ChesterRep Collection:
http://hdl.handle.net/10034/6981
Sat, 22 Jul 2017 06:37:38 GMT2017-07-22T06:37:38ZA Note on the Well-Posedness of Terminal Value Problems for Fractional Differential Equations.
http://hdl.handle.net/10034/620550
Title: A Note on the Well-Posedness of Terminal Value Problems for Fractional Differential Equations.
Authors: Diethelm, Kai; Ford, Neville J.
Abstract: This note is intended to clarify some im-
portant points about the well-posedness of terminal value
problems for fractional di erential equations. It follows the
recent publication of a paper by Cong and Tuan in this jour-
nal in which a counter-example calls into question the earlier
results in a paper by this note's authors. Here, we show in
the light of these new insights that a wide class of terminal
value problems of fractional differential equations is well-
posed and we identify those cases where the well-posedness
question must be regarded as open.Sat, 17 Jun 2017 00:00:00 GMThttp://hdl.handle.net/10034/6205502017-06-17T00:00:00ZOrthogonality for a class of generalised Jacobi polynomial $P^{\alpha,\beta}_{\nu}(x)$
http://hdl.handle.net/10034/620536
Title: Orthogonality for a class of generalised Jacobi polynomial $P^{\alpha,\beta}_{\nu}(x)$
Authors: Ford, Neville J.; Moayyed, H.; Rodrigues, M. M.
Abstract: This work considers g-Jacobi polynomials, a fractional generalisation of the classical Jacobi polynomials. We discuss the polynomials and compare some of their properties to the classical case. The main result of the paper is to show that one can derive an orthogonality property for a sub-class of g-Jacobi polynomials $P^{\alpha,\beta}_{\nu}(x)$ The paper concludes with an application in modelling of ophthalmic surfaces.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6205362017-01-01T00:00:00ZOn the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system
http://hdl.handle.net/10034/620434
Title: On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system
Authors: Kavallaris, Nikos I.; Suzuki, Takashi
Abstract: The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.
Description: This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/0951-7715Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6204342017-01-01T00:00:00ZDiscontinuous Galerkin time stepping method for solving linear space fractional partial differential equations
http://hdl.handle.net/10034/620426
Title: Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equations
Authors: Liu, Yanmei; Yan, Yubin; Khan, Monzorul
Abstract: 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.Mon, 23 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6204262017-01-23T00:00:00Z