Weight Function Approach to q-Difference Equations for the q-Hypergeometric Polynomials

In this paper we define a new algebra generated by the difference operators D _q and D _q-1 with two analytic functions (x) and (x). Also, we define an operator M = J ₁ J ₂–J ₃ J ₄ s.t. all q-hypergeometric orthogonal polynomials Y _n(x), x cos(), are eigenfunctions of the operator M with eigenvalues _q [n] _q . The choice of (x) and (x) depend on the weight function of Y _n (x).     

Auto-Bäcklund transformations for a matrix partial differential equation

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.     

Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part I. Derivation of different formulations and constant density limit

The paper presents various formulations of characteristics-based schemes in the framework of the artificial-compressibility method for variable-density incompressible flows. In contrast to constant-density incompressible flows, where the characteristics-based variables reconstruction leads to a single formulation, in the case of variable density flows three different schemes can be obtained henceforth labeled as: transport, conservative and hybrid schemes. The conservative scheme results in pseudo-compressibility terms in the (multi-species) density reconstruction. It is shown that in the limit of constant density, the transport scheme becomes the (original) characteristics-based scheme for incompressible flows, but the conservative and hybrid schemes lead to a new characteristics-based variant for constant density flows. The characteristics-based schemes are combined with second and third-order interpolation for increasing the computational accuracy locally at the cell faces of the control volume. Numerical experiments for constant density flows reveal that all the characteristics-based schemes result in the same flow solution, but they exhibit different convergence behavior. The multigrid implementation and numerical studies for variable density flows are presented in Part II of this study.     

关于麦克斯韦方程组的讨论

本文讨论了麦克斯韦方程组的来龙去脉、形式和地位，对麦克斯韦主程组的教学具有一定的参考价值。     

Stability analysis using multiple scales homotopy approach of coupled cylindrical interfaces under the influence of periodic electrostatic fields

The influence of axial periodic electric fields on streaming flows through three coaxial infinitely vertical cylinders is considered. The three fluid layers are assumed to be incompressible, dielectric, viscous and saturated through porous media. To relax the mathematical manipulation of the problem, the viscous potential theory is considered. Through the current work, the stability analysis of the coupled Mathieu equations is developed in the analogy of the multiple scales homotopy technique. Away from the symmetric and anti-symmetric modes, the present study investigates a general case of the surface waves deflections. To overcome the lengthy of the algebraic calculations, the matrices concept is utilized. The stability analysis reveals the resonance as well as non-resonance cases. A set of graphs are depicted to indicate some resonance cases for a chosen sample through a dimensionless system. Therefore, the influence of some physical parameters on the stability picture is indicated. In addition, the perturbed solutions of the governed Mathieu equations are graphed.     

Novel Wronskian Solutions to Boussinesq Equation

The new method for constructing the Wronskian entries is applied to the Boussinesq equation. The novel Wronskian solutions to it are obtained, including solitons, rational solutions, Matveev solutions, and complexitons.     

An analysis of quantum chromodynamic structure function beyond leading order

We obtain approximate solutions of Altarelli-Parisi equations beyond leading logarithmic approximation. Our results suggest quantitative utility of higher order terms in the structure function analysis of deep inelastic scattering.     

A kind of distortion of mean velocity profile in pipe Poiseuille flow and its stability behaviour

This paper presents a kind of distortion of Hagen-Poiseuille velocity profile in pipe Poiseuille flow. This distortion can be regarded as a general expression of the influence on the mean flow by nonlinear interaction of various components of axisymmetric perturbations. Through the investigation of the stability behaviour of this velocity profile, this paper obtains unstable result induced by axisymmetric perturbations for the first time, and thus presents a new possible approach which leads to instability of Hagen-Poiseuille flow.