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error estimates (6)

finite element method (2)stability (2)time fractional partial differential equations (2)2-D Schrodinger equation (1)Chernoff formula (1)Crank-Nicolson time stepping (1)cubic splines (1)degenerate parabolic equation (1)dynamical boundary condition (1)View MoreJournalESAIM M2AN - Mathematical Modelling and Numerical Analysis (1)Fractional Calculus and Applied Analysis (1)Fractional calculus and applied analysis (1)Journal of Computational Analysis and Applications (1)Journal of Scientific Computing (1)View MoreAuthorsYan, Yubin (5)Li, Zhiqiang (2)Antonopoulou, Dimitra (1)Ford, Neville J. (1)Kavallaris, Nikos I. (1)Khan, Monzorul (1)Liang, Zongqi (1)Liu, Fang (1)Xiao, Jingyu (1)Types
Article (6)

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Galerkin methods for a Schroedinger-type equation with a dynamical boundary condition in two dimensions

Antonopoulou, Dimitra (EDP Sciences / SMAI, 2015-06-30)

In this paper, we consider a two-dimensional Schodinger-type equation with a dynamical boundary condition. This model describes the long-range sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a Crank-Nicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution.

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

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.

Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noises

Liu, Fang; Yan, Yubin; Khan, Monzorul (Springer, 2018-02)

Fourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive 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. The space-time noise is approximated by the piecewise constant functions in the time direction and by some appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. For the linear problem, we obtain the precise error estimates in the $L_{2}$ norm and find the relations between the error bounds and the fractional powers. For the nonlinear problem, we introduce the numerical algorithms and MATLAB codes based on the FFT transforms. Our numerical algorithms can be adapted easily to solve other stochastic space fractional partial differential equations with multiplicative noises. Numerical examples for the semilinear stochastic space fractional partial differential equations are given.

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.

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.

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