Singular Integral Operators with Cauchy Kernel in Fractional Spaces

We single out the Besov spaces that embed into the class of continuous functions and enjoy the Fredholm theory of linear singular integral equations with Cauchy kernel. We give basic results of this theory in the class of continuous (rather than Holder continuous) functions in terms of Besov spaces. Alongside elliptic operators we consider violations of ellipticity: the degeneration of the symbol of an operator at finitely many points.     

具有超解析Cauchy核的奇异积分方程

We investigate the solvability and the solving methods of singular integral equation with hyperanalytic Cauchy kernel(?) where α,k are hypercomplex functions in the sense of A. Douglis, φ is the unknown function. We prove that the equation Kφ=f , under certain conditions, is Noetherian. And a direct method to solve Kφ = f whenα and k are the boun dary values of hyperanalytic functions is given.     

非线性分数阶微分方程的奇摄动

研究了—类奇摄动非线性分数阶微分方程Cauchy问题．在适当的条件下,首先求出了原问题的外部解,然后利用伸长变量、合成展开法和幂级数展开理论构造出解的初始层项,并由此得到解的形式渐近展开式．最后利用微分不等式理论,讨论了问题解的渐近性态,得到了原问题解的一致有效的渐近估计式．     

cauchy核奇异积分方程关于积分曲线的稳定性

当E为复平面上的有界连通区域,所有已知函数在E上满足Holder条件,光滑封闭曲线гсE时,借助广义逆,讨论了正则型Cauchy核奇异积分方程在г发生某种光滑扰动时的稳定性问题,给出了相应的误差估计,并建立了收敛性定理.     

Hermite-Type Method for Volterra Integral Equation with Certain Weakly Singular Kernel

1.IntroductionThispaPerconsidersthenumericalsolutionofthesecondkindVolterraintegralequationy(t)+(Ky)(t)=g(t),(1.1)wherey(t)istheunknownsolution,g(t)isagivenfUnctionandKistheintegraJoperatorforsomegivenkernelfunctionK,(Ky)(t)=l'K(f)y(8)ids.(1.2)Suchequationsarisefromcertaindiffusionproblems.BecauseKisnotcompact,sothestandaxdstabilityproofSfornumericaJmethodsdonotfit.ManypeoplehaveworkedonHermite-typecollocationmethodsforsecond-kindVolterraintegralequationswithsmoothkernels[3,4,5'6],butver…     

Boundedness for the Singular Integral with Variable Kernel and Fractional Differentiation on Weighted Morrey Spaces

Let T be the singular integral operator with variable kernel, T*be the adjoint of T and T~#be the pseudo-adjoint of T. Let T_1T_2 be the product of T_1 and T_2, T_1? T_2 be the pseudo product of T_1 and T_2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator Dγon the weighted Morrey spaces.     

Lie Point Symmetry Analysis of the Harry-Dym Type Equation with Riemann-Liouville Fractional Derivative

In this paper, Lie point symmetry group of the Harry-Dym type equation with Riemann-Liouville fractional derivative is constructed. Then complete subgroup classification is obtained by means of the optimal system method. Finally, corresponding group-invariant solutions with reduced fractional ordinary differential equations are presented via similarity reductions.     

非线性分数阶微分方程Robin型边值问题

In this paper,the existence and multiplicity of positive solutions for Robin type boundary value problem of differential equation involving the Riemann-Liouville fractional order derivative are established.     

一类具有Riemann-Liouville分数阶积分边值条件的奇异分数阶微分方程解的存在性

研究一类具有Riemann-Liouville分数阶积分边值条件的奇异分数阶微分方程解的存在性,其非线性项包含Caputo型分数阶导数,且在t=0具有奇异性.应用Schauder不动点定理获得了解的存在性定理,并给出了应用实例.     

带弱奇异核非线性积分微分方程的有限元分析

讨论了带弱奇异核的非线性抛物积分微分方程的Hermite型各向异性矩形元逼近.在各向异性网格下导出了关于Riesz投影的L~2和H~1模的误差估计.在半离散和向后欧拉全离散格式下,基于Riesz投影的性质并利用平均值技巧,分别得到了L~2模意义下的最优误差估计.     

A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.     

L~1空间带弱奇异核的第二类Fredholm积分方程解法探究

针对带有弱奇异核的第二类Fredholm积分方程数值解法问题,介绍了两种方法.一种方法是直接用L~1空间中的离散化方法求其数值解;另一种方法是将弱奇异核通过迭代变为连续核,再用L~1空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出直接用L~1空间中离散化方法更好.     

带弱奇异核的第二类Fredholm积分方程数值解法比较

针对带有弱奇异核的第二类Fredholm积分方程数值解法问题,介绍了两种方法.一种方法是泰勒级数展开法;另一种方法是将弱奇异核通过迭代变为连续核,再用L¹空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出L1空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出L¹空间中离散化方法更好.     

A Compact Difference Scheme for an Evolution Equation with a Weakly Singular Kernel

This paper is concerned with a compact difference scheme with the truncation error of order 3/2 for time and order 4 for space to an evolution equation with a weakly singular kernel. The integral term is treated by means of the second order convolution quadrature suggested by Lubich. The stability and convergence are proved by the energy method. A numerical experiment is reported to verify the theoretical predictions.     

Boundary Value Problem for a First-Order Partial Differential Equation with Liouville Fractional Derivative

Differential Equations - We consider a first-order partial differential equation with a fractional derivative with respect to one of two independent variables. Fractional differentiation is given...     

Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation

Mathematical Notes - The paper investigates an inverse problem for finding the order of the fractional derivative in the sense of Gerasimov–Caputo in the wave equation with an arbitrary...