scholarly journals The solution of dual series and dual integral equations

1964 ◽  
Vol 6 (3) ◽  
pp. 123-129 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Integral Equations ◽  
General Type ◽  
Bessel Functions ◽  
Fractional Integration ◽  
Dual Integral Equations

There exist several different approaches to the problem of solving dual integral equations involving Bessel Functions [1, 2, 3, 4, 5, 6,7], and Erdelyi and Sneddon in a recent paper [8] have shown that the introduction of certain operators occurring in the theory of fractional integration enables the relationships between the various methods to be clearly demonstrated. For dual integral equations other than those involving Bessel Functions the operators introduced by Erdélyi and Sneddon are not always the appropriate ones to use and it seems to be of interest to consider this more general type of situation.

scholarly journals The Solution of Certain Dual Integral Equations

1961 ◽  
Vol 12 (4) ◽  
pp. 213-216 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Integral Equations ◽  
Bessel Functions ◽  
Dual Integral Equations ◽  
Mellin Transforms ◽  
Formal Application

SummaryA class of dual integral equations involving Bessel functions is solved by formal application of Mellin transforms.

A formal solution of certain dual integral equations involving H-functions

1967 ◽  
Vol 63 (1) ◽  
pp. 171-178 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Formal Solution ◽  
Fractional Integration ◽  
Dual Integral Equations ◽  
First Case ◽  
The Common ◽  
The Given ◽  
Fourier Kernel

AbstractThe problem discussed is the formal solution of certain dual integral equations involving H-functions. The method followed is that of fractional integration. The given dual integral equations have been transformed, by the application of fractional integration operators, into two others with a common kernel and the problem is then reduced to solving one integral equation. In the first case the common kernel comes out to be a symmetrical Fourier kernel given earlier by Fox and the formal solution is then immediate. In the second case the common kernel is a generalized Fourier kernel and dual integral equations of this type have recently been studied by Fox.

Fractional Integration and Dual Integral Equations

1962 ◽  
Vol 14 ◽  
pp. 685-693 ◽  
Author(s):  
A. Erdélyi ◽  
I. N. Sneddon
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Fractional Integration ◽  
Direct Solution ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Symmetrical Form ◽  
Image Position

In the analysis of mixed boundary value problems by the use of Hankel transforms we often encounter pairs of dual integral equations which can be written in the symmetrical form(1.1)Equations of this type seem to have been formulated first by Weber in his paper (1) in which he derives (by inspection) the solution for the case in which α — β = ½, v = 0, F ≡ 1, G ≡ 0.The first direct solution of a pair of equations of this type was given by Beltrami (2) for the same values of α— β and v with G(p) ≡ 0 but with F(ρ) arbitrary.

scholarly journals A note on Bessel function dual integral equation with weight function

1988 ◽  
Vol 11 (3) ◽  
pp. 543-549 ◽  
Author(s):  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Weight Function ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Bessel Functions ◽  
Dual Integral Equation ◽  
Dual Integral Equations ◽  
Arbitrary Weight

An elementary procedure based on Sonine's integrals has been used to reduce dual integral equations with Bessel functions of different orders as kernels and an arbitrary weight function to a Fredholm integral equation of the second kind. The result obtained here encompasses many results concerning dual integral equations with Bessel functions as kernels known in the literature.

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:

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.

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