Dual and Triple Integral Equations

2000 ◽  
pp. 375-412
Author(s):  
Ricardo Estrada ◽  
Ram P. Kanwal
Keyword(s):  
Integral Equations ◽  
Triple Integral Equations

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 ◽  
Image Position

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 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 The Study of Triple Integral Equations with Generalized Legendre Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
B. M. Singh ◽  
J. Rokne ◽  
R. S. Dhaliwal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Triple Integral Equations

A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.

scholarly journals The solution of some integral equations

1987 ◽  
Vol 29 (2) ◽  
pp. 203-215
Author(s):  
John F. Ahner ◽  
John S. Lowndes
Keyword(s):  
Wave Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Laplace Equation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Triple Integral Equations

AbstractAlgorithms are developed by means of which certain connected pairs of Fredholm integral equations of the first and second kinds can be converted into Fredholm integral equations of the second kind. The methods are then used to obtain the solutions of two different sets of triple integral equations tht occur in mixed boundary value problems involving Laplace' equation and the wave equation respectively.

Slow viscous flow due to motion of an annular disk; pressure-driven extrusion through an annular hole in a wall

1991 ◽  
Vol 231 ◽  
pp. 51-71 ◽  
Author(s):  
A. M. J. Davis
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Viscous Flow ◽  
Integral Equations ◽  
Bessel Functions ◽  
Annular Disk ◽  
Slow Viscous Flow ◽  
Triple Integral Equations ◽  
Flow Through ◽  
Pressure Driven

The description of the slow viscous flow due to the axisymmetric or asymmetric translation of an annular disk involves the solution of respectively one or two sets of triple integral equations involving Bessel functions. An efficient method is presented for transforming each set into a Fredholm integral equation of the second kind. Simple, regular kernels are obtained and the required physical constants are readily available. The method is also applied to the pressure-driven extrusion flow through an annular hole in a wall. The velocity profiles in the holes are found to be flatter than expected with correspondingly sharper variation near a rim. For the sideways motion of a disk, an exact solution is given with bounded velocities and both components of the rim pressure singularity minimized. The additional drag experienced by this disk when the fluid is bounded by walls parallel to the motion is then determined by solving a pair of integral equations, according to methods given in an earlier paper.

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.

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