ChesterRep Collection:
http://hdl.handle.net/10034/6981
Sat, 21 Apr 2018 17:48:16 GMT2018-04-21T17:48:16ZUnits of the group algebra of the group $C_n\times D_6$ over any finite field of characteristic $3$
http://hdl.handle.net/10034/620840
Title: Units of the group algebra of the group $C_n\times D_6$ over any finite field of characteristic $3$
Authors: Gildea, Joe; Taylor, Rhian
Abstract: In this paper, we establish the structure of the unit group of the group algebra ${\FF}_{3^t}(C_n\times D_6)$ for $n \geq 1$.Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10034/6208402018-01-01T00:00:00ZA novel high-order algorithm for the numerical estimation of fractional differential equations
http://hdl.handle.net/10034/620809
Title: A novel high-order algorithm for the numerical estimation of fractional differential equations
Authors: Asl, Mohammad S.; Javidi, Mohammad; Yan, Yubin
Abstract: This paper uses polynomial interpolation to design a novel high-order algorithm for the numerical estimation of fractional differential equations. The Riemann-Liouville fractional derivative is expressed by using the Hadamard finite-part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O(h4−a). Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed
algorithm.Tue, 09 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10034/6208092018-01-09T00:00:00ZA higher order numerical method for time fractional partial differential equations with nonsmooth data
http://hdl.handle.net/10034/620810
Title: A higher order numerical method for time fractional partial differential equations with nonsmooth data
Authors: Xing, Yanyuan; Yan, Yubin
Abstract: Gao et al. (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu (2016), where $k$ is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ uniformly with respect to the time variable $t$. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ uniformly with respect to the time variable $t$. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ as in Gao \et \cite{gaosunzha} (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ for any fixed $t_{n}>0$ for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.Tue, 02 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10034/6208102018-01-02T00:00:00ZConstructions for Self-Dual Codes Induced from Group Rings
http://hdl.handle.net/10034/620815
Title: Constructions for Self-Dual Codes Induced from Group Rings
Authors: Gildea, Joe; Kaya, Abidin; Taylor, Rhian; Yildiz, Bahattin
Abstract: In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68.Sat, 03 Feb 2018 00:00:00 GMThttp://hdl.handle.net/10034/6208152018-02-03T00:00:00Z