Dual integral equations involving inverse Mellin transforms

1986 ◽

Vol 9 (2) ◽

pp. 293-300 ◽

Cited By ~ 2

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.

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Some dual integral equations involving inverse finite Mellin transforms

Glasgow Mathematical Journal ◽

10.1017/s0017089500001932 ◽

1973 ◽

Vol 14 (2) ◽

pp. 179-184 ◽

Cited By ~ 2

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.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100036987 ◽

1963 ◽

Vol 59 (2) ◽

pp. 351-362 ◽

Cited By ~ 125

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.

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The Solution of Certain Dual Integral Equations

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500025451 ◽

1961 ◽

Vol 12 (4) ◽

pp. 213-216 ◽

Cited By ~ 11

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.

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On Certain Dual Integral Equations

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500034249 ◽

1961 ◽

Vol 5 (1) ◽

pp. 21-24 ◽

Cited By ~ 98

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.

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The elementary solution of dual integral equations

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500034006 ◽

1960 ◽

Vol 4 (3) ◽

pp. 108-110 ◽

Cited By ~ 43

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.

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