Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation

On the Hilbert space \(\widetilde{L}_{2}(\mathbb {T})\) the singular integral operator with non-Carleman shift and conjugation \(K=P_{+}+(aI+AC)P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(A=\sum \nolimits _{j=0}^{m}a_{j}U^{j}\), \(a,a_{j}\), \(j=\overline{1,m}\), are continuous functions on the unit circle \(\mathbb {T}\), U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.     

Note on a Singular Integral Equation

A simple direct relationship is.established between solutionsof a singular integral equation encountered in wave-guide theoryand in the theory of dislocations and the solutions of the singularCarleman integral equation.     

A Note on a Singular Integral Equation

A direct method is presented for solving a singular integralequation which is a generalization of one occurring in wave-guidetheory and in the theory of dislocations. The present methodavoids elaborate application of the complex variable methodsnormally associated with such singular integral equations.     

Factorization of a Symbol Corresponding to the Sum of a Finite Number of Singular Integral Operators with Non-Carleman Shifts

An equation containing a finite sum of singular integral operators with non-Carleman shifts is considered. The unique solvability of this equation in the Hölder classes under certain constraints imposed on the coefficients is proved. It is shown that the solution can be written in quadratures.     

Spectrum Problems for Singular Integral Operators with Carleman Shift

In this paper we are concerned with the complete spectral analysis for the operator 𝒯 = 𝒳𝒮𝒰 in the space L_p(𝕋) (𝕋 denoting the unit circle), where 𝒳 is the characteristic function of some arc of 𝕋, 𝒮 is the singular integral operator with Cauchy kernel and 𝒰 is a Carleman shift operator which satisfies the relations 𝒰² = I and 𝒮𝒰 = ±𝒰𝒮, where the sign + or — is taken in dependence on whether 𝒰 is a shift operator on L_p(𝕋) preserving or changing the orientation of 𝕋. This includes the identification of the Fredholm and essential parts of the spectrum of the operator 𝒯, the determination of the defect numbers of 𝒯 — λI, for λ in the Fredholm part of the spectrum, as well as a formula for the resolvent operator.     

一类带约束的二维弱奇异积分方程的解*

本文找出二维弱奇异第一类积分方程作用着约束方程的解p.p=p(r,θ)={2/[π2k(φ₀]}√F(r,θ)-c_*(0≤r≤r_*)其中是(s,φ)原点在M(r,θ)的局部极坐标,(r,θ)是原点在O(0,0)的总体极坐标:k和F是给出的连续函数;φ₀是一常数;F(_*,θ)=c_*(常数)是研究域Q的边界围线。所用方法可推广到三维情形。     

无界区域上Cauchy型奇异积分算子的性质

On the basis of introducing the modified Cauchy kernel, we discuss the Hoelder continuity of the Cauchy-type singular integral operator on unbounded domains for regular functions by dividing into the following three cases： two points are on the boundary of region; one point is on the boundary and another point is in the interior（or exterior） of the region; two points are in the interior （or exterior） of the region.     

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.     

一类具对流项的奇异扩散方程解的逼近性质

讨论了一类带对流项的奇异扩散方.程的Neumann边值问题,证明了整体解的存在唯一性;讨论了带对流项非线性间题解的线性逼近,得到了逼近的显式表示式;同时还对││u-(u)││L2(0,1)进行了估计,得到了解关于时间t充分大时的渐近性态,其中(u)＝∫o/1udx.     

奇异积分中的一些问题

设n≥2,Ω为R~n中单位球面S~(n-1)上的可积函数且Ω在S~(n-1)上的平均值为零,即∫_S~(n1)~Ω(x)dσ(x)=0.其中dσ为S~(n-1)上的体积元.定义奇异积分算子T_0,和相应的极大算子T~*,其中h∈L~∞(R~+).关于算子T和T~*已有许多研究([1]-[6]等).在1986年,Namazi利用Fourier变换的Hausdorff-Young不等式证明了     

Fredholm Theory for a Class of Singular Integral Operators with Carleman Shift and Unbounded Coefficients

A criterion for the Fredholmness of singular integral operators with Carleman shift in L_P(Γ) is obtained, where Γ is either the unit circle or the real line. The approach allows to consider unbounded coefficients in a class related to that of quasicontinuous functions. Applications to Wiener-Hopf-Hankel type operators and operators with linear fractional Carleman shift on IR are included.     

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…     

具积分边值条件的迁移方程的性质

在L~1空间上研究了一类增生的细菌群体中具积分边界条件的迁移方程.得出迁移算子是预解正算子,微分算子的共轭算子及共轭算子的定义域.证明了迁移算子的共轭算子定义域的正锥在共轭空间的正锥中共尾.最后证明了迁移算子的增长界等于其谱界.