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SubjectsError estimates (9)error estimates (6)Caputo fractional derivative (4)Finite difference method (4)stability (4)finite element method (3)Laplace transform (3)numerical schemes (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 (33)

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 (30)Book chapter (2)Meetings and Proceedings (1)

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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.

High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

Li, Zhiqiang; Liang, Zongqi; Yan, Yubin (Springer Link, 2016-11-15)

In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(τ^(3−α) +h^2 ),0

Fourier spectral methods for some linear stochastic space-fractional partial differential equations

Liu, Yanmei; Khan, Monzorul; Yan, Yubin (MDPI, 2016-07-01)

Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in one-dimensional case are introduced and analyzed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in $L^{2}$- norm are obtained. 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 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.

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.

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.

Analysis of transient Rivlin-Ericksen fluid and irreversibility of exothermic reactive hydromagnetic variable viscosity

Olakunle, Salawu; Kareem, Rasaq; Yan, Yubin (Shahid Chamran University of Ahvaz, 2019-03-15)

The study analysed unsteady Rivlin-Ericksen fluid and irreversibility of exponentially temperature dependent variable viscosity of hydromagnetic two-step exothermic chemical reactive flow along the channel axis with walls convective cooling. The non-Newtonian Hele-Shaw flow of Rivlin-Erickson fluid is driven by bimolecular chemical kinetic and unvarying pressure gradient. The reactive fluid is induced by periodic changes in magnetic field and time. The Newtons law of cooling is satisfied by the constant heat coolant convection exchange at the wall surfaces with the neighboring regime. The dimensionless non-Newtonian reactive fluid equations are numerically solved using a convergent and consistence semi-implicit finite difference technique which are confirmed stable. The response of the reactive fluid flow to variational increase in the values of some entrenched fluid parameters in the momentum and energy balance equations are obtained. A satisfying equations for the ratio of irreversibility, entropy generation and Bejan number are solved with the results presented graphically and discussed quantitatively. From the study, it was obtained that the thermal criticality conditions with the right combination of thermo-fluid parameters, the thermal runaway can be prevented. Also, the entropy generation can minimize by at low dissipation rate and viscosity.

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

Yanzhi, Liu; Roberts, Jason A.; 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.

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