Two dimensional mixed boundary-value problems in a wedge-shaped region

1968 ◽  
Vol 64 (2) ◽  
pp. 503-505 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Two Dimensional ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Mellin Transforms

In a recent paper Srivastav (2) considered the solution of certain two-dimensional mixed boundary-value problems in a wedge-shaped region. The problems were formulated as dual integral equations involving Mellin transforms and were reduced to the solution of a Fredholm integral equation of the second kind. In this paper it will be shown that a closed form solution to the problems treated in (2) may be obtained by elementary means.

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 A Note on an asymmetric mixed boundary value problem for a half space with a cylindrical cavity

1965 ◽  
Vol 7 (1) ◽  
pp. 45-47
Author(s):  
Prem Narain
Keyword(s):  
Boundary Value Problems ◽  
Half Space ◽  
Cylindrical Cavity ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Subsequent Paper ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Asymmetric Problem ◽  
Type Boundary

In recent years interest in the mixed boundary value problems of mathematical physics has increased appreciably because of various applications. The mixed boundary value problems for simply-connected regions have been investigated widely and it can reasonably be hoped that within a short time the theory will reach a satisfactory stage. It appears, however, that very few problems for multiply-connected domains have been solved. Recently Srivastav [2] has considered the problem of rinding an axisymmetric potential function for a half space with a cylindrical cavity subject to mixed type boundary conditions. In a subsequent paper [1], Srivastav extends the analysis to the asymmetric problem and formulates the problem in terms of dual integral equations involving Bessel functions of the first and second kinds whose solution leads to the solution of the potential problem. The latter paper, however, involves heavy manipulations and complicated contour integrals.

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.

Asymmetric Mixed Boundary-Value Problems of the Elastic Half-Space

1965 ◽  
Vol 32 (2) ◽  
pp. 411-417 ◽  
Author(s):  
R. A. Westmann
Keyword(s):  
Boundary Value Problems ◽  
Half Space ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Elastic Half Space ◽  
Integral Solution ◽  
Displacement Fields ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Stress And Displacement Fields

Solutions are presented, within the scope of classical elastostatics, for a class of asymmetric mixed boundary-value problems of the elastic half-space. The boundary conditions considered are prescribed interior and exterior to a circle and are mixed with respect to shears and tangential displacements. Using an established integral-solution form, the problem is reduced to two pairs of simultaneous dual integral equations for which the solution is known. Two illustrative examples, motivated by problems in fracture mechanics, are presented; the resulting stress and displacement fields are given in closed form.

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