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SubjectsError estimates (9)error estimates (6)Caputo fractional derivative (4)Finite difference method (4)finite element method (3)Laplace transform (3)numerical schemes (3)stability (3)Caputo derivative (2)discrete equations (2)View MoreJournalJournal of Computational and Applied Mathematics (4)Computational Methods in Applied Mathematics (3)Applied Numerical Mathematics (2)Fractional Calculus and Applied Analysis (2)Journal of Computational Physics (2)View MoreAuthors

Yan, Yubin (32)

Ford, Neville J. (11)Khan, Monzorul (4)Liang, Zongqi (4)Liu, Fang (4)Pal, Kamal (4)Li, Zhiqiang (3)Xiao, Jingyu (3)Liu, Yanmei (2)Malique, Md A. (2)View MoreTypesArticle (29)Book chapter (2)Meetings and Proceedings (1)

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A high order numerical method for solving nonlinear fractional differential equation with non-uniform meshes

Fan, Lili; Yan, Yubin (Springer Link, 2019-01-18)

We introduce a high-order numerical method for solving nonlinear fractional differential equation with non-uniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N-1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j-1}, t_{j}, t_{j+1}$. A high-order numerical method is obtained. Then we apply this numerical method with the non-uniform meshes with the step size $\tau_{j}= t_{j+1}- t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the non-uniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same non-uniform meshes.

Stability of a numerical method for a fractional telegraph equation

Yan, Yubin; Xiao, Jingyu; Ford, Neville J. (De Gruyter, 2012-03)

In this paper, we introduce a numerical method for solving the time-space fractional telegraph equations. The numerical method is based on a quadrature formula approach and a stability condition for the numerical method is obtained. Two numerical examples are given and the stability regions are plotted.

A finite element method for time fractional partial differential equations

Ford, Neville J.; Xiao, Jingyu; Yan, Yubin (2011)

This article considers the finite element method for time fractional differential equations.

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.

Error estimates of high-order numerical methods for solving time fractional partial differential equations

Li, Zhiqiang; Yan, Yubin (De Gruyter, 2018-07-12)

Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3- \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Ford, Neville J.; Yan, Yubin (De Gruyter, 2017-10-31)

In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Stabilizing a mathematical model of plant species interaction

Yan, Yubin; Ekaka-A, Enu-Obari N. (Elsevier, 2011-09-03)

In this paper, we will consider how to stabilize a mathematical model of plant species interaction which is modelled by using Lotka-Volterra system. We first identify the unstable steady states of the system, then we use the feedback control based on the solutions of the Riccati equation to stabilize the linearized system. We further stabilize the nonlinear system by using the feedback controller obtained in the stabilization of the linearized system. We introduce the backward Euler method to approximate the feedback control nonlinear system and obtain the error estimates. Four numerical examples are given which come from the application areas.

Error estimates of a high order numerical method for solving linear fractional differential equations

Li, Zhiqiang; Yan, Yubin; Ford, Neville J. (Elsevier, IMACS, 2016-04-29)

In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O(∆t 2−α ), 0 < α < 1, where α is the order of the fractional derivative and ∆t is the step size. We then use the similar idea to prove the error estimates of a high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O(∆t 3−α ), 0 < α < 1. The numerical examples are given to show that the numerical results are consistent with the theoretical results.

Detailed error analysis for a fractional adams method with graded meshes

Liu, Yanzhi; Roberts, Jason A.; Yan, Yubin (Springer, 2017-09-21)

We consider a fractional Adams method for solving the nonlinear fractional differential equation $\, ^{C}_{0}D^{\alpha}_{t} y(t) = f(t, y(t)), \, \alpha >0$, equipped with the initial conditions $y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots, \lceil \alpha \rceil -1$. Here $\alpha$ may be an arbitrary positive number and $ \lceil \alpha \rceil$ denotes the smallest integer no less than $\alpha$ and the differential operator is the Caputo derivative. Under the assumption $\, ^{C}_{0}D^{\alpha}_{t} y \in C^{2}[0, T]$, Diethelm et al. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes $t_{n}= T (n/N), n=0, 1, 2, \dots, N$ and proved that this method has the optimal convergence order uniformly in $t_{n}$, that is $O(N^{-2})$ if $\alpha > 1$ and $O(N^{-1-\alpha})$ if $\alpha \leq 1$. They also showed that if $\, ^{C}_{0}D^{\alpha}_{t} y(t) \notin C^{2}[0, T]$, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well known that for $y \in C^{m} [0, T]$ for some $m \in \mathbb{N}$ and $ 0 < \alpha 1$, we show that the optimal convergence order of this method can be recovered uniformly in $t_{n}$ even if $\, ^{C}_{0}D^{\alpha}_{t} y$ behaves as $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Numerical Approximation of Stochastic Time-Fractional Diffusion

Yan, Yubin; Jin, Bangti; Zhou, Zhi

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive
fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo
fractional derivative of order $\alpha\in(0,1)$, and fractionally integrated Gaussian noise (with a
Riemann-Liouville fractional integral of order $\gamma \in[0,1]$ in the front). The numerical scheme
approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements
and in time by the classical Gr\"unwald-Letnikov method, and the noise by the $L^2$-projection. Sharp
strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the
deterministic counterpart. One- and two-dimensional numerical results are presented to support the
theoretical findings.

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