Solution of boundary value problems for Laplace’s equation in a piecewise homogeneous plane with a parabolic crack (screen)

2009 ◽  
Vol 49 (11) ◽  
pp. 1847-1852
Author(s):  
S. E. Kholodovskii
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Parabolic Crack ◽  
Homogeneous Plane

XII.—Paraboloidal Co-ordinates and Laplace's Equation

1963 ◽  
Vol 66 (3) ◽  
pp. 129-139
Author(s):  
F. M. Arscott
Keyword(s):  
Wave Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Subsequent Paper ◽  
Laplace’S Equation ◽  
Mathieu Functions ◽  
Laplace's Equation ◽  
Normal Surfaces ◽  
Special Cases ◽  
Hyperbolic Paraboloids

SynopsisIn this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.

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