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SubjectsCaputo fractional derivative (1)delay equations (1)discrete equations (1)Error estimates (1)finite element method (1)Fractional differential equation (1)fractional telegraph equation (1)functional differential equations (1)Hadamard finite-part integral (1)numerical analysis (1)View MoreJournal

Journal of Computational and Applied Mathematics (4)

Authors
Yan, Yubin (4)

Ford, Neville J. (2)Asl, Mohammad S. (1)Javidi, Mohammad (1)Liang, Zongqi (1)Liu, Fang (1)Malique, Md A. (1)Rodrigues, M. M. (1)Xiao, Jingyu (1)Types
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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 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.

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.

Numerical treatment of oscillary functional differential equations

Ford, Neville J.; Yan, Yubin; Malique, Md A. (Elsevier, 2010-09-01)

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

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