ChesterRep Collection:
http://hdl.handle.net/10034/6981
Mon, 30 Nov 2015 17:01:25 GMT2015-11-30T17:01:25ZMotion of a droplet for the Stochastic mass conserving Allen-Cahn equation
http://hdl.handle.net/10034/582725
Title: Motion of a droplet for the Stochastic mass conserving Allen-Cahn equation
Authors: Antonopoulou, Dimitra; Bates, P.; Bloemker, D.; Karali, G.
Abstract: We study the stochastic mass-conserving Allen-Cahn equation posed on a smoothly bounded
domain of R2 with additive, spatially smooth, space-time noise. This equation describes the stochastic
motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point
of locally maximum curvature. We apply It^o calculus to derive the stochastic dynamics of the center of
the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco [2]
for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by
the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption
of a su ciently small noise strength, we establish stochastic stability of a neighborhood of the manifold of
boundary droplet states in the L2- and H1-norms, which means that with overwhelming probability the
solution stays close to the manifold for very long time-scales.Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10034/5827252015-01-01T00:00:00ZExistence of time periodic solutions for a class of non-resonant discrete wave equations
http://hdl.handle.net/10034/582661
Title: Existence of time periodic solutions for a class of non-resonant discrete wave equations
Authors: Zhang, Guang; Feng, Wenying; Yan, Yubin
Abstract: 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.Fri, 17 Apr 2015 00:00:00 GMThttp://hdl.handle.net/10034/5826612015-04-17T00:00:00ZFinite Difference Method for Two-Sided Space-Fractional Partial Differential Equations
http://hdl.handle.net/10034/582252
Title: Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations
Authors: Pal, Kamal; Liu, Fang; Yan, Yubin; Roberts, Graham
Abstract: Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order O(Δx^(3−α )),1<α<2 is obtained. A shifted implicit finite difference method is introduced for solving two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O(Δt+Δx^( min(3−α,β)) ),1<α<2,β>0 , where Δt,Δx denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.Mon, 01 Jun 2015 00:00:00 GMThttp://hdl.handle.net/10034/5822522015-06-01T00:00:00ZHigher order numerical methods for solving fractional differential equations
http://hdl.handle.net/10034/580035
Title: Higher order numerical methods for solving fractional differential equations
Authors: Yan, Yubin; Pal, Kamal; Ford, Neville J.
Abstract: 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.Sat, 05 Oct 2013 00:00:00 GMThttp://hdl.handle.net/10034/5800352013-10-05T00:00:00Z