scholarly journals On Certain Integral Equations of Convolution Type with Bessel-Function Kernels

1966 ◽  
Vol 15 (2) ◽  
pp. 111-116 ◽  
Author(s):  
R. P. Srivastav
Keyword(s):  
Bessel Function ◽  
Integral Equations ◽  
Mellin Transform ◽  
Fractional Integration ◽  
Convolution Type ◽  
Image Position ◽  
Limiting Case

In this paper we first obtain elementary solutions of the integral equationsandUsing these solutions we then define operators of fractional integration. These operators may be regarded as a generalisation of the operators of fractional integration introduced by Sneddon (1) as a modification of Erdé1yi–Kober operators. In fact Erdélyi–Kober–Sneddon operators may be obtained by multiplying both sides of the equations by α-1½β and considering the limiting case α→0. We employ these operators to find a generalisation of the Mellin transform.

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)⌉.

Integral transforms based upon fractional integration

1963 ◽  
Vol 59 (1) ◽  
pp. 63-71 ◽  
Author(s):  
Charles Fox
Keyword(s):  
Functional Equation ◽  
Mellin Transform ◽  
Fourier Transforms ◽  
Fractional Integration ◽  
Integral Transforms ◽  
Image Position

AbstractThe theory of Fourier transformscan be developed from the functional equation K(s) K(1 – s) = 1, where K(s) is the Mellin transform of the kernel k(x).In this paper I show that reciprocities can be obtained which are analogous to the Fourier transforms above but which develop from the much more general functional equationThe reciprocities are obtained by using fractional integration. In addition to the reciprocities we have analogues of the Parseval theorem and of the discontinuous integrals usually associated with Fourier transforms.In order to simplify the analysis I confine myself to the case n = 1 and to L2 space.

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 some fractional integrals and their applications

1985 ◽  
Vol 28 (1) ◽  
pp. 97-105
Author(s):  
J. S. Lowndes
Keyword(s):  
Bessel Function ◽  
Fractional Integration ◽  
Fractional Integrals ◽  
Modified Bessel Function ◽  
Image Position

In previous papers [3, 4] the author has discussed the symmetric generalised Erdélyi–Kober operators of fractional integration defined bywhere α>0, γ≧0 and the operators ℑiγ(η,α) and defined as in equations (1) and (2) respectively but with Jα−1, the Bessel function of the first kind replaced by Iα−1, the modified Bessel function of the first kind.

A Wiener-Hopf solution to the triple integral equations for the electrified disc in a coplanar gap

1970 ◽  
Vol 68 (2) ◽  
pp. 529-545 ◽  
Author(s):  
D. A. Spence
Keyword(s):  
Integral Equations ◽  
Mellin Transform ◽  
Potential Problem ◽  
Asymptotic Expression ◽  
Hankel Transform ◽  
Fredholm Equation ◽  
Fractional Operators ◽  
Triple Integral Equations ◽  
Image Position ◽  
Gap Width

AbstractThe axisymmetric potential problem for a plane circular electrode of radius a in a concentric hole of radius b in a coplanar earthed sheet is formulated in terms of triple integral equations for the Hankel transform of the potential, and reduced to a single Fredholm equation by use of the Erdélyi-Kober fractional operators.In the limit of small gap width (b − a)/b, the equation takes the formwhich is solved by applying the Wiener-Hopf technique to the Mellin transform of f(x). This leads to the asymptotic expressionfor the capacity of the disc; for the opposite limit the expressionis derived. Numerical integration of the governing Fredholm equation has been carried out for a range of intermediate values of b/a.

scholarly journals On an integral transform

1986 ◽  
Vol 9 (2) ◽  
pp. 283-292 ◽  
Author(s):  
D. Naylor
Keyword(s):  
Bessel Function ◽  
Integral Equations ◽  
Inversion Formula ◽  
Integral Transform ◽  
Zero Order ◽  
Convolution Type ◽  
Type Integral ◽  
Related Integral

This paper establishes properties of a convolution type integral transform whose kernel is a Macdonald type Bessel function of zero order. An inversion formula is developed and the transform is applied to obtain the solution of some related integral equations.

An inversion formula for the Varma transform

1966 ◽  
Vol 62 (3) ◽  
pp. 467-471 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
Integral Equation ◽  
Bessel Function ◽  
Inversion Formula ◽  
Fractional Integration ◽  
Integration Theory ◽  
Modified Bessel Function ◽  
Laplace Integral ◽  
Image Position

AbstractRecently Fox ((5)) has given an inversion formula for the transform whose kernel is xνKν(x), where Kν(x) is the modified Bessel function of the second kind, by the application of fractional integration theory. In the present paper it has been shown that the integral equationcan be thrown into the form of a Laplace integral, with the help of fractional integration, which can be solved by known methods.

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 Quadruple integral equations and operators of fractional integration

1971 ◽  
Vol 12 (1) ◽  
pp. 60-64 ◽  
Author(s):  
M. Iftikhar Ahmad
Keyword(s):  
Integral Equations ◽  
Fractional Integration ◽  
Triple Integral Equations ◽  
Image Position

Cooke [1] modified a technique used by Erdelyi and Sneddon [2] to solve triple integral equations of a certain type. In this paper, we extend this method to solve the quadruple integral equationswhere F1, G2, F3 and G4 are prescribed functions of p and ψ(ξ) is to be determined. With no loss of generality we shall assume that G2(p)≡0, G4(p)≡0.

scholarly journals The solution of certain triple integral equations involving inverse Mellin transforms

1973 ◽  
Vol 14 (1) ◽  
pp. 65-72 ◽  
Author(s):  
J. Tweed
Keyword(s):  
Positive Integer ◽  
Integral Equations ◽  
Mathematical Theory ◽  
Mellin Transform ◽  
Theory Of Elasticity ◽  
Mellin Transforms ◽  
Triple Integral Equations ◽  
Image Position

In this paper, we shall be concerned with the solution of triple integral equations of the typewhere M−1 is the inverse Mellin transform, nis a positive integer, and — 1 < Res < 0. The use of these equations will be illustrated by their application to two well-known problems in the mathematical theory of elasticity and further applications will be reported later.

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