Integral Transforms and Dual Integral Equations to Solve Heat Equation with Mixed Conditions

2014 ◽  
Vol 7 (4) ◽  
pp. 15-21 ◽  
Author(s):  
Naser Hoshan
Keyword(s):  
Heat Equation ◽  
Integral Equations ◽  
Integral Transforms ◽  
Dual Integral Equations

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.

Ian Naismith Sneddon, O.B.E. 8 December 1919 – 4 November 2000

2002 ◽  
Vol 48 ◽  
pp. 417-437 ◽  
Author(s):  
P. Chadwick
Keyword(s):  
World War Ii ◽  
Integral Equations ◽  
Linear Theory ◽  
Analytical Techniques ◽  
Integral Transforms ◽  
Theory Of Elasticity ◽  
World War ◽  
Dual Integral Equations ◽  
Cultural Life ◽  
The University

Ian Sneddon was introduced to problems in the linear theory of elasticity involving the indentation of a surface and the extension of a crack through his work for the Ministry of Supply during World War II. He maintained a lifelong research interest in this area and also made distinguished contributions to a range of related analytical techniques, notably the application of integral transforms and the solution of dual integral equations. His many books, extensive travels and engaging personality made him very well known internationally, and he developed particularly fruitful contacts in the USAand Poland. He gave devoted service to the University of Glasgow, where the bulk of his career was spent, and played an active part in the cultural life of Scotland.

Dynamic Analysis of a Piezoelectric Material with an Impermeable Penny-Shaped Crack under Pure Torsional Wave

2007 ◽  
Vol 345-346 ◽  
pp. 433-436
Author(s):  
Zeng Tao Chen
Keyword(s):  
Integral Equations ◽  
Three Dimensional ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Fredholm Integral Equations ◽  
Dual Integral Equations ◽  
Transient Wave ◽  
Total Displacement ◽  
Penny Shaped Crack ◽  
Shaped Crack

A torsional transient wave was assumed acting at infinity on the piezoelectric body with an embedded penny-shaped crack. Appropriate governing equations and boundary conditions have been built within the three-dimensional electroelasticity. The total displacement field was simply considered as the combination of two parts, one related to incident waves inducing an oscillating motion, and another with the scattered waves. An electric impermeable crack was assumed to simplify the mathematical analysis. The problem was formulated in terms of integral transforms techniques. Hankel transform were applied to obtain the dual integral equations, which were then expressed to Fredholm integral equations of the second kind.

Dual Integral Equations

1963 ◽  
Vol 15 ◽  
pp. 631-640 ◽  
Author(s):  
E. R. Love
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Electrostatic Field ◽  
Fredholm Integral Equation ◽  
Integral Transforms ◽  
Dual Integral Equations ◽  
Image Position

Erdélyi and Sneddon (4) have reduced the dual integral equations (4, (1.4))where Ψ is unknown, to a single Fredholm integral equation (4, (4.4)), from the solution of which Ψ is explicitly obtainable. Their work extended and clarified an investigation by Cooke (1), placing it in a context of standard integral transforms. Cooke's reduction was obtained after consideration of the Fredholm integral equation obtained by Love (8) in discussing Nicholson's problem of the electrostatic field of two equal circular coaxial conducting disks (9).

scholarly journals DUAL INTEGRAL EQUATIONS TECHNIQUE IN ELECTROMAGNETIC WAVE SCATTERING BY A THIN DISK

2009 ◽  
Vol 16 ◽  
pp. 107-126 ◽  
Author(s):  
Mikhail V. Balaban ◽  
Ronan Sauleau ◽  
Trevor Mark Benson ◽  
Alexander I. Nosich
Keyword(s):  
Electromagnetic Wave ◽  
Integral Equations ◽  
Wave Scattering ◽  
Thin Disk ◽  
Dual Integral Equations ◽  
Electromagnetic Wave Scattering

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.

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