A numerical method is given for integral equations with singular kernels. The method modifies the ideas of product integration contained in [3], and it is analyzed using the general schema of [1]. The emphasis is on equations which were not amenable to the method in [3]; in addition, the method tries to keep computer running time to a minimum, while maintaining an adequate order of convergence. The method is illustrated extensively with an integral equation reformulation of boundary value problems forΔ u?P(r 2)u=0; see [9].

background or methods

The Construction of Solutions for Boundary Value Problems by Function Theoretic Methods

The numerical solution of Fredholm integral equations of the second kind with singular kernels

Integral operators and the first initial boundary value problem for pseudoparabolic equations with analytic coefficients

The construction of solutions to pseudoparabolic equations in noncylindrical domains

8.—Constructive Methods for Solving the Exterior Neumann Problem for the Reduced Wave Equation in a Spherically Symmetric Medium

This paper is concerned with special Bergman operators for linear partial differential equations. E. Kreyszig introduced a class of equations which possess polynomials as generating functions for Bergman operators. These generating functions permit an integral-free representation of the solutions which can also be obtained by K. W. Bauer's and E. Peschl's differential operators, as was shown by M. Kracht and E. Kreyszig. For an extension of Kreyszig's class by W. Watzlawek we state conditions that are necessary and sufficient for the existence of a polynomial generating function “of the first kind”. We give an explicit characterization of the corresponding subclass and a representation of those generating functions, which are important for developing a function theory of the equations considered.

The Construction of Solutions for Boundary Value Problems by Function Theoretic Methods

Citations

The numerical solution of Fredholm integral equations of the second kind with singular kernels

Integral operators and the first initial boundary value problem for pseudoparabolic equations with analytic coefficients

The construction of solutions to pseudoparabolic equations in noncylindrical domains

8.—Constructive Methods for Solving the Exterior Neumann Problem for the Reduced Wave Equation in a Spherically Symmetric Medium

Bergmansche Polynom-Erzeugende Erster Art

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