A formal solution of certain dual integral equations involving H-functions

1967 ◽  
Vol 63 (1) ◽  
pp. 171-178 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Formal Solution ◽  
Fractional Integration ◽  
Dual Integral Equations ◽  
First Case ◽  
The Common ◽  
The Given ◽  
Fourier Kernel

AbstractThe problem discussed is the formal solution of certain dual integral equations involving H-functions. The method followed is that of fractional integration. The given dual integral equations have been transformed, by the application of fractional integration operators, into two others with a common kernel and the problem is then reduced to solving one integral equation. In the first case the common kernel comes out to be a symmetrical Fourier kernel given earlier by Fox and the formal solution is then immediate. In the second case the common kernel is a generalized Fourier kernel and dual integral equations of this type have recently been studied by Fox.

A method for the exact solution of a class of two-dimensional diffraction problems

1951 ◽  
Vol 205 (1081) ◽  
pp. 286-308 ◽  
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Integral Equations ◽  
Scattered Field ◽  
Plane Waves ◽  
Angular Spectrum ◽  
Diffraction Theory ◽  
General Interest ◽  
Two Dimensional ◽  
Dual Integral Equations

In the last few years Copson, Schwinger and others have obtained exact solutions of a number of diffraction problems by expressing these problems in terms of an integral equation which can be solved by the method of Wiener and Hopf. A simpler approach is given, based on a representation of the scattered field as an angular spectrum of plane waves, such a representation leading directly to a pair of ‘dual’ integral equations, which replaces the single integral equation of Schwinger’s method. The unknown function in each of these dual integral equations is that defining the angular spectrum, and when this function is known the scattered field is presented in the form of a definite integral. As far as the ‘radiation’ field is concerned, this integral is of the type which may be approximately evaluated by the method of steepest descents, though it is necessary to generalize the usual procedure in certain circumstances. The method is appropriate to two-dimensional problems in which a plane wave (of arbitrary polarization) is incident on plane, perfectly conducting structures, and for certain configurations the dual integral equations can be solved by the application of Cauchy’s residue theorem. The technique was originally developed in connexion with the theory of radio propagation over a non-homogeneous earth, but this aspect is not discussed. The three problems considered are those for which the diffracting plates, situated in free space, are, respectively, a half-plane, two parallel half-planes and an infinite set of parallel half-planes; the second of these is illustrated by a numerical example. Several points of general interest in diffraction theory are discussed, including the question of the nature of the singularity at a sharp edge, and it is shown that the solution for an arbitrary (three-dimensional) incident field can be derived from the corresponding solution for a two-dimensional incident plane wave.

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 Unauthorized medical intervention and informed consent in the common law of Canada prior to the Supreme Court’s decision of Reibl v. Hughes (1899–1980)

2020 ◽  
pp. 260-282
Author(s):  
Anatoliy Lytvynenko
Keyword(s):  
Informed Consent ◽  
Medical Treatment ◽  
Common Law ◽  
Case Law ◽  
Medical Intervention ◽  
Invasive Treatment ◽  
First Case ◽  
The Common ◽  
The Right ◽  
The Given

The given article deals with the Canadian legacy of civil actions on negligence and technical assault or battery involving an unauthorizedmedical interference to plaintiff. In modern doctrine and case-law, the given concept is named “informed consent”, upon whichthe patient is not a mere subject of medical treatment, but has a substantial set of patient rights, involving the informational ones, whichincludes his right to be informed on further invasive treatment and thus to be able to assent or decline it. The doctrine of informed consent,arising from actions on unauthorized medical treatment in both common law and civil law jurisdictions, has a centuryfold historyin the jurisprudence. In the common-law world, it was bred in the end of the 19th century primarily in the jurisprudence of Americancourts, but still has its distinct peculiarities in the common law of Canada throughout the twentieth century. The span on the researchedjurisprudence embraces the time period of 1899 (judgment of Parnell, which was the first case to deal with the subject) to 1980 (caseof Reibl v. Hughes), where the Canadian Supreme Court has firmly recognized the principle of informed consent in the acting commonlaw. In the 1990s, the principles of informed consent had been codified. The author has investigated on the evolvement of the conceptof patient’s right to autonomy in the state from the very beginning to the judgment of Reibl v. Hughes in 1980, and has researched theroots of the “right to autonomy” as an extension of the right to privacy, which has penumbrally existed in Canadian jurisprudence forover a century, despite having been recognized as such relatively recently, despite an existence of various early case-law legacy. Apartfrom the abovesaid, the author aimed to define the authorities used by Canadian courts in the earlier cases dealing with unconsentedsurgery, which involves judgments from other jurisdictions as well as professional legal and medical textbooks.

scholarly journals On the solution of certain dual integral equations

1963 ◽  
Vol 6 (1) ◽  
pp. 39-44 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Integral Equations ◽  
Formal Solution ◽  
Direct Solution ◽  
Dual Integral Equations ◽  
Image Position

1. In this note we consider the formal solution of the dual integral equationswhere f(x) and g(x) are given and χ(x) is to be found. The direct solution of these equations has been given by Noble [1] but we shall show that they may be solved more easily if they are first reduced to a form in which g(x) ≡ 0.

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.

Torsional Impact Response of a Thick-Walled Cylinder With a Circumferential Edge Crack

1990 ◽  
Vol 112 (4) ◽  
pp. 367-373 ◽  
Author(s):  
Y. Shindo ◽  
W. Li
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Edge Crack ◽  
Dynamic Stress Intensity Factor ◽  
Impact Response ◽  
Cauchy Kernel ◽  
Dual Integral Equations ◽  
Singular Stress ◽  
Singular Stress Field ◽  
Walled Cylinder

This paper considers the torsional impact response of a long thick-walled cylinder containing an internal or external circumferential edge crack. Laplace and Hankel transforms are used to reduce the elastodynamic problem to a pair of dual integral equations. The dual integral equations are solved by using the standard transform technique, and the result is expressed in terms of an integral equation which has a generalized Cauchy kernel as the dominant part. The kernel of the integral equation is improved in order that the calculation may be made easy. A numerical Laplace inversion technique is used to recover the time dependence of the solution. The dynamic singular stress field is determined, and the numerical results on the dynamic stress intensity factor are obtained to show the influence of inertia, geometry, and their interactions.

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 On a dual integral equation with a trigonometric kernel

1977 ◽  
Vol 18 (2) ◽  
pp. 175-177 ◽  
Author(s):  
D. C. Stocks
Keyword(s):  
Integral Equation ◽  
Elasticity Theory ◽  
Integral Equations ◽  
Crack Problem ◽  
Dual Integral Equation ◽  
Dual Integral Equations ◽  
Cosine Transform ◽  
Fourier Cosine ◽  
Fourier Cosine Transform ◽  
Image Position

In this note we formally solve the following dual integral equations:where h is a constant and the Fourier cosine transform of u–1 φ(u) is assumed to exist. These dual equations arise in a crack problem in elasticity theory.

scholarly journals A note on dual integral equations involving inverse associated Weber-Orr transforms

1996 ◽  
Vol 19 (1) ◽  
pp. 161-169
Author(s):  
Nanigopal Mandal ◽  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Dual Integral Equations ◽  
Elementary Methods ◽  
Special Case

We consider dual integral equations involving inverse associated Weber-Orr transforms. Elementary methods have been used to reduce dual integral equations to a Fredholm integral equation of second kind. Some known results are obtained as special case.

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