Boundary regularity for the Navier-Stokes equations in a half-space

Weak solutions to the nonstationary Navier-Stokes equations in a half-space are locally bounded at the boundary except for a closed set with finite one-dimensional Hausdorff measure.This research was supported in part by the National Science Foundation Grant MCS-81-02737     

Global attractors for degenerate partial differential equations from nonlinear viscoelasticity

We study several problems for the forced motion of light, uniform, nonlinearly viscoelastic bodies carrying heavy attachments. A ‘reduced’ problem for such motions is obtained by setting the ratio of the inertia of the viscoelastic body to the inertia of the attachment equal to zero. Using methods from infinite-dimensional dynamical systems theory, we prove that the degenerate partial differential equation of this reduced problem has an attractor and that this attractor is contained in an invariant two-dimensional manifold on which solutions are governed by the classical ordinary differential equation for the forced motion of a particle on a massless spring.     

Local behavior of solutions of some elliptic equations

We study the local behavior of solutions of some nonlinear elliptic equations. These equations are of interest in differential geometry and mathematical physics.     

Exact solutions of some nonlinear equations

We obtain exact solutions of three nonlinear diffusive equations and of the KdV-Burger equation by making an ansatz for the solution in each case.     

A splitting extrapolation for solving nonlinear elliptic equations with d-quadratic finite elements

Nonlinear elliptic partial differential equations are important to many large scale engineering and science problems. For this kind of equations, this article discusses a splitting extrapolation which possesses a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than Richardson extrapolation. According to the problems, some domain decompositions are constructed and some independent mesh parameters are designed. Multi-parameter asymptotic expansions are proved for the errors of approximations. Based on the expansions, splitting extrapolation formulas are developed to compute approximations with high order of accuracy on a globally fine grid. Because these formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems.     

On the solutions in elliptic functions of integrable nonlinear equations

A new solution in elliptic functions for the KdV equation is constructed using the method proposed by Belokolos and the author.     

Solitary wave solutions for some nonlinear time-fractional partial differential equations

In this paper, we have studied the hybrid projective synchronisation for incommensurate, integer and commensurate fractional-order financial systems with unknown disturbance. To tackle the problem of unknown bounded disturbance, fractional-order disturbance observer is designed to approximate the unknown disturbance. Further, we have introduced simple sliding mode surface and designed adaptive sliding mode controllers incorporating with the designed fractional-order disturbance observer to achieve a bounded hybrid projective synchronisation between two identical fractional-order financial model with different initial conditions. It is shown that the slave system with disturbance can be synchronised with the projection of the master system generated through state transformation. Simulation results are presented to ensure the validity and effectiveness of the proposed sliding mode control scheme in the presence of external bounded unknown disturbance. Also, synchronisation error for commensurate, integer and incommensurate fractional-order financial systems is studied in numerical simulation.     

Computational study of some nonlinear shallow water equations

The shallow water equations have wide applications in ocean, atmospheric modeling and hydraulic engineering, also they can be used to model flows in rivers and coastal areas. In this article we obtained exact solutions of three equations of shallow water by using $\frac{{G'}} {G} $ -expansion method. Hyperbolic and triangular periodic solutions can be obtained from the $\frac{{G'}} {G} $ -expansion method.     

Exact travelling wave solutionsof some nonlinear evolutionary equations

A direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions.     

The modified simple equation method for solving some fractional-order nonlinear equations

Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinear fractional Klein–Gordon equation by using the modified simple equation method.     

On smoothness of solutions of some elliptic functional-differential equations with degeneration

We consider second-order differential-difference equations in bounded domains in the case where several degenerate difference operators enter the equation. The degeneration leads to the fact that the multiplicity of the zero eigenvalue for the corresponding differential-difference operator becomes infinite. Regularity of generalized solutions for such equations is known to fail in the interior of the domain. However, we prove that the projections of solutions onto the orthogonal complement to the kernel of the “leading” difference operator remain regular in certain subdomains which form a decomposition of the original domain.     

Pure soliton solutions of some nonlinear partial differential equations

A general approach is given to obtain the system of ordinary differential equations which determines the pure soliton solutions for the class of generalized Korteweg-de Vries equations (cf. [6]). This approach also leads to a system of ordinary differential equations for the pure soliton solutions of the sine-Gordon equation.     

Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method

Based on the Exp-function method, exact solutions for some nonlinear evolution equations are obtained. The KdV equation, Burgers' equation and the combined KdV–mKdV equation are chosen to illustrate the effectiveness of the method.     

Orbital stability of standing waves for some nonlinear Schrödinger equations

We present a general method which enables us to prove the orbital stability of some standing waves in nonlinear Schrödinger equations. For example, we treat the cases of nonlinear Schrödinger equations arising in laser beams, of time-dependent Hartree equations ....     

Using the seventh-order numerical method to solve first-order nonlinear coupled-wave equations for degenerate two-wave and four-wave mixing

Using a new seventh-order numerical method [theO(h ⁷) method] for solving two-point boundary value problems, numerical solutions of the first-order nonlinear coupledwave equations for degenerate two-wave and four-wave mixing in a reflection geometry have been obtained. A computer program employing the Gauss-Jordan elimination technique has also been adopted to effectively solve the resultant large, sparse and unsymmetric matrix, obtained from theO(h ⁷) method and the Newton-Raphson iteration method. Numerical results from the computer calculations are presented graphically. A comparison between thisO(h ⁷) method and the shooting method, mainly from the viewpoint of computational efficiency, is also made.     

Lie-Bäcklund transformation and ordinary differential equations associated with some nonlinear equations

Lie-Bäcklund-type analysis have been performed for one nonlinear partial differential equation, which is somewhat different from those usually studied. We consider the KdV equation with an explicit x dependence. In this case we show the form of symmetry generators and find the ordinary differential equation connected with them having no movable critical points. This clearly extends the class of equations analysed by Ablowitz.     

Fourier transform and smoothness of solutions of a class of semilinear elliptic degenerate equations with double characteristics

This paper is a continuation of our earlier note [V. T. T. Hien and N. M. Tri, “Analyticity of Solutions of Semililnear Equations with Double Characteristics,” J. Math. Anal. Appl. 337, 1249–1260 (2008)]. Here we prove the analyticity of solutions of a class of semilinear elliptic degenerate equations with double characteristics by using the Fourier transform.     

A higher order lattice BGK model for simulating some nonlinear partial differential equations

In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux + βunux - γuxx + δuxxx = F(u). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.     

Soliton solutions of some nonlinear evolution equations with time-dependent coefficients

In this paper, we obtain exact soliton solutions of the modified KdV equation, inhomogeneous nonlinear Schrödinger equation and G(m, n) equation with variable-coefficients using solitary wave ansatz. The constraint conditions among the time-dependent coefficients turn out as necessary conditions for the solitons to exist. Numerical simulations for dark and bright soliton solutions for the mKdV equation are also given.     

Exact and explicit solitary wave solutions to some nonlinear equations

Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative Φ⁴-model equation, the generalized Fisher equation, and the elastic-medium wave equation.