An advanced meshless method for time fractional diffusion equation

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations

A node-based smoothed conforming point interpolation method (NS-CPIM) for elasticity problems

This paper formulates a node-based smoothed conforming point interpolation method (NS-CPIM) for solid mechanics. In the proposed NS-CPIM, the higher order conforming PIM shape functions (CPIM) have been constructed to produce a continuous and piecewise quadratic displacement field over the whole problem domain, whereby the smoothed strain field was obtained through smoothing operation over each smoothing domain associated with domain nodes. The smoothed Galerkin weak form was then developed to create the discretized system equations. Numerical studies have demonstrated the following good properties: NS-CPIM (1) can pass both standard and quadratic patch test; (2) provides an upper bound of strain energy; (3) avoid the volumetric locking; (4) provides the higher accuracy than those in the node-based smoothed schemes of the original PIMs

Extraordinary variability and sharp transitions in a maximally frustrated dynamic network

Using Monte Carlo and analytic techniques, we study a minimal dynamic network involving two populations of nodes, characterized by different preferred degrees. Reminiscent of introverts and extroverts in a population, one set of nodes, labeled \textit{introverts} ($I$), prefers fewer contacts (a lower degree) than the other, labeled \textit{extroverts} ($E$). As a starting point, we consider an \textit{extreme} case, in which an $I$ simply cuts one of its links at random when chosen for updating, while an $E$ adds a link to a random unconnected individual (node). The model has only two control parameters, namely, the number of nodes in each group, $N_{I}$ and $N_{E}$). In the steady state, only the number of crosslinks between the two groups fluctuates, with remarkable properties: Its average ($X$) remains very close to 0 for all $N_{I}>N_{E}$ or near its maximum ($\mathcal{N}\equiv N_{I}N_{E}$) if $N_{I}<N_{E}$. At the transition ($N_{I}=N_{E}$), the fraction $X/\mathcal{N}$ wanders across a substantial part of $[0,1]$, much like a pure random walk. Mapping this system to an Ising model with spin-flip dynamics and unusual long-range interactions, we note that such fluctuations are far greater than those displayed in either first or second order transitions of the latter. Thus, we refer to the case here as an `extraordinary transition.' Thanks to the restoration of detailed balance and the existence of a `Hamiltonian,' several qualitative aspects of these remarkable phenomena can be understood analytically.Comment: 6 pages, 3 figures, accepted for publication in EP

Lattice dynamics and electron-phonon interaction in (3,3) carbon nanotubes

We present a detailed study of the lattice dynamics and electron-phonon coupling for a (3,3) carbon nanotube which belongs to the class of small diameter based nanotubes which have recently been claimed to be superconducting. We treat the electronic and phononic degrees of freedom completely by modern ab-initio methods without involving approximations beyond the local density approximation. Using density functional perturbation theory we find a mean-field Peierls transition temperature of approx 40K which is an order of magnitude larger than the calculated superconducting transition temperature. Thus in (3,3) tubes the Peierls transition might compete with superconductivity. The Peierls instability is related to the special 2k_F nesting feature of the Fermi surface. Due to the special topology of the (n,n) tubes also a q=0 coupling between the two bands crossing the Fermi energy at k_F is possible which leads to a phonon softening at the Gamma point.Comment: 4 pages, 3 figures; to be published in Phys. Rev. Let

Two-stage Turing model for generating pigment patterns on the leopard and the jaguar

Based on the results of phylogenetic analysis, which showed that flecks are the primitive pattern of the felid family and all other patterns including rosettes and blotches develop from it, we construct a Turing reaction-diffusion model which generates spot patterns initially. Starting from this spotted pattern, we successfully generate patterns of adult leopards and jaguars by tuning parameters of the model in the subsequent phase of patterning

Understanding the different rotational behaviors of 252^{252}No and 254^{254}No

Total Routhian surface calculations have been performed to investigate rapidly rotating transfermium nuclei, the heaviest nuclei accessible by detailed spectroscopy experiments. The observed fast alignment in $^{252}$No and slow alignment in $^{254}$No are well reproduced by the calculations incorporating high-order deformations. The different rotational behaviors of $^{252}$No and $^{254}$No can be understood for the first time in terms of $\beta_6$ deformation that decreases the energies of the $\nu j_{15/2}$ intruder orbitals below the N=152 gap. Our investigations reveal the importance of high-order deformation in describing not only the multi-quasiparticle states but also the rotational spectra, both providing probes of the single-particle structure concerning the expected doubly-magic superheavy nuclei.Comment: 5 pages, 4 figures, the version accepted for publication in Phys. Rev.

Oscillatory Turing Patterns in a Simple Reaction-Diffusion System

Turing suggested that, under certain conditions, chemicals can react and diffuse in such a way as to produce steady-state inhomogeneous spatial patterns of chemical concentrations. We consider a simple two-variable reaction-diffusion system and find there is a spatio-temporally oscillating solution (STOS) in parameter regions where linear analysis predicts a pure Turing instability and no Hopf instability. We compute the boundary of the STOS and spatially non-uniform solution (SSNS) regions and investigate what features control its behavior