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.

scholarly journals On dual integral equations with Hankel kernel and an arbitrary weight function

1986 ◽  
Vol 9 (2) ◽  
pp. 293-300 ◽  
Author(s):  
C. Nasim
Keyword(s):  
Weight Function ◽  
Integral Equations ◽  
Single Equation ◽  
Dual Integral Equations ◽  
Fredholm Type ◽  
Operational Procedure ◽  
Mellin Transforms ◽  
Special Cases ◽  
General Order ◽  
Arbitrary Weight

In this paper we deal with dual integral equations with an arbitrary weight function and Hankel kernels of distinct and general order. We propose an operational procedure, which depends on exploiting the properties of the Mellin transforms, and readily reduces the dual equations to a single equation. This then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function. Most of the known results are then derived as special cases of our general result.

scholarly journals Some dual integral equations involving inverse finite Mellin transforms

1973 ◽  
Vol 14 (2) ◽  
pp. 179-184 ◽  
Author(s):  
John Tweed
Keyword(s):  
Integral Equations ◽  
Dual Integral Equations ◽  
Mellin Transforms

The object of this paper is to find the solutions of some dual equations involving the inverses of certain Mellin type transforms that were first introduced by D. Naylor in his paper [1]. Because these transforms are relatively unknown we shall begin by defining them and making a note of some of their properties. The main result is contained in the following theorem.

The solution of Bessel function dual integral equations by a multiplying-factor method

1963 ◽  
Vol 59 (2) ◽  
pp. 351-362 ◽  
Author(s):  
B. Noble
Keyword(s):  
Analytic Continuation ◽  
Bessel Function ◽  
Integral Equations ◽  
Complex Plane ◽  
Real Variable ◽  
Dual Integral Equations ◽  
Variable Technique ◽  
Mellin Transforms ◽  
Factor Method ◽  
Image Position

In this paper we first of all consider the dual integral equationswhere f(ρ), g(ρ) are given, A(t) is unknown, and α is a given constant. This system, with g(ρ) = 0, was originally considered by Titchmarsh ((13), p. 337), and Busbridge (1), who obtained a solution by the use of Mellin transforms and analytic continuation in the complex plane. The method described in this paper involves the application of certain multiplying factors to the equations. In the present case it is relatively easy to guess the multiplying factors and then the method is essentially a real-variable technique. It is presented in this way in § 2 below.

scholarly journals On Certain Dual Integral Equations

1961 ◽  
Vol 5 (1) ◽  
pp. 21-24 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Mellin Transforms ◽  
Abel Integral Equation ◽  
Image Position ◽  
Fourier Integrals

In his book on Fourier Integrals, Titchmarsh [l] gave the solution of the dual integral equationsfor the case α > 0, by some difficult analysis involving the theory of Mellin transforms. Sneddon [2] has recently shown that, in the cases v = 0, α = ±½, the problem can be reduced to an Abel integral equation by making the substitutionorIt is the purpose of this note to show that the general case can be dealt with just as simply by puttingThe analysis is formal: no attempt is made to supply details of rigour.

scholarly journals The elementary solution of dual integral equations

1960 ◽  
Vol 4 (3) ◽  
pp. 108-110 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Integral Equations ◽  
Present Note ◽  
Mixed Boundary ◽  
Formal Solution ◽  
Mathematical Physics ◽  
Elementary Solution ◽  
Dual Integral Equations ◽  
Mellin Transforms ◽  
Method Of Solution ◽  
Image Position

When the theory of Hankel transforms is applied to the solution of certain mixed boundary value problems in mathematical physics, the problems are reduced to the solution of dual integral equations of the typewhere α and ν are prescribed constants and f(ρ) is a prescribed function of ρ [1]. The formal solution of these equations was first derived by Titchmarsh [2]. The method employed by Titchmarsh in deriving the solution in the general case is difficult, involving the theory of Mellin transforms and what is essentially a Wiener-Hopf procedure. In lecturing to students on this subject one often feels the need for an elementary solution of these equations, especially in the cases α = ± 1, ν = 0. That such an elementary solution exists is suggested by Copson's solution [3] of the problem of the electrified disc which corresponds to the case α = –l, ν = 0. A systematic use of a procedure similar to Copson's has in fact been made by Noble [4] to find the solution of a pair of general dual integral equations, but again the analysis is involved and long. The object of the present note is to give a simple solution of the pairs of equations which arise most frequently in physical applications. The method of solution was suggested by a procedure used by Lebedev and Uflyand [5] in the solution of a much more general problem.

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.

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.

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