (1)Discontinuous Galerkin method (1)Finite element method (1)Fractional diffusion problem (1)View MoreJournalComputational Methods in Applied Mathematics (2)Applied Numerical Mathematics (1)BIT Numerical Mathematics (1)International Journal of Computer Mathematics (1)Journal of Computational and Applied Mathematics (1)View MoreAuthorsYan, Yubin (8)Ford, Neville J. (4)Khan, Monzorul (2)Pal, Kamal (2)Asl, Mohammad S. (1)Javidi, Mohammad (1)Liu, Yanmei (1)Roberts, Jason (1)Xing, Yanyuan (1)Yang, Yan (1)View MoreTypesArticle (8)

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A higher order numerical method for time fractional partial differential equations with nonsmooth data

Xing, Yanyuan; Yan, Yubin (Elsevier, 2018-01-02)

Gao et al. (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu (2016), where $k$ is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ uniformly with respect to the time variable $t$. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ uniformly with respect to the time variable $t$. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ as in Gao \et \cite{gaosunzha} (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ for any fixed $t_{n}>0$ for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

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.

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

Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equations

Liu, Yanmei; Yan, Yubin; Khan, Monzorul (Elsevier, 2017-01-23)

In this paper, we consider the discontinuous Galerkin time stepping method for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinuous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in $t$ of degree at most $q-1, q \geq 1$, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given.

A novel high-order algorithm for the numerical estimation of fractional differential equations

Asl, Mohammad S.; Javidi, Mohammad; Yan, Yubin (Elsevier, 2018-01-09)

This paper uses polynomial interpolation to design a novel high-order algorithm for the numerical estimation of fractional differential equations. The Riemann-Liouville fractional derivative is expressed by using the Hadamard finite-part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O(h4−a). Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm.

A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes

Yanzhi, Liu; Roberts, Jason; Yan, Yubin (Taylor & Francis, 2017-10-09)

We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth, Li et al. \lq \lq Finite difference methods with non-uniform meshes for nonlinear fractional differential equations\rq\rq, Journal of Computational Physics, 316(2016), 614-631, obtained the error estimates of finite difference methods with non-uniform meshes. However the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with non-uniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.

Some time stepping methods for fractional diffusion problems with nonsmooth data

Yang, Yan; Yan, Yubin; Ford, Neville J. (De Gruyter, 2017-09-02)

We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data

Yan, Yubin; Khan, Monzorul; Ford, Neville J. (Society for Industrial and Applied Mathematics, 2018-01-11)

We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197-221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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