Convergence analysis of a LDG method for tempered fractional convection–diffusion equations

This paper proposes a local discontinuous Galerkin method for tempered fractional convection–diffusion equations. The tempered fractional convection–diffusion is converted to a system of the first order of differential/integral equation, then, the local discontinuous Galerkin method is employed to solve the system. The stability and order of convergence of the method are proven. The order of convergence O(hk+1) depends on the choice of numerical fluxes. The provided numerical examples confirm the analysis of the numerical scheme.

Existence of travelling pulses in a neural model

In 1992 Ermentrout and McLeod published in this journal a landmark study of travelling wavefronts for a differential–integral equation model of a neural network. Since then a number of authors have extended the model by adding an additional equation for a ‘recovery variable’, thus allowing the possibility of travelling-pulse-type solutions. In a recent paper, Faye gave perhaps the first rigorous proof of the existence (and stability) of a travelling-pulse solution for a model of this type, treating a simplified version of equations originally developed by Kilpatrick and Bressloff. The excitatory weight function J used in this work allowed the system to be reduced to a set of four coupled ordinary differential equations (ODEs), and a specific firing-rate function S, with parameters, was considered. The method of geometric singular perturbation was employed, together with blow-ups. In this paper we extend Faye's results on existence by dropping one of his key hypotheses, proving the existence of pulses at least two different speeds, and, in a sense, allowing a wider range of the small parameter in the problem. The proofs are classical and self-contained aside from standard ODE material.

Analytical Approximate Prediction of Thermal Post-Buckling Behavior of the Spring-Hinged Beam

This paper is concerned with thermal post-buckling of uniform isotropic beams with axially immovable spring-hinged ends. The ends of the beam with elastic rotational restraints represent the actual practical support conditions and the classical hinged and clamped conditions can be achieved as the limiting cases of the rotational spring stiffness. The governing differential–integral equation is solved by assuming suitable admissible function for lateral displacement and by employing the Galerkin method. A brief and explicit analytical approximate formulation is established to predict the thermal post-buckling behavior of the beam. The present analytical approximate expressions show excellent agreement with the corresponding numerical solutions based on the shooting method. This confirms the effectiveness and verifies the accuracy of the formulas established.

An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials

An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient.

Large Amplitude Free Vibration Analysis of Nanotubes Using Variational and Homotopy Methods

Linear theories are basically unable to model the dynamic behavior of nanotubes due to the large deflection/dimension ratios. In this paper the closed form expressions are obtained for the large-amplitude free vibration of nanotubes. The nonlinear governing differential-integral equation of motion is derived and solved using the Galerkin approach. The derived nonlinear differential equation is then solved using the Variational Approach (VA) and the Homotopy Analysis Method (HAM). The fundamental harmonic as well as higher-order harmonics are analytically obtained. The approximate solutions are compared with those of the numerical responses and accordingly a numerical analysis is carried out. A parametric sensitivity analysis is carried out and different effects of the physical parameters and initial conditions on the natural frequencies are examined. It is found that both the variational analysis and homotopy method are quite consistent and satisfactory techniques to analyze the vibration of nanotubes.

A Model of Batch Grinding of Talc Powders

Grinding of talc powders has been studied both theoretically and experimentally. The specific rates of breakage of talc powders were measured based on the first-order breakage kinetics model and the cumulative breakage distribution parameters of talc powders were measured from primary breakage products. Based on the measurement results, the specific rate of breakage and cumulative breakage distribution functions were correlated with particle size asand , repectively. A differential-integral equation was thus build to describe grinding as a rate process and was integrated numerically. Comparisons on size distribution showed that the specific rate of breakage of talc powders increased with grinding time at an increase rateabout 0.0066min-2.