Note on the reduction of dual and triple series equations to dual and triple integral equations

1963 ◽  
Vol 59 (4) ◽  
pp. 731-734 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Theory ◽  
Bessel Functions ◽  
Circular Disc ◽  
Legendre Functions ◽  
Dual Integral Equations ◽  
Spherical Cap ◽  
Triple Integral Equations ◽  
Dual Series Equations

Dual integral equations involving Bessel functions occur in the solution of some boundary-value problems in potential theory with conditions prescribed on a circular disc and a considerable amount of attention has been given to the solution of such equations (cf. (1)). The method of solving these dual integral equations is very similar to that employed in the solution of certain dual series equations involving Legendre functions. Equations of this type occur in problems in potential theory with conditions prescribed on a spherical cap and their solution has been obtained by Collins (2). No definite mathematical connexion has, however, been established between these dual series and dual integral equations and the object of this note is to establish such a connexion.

scholarly journals On the solution of the Helmholtz equation on regions with corners

2016 ◽  
Vol 113 (33) ◽  
pp. 9171-9176 ◽  
Author(s):  
Kirill Serkh ◽  
Vladimir Rokhlin
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Theory ◽  
Bessel Functions ◽  
Boundary Integral Equations ◽  
Numerical Algorithms ◽  
Boundary Integral ◽  
Numerical Examples ◽  
Polygonal Domains

In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

scholarly journals A Pair of Dual Integral Equations Involving Bessel Functions of the First and the Second Kind

1964 ◽  
Vol 14 (2) ◽  
pp. 149-158 ◽  
Author(s):  
R. P. Srivastav
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Function ◽  
Cylindrical Cavity ◽  
Bessel Functions ◽  
Boundary Value ◽  
Regularity Conditions ◽  
Dual Integral Equations ◽  
Image Position

In this paper we give a method for the solution of the dual integral equationswhere Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation forsubject to the usual regularity conditions and the following boundary conditions:

scholarly journals Some dual series and triple integral equations

1969 ◽  
Vol 16 (4) ◽  
pp. 273-280 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Jacobi Polynomial ◽  
Triple Integral Equations ◽  
Dual Series Equations ◽  
Image Position

In this paper we first of all solve the dual series equationswhere ƒ(ρ) and g(ρ) are prescribed functions,is the Jacobi polynomial (2).

scholarly journals The Approximate Solution of Dual Integral Equations by Variational Methods

1958 ◽  
Vol 11 (2) ◽  
pp. 115-126 ◽  
Author(s):  
B. Noble
Keyword(s):  
Approximate Solution ◽  
Variational Methods ◽  
Integral Equations ◽  
Separation Of Variables ◽  
Circular Disc ◽  
Cylindrical Coordinates ◽  
Dual Integral Equations ◽  
Particular Solution ◽  
Image Position

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution ofsuch that on z = 0Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution.

scholarly journals The Study of Triple Integral Equations with Generalized Legendre Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
B. M. Singh ◽  
J. Rokne ◽  
R. S. Dhaliwal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Triple Integral Equations

A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.

scholarly journals The solution of some integral equations

1987 ◽  
Vol 29 (2) ◽  
pp. 203-215
Author(s):  
John F. Ahner ◽  
John S. Lowndes
Keyword(s):  
Wave Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Laplace Equation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Triple Integral Equations

AbstractAlgorithms are developed by means of which certain connected pairs of Fredholm integral equations of the first and second kinds can be converted into Fredholm integral equations of the second kind. The methods are then used to obtain the solutions of two different sets of triple integral equations tht occur in mixed boundary value problems involving Laplace' equation and the wave equation respectively.

The solution of Bessel function dual integral equations by converting to the first kind Fredholm Integral Equation: an extension of Noble’s solution

2018 ◽  
Vol 24 (8) ◽  
pp. 2536-2557
Author(s):  
S Cheshmehkani ◽  
M Eskandari-Ghadi
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Integral Transforms ◽  
Original Method ◽  
Fredholm Integral Equations ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Factor Method

In certain mixed boundary value problems, Hankel integral transforms are applied and subsequently dual integral equations involving Bessel functions have to be solved. In the literature, if possible by employing the Noble’s multiplying factor method, these dual integral equations are usually converted to the second kind Fredholm Integral Equations (FIEs) and solved either analytically or numerically, respectively, for simple or complicated kernels. In this study, the multiplying factor method is extended to convert the dual integral equations both to the first and the second kind FIEs, and the conditions for converting to each kind of FIE are discussed. Furthermore, it is shown that under some simple circumstances, many mixed boundary value problems arising from either elastostatics or elastodynamics can be converted to the well-posed first kind FIE, which may be solved analytically or numerically. Main criteria for well-posedness of FIEs of the first kind in such problems are also presented. Noble’s original method is restricted to some limited conditions, which are extended here for both first and second kind FIEs to cover a wider range of dual integral equations encountered in engineering mixed boundary value problems.

scholarly journals On the Solution of some Axisymmetric Boundary Value Problems by Means of Integral Equations. IV. The Electrostatic Potential due to a Spherical Cap between Two Infinite Conducting Planes

1960 ◽  
Vol 12 (2) ◽  
pp. 95-106 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Electrostatic Potential ◽  
Potential Problem ◽  
Boundary Value ◽  
Circular Disk ◽  
Spherical Cap ◽  
Potential Problems ◽  
Spherical Caps ◽  
Circular Disks

This paper is a sequel to previous papers (1, 2, 3) on the solution of axisymmetric potential problems for circular disks and spherical caps by means of integral equations and applies the methods developed in these papers to the electrostatic potential problem for a perfectly conducting thin spherical cap or circular disk between two infinite earthed conducting planes.

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