ChesterRep Collection:
http://hdl.handle.net/10034/6981
Sun, 01 Jan 2017 01:36:58 GMT2017-01-01T01:36:58ZHigh-Order Numerical Methods for Solving Time Fractional Partial Differential Equations
http://hdl.handle.net/10034/620273
Title: High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations
Authors: Li, Zhiqiang; Liang, Zongqi; Yan, Yubin
Abstract: 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<α<1
O(τ^(3−α)+h^2),0<α<1
are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699–A2724, 2016), where τ and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.
Description: The final publication is available at Springer via http://dx.doi.org/10.1007/s10915-016-0319-1Tue, 15 Nov 2016 00:00:00 GMThttp://hdl.handle.net/10034/6202732016-11-15T00:00:00ZThe sharp interface limit for the stochastic Cahn-Hilliard Equation
http://hdl.handle.net/10034/620253
Title: The sharp interface limit for the stochastic Cahn-Hilliard Equation
Authors: Antonopoulou, Dimitra; Bloemker, Dirk; Karali, Georgia
Abstract: We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp
interface limit, where the positive parameter \eps tends to zero, which measures the width of transition
layers generated during phase separation. We also couple the noise strength to this parameter.
Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the
solutions of stochastic Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic
forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a
deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to
extend the rigorous result to larger noise strength.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6202532017-01-01T00:00:00ZOn a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics
http://hdl.handle.net/10034/620247
Title: On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics
Authors: Kavallaris, Nikos I.; Lankeit, Johannes; Winkler, Michael
Abstract: We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for
\[
u_t = u \Delta u + u \int_\Omega |\nabla u|^2
\]
in bounded domains $\Om\sub\R^n$ which arises in game theory. We prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with $\overline{\Omega}$, i.e. the finite-time blow-up is global.Tue, 01 Nov 2016 00:00:00 GMThttp://hdl.handle.net/10034/6202472016-11-01T00:00:00ZNumerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method
http://hdl.handle.net/10034/620241
Title: Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method
Authors: Morgado, Maria L.; Rebelo, Magda S.; Ferras, Luis L.; Ford, Neville J.
Abstract: In this work we present a new numerical method for the solution of the distributed order time fractional
diffusion equation. The method is based on the approximation of the solution by a double
Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of
equations at suitable collocation points. An error analysis is provided and a comparison with other
methods used in the solution of this type of equation is also performed.Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10034/6202412016-01-01T00:00:00Z