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CMSC 858M/AMSC 698R
Fast Multipole Methods
Nail A. Gumerov & Ramani Duraiswami
Lecture 5
Outline
• Laplace equation in 3D (continued)
• Helmholtz equation in 2D
• Helmholtz equation in 3D
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Addition Theorem for Spherical
HarmonicsSpherical Harmonics
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s1
s2
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s1
s2
degree
order
Vector form of the addition theorem
S- and R- expansions of
Fundamental Solution
Multipole (!)
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r
rir
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Helmholtz Equation
• Wave equation in frequency domain– Acoustics
– Electromagneics (Maxwell equations)
– Diffusion/heat transfer/boundary layers
– Telegraph, and related equations
– k can be complex
• Quantum mechanics– Klein-Gordon equation
– Shroedinger equation
• Relativistic gravity (Yukawa potentials, k is purely imaginary)
• Molecular dynamics (Yukawa)
• Appears in many other models
2D Helmholtz Equation
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2D Helmholtz Equation
Helmholtz equation:
Wave equation:
Time harmonic
solution
Helmholtz equation:
Fourier transform:
circular
frequency
wave speed
wavenumber
Construction of basis for 2D
Helmholtz equation
Polar coordinates
Helmholtz
equation
Elementary solution
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Construction of basis for 2D Helmholtz
equation (2): Angular dependence
Periodicity condition
Discontinuous at the ends
(use Analytical continuation)
Construction of basis for 2D Helmholtz
equation (3): Radial dependence
Bessel equation:
Bessel functions
of order n
First kind Second kind
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Bessel functions of integer order
Learn properties of Bessel functions! Abramowitz & Stegun (1964)
(Handbook on special functions: available online!)
Construction of basis for 2D
Helmholtz equation (4)
Usual basis functions
Regular expansion
Singular (radiating)
expansion
Hankel functionsfirst kind
second kind
Sommerfeld radiation
condition
Radiating
functions
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Reference
N.A. Gumerov & R. Duraiswami
Fast Multipole Methods for the Helmholtz Equation in Three
Dimensions
Academic Press, Oxford, UK (2004).
Spherical Basis Functions
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O
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r
rir
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Spherical Coordinates
Regular Basis Functions
Singular Basis Functions
Spherical Harmonics
Associated Legendre Functions
Spherical Bessel Functions
Spherical Hankel Functions
of the First Kind