In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundary value problems of fractional-order with variable coefficients. Here the approximation is based on shifted Legendre polynomials and the quadrature rule is treated on shifted Legendre Gauss-Lobatto points. We also present a Gauss-Lobatto shifted Legendre collocation method for solving nonlinear multi-order FDEs with multi-point boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. Thus we can find directly the spectral solution of the proposed problem. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms.

We propose a new mathematical model to investigate the recent outbreak of the coronavirus disease (COVID-19). The model is studied qualitatively using stability theory of differential equations and the basic reproductive number that represents an epidemic indicator is obtained from the largest eigenvalue of the next-generation matrix. The global asymptotic stability conditions for the disease free equilibrium are obtained. The real COVID-19 incidence data entries from 01 July, 2020 to 14 August, 2020 in the country of Pakistan are used for parameter estimation thereby getting fitted values for the biological parameters. Sensitivity analysis is performed in order to determine the most sensitive parameters in the proposed model. To view more features of the state variables in the proposed model, we perform numerical simulations by using different values of some essential parameters. Moreover, profiles of the reproduction number through contour plots have been biologically explained.

We show that the double Ᏸ of the nontrivially associated tensor category constructed from left coset representatives of a subgroup of a finite group X is a modular category. Also we give a definition of the character of an object in this category as an element of a braided Hopf algebra in the category. This definition is shown to be adjoint invariant and multiplicative on tensor products. A detailed example is given. Finally, we show an equivalence of categories between the nontrivially associated double Ᏸ and the trivially associated category of representations of the Drinfeld double of the group D(X).2000 Mathematics Subject Classification: 18D10, 16W30.1. Introduction. This paper will make continual use of formulae and ideas from [2], and these definitions and formulae will not be repeated, as they would add very considerably to the length of the paper. The paper [2] is itself based on the papers [3,4], but is mostly self-contained in terms of notation and definitions. The book [6] has been used as a standard reference for Hopf algebras, and [1,8] as references for modular categories.In [2], there is a construction of a nontrivially associated tensor category Ꮿ from data which is a choice of left coset representatives M for a subgroup G of a finite group X. This introduces a binary operation "·" and a G-valued "cocycle" τ on M. There is also a double construction where X is viewed as a subgroup of a larger group. This gives rise to a braided category Ᏸ, which is the category of reps of an algebra D, which is itself in the category, and it is the category that we concentrate on in this paper.It is our aim to show that the nontrivially associated algebra D has reps which have characters in the same way that the reps of a finite group have characters, and also that the category of its representations has a modular structure in the same way that the category of reps of the double of a group has a modular structure.We begin by describing the indecomposable objects in Ꮿ, in a manner similar to that used in [4]. A detailed example is given using the group D 6 . Then we show how to find the dual objects in the category, and again illustrate this with an example.Next, we show that the rigid braided category Ᏸ is a ribbon category. The ribbon maps are calculated for the indecomposable objects in our example category.In the next section, we explicitly evaluate in Ᏸ the standard diagram for trace in a ribbon category [6]. Then we define the character of an object in Ᏸ as an element of the dual of the braided Hopf algebra D. This element is shown to be right adjoint invariant. Also we show that the character is multiplicative for the tensor product of