Hybrid,numerical solutions,for three-dimensional,compressible Navier-Stokes layer

The author proposes new hybrid solutions for the three-dimensional, compressible Navier-Stokes layer (NSL) over a flying configuration (FC), which use the analytical potential solutions, of the same FC, two times, namely: to reinforce the numerical solutions by multiplying them with these analytical solutions and as outer flow (instead of the parallel flow, used by Prandtl in his boundary layer theory). These hybrid solutions fulfill the last behavior, have correct jumps along the singular lines (like subsonic leading edges, junction lines wing-fuselage, etc.), are split, accurate and rapid convergent. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time dependent linear transport I. Existence,uniqueness, and continuous dependence

This paper concerns the time dependent linear transport equation posed in a multidimensional rectangular parallelepiped with partially reflecting walls. We consider the continuous transport equation and the discrete ordinate equations simultaneously. Our boundary condition, partial specular reflection, includes both vacuum and reflecting boundaries as special cases. We define strong and weak solutions of the problem, strong solutions being solutions in the ordinary sense and weak solutions being distributions, and show that a weak solution is a strong solution if it has space and time derivatives almost everywhere. For weak solutions we establish existence, uniqueness, and continuous dependence upon the initial data and the other functions which define the problem.     

Convergence,boundedness, and ergodicity of regime-switching diusion processes with infinite memory

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions X_t: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (X_t,Λ(t)); and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.     

Nonlinear diffusion from a delocalized source: affine self-similarity,time reversal, & nonradial focusing geometries

A family of explicit solutions is described, to the porous medium equation in its full range of nonlinearities (plus some analogous fourth-order diffusions), in which the pressure is given by a quadratic function of space at each instant in time. These include spreading solutions whose source is concentrated on any conic region of dimension lower than the ambient space, and solutions which focus at conic regions. The singular limiting distributions are affine projections of Barenblatt type solutions (with arbitrary signature) onto lower dimensional subspaces. All affine images of Barenblatt solutions form an invariant space on which the dynamics can be integrated explicitly. A time-reversal symmetry is revealed for the pressure equation which transforms spreading solutions to focusing solutions, and vice-versa. This yields new information about the long and short time asymptotics of finite-mass solutions, about the instability of focusing, and about singularity geometry.     

First-Order and Second-Order,Chaos-Free,Finite Difference Schemes for Fisher Equation

A new class of finite difference schemes is constructed for Fisher partial differential equation i.e. the reaction-diffusion equation with stiff source term: $au(1-u)$. These schemes have the properties that they reduce to high fidelity algorithms in the diffusion-free case namely in which the numerical solutions preserve the properties in the exact solutions for arbitrary time step-size and reaction coefficient α＞0 and all nonphysical spurious solutions including bifurcations and chaos that normally appear in the standard discrete models of Fisher partial differential equation will not occur. The implicit schemes so developed obtain the numerical solutions by solving a single linear algebraic system at each step. The boundness and asymptotic behaviour of numerical solutions obtained by all these schemes are given. The approach constructing the above schemes can be extended to reaction-diffusion equations with other stiff source terms.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Cone convexity,cone extreme points,and nondominated solutions in decision problems with multiobjectives

Although there is no universally accepted solution concept for decision problems with multiple noncommensurable objectives, one would agree that agood solution must not be dominated by the other feasible alternatives. Here, we propose a structure of domination over the objective space and explore the geometry of the set of all nondominated solutions. Two methods for locating the set of all nondominated solutions through ordinary mathematical programming are introduced. In order to achieve our main results, we have introduced the new concepts of cone convexity and cone extreme point, and we have explored their main properties. Some relevant results on polar cones and polyhedral cones are also derived. Throughout the paper, we also pay attention to an important special case of nondominated solutions, that is, Pareto-optimal solutions. The geometry of the set of all Pareto solutions and methods for locating it are also studied. At the end, we provide an example to show how we can locate the set of all nondominated solutions through a derived decomposition theorem.     

Breather wave,periodic, and cross-kink solutions to the generalized Bogoyavlensky-Konopelchenko equation

The present article deals with multi-waves and breather wave solutions of the generalized Bogoyavlensky-Konopelchenko equation by virtue of the Hirota bilinear operator method and the semi-inverse variational principle. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic, and cross-kink solutions in which have been investigated by the approach of the bilinear method. With certain parameter constraints in the multi-waves and breather, all cases of the periodic and cross-kink solutions can be captured from the one and two soliton(s). The obtained solutions are extended with numerical simulation to analyze graphically, which results into 1- and 2-soliton solutions and also periodic and cross-kink solutions profiles, that will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.     

Symmetries,the current function,and exact solutions for Broadwell’s two-dimensional stationary kinetic model

We investigate the Broadwell stationary kinetic model for four velocities on a plane using the current function that satisfies a partial differential equation. For this equation, we evaluate the algebras of classical and nonclassical symmetries and then construct invariant solutions. All classes of solutions describe nonpotential flows. We consider the relation between nonclassical symmetries and previously obtained solutions.     

Polynomial solutions of the classical equations of Hermite,Legendre, and Chebyshev

The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x ^M on the right-hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.     

QUASI－CONVEX MULTIOBJECTIVE GAME－SOLUTION CONCEPTS，EXISTENCE AND SCALARIZATION

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Exponential sums,the geometry of hyperplane sections,and some diophantine problems

We estimate exponential sums with additive character along an affine variety given by a system of homogeneous equations, with a homogeneous function in the exponent. The proof uses the results of Deligne’s Weil Conjectures II and a generalization of Lefschetz hyperplane theorem to singular varieties. We apply our estimate to obtain an upperbound for the number of integer solutions of a system of homogeneous equations in a box. Another application is devoted to uniform distribution of solutions of a system of homogeneous congruences modulo a prime in the following sense: the portion of solutions in a box is proportional to the volume of the box, provided the box is not very small.     

2D steady-state general solution and fundamental solution for fluid-saturated,orthotropic, poroelastic materials

The 2D steady-state solutions regarding the expressions of stress and strain for fluid-saturated, orthotropic, poroelastic plane are derived in this paper. For this object, the general solutions of the corresponding governing equation are first obtained and expressed in harmonic functions. Based on these compact general solutions, the suitable harmonic functions with undetermined constants for line fluid source in the interior of infinite poroelastic body and a line fluid source on the surface of semi-infinite poroelastic body are presented, respectively. The fundamental solutions can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions, equilibrium conditions and boundary conditions.     

Stability of multi-compacton solutions and Backlund transformation in K(m,n,1)

We introduce a fifth-order K(m,n,1) equation with nonlinear dispersion to obtain multi-compacton solutions by Adomian decomposition method. Using the homogeneous balance (HB) method, we derive a Backlund transformation of a special equation K(2,2,1) to determine some solitary solutions of the equation. To study the stability of multi-compacton solutions in K(m,n,1) and to obtain some conservation laws, we present a similar fifth-order equation derived from Lagrangian. We finally show the linear stability of all obtained multi-compacton solutions.     

K(n,2n,-n)方程行波解的分支

本文研究了K(n,2n,-n)方程行波解与参数a,b,c,g,n等的关系.利用动力系统分支理论,得到了孤立波、扭结和反扭结波解,以及不可数无穷多光滑周期波解的存在性.本文推广了文献[1]中的结果.     

New sets of solitary wave solutions to the KdV,mKdV, and the generalized KdV equations

In this work we introduce new schemes, each combines two hyperbolic functions, to study the KdV, mKdV, and the generalized KdV equations. It is shown that this class of equations gives conventional solitons and periodic solutions. We also show that the proposed schemes develop sets of entirely new solitary wave solutions in addition to the traditional solutions. The analysis can be used to a wide class of nonlinear evolutions equations.     

The cost of deterministic,adaptive, automatic algorithms: Cones,not balls

Automatic numerical algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. The computational cost is often determined adaptively by the algorithm based on the function values sampled. While adaptive, automatic algorithms are widely used in practice, most lack guarantees, i.e., conditions on input functions that ensure that the error tolerance is met.     

Keplerian Action,Convexity Optimization,and the 4-Body Problem

In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints. Our approach is based on some novel features of the Keplerian action functional, constraint convex optimization techniques, and variational methods. We demonstrate the strength of this method by constructing relative periodic solutions for the planar four-body problem within a special topological class, and our results hold for an open set of masses.     

A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities

A priori bounds for solutions of a wide class of quasilinear degenerate elliptic inequalities are proved. As an outcome we deduce sharp Liouville theorems. Our investigation includes inequalities associated to p-Laplacian and the mean curvature operators in Carnot groups setting. No hypotheses on the solutions at infinity are assumed. General results on the sign of solutions for quasilinear coercive/noncoercive inequalities are considered. Related applications to population biology and chemical reaction theory are also studied.     

L1 asymptotic behavior of compressible,isentropic, viscous 1-D flow

We study the large time behavior in L¹ of the compressible, isentropic, viscous 1-D flow. Under the assumption that the initial data are smooth and small, we show that the solutions are approximated by the solutions of a parabolic system, and in turn by diffusion waves, which are solutions of Burgers equations. Decay rates in L¹ are obtained. Our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system. © 1994 John Wiley & Sons, Inc.