The Functional Laplace Operator and Classical Diffusion Equations. Boundary Value Problems for Uniform Domains. Harmonic Controlled Systems

1988 ◽  
pp. 85-130
Author(s):  
Efim M. Polishchuk
Keyword(s):  
Boundary Value Problems ◽  
Laplace Operator ◽  
Boundary Value ◽  
Diffusion Equations ◽  
Classical Diffusion ◽  
Controlled Systems ◽  
Uniform Domains

Note on the generalized Mehler transform

1964 ◽  
Vol 60 (1) ◽  
pp. 57-59 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Boundary Value Problems ◽  
Inversion Formula ◽  
Integral Transform ◽  
Legendre Function ◽  
Boundary Value ◽  
Diffusion Equations ◽  
Legendre Functions ◽  
Image Position ◽  
And Diffusion ◽  
Associated Legendre Function

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined bywhere ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.

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