A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

1995 ◽  
Vol 46 (1) ◽  
pp. 107-121 ◽  
Author(s):  
Mujibur Rahman
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Polynomial Solution ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems

The solution of Bessel function dual integral equations by converting to the first kind Fredholm Integral Equation: an extension of Noble’s solution

2018 ◽  
Vol 24 (8) ◽  
pp. 2536-2557
Author(s):  
S Cheshmehkani ◽  
M Eskandari-Ghadi
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Integral Transforms ◽  
Original Method ◽  
Fredholm Integral Equations ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Factor Method

In certain mixed boundary value problems, Hankel integral transforms are applied and subsequently dual integral equations involving Bessel functions have to be solved. In the literature, if possible by employing the Noble’s multiplying factor method, these dual integral equations are usually converted to the second kind Fredholm Integral Equations (FIEs) and solved either analytically or numerically, respectively, for simple or complicated kernels. In this study, the multiplying factor method is extended to convert the dual integral equations both to the first and the second kind FIEs, and the conditions for converting to each kind of FIE are discussed. Furthermore, it is shown that under some simple circumstances, many mixed boundary value problems arising from either elastostatics or elastodynamics can be converted to the well-posed first kind FIE, which may be solved analytically or numerically. Main criteria for well-posedness of FIEs of the first kind in such problems are also presented. Noble’s original method is restricted to some limited conditions, which are extended here for both first and second kind FIEs to cover a wider range of dual integral equations encountered in engineering mixed boundary value problems.

scholarly journals Dual integral equations with trigonometrical kernels

1962 ◽  
Vol 5 (3) ◽  
pp. 147-152 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
General Solution ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Image Position

In the analysis of mixed boundary value problems in the plane, we encounter dual integral equations of the typeIf we make the substitutions cos we obtain a pair of dual integral equations of the Titchmarsh type [1, p. 334] with α = − 1, v = − ½ (in Titchmarsh's notation). This is a particular case which is not covered by Busbridge's general solution [2], so that special methods have to be derived for the solution.

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 The solution of certain simultaneous pairs of dual integral equations

1968 ◽  
Vol 9 (2) ◽  
pp. 92-102 ◽  
Author(s):  
M. Lowengrub
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Unknown Function ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Recent Book ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Infinite Domains ◽  
Image Position

In the analysis of mixed boundary value problems by Hankel transforms, one often encounters dual integral equations of the formwhere I1 = (0, 1), I2 = (1, ∞); w1(x), w2(x) are weight functions, ψ(x) is the unknown function, and f(y), g(y) are functions continuously differentiate on I1 and I2 respectively. Many successful attempts have been made to solve (1.1) and (1.2). These are all discussed in a recent book by Sneddon [7]. As pointed out in a recent paper by Erdogan and Bahar [4], in mixed boundary value problems of semi-infinite domains involving more than one unknown function such as those arising in elastostatics, viscoelasticity, and electrostatics, the formulation will lead to a system of simultaneous dual integral equations which is a generalization of (1.1) and (1.2). These equations may be expressed as follows:with i = 1, 2, …, n, where we use the notation

scholarly journals Dual integral equations with Fox'sH-function kernel

1984 ◽  
Vol 7 (2) ◽  
pp. 257-262 ◽  
Author(s):  
R. N. Kalia
Keyword(s):  
Boundary Value Problems ◽  
Bessel Function ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Direct Method ◽  
Formal Solution ◽  
Boundary Value ◽  
General Category ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems

The dual integral equations involving Bessel function kernels were first considered by Weber in 1873. The problem comprised of finding potential of an electrified disc which belongs to a general category of mixed boundary value problems. Titchmarsh gave the formal solution using Wiener-Hopf procedure. We use this direct method as improvised by Busbridge to solve a class of dual integral equations which can be reduced to other known kernels by particularizing the parameters in the Fox'sH-function.

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.

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