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SubjectsError estimates (9)error estimates (5)Caputo fractional derivative (4)Finite difference method (3)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 (30)

Ford, Neville J. (11)Khan, Monzorul (4)Liang, Zongqi (4)Li, Zhiqiang (3)Xiao, Jingyu (3)Liu, Fang (2)Liu, Yanmei (2)Malique, Md A. (2)Pal, Kamal (2)View MoreTypes
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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

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.

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.

A Dufort-Frankel Difference Scheme for Two-Dimensional Sine-Gordon Equation

Liang, Zongqi; Yan, Yubin; Cai, Guorong (Hindawi Publishing Corporation, 2014-10)

A standard Crank-Nicolson finite-difference scheme and a Dufort-Frankel finite-difference scheme are introduced to solve two-dimensional damped and undamped sine-Gordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictor-corrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Existence of time periodic solutions for a class of non-resonant discrete wave equations

Zhang, Guang; Feng, Wenying; Yan, Yubin (Springer, 2015-04-17)

In this paper, a class of discrete wave equations with Dirichlet boundary conditions are obtained by using the center-difference method. For any positive integers m and T, when the existence of time mT-periodic solutions is considered, a strongly indefinite discrete system needs to be established. By using a variant generalized weak linking theorem, a non-resonant superlinear (or superquadratic) result is obtained and the Ambrosetti-Rabinowitz condition is improved. Such a method cannot be used for the corresponding continuous wave equations or the continuous Hamiltonian systems; however, it is valid for some general discrete Hamiltonian systems.

Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances

Ferreira, José M.; Ford, Neville J.; Malique, Md A.; Pinelas, Sandra; Yan, Yubin (Elsevier, 2011-04-12)

This article discusses the oscillatory behaviour of the differential equation of mixed type.

Numerical treatment of oscillary functional differential equations

Ford, Neville J.; Yan, Yubin; Malique, Md A. (Elsevier, 2010-09)

This preprint is concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear v-methods will also be oscillatory. We conclude with some general theory

Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

Liu, Fang; Liang, Zongqi; Yan, Yubin (Elsevier, 2018-12-17)

We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding

Kavallaris, Nikos I.; Yan, Yubin (Cambridge University Press, 2015-01-31)

In the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm.

Numerical analysis of a two-parameter fractional telegraph equation

Ford, Neville J.; Rodrigues, M. M.; Xiao, Jingyu; Yan, Yubin (Elsevier, 2013-09-26)

In this paper we consider the two-parameter fractional telegraph equation of the form
$$-\, ^CD_{t_0^+}^{\alpha+1} u(t,x) + \, ^CD_{x_0^+}^{\beta+1} u (t,x)- \, ^CD_{t_0^+}^{\alpha}u (t,x)-u(t,x)=0.$$ Here
$\, ^CD_{t_0^+}^{\alpha}$, $\, ^CD_{t_0^+}^{\alpha+1}$, $\, ^CD_{x_0^+}^{\beta+1}$ are
operators of the Caputo-type
fractional derivative, where $0\leq \alpha < 1$ and $0 \leq \beta < 1$. The existence
and uniqueness of the equations are proved by using the Banach
fixed point theorem. A numerical method is introduced to solve this
fractional telegraph equation and stability conditions for the numerical
method are obtained. Numerical examples are given in the final section of the
paper.

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