Application of the singular function expansion to an integral equation for scattering

ABSTRACTA simple derivation (based on a derived identity of integral equation theory) of fully separable, non–muffin–tin multiple scattering theory is presented. Three multipolar representations have been proposed for use in this theory: the original representation by Williams and Morgan, a more recent one by Brown and Ciftan, and a “Bloch periodic” representation by Badralexe and Freeman (which was addressed in earlier work). We study the properties of the Williams and Morgan representation in the context of the 2D empty square lattice, where the representation of Brown and Ciftan is manifestly exact and correct (so that MST derived in terms of it is a “natural” theory). We show that the representation of Williams and Morgan contains an explict divergence predicted by its implicit use of a wrong-order Green's function expansion but that MST expressed in terms of this representation is still a fairly good asymptotic approximation to the formally exact theory.

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Analytic Three-Dimensional Model of a Double-Shielded Giant Magnetoresistive Perpendicular Head Using a Singular Function Expansion

IEEE Transactions on Magnetics ◽

10.1109/tmag.2007.896514 ◽

2007 ◽

Vol 43 (7) ◽

pp. 3305-3314 ◽

Cited By ~ 2

Author(s):

P.M. Jermey ◽

H.A. Shute ◽

D.T. Wilton

Keyword(s):

Three Dimensional ◽

Singular Function ◽

Dimensional Model ◽

Three Dimensional Model ◽

Function Expansion

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Collocation with Chebyshev polynomials for Symm's integral equation on an interval

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000008729 ◽

1992 ◽

Vol 34 (2) ◽

pp. 199-211 ◽

Cited By ~ 15

Author(s):

I. H. Sloan ◽

E. P. Stephan

Keyword(s):

Integral Equation ◽

Chebyshev Polynomials ◽

Basis Functions ◽

Singular Function ◽

Polynomial Space ◽

The Novel ◽

Negative Power ◽

Collocation Points ◽

Chebyshev Points ◽

Symm’S Integral Equation

AbstractA collocation method for Symm's integral equation on an interval (a first-kind integral equation with logarithmic kernel), in which the basis functions are Chebyshev polynomials multiplied by an appropriate singular function and the collocation points are Chebyshev points, is analysed. The novel feature lies in the analysis, which introduces Sobolev norms that respect the singularity structure of the exact solution at the ends of the interval. The rate of convergence is shown to be faster than any negative power of n, the degree of the polynomial space, if the driving term is smooth.

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