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 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 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.

scholarly journals Fixed point theorems for Geraghty-type mappings applied to solving nonlinear Volterra-Fredholm integral equations in modular G-metric spaces

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Godwin Amechi Okeke ◽  
Daniel Francis
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Content Type ◽  
Special Cases ◽  
Type Integral ◽  
Social Applications

PurposeThe authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.Design/methodology/approachThe results of this paper are theoretical and analytical in nature.FindingsThe authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.Research limitations/implicationsThe results are theoretical and analytical.Practical implicationsThe results were applied to solving nonlinear integral equations.Social implicationsThe results has several social applications.Originality/valueThe results of this paper are new.

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 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 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 Dual integral equations involving Weber-Orr transforms

1991 ◽  
Vol 14 (1) ◽  
pp. 163-176 ◽  
Author(s):  
C. Nasim
Keyword(s):  
General Solution ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Special Cases ◽  
Elementary Methods

In this paper we consider dual integral equations involving inverse Weber-Orr transforms of the typeWν−k,ν−1[;],k=0,1,…. A general solution is established using elementary methods. Many known results are derived as special cases.

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