矩阵方程X-A*X-q A=Q当q＞1时的Hermite正定解

讨论了矩阵方程X-A*X-qA=Q在q＞1时的Hermite正定解的存在性和解的性质,并且构造了两种数值求解的迭代方法.利用数值例子对以上结果进行了说明.     

Generalized finite differences using fundamental solutions

It is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions. The fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a Dirac delta distribution. Fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary‐only global approximations in the domain of interest. In this work the same characteristic of the fundamental solutions, that of solving the homogeneous equation exactly, is used but not to build a global approximation. On the contrary, local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid. Copyright © 2009 John Wiley & Sons, Ltd.     

一类具有非线性边界流的双重退化抛物型方程的临界指标

本文研究了一类具有非线性边界流的双重退化抛物型方程,该方程可用来描述多孔介质中的非牛顿渗流现象,可以描述气体或液体在多孔介质中的流动,具有广泛的实际背景.通过构造不同的自相似上、下解得到了方程的临界指标,即整体存在指标po和临界Fujita指标pc.主要结果为:当0＜p≤po时,方程存在整体解;当po＜p＜pc时,方程...     

广义Tanh函数法中Riccati方程和sine-Gordon方程的新解及其新应用

非线性偏微分方程的显式解析解,特别是行波解,蕴含了方程的丰富信息,对于描述各种现象的发展规律起着至关重要的作用．本文尝试构造 KdV 方程多种形式的新显式行波解．首先,利用试探函数法和 Matlab计算给出了 Riccat 方程的许多新显式解析解．其次,运用广义 Tanh 函数法以及 Riccati 方程的新解得到了 sine-Gordon 方程的许多新显式解析解．最后,作为新的应用,把三角函数法结合 sine-Gordon 方程的新显式解析解并利用简化的变换形式进一步找到了 KdV 方程的许多新显式行波解．这些结果推广和补充了以往的相关研究成果,特别地,这些方法和新的结果可以用于求解许多非线性偏微分方程的新显式行波解．     

Numerical solution for degenerate scale problem for exterior multiply connected region

Based on some previous publications, this paper investigates the numerical solution for degenerate scale problem for exterior multiply connected region. In the present study, the first step is to formulate a homogenous boundary integral equation (BIE) in the degenerate scale. The coordinate transform with a magnified factor, or a reduced factor h is performed in the next step. Using the property ln(hx)=ln(x)+lg(h), the new obtained BIE equation can be considered as a non-homogenous one defined in the transformed coordinates. The relevant scale in the transformed coordinates is a normal scale. Therefore, the new obtained BIE equation is solvable. Fundamental solutions are introduced. For evaluating the fundamental solutions, the right-hand terms in the non-homogenous equation, or a BIE, generally take the value of unit or zero. By using the obtained fundamental solutions, an equation for evaluating the magnified factor “h” is obtained. Finally, the degenerate scale is obtainable. Several numerical examples with two ellipses in an infinite plate are presented. Numerical solutions prove that the degenerate scale does not depend on the normal scale used in the process for evaluating the fundamental solutions.     

On the derivation of first integrals for similarity solutions

In two recent papers the authors have obtained a number of first integrals for similarity solutions of nonlinear diffusion and of general high-order nonlinear evolution equation. Such integrals exist only for special parameter values and are obtained via integration of the ordinary differential equation, which results when the functional form of the solution is substituted into the governing partial differential equation. In this paper we show that these special parameter values also occur in a natural way when we utilize the first order partial differential equation instead of the explicit functional form and we ask under what conditions can a first integral with respect to either of the independent variables x or t be deduced. This simple procedure generates all previous results and presents the idea of similarity solutions in an entirely new light. That is, the significant features of similarity solutions for partial differential equations are not necessarily the explicit functional form and subsequent reduction to an ordinary differential equation but rather that the solutions sort are common to two partial differential equations. The process is illustrated with reference to an extensive number of examples including nonlinear diffusion, general diffusion equations containing a number of parameters and high-order nonlinear evolution equations. In addition a new exact solution for nonlinear diffusion is obtained which is illustrated graphically.     

Methods of fundamental solutions for time‐dependent heat conduction problems

A time‐dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified Helmholtz equation or the fundamental solution to the heat equation. This paper presents solutions using both formulations in terms of initial and boundary conditions. Such formulations enable calculation of errors and variance, which indicates sensitivities of solutions to uncertainties in initial and boundary conditions. Both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Converging spherical and cylindrical shock waves

The self-similar solutions for converging spherical and cylindrical strong shock waves in a non-ideal gas satisfying the equation of state of the Mie-Gruneisen type are investigated. The equations governing the flow, which are highly non-linear hyperbolic partial differential equations, are first reduced to a Poincaré-type ordinary differential equation with suitable approximation. Such an approximation helps in obtaining the self-similar solutions and the similarity exponent numerically by phase-plane analysis.     

Soliton-like solutions and periodic form solutions for two variable-coefficient evolution equations using symbolic computation

Summary. Some variable-coefficient generalizations of some nonlinear evolution equations (NLEEs) bear more realistic physical importance. By means of a generalized Riccati equation expansion (GREE) method and a symbolic computation system – Maple – we investigate the variable-coefficient Fisher-type equation and the nearly concentric KdV equation. As a result, rich families of exact analytic solutions for these two equations, including the non-travelling waves and coefficient functions soliton-like solutions, singular soliton-like solutions, and periodic form solutions, are obtained.     

Numerical examination for degenerate scale problem for ellipse‐shaped ring region in BIE

This paper investigates the degenerate scale problem for an ellipse‐shaped ring region in boundary integral equation (BIE). A homogenous integral equation is introduced. The integral equation is reduced to an algebraic equation after discretization. The critical value for the degenerate scale can be obtained from the vanishing condition of a determinant. It is proved that there are two critical values for the degenerate scale, rather than one. This finding is first proposed in the paper. Two particular problems with known solutions are examined numerically. The loadings applied on the exterior boundary may result in a resultant force in the x‐direction or in the y‐direction. The improper numerical solutions have been found once the real size approaches the critical value. Two techniques for avoiding the improper solutions are suggested. The techniques depend on the appropriate choice of the used size or adding a constant in a kernel of the integral equation. It is proved that both techniques will give accurate numerical results. Numerical examinations for the problem are emphasized in the paper. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of nonlinear partial differential equations using the Chebyshev spectral method

The spectral method is a powerful numerical technique for solving engineering differential equations. The method is a specialization of the method of weighted residuals. Trial functions that are easily and exactly differentiable are used. Often the functions used also satisfy an orthogonality equation, which can improve the efficiency of the approximation. Generally, the entire domain is modeled, but multiple sub-domains may be used. A Chebyshev-Collocation Spectral method is used to solve a two-dimensional, highly nonlinear, two parameter Bratu's equation. This equation previously assumed to have only symmetric solutions are shown to have regions where solutions that are non-symmetric in x and y are valid. Away from these regions an accurate and efficient technique for tracking the equation's multi-valued solutions was developed.It is found that the accuracy of the present method is very good, with a significant improvement in computer time.     

A relation between the solutions of the half-space Dirichlet problems for Helmholtz's equation in ℝ n and Laplace's equation in ℝ n+1

Summary Multiple Fourier transforms are used to derive the solutions of the half-space Dirichlet problems for Helmholtz's equation in ℝ ⁿ and Laplace's equation in ℝ ⁿ⁺¹ and to exhibit the relation between the two solutions.     

A nonlinear eigenvalue problem from thin-film flow

Steady solutions of a fourth-order partial differential equation modeling the spreading of a thin film including the effects of surface shear, gravity, and surface tension are considered. The resulting fourth-order ordinary differential equation is transformed into a canonical third-order ordinary differential equation. When transforming the problem into standard form the position of the contact line becomes an eigenvalue of the physical problem. Asymptotic and numerical solutions of the resulting eigenvalue problem are investigated. The eigenvalue formulation of the steady problem yields a maximum value of the contact angle of 63.4349^?.     

Exact solutions of a Navier–Stokes equation in the form of polynomials

Exact polynomial solutions of a Navier–Stokes equation that describe flow of a Newtonian incompressible fluid are classified. Four classes of such solutions are recognized: parametric, time, and coordinate solutions and polynomials in inverse powers of the Reynolds number. The procedure of finding the exact binomial solutions is discussed. An example of nonseparating flow about a circle is given.     

Green’s functions for non-self-adjoint problems in heat conduction with steady motion

Heat conduction in a rectangular parallelepiped that is in steady motion relative to a fluid is studied in this paper. The governing equation consists of the standard heat equation plus lower-order derivative terms with the space variables that represent the effects of the solid flow. The presence of the first-order-derivative terms with the space variables renders the spatial part of the governing differenial equation non-self-adjoint and care must be exercised in defining the new Green’s functions to be used in representing the solutions of initial- and boundary-value problems. It is illustrated how the Green’s functions may be constructed and how solutions of initial- and boundary-value problems may be obtained that lead to numerical results. Convergence properties of the solutions are also discussed.     

Traveling wave solutions of nonlinear partial differential equations

Elementary transformations are utilized to obtain traveling wave solutions of some diffusion and wave equations, including long wave equations and wave equations the nonlinearity of which consists of a linear combination of periodic functions, either trigonometric or elliptic. In particular, a theorem is established relating the solutions of a single cosine equation and a double sine-cosine equation. It is shown that the latter admits a Bäcklund Transformation.     

Localization of Resonant Spherical Waves

This paper treats radial spherical resonant waves excited in the transresonant regime. An approximate general solution of a perturbed-wave equation is presented here, which takes into account nonlinear, spatial, and dissipative effects. Then the boundary problem reduces to the perturbed compound Burgers–Korteweg–de Vries equation (BKdV) in time. Several solutions to this equation are constructed. Shock waves may be excited near resonance according to the solutions for an inviscid medium. However, both viscosity and spatial dispersion begin to be important very close to resonance and prevent the formation of shock discontinuity. As a result, periodic localized excitations are generated in resonators instead of shock waves.     

(2+1)维Nizhnik方程的Jacobi椭圆函数周期解

利用最近提出的F-展开法,导出了(2 1)维Nizhnik方程的由Jacobi椭圆函数表示的周期解,并且在极限情况下,可以推得(2 1)维Nizhnik方程的孤波解以及其他形式解。     

Some results for an $${\mathcal{N}}$$-dimensional nonlinear diffusion equation with radial symmetry

The solutions of a nonlinear diffusion equation by considering the radially symmetric N{\mathcal{N}}-dimensional case are investigated. This equation has the nonlinearity present in the diffusive term and external force. The solutions are obtained by using a similarity method and connected to the q-exponential and q-logarithmic functions which emerge from the Tsallis formalism. In addition, the results obtained here may be useful to investigate a rich class of situations related to anomalous diffusion.     

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.