Solution of certain weakly singular integral equations

^** Email: alok{at}math.iisc.ernet.in Direct function theoretic methods are developed to handle twoweakly singular integral equations with their kernels havinglogarithmic singularity. The present methods avoid the occurrenceof higher-order (or strong) singularities, like the Cauchy typesingularity in the representation of the solutions of such integralequations.     

Stability of characteristic values of singular integral equations

A study is made of the problem concerning existence of nontrivial solutions of the equation (A+C)f=f for all compact (or small in norm) operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 136–142, 1987.     

Entropy and eigenvalues of weakly singular integral operators

In this paper the asymptotic behaviour of entropy numbers and eigenvalues of weakly singular integral operators acting between L_p-spaces over arbitrary bounded open domains in ^N are investigated. The singularities contain not only power but also logarithmic terms. In particular, we consider the case where the (power) order of the singularity equals half of the dimension of the domain, thus refining earlier results of H. König [6].     

Solution of certain weakly singular integral equations of the first kind with indefinite parameters

In this paper we study a two-dimensional weakly singular integral equation of the first kind with logarithmic kernel. We construct a pair of spaces of the desired elements and the right-hand sides, where we prove the correctness of the problem under consideration and obtain inversion formulas for the integral operator.     

Systems of singular integral equations of certain mixed boundary-value problems of mathematical physics

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 43, pp. 33–42, 1985.     

Stability theorems for singular perturbation of eigenvalues

Two stability, i.e., convergence in multiplicity, theorems for eigenvalues of non-self-adjoint singular perturbation problems are proved in this paper. Use is made of quadratic interpolation between Hubert spaces to obtain an extension of the notion of a holomorphic family of type (B). Applications to singular perturbation problems for non-self-adjoint elliptic partial differential operators, and to singular potential perturbation problems with complex coupling constants, are presented.This research was partially supported by NSF Grant 02 MCS-7902663     

A note on singular integral equations

The class of functions for which certain singular integral equations hold good is discussed and the method of obtaining appropriate solutions of such equations is explained through equations of simple form. A discussion of this type is felt to be essential for beginners.     

Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations

Summary We consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp _gw Au=f on a cube =(0, 1) ^m . Utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside . For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method inL ²(). Afterwards, form2 and a right hand sidefH ⁸(),s>m/2, we prove an asymptotic convergence estimate.The author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant number Ko 634/32-1     

Computing the singular behavior of solutions of the cauchy singular integral equations

Stenger's formula for numerical computation of principal value integrals isused to determine the singular behavior of solutions of homogeneous Cauchy singular integral equations near the end-points of the domain of integration.     

The approximate solution of singular integral equations

A computational scheme of collocation type is proposed for a singular linear integral equation with a power singularity in the regular integral and the justification is given. The results obtained are used to justify the approximate solution of the singular integral equation $$Kx \equiv a(t)x(t) + \frac{{b(t)}}{{\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{x(\tau )d\tau }}{{\tau - t}} + \frac{1}{{2\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{h|t,\tau )x(\tau )}}{{\left| {\tau - t} \right|^\delta }}d\tau = f(t)$$ by a modification of the method of minimal residuals.     

Information complexity of weakly singular integral equations

We establish the exact power order of information complexity for integral equations whose kernels have power singularities and free terms belong to the corresponding Hölder space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.46, No. 11, pp. 1527–1533, November, 1994.This work was supported by the Foundation for Fundamental Researches of the Ukrainian State Committee on Science and Technology.