Sobolev type embedding and weak solutions with a prescribed singular set

New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.     

闭光滑流形上的高阶线性微-积分方程

利用积分变换技巧,作者给出了C~n中闭光滑可定向流形上一个新的Bochner-Martinelli型积分的高阶偏导数的奇异积分的Hadamard主值,获得了高阶奇异积分的Plemelj公式和合成公式,还讨论了相应的变系数线性微分积分方程的正则化,证明其可转化为一类等价的Fredholm方程。并且指出其特征方程当给出一组适当的边值条件时,在L~*中存在唯一解。     

Solutions with moving singularities for a semilinear parabolic equation

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.     

Singular Integral Equation Equivalent in the Space of Smooth Functions to an Ordinary Differential Equation,Method of Successive Approximations for the Construction of Its Smooth Solutions and Its Nonsmooth Solutions

Ukrainian Mathematical Journal - We propose a singular integral equation whose definition is extended to a singular point by additional conditions. In the space of smooth functions, this equation...     

Explicit Peakon solutions to a family of wave-breaking equations

The singular traveling wave solutions of a general 4-parameter family equation which unifies the Camass-Holm equation, the Degasperis-Procesi equation and the Novikov equation are investigated in this paper. At first, we obtain the explicit peakon solutions for one of its specific case that $a=(p+2)c$, $b=(p+1)c$ and $c=1$, which is referred to a generalized Camassa-Holm-Novikov (CHN) equation, by reducing it to a second-order ordinary differential equation (ODE) and solving its associated first-order integrable ODE. By observing the characteristics of peakon solutions to the CHN equation, we construct the peakon solutions for the general 4-parameter breaking wave equation. It reveals that singularities of the peakon solutions come up only when the solutions attain singular points of the equation, which might be a universal principal for all singular traveling wave solutions for wave breaking equations.     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

Approximated Solution of a Nonlinear Singular Equation of Prandtl's Type

A nonlinear strongly singular integral equation, which can be reduced to a nonlinear singular integro-differential equation of Prandtl's type, is considered. A collocation method for solution is treated and the convergence of the approximated solution to the unique solution of the nonlinear integral equation is proved.     

A singular solution with smooth initial data for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux-Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.     

Singular integral equation with Cauchy kernel on the real axis

We find closed-form formulas for the solution of the simplest singular integral equation with Cauchy kernel on the real axis and use them to reduce the full singular integral equation considered in the paper to a Fredholm equation. We construct numerical schemes for the above-mentioned equations and estimate the accuracy order of the approximate solution.     

Singular periodic waves of an integrable equation from short capillary-gravity waves

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.     

具有周期边界的守恒型方程的守恒型差分格式

研究守恒型奇摄动方程的周期边界问题,构造了一个守恒型差分格式,利用分解解的奇性项的方法,结合问题的渐近展开,证明所构造的差分格式为一阶一致收敛。     

偏微分方程边值反问题的数值方法研究

本文研究了奇异积分方程在反边值问题中的应用问题.利用圆周上的自然积分方程及其反演公式,把Laplace方程的边值反问题转化为一对超奇异积分方程和弱奇异积分方程的组合,通过选取三角插值近似奇异积分的计算并构造相应的配置格式,并使用Tikhonov正则化方法求解所得到的线性方程组.数值实验表明了该方法的有效性.     

热载荷作用下平面裂纹问题的奇异积分方程与边界无法

从边界积分方程出发,导出了二维裂纹体热传导问题及热弹性问题的积分方程组,继而使用奇异积分方程与边界元相结合的方法,为其建立了相应的数值求解方法。此外,利用奇异积分方程的主部分析法,严格地证明了裂纹尖端温度梯度场的1/√r 奇异性,并且给出了奇性温度梯度场的精确解。最后。对一些典型例子,做了数值计算。     

A probabilistic-informational algorithm for approximate solution of singular integral equations. The method of reduction to Riemann-Hilbert problems

We propose a numerical-analytic method of solving singular integral equations with a singular kernel of Cauchy type on an interval. The method relies on the construction of a regular operator of special form, whose action on the original singular equation leads to an integral equation that admits an explicit solution. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 36–40.     

Riemann-Hilbert approach to TD equation with nonzero boundary condition

We extend the Riemann-Hilbert approach to the TD equation, which is a highly nonlinear differential integrable equation. Zero boundary condition at infinity for the TD equation is not suitable. Inverse scattering transform for this equation involves the singular Riemann-Hilbert problem, which means that the sectionally analytic functions have singularities on the boundary curve. Regularization procedures of the singular Riemann-Hilbert problem for two cases, the general case and the case for reflectionless potentials, are considered. Solitonic solutions to the TD equation are given.     

Collision of delta-waves in a turbulent model studied via a distribution product

By using a product of distributions, the existence and collision of soliton delta-waves for a singular perturbation of the Burgers conservative equation are established. We also prove that singular solitons under collision behave as in classical soliton collision (for example, as described by the Korteweg-de Vries equation). The impossibility of two delta-wave collisions for the inviscid Burgers conservative equation is also verified. The introduction is dedicated to a motivation for our study.     

一类拟线性椭圆型方程的奇摄动

本文讨论了一类二阶拟线性椭圆型方程的奇摄动问题,给出了外部解和边界层项的N阶递推方程,并对余项进行了估计,从而导得了解的渐近展开式和摄动问题解的存在唯一性.     

Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.