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 the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

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 Multiplying Factor Method for the Solution of Wiener-Hopf Equations

1961 ◽  
Vol 12 (3) ◽  
pp. 119-122 ◽  
Author(s):  
B. Noble ◽  
A. S. Peters
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Factor Method ◽  
Single Integral ◽  
Image Position ◽  
Related Technique

In (1), § 6.2, a multiplying factor method has been used to solve certain dual integral equations. The results are then used to solve a single integral equation of the Wiener-Hopf type. In this note we indicate how a related technique can be used to solve Wiener-Hopf integral equations directly. ConsiderwhereDefinewhere α = σ+iτ, and F+(α) is regular for τ>q; K(α) is regular and non-zero in −p < τ < p. For simplicity we restrict ourselves to the case where

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 Dual Integral Equations with Trigonometric Kernels

1966 ◽  
Vol 15 (1) ◽  
pp. 73-74
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Dual Integral Equations ◽  
Image Position

Consider the dual equationswhere

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.

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 A Pair of Dual Integral Equations Occurring in Diffraction Theory

1962 ◽  
Vol 13 (2) ◽  
pp. 179-187 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Integral Equations ◽  
Diffraction Theory ◽  
Dual Integral Equations ◽  
Special Cases ◽  
Image Position

Dual integral equations of the formwhere f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.

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.

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