一类非线性发展方程的精确解

利用hirota双线性法,得到(3+1)维孤子方程、(3+1)维KP-Boussinesq方程、(2+1)维修正Caudrey-Dodd-Gibbon-Kotera-S awada方程、Hirota-Satsuma浅水波方程的精确解,并做出一部分解的图形,进一步研究解的结构和性质.     

Exact Solution of the Time Fractional Variant Boussinesq-Burgers Equations

Applications of Mathematics - In the present article, we consider a nonlinear time fractional system of variant Boussinesq-Burgers equations. Using Lie group analysis, we derive the infinitesimal...     

Solution of Goursat problem for the Maxwell-Bloch system

A formal method of solving the initial—boundary-value problem in the quadrantx0,t0 for the Maxwell-Bloch system of equations is presented in terms of the inverse scattering problem.Institute of Mathematics, Urals Branch, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 29–37, January, 1994.     

Klein-Gordon-Schrodinger方程组的精确孤立波解

利用齐次平衡原则导出Klein-Gordom-Schrodinger方程组的精确孤立波解.该解在形式上比文献中纯理论的存在性证明的结果更一般,文献中的解的形式是该结果的特殊情形.     

关于某些非线性偏微分方程的准确解

以分层理论为基础，对偏微分方程给以一种全新的分类．     

On the Solution of Non-linear Volterra Integral Equations Using Two-point Rational Approximates

In recent years methods have been developed for obtaining asymptoticsolutions to non-linear Volterra integral equations using Mellintransform techniques which enable series solutions for largeand small values of the argument to be found. This paper indicateshow global approximations to the solutions can be constructedby the use of two-point rational approximants in the form ofcontinued fractions.     

An Algorithm for Minimax Solution of Overdetermined Systems of Non-linear Equations

The problem of minimizing the maximum residual of a system ofnon-linear equations is studied in the case where the numberof equations is larger than the number of unknowns. It is supposedthat the functions defining the problem have continuous firstderivatives, and the algorithm is based on successive linearapproximations to these functions. The resulting linear systemsare solved in the minimax sense, subject to bounds on the solutions,the bounds being adjusted automatically, depending on the goodnessof the linear approximations. It is proved that the method alwayshas sure convergence properties. Some numerical examples aregiven.     

A Collocation Method for the Numerical Solution of Non-linear Parabolic Partial Differential Equations

Lagrange interpolation formulae are used to obtain a new algorithmfor the approximate polynomial solution of linear and non-linearparabolic equations. The algorithm is described for problemsin one space dimension, although it is applicable to problemsdefined in two or more dimensions. It is also shown how thealgorithm may be adapted to solve a moving boundary (Stefan)problem.     

三重介质弹性渗流方程组的精确解

本文用分解的方法求得了有限封闭地层中三重介质弹性渗流的精确解.它不仅概括了已有的双重介质弹性渗流的主要结果,而且给出各重介质弹性渗流的基本特征.     

Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations

Journal of Nonlinear Science - For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By...     

圆薄膜在集中力作用下大变形基本方程的精确解

利用一种新型的简易方法,获得圆薄膜在中心集中力作用下的基本方程与边界条件下非线性边值问题的精确解;并利用现代不动点定理讨论了该问题解的存在唯一性．虽然求解的是圆薄膜在中心集中力作用下的非线性问题,但此原理亦可应用在其它类似的非线性问题．     

一类分数阶非线性波方程的精确解及应用

基于分离变量的思想构造了分数阶非线性波方程含常系数的解的形式.在用待定系数法求解时,根据原方程确定假设解中的待定参数,得到具体解的表达式.利用该方法求解了3个非线性波方程,即分数阶CH(Camassa-Holm)方程、时间分数阶空间五阶Kdv-like方程、分数阶广义Ostrovsky方程.比较简便地得到了这些方程的精确解.文献中关于整数阶非线性波方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.对能够通过待定系数法求出精确解的分数阶微分方程所应满足的条件进行了阐述.     

Boltzmann方程的精确解

本文建立了非线性Boltzmann输运方程与线性AKNS方程之间的直接联系,从而以不同于Chapman,Enskog和Grad的方式,使Boltzmann方程化归为Dirac方程的求解。在没有其他外场作用的情形中,Boltzmann方程的精确解可以用逆散射方法来求得。     

Lipschitz Operators and the Solvability of Non-linear Operator Equations

Let U and V be Banach spaces, L and N be non-linear operators from U into V. L is said to be Lipschitz if LI(L) := sup{‖Lx - Ly‖. ‖x - y‖^-1 : x ≠ y} is finite. In this paper, we give some basic properties of Lipschitz operators and then discuss the unique solvability, exact solvability,approximate solvability of the operator equations Lx = y and Lx Nx = y. Under some conditions we prove the equivalence of these solvabilities. We also give an estimation for the relative-errors of the solutions of these two systems and an application of our method to a non-linear control system.     

Exact Solutions of Extended Boussinesq Equations

The problem addressed in this paper is the verification of numerical solutions of nonlinear dispersive wave equations such as Boussinesq-like system of equations. A practical verification tool for numerical results is to compare the numerical solution to an exact solution if available. In this work, we derive some exact solitary wave solutions and several invariants of motion for a wide range of Boussinesq-like equations using Maple software. The exact solitary wave solutions can be used to specify initial data for the incident waves in the Boussinesq numerical model and for the verification of the associated computed solution. The invariants of motions can be used as verification tools for the conservation properties of the numerical model.     

Exact Internal Controllability of Maxwell's Equations

In this paper we obtain two exact internal controllability results of Maxwell's equations in a general region by using multiplier techniques. The first one is exact controllability in a short time, in which we obtain the ``optimal" (observability) estimates when the location and the shape of the controller is fixed. What happens if we allow the controller to change? Under some conditions, we show that by doing that the system can be exactly controllable within any given time duration, which is our second exact controllability result. Accepted 30 September 1998     

Exact Closed Form Solution for Two Coupled Inhomogeneous First Order Difference Equations with Variable Coefficients

We present an exact closed form solution for two coupled, homogeneous as well as inhomogeneous, first order difference equations with variable coefficients. The solution is obtained by using the graph theoretic, discrete path formalism. The central parameters in the solution are the crossing index and the crossing number. The transition from an enumerative graph theoretic solution to a closed form combinatorial solution is made possible by an isomorphism in-between paths on the signal flow graph, and n -tuplets of binary numbers.     

Construction of a Multisoliton Blowup Solution to the Semilinear Wave Equation in One Space Dimension

We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blowup solution with a characteristic point, we refine the blowup behavior first derived by Merle and Zaag. We also refine the geometry of the blowup set near a characteristic point and show that, except for perhaps one exceptional situation, it is never symmetric with respect to the characteristic point. Then, we show that all blowup modalities predicted by those authors do occur. More precisely, given any integer k ≥ 2 and $\zeta _0 \in {\cal R}$ , we construct a blowup solution with a characteristic point a such that the asymptotic behavior of the solution near (a,T(a)) shows a decoupled sum of k solitons with alternate signs whose centers (in the hyperbolic geometry) have ζ₀ as a center of mass for all times. © 2013 Wiley Periodicals, Inc.