Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy

We develop a method for constructing algebro-geometric solutions of the Blaszak–Marciniak (BM) lattice hierarchy based on the theory of trigonal curves. We first derive the BM lattice hierarchy associated with a discrete (3×3)-matrix spectral problem using Lenard recurrence relations. Using the characteristic polynomial of the Lax matrix for the BM lattice hierarchy, we introduce a trigonal curve with two infinite points, which we use to establish the associated Dubrovin-type equations. We then study the asymptotic properties of the algebraic function carrying the data of the divisor and the Baker–Akhiezer function near the two infinite points on the trigonal curve. We finally obtain algebro-geometric solutions of the entire BM lattice hierarchy in terms of the Riemann theta function.     

A lattice Boltzmann model for Maxwell’s equations

In this paper, a lattice Boltzmann model for the Maxwell’s equations is proposed by taking separate sets of distribution functions for the electric and magnetic fields, and a lattice Boltzmann model for the Maxwell vorticity equations with third order truncation error is proposed by using the higher-order moment method. At the same time, the expressions of the equilibrium distribution function and the stability conditions for this model are given. As numerical examples, some classical electromagnetic phenomena, such as the electric and magnetic fields around a line current source, the electric field and equipotential lines around an electrostatic dipole, the electric and magnetic fields around oscillating dipoles are given. These numerical results agree well with classical ones.     

A lattice Boltzmann model for the eddy–stream equations in two-dimensional incompressible flows

A lattice Boltzmann model for two-dimensional incompressible flows with eddy–stream equations is proposed. By using two kinds of distribution functions and employing several higher-order moments of equilibrium distribution functions, the eddy equation and stream function equation with the second-order truncation error are obtained. In the numerical examples, we compared the numerical results of this scheme with those obtained by other classical method. The numerical results agree well with the classical ones.     

Integrable Extended Blaszak–Marciniak Lattice and Another Extended Lattice with Their Lax Pairs

Two new lattices are given. One is an extended system of the Blaszak–Marciniak lattice, and the other is the extended system of the lattice given by Wu and Hu. Lax pairs for these two lattices are derived from their corresponding bilinear Bäcklund transformation.     

Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations

In this work, we present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential–difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential–difference equations in mathematical physics via the lattice equation. The proposed method is more effective and powerful for obtaining the exact solutions for nonlinear differential–difference equations.     

Cauchy matrix scheme for semidiscrete lattice Korteweg–de Vries-type equations

Theoretical and Mathematical Physics - Based on a determining equation set and master function, we consider a Cauchy matrix scheme for three semidiscrete lattice Korteweg–de Vries-type...     

A new regularity criterion for weak solutions to the Navier–Stokes equations

In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier–Stokes equations. We show that if any one component of the velocity field belongs to $L^{\alpha }([0,T);L^{\gamma }({\mathbb{R}}^3))$ with $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.⩽\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, $6<\gamma ⩽\infty$, then the weak solution actually is regular and unique.     

A 8-neighbor model lattice Boltzmann method applied to mathematical–physical equations

A lattice Boltzmann method (LBM) 8-neighbor model (9-bit model) is presented to solve mathematical–physical equations, such as, Laplace equation, Poisson equation, Wave equation and Burgers equation. The 9-bit model has been verified by several test cases. Numerical simulations, including 1D and 2D cases, of each problem are shown, respectively. Comparisons are made between numerical predictions and analytic solutions or available numerical results from previous researchers. It turned out that the 9-bit model is computationally effective and accurate for all different mathematical–physical equations studied. The main benefits of the new model proposed is that it is faster than the previous existing models and has a better accuracy.     

A new Jacobi rational–Gauss collocation method for numerical solution of generalized pantograph equations

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational–Gauss collocation points. The proposed Jacobi rational–Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line.     

A mixed problem for the steady Navier–Stokes equations

We consider a mixed boundary problem for the Navier–Stokes equations in a bounded Lipschitz two-dimensional domain: we assign a Dirichlet condition on the curve portion of the boundary and a slip zero condition on its straight portion. We prove that the problem has a solution provided the boundary datum and the body force belong to a Lebesgue’s space and to the Hardy space respectively.