scholarly journals Some triple integral equations

1960 ◽  
Vol 4 (4) ◽  
pp. 200-203 ◽  
Author(s):  
C. J. Tranter
Keyword(s):  
Boundary Condition ◽  
Closed Form ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Triple Integral Equations ◽  
Special Cases ◽  
Potential Problems ◽  
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Different Parts

Potential problems in which different conditions hold over two different parts of the same boundary can often be conveniently reduced to the solution of a pair of dual integral equations. In some problems, however, the boundary condition is such that different conditions hold over three different parts of the boundary and, in such cases, the integral equations involved are frequently of the formwhere f(r), g(r) are specified functions of r, p = ± ½ and ø(u) is to be found. Such equations might well be called triple integral equations and, in this note, I point out certain special cases which I have found to be capable of solution in closed form.

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 series and triple integral equations

1969 ◽  
Vol 16 (4) ◽  
pp. 273-280 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Jacobi Polynomial ◽  
Triple Integral Equations ◽  
Dual Series Equations ◽  
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In this paper we first of all solve the dual series equationswhere ƒ(ρ) and g(ρ) are prescribed functions,is the Jacobi polynomial (2).

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 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 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 Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
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Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

scholarly journals Closed-form solution for piezoelectric layer with two collinear cracks parallel to the boundaries

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
B. M. Singh ◽  
J. Rokne ◽  
R. S. Dhaliwal
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Piezoelectric Layer ◽  
Electric Displacement ◽  
Closed Form Solution ◽  
Finite Width ◽  
Intensity Factors ◽  
Collinear Cracks ◽  
Triple Integral Equations ◽  
Electromechanical Loading

We consider the problem of determining the stress distribution in an infinitely long piezoelectric layer of finite width, with two collinear cracks of equal length and parallel to the layer boundaries. Within the framework of reigning piezoelectric theory under mode III, the cracked piezoelectric layer subjected to combined electromechanical loading is analyzed. The faces of the layers are subjected to electromechanical loading. The collinear cracks are located at the middle plane of the layer parallel to its face. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. The triple integral equations are solved exactly. Closed form analytical expressions for stress intensity factors, electric displacement intensity factors, and shape of crack and energy release rate are derived. As the limiting case, the solution of the problem with one crack in the layer is derived. Some numerical results for the physical quantities are obtained and displayed graphically.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

scholarly journals Simultaneous dual integral equations

1973 ◽  
Vol 14 (1) ◽  
pp. 73-76
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Inversion Formula ◽  
Hankel Transform ◽  
Dual Integral Equations ◽  
Image Position

Lowengrub [l] has considered simultaneous dual integral equations of the formwhere i = 1,2 …n, I1= {x:0 ≦ x x <1}, I2 {= x:0 ≦ x >1}, the cIJ are constants, the f1(x) are known functions and the functions φ(x) are to be determined.denotes the modified operator of the Hankel transform with the inversion formula

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