Mathematicshttp://hdl.handle.net/10034/69812021-10-02T11:05:47Z2021-10-02T11:05:47ZOscillatory and stability of a mixed type difference equation with variable coefficientsYan, YubinPinelas, SandraRamdani, NedjemYenicerioglu, Ali Fuathttp://hdl.handle.net/10034/6260042021-10-02T01:37:01Z2021-08-12T00:00:00ZOscillatory and stability of a mixed type difference equation with variable coefficients
Yan, Yubin; Pinelas, Sandra; Ramdani, Nedjem; Yenicerioglu, Ali Fuat
The goal of this paper is to study the oscillatory and stability of the mixed type difference equation with variable coefficients \[
\Delta x(n)=\sum_{i=1}^{\ell}p_{i}(n)x(\tau_{i}(n))+\sum_{j=1}^{m}q_{j}(n)x(\sigma_{i}(n)),\quad n\ge n_{0}, \] where $\tau_{i}(n)$ is the delay term and $\sigma_{j}(n)$ is the advance term and they are positive real sequences for $i=1,\cdots,l$ and $j=1,\cdots,m$, respectively, and $p_{i}(n)$ and $q_{j}(n)$ are real functions. This paper generalise some known results and the examples illustrate the results.
2021-08-12T00:00:00ZSpatial discretization for stochastic semilinear subdiffusion driven by integrated multiplicative space-time white noiseYan, YubinHoult, JamesWang, Junmeihttp://hdl.handle.net/10034/6260032021-10-02T01:36:45Z2021-08-12T00:00:00ZSpatial discretization for stochastic semilinear subdiffusion driven by integrated multiplicative space-time white noise
Yan, Yubin; Hoult, James; Wang, Junmei
Spatial discretization of the stochastic semilinear subdiffusion driven by integrated multiplicative space-time white noise is considered. The spatial discretization scheme discussed in Gy\"ongy \cite{gyo_space} and Anton et al. \cite{antcohque} for stochastic quasi-linear parabolic partial differential equations driven by multiplicative space-time noise is extended to the stochastic subdiffusion. The nonlinear terms $f$ and $\sigma$ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the integrated multiplicative space-time white noise are discretized by using finite difference methods. Based on the approximations of the Green functions which are expressed with the Mittag-Leffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under the suitable smoothness assumptions of the initial values.
2021-08-12T00:00:00ZError estimates of a continuous Galerkin time stepping method for subdiffusion problemYan, YubinYan, YuyuanLiang, ZongqiEgwu, Bernardhttp://hdl.handle.net/10034/6260022021-10-02T03:21:03Z2021-07-29T00:00:00ZError estimates of a continuous Galerkin time stepping method for subdiffusion problem
Yan, Yubin; Yan, Yuyuan; Liang, Zongqi; Egwu, Bernard
A continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \, \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and L-type methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.
The final publication is available at Springer via http://dx.doi.org/10.1007/s10915-021-01587-9.
CC Licence not permitted for AM version see https://www.springernature.com/gp/open-research/policies/journal-policies
2021-07-29T00:00:00ZLayer Dynamics for the one dimensional $\eps$-dependent Cahn-Hilliard / Allen-Cahn EquationAntonopoulou, DimitraKarali, GeorgiaTzirakis, Konstantinoshttp://hdl.handle.net/10034/6256042021-09-07T09:59:58Z2021-08-27T00:00:00ZLayer Dynamics for the one dimensional $\eps$-dependent Cahn-Hilliard / Allen-Cahn Equation
Antonopoulou, Dimitra; Karali, Georgia; Tzirakis, Konstantinos
We study the dynamics of the one-dimensional ε-dependent Cahn-Hilliard / Allen-Cahn equation within a neighborhood of an equilibrium of N transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a mass-conservation constraint in place of one of the zero-mass flux boundary conditions at
x = 1. Motivated by the study of Carr and Pego on the layered metastable patterns of Allen-Cahn in [10], and by this of Bates and Xun in [5] for the Cahn-Hilliard equation, we implement an N-dimensional, and a mass-conservative N−1-dimensional manifold respectively; therein, a metastable state with N transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized Cahn-Hilliard / Allen-Cahn operator, and specify wide families of ε-dependent weights δ(ε), µ(ε), acting at each part of the operator, for which the dynamics are stable and rest exponentially small in ε. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to Allen-Cahn, or, when the mass is conserved, close to the Cahn-Hilliard solution.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00526-021-02085-4
2021-08-27T00:00:00Z