scholarly journals On some results concerning integral equations

1913 ◽  
Vol 32 ◽  
pp. 19-29
Author(s):  
Pierre Humbert
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Large Class ◽  
Integral Equations ◽  
Singular Integral ◽  
General Type ◽  
Singular Integral Equations ◽  
General Function ◽  
Linear Differential ◽  
Image Position

It is proposed in this paper to show now the well-known Laplace's transformation,which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such aswhich will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.

Solution of a Linear Integral Equation of Third Kind

2002 ◽  
Vol 9 (1) ◽  
pp. 179-196
Author(s):  
D. Shulaia
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Fredholm Theory ◽  
Linear Integral Equation ◽  
Hölder Functions ◽  
Linear Integral ◽  
Necessary And Sufficient

Abstract The aim of this paper is to study, in the class of Hölder functions, a nonhomogeneous linear integral equation with coefficient cos 𝑥. Necessary and sufficient conditions for the solvability of this equation are given under some assumptions on its kernel. The solution is constructed analytically, using the Fredholm theory and the theory of singular integral equations.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

Two singular integral equations involving confluent hypergeometric functions

1969 ◽  
Vol 66 (1) ◽  
pp. 71-89 ◽  
Author(s):  
Tilak Raj Prabhakar
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Singular Integral Equations ◽  
Whittaker Functions ◽  
Confluent Hypergeometric Functions ◽  
The Laplace Transform ◽  
Similar Transform ◽  
Image Position

Widder(1) obtained an inversion of the convolution transformby the method of the Laplace transform, Ln(x) being the Laguerre polynomial. Buschman (2) inverted a similar transform with a generalized Laguerre polynomial as kernel and also solved (3) the singular integral equationusing Mikusinski operators. Srivastava(4, 4a) solved singular integral equations with kernels involving and Whittaker functions Mk,μ(x).

scholarly journals Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 140
Author(s):  
Liudmila Nickelson ◽  
Raimondas Pomarnacki ◽  
Tomyslav Sledevič ◽  
Darius Plonis
Keyword(s):  
Differential Equations ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Integral Form ◽  
Hankel Function ◽  
Cauchy Type ◽  
Constitutive Parameters

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper.

Fast algorithms using orthogonal polynomials

Acta Numerica ◽  
2020 ◽  
Vol 29 ◽  
pp. 573-699
Author(s):  
Sheehan Olver ◽  
Richard Mikaël Slevinsky ◽  
Alex Townsend
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Low Rank ◽  
Quadrature Rules ◽  
Efficient Computation ◽  
Optimal Complexity ◽  
Wide Range

We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications.

Regularization Method for Complete Singular Integral Equation with Hilbert Kernel on Open Arcs

2013 ◽  
Vol 765-767 ◽  
pp. 643-646
Author(s):  
Li Xia Cao
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Regularization Method ◽  
Singular Integral Equations ◽  
Noether Theorem ◽  
Hilbert Kernel

We considered the regularization method for a kind of complete singular integral equation with Hilbert kernel on open arcs lying in a period strip. And based on this, we obtained the solvable Noether theorem for this kind of complete singular integral equations.

Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations

2018 ◽  
Vol 68 (1) ◽  
pp. 77-88
Author(s):  
Marcin Borkowski ◽  
Daria Bugajewska
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Integral Equations ◽  
Bounded Variation ◽  
Volterra Integral Equation ◽  
Functions Of Bounded Variation ◽  
Hammerstein Integral Equation ◽  
Linear Differential ◽  
Unbounded Intervals ◽  
Differential And Integral Equations

Abstract In this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

scholarly journals XXVI.—On a Group of Linear Differential Equations of the 2nd Order, including Professor Chrystal's Seiche-equations

1906 ◽  
Vol 41 (3) ◽  
pp. 651-676 ◽  
Author(s):  
J. Halm
Keyword(s):  
Differential Equations ◽  
Stokes Equation ◽  
Mathematical Theory ◽  
General Type ◽  
The Other ◽  
Special Cases ◽  
Linear Differential ◽  
Image Position ◽  
Special Case ◽  
The Stokes Equation

It is readily seen that the two differential equationswhich play an important rôle in Professor Chrystal's mathematical theory of the Seiches, are special cases of the more general typeWith regard to the first, the Seiche-equation, this becomes at once apparent by writing a= − ½. Equation (2), on the other hand, which we may briefly call the Stokes equation [see Professor Chrystal's paper on “Some further Results in the Mathematical Theory of Seiches,” Proc. Roy. Soc. Edin., vol. xxv.] will be recognised as a special case (a = + 1) of the equationwhich is transformed into (3) by the substitution .

scholarly journals Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 872 ◽  
Author(s):  
◽  
Shuhuang Xiang ◽  
Guidong Liu
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Kernel Function ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Cauchy Type ◽  
Error Analyses ◽  
Highly Oscillatory

This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate thesolution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind witha highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed methodgives more accurate results than the scheme introduced in Dezhbord at el. (2016) and Eshkuvatovat el. (2009) for small values of N. Finally, this paper illustrates some error analyses and numericalresults for CSIEs.

On an approximate solution of a class of surface singular integral equations of the first kind

2020 ◽  
Vol 27 (1) ◽  
pp. 97-102 ◽  
Author(s):  
Elnur H. Khalilov
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problems

AbstractIn this work, a method for calculating an approximate solution of a singular integral equation of first kind is presented for the Neumann boundary value problems for the Helmholtz equation.

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