scholarly journals The iterated equation of generalized axially symmetric potential theory, VI General solutions

1974 ◽  
Vol 18 (3) ◽  
pp. 318-327
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Polar Coordinates ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
General Solutions ◽  
Real Value ◽  
Image Position

The iterated equation of generalized axially symmetric potential theory [1] is the equation where, in its simplest form, the operator Lk is defined by the function f f(x, y) being assumed to belong to the class of C2n functions and the parameter l to take any real value. In appropriate circumstances, which will be indicated later, the operator can be generalized but as this can be done without altering the methods used, the operator will be taken in the form where r, θ are polar coordinates such that x = r cos θ, y = r sin μ = cosθ.

scholarly journals The iterated equation of generalized axially symmetric potential theory. III. Conjugate general solutions

1967 ◽  
Vol 7 (3) ◽  
pp. 290-300
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
General Solutions ◽  
Image Position

The iterated equation of generalized axially symmetric potential theory (GASPT), in the notation of the first paper of this series [1] which will be designated I, is the equation where the operator Lk is defined by

scholarly journals The iterated equation of generalized axially symmetric potential theory. II. General solutions of Weinstein's type

1967 ◽  
Vol 7 (3) ◽  
pp. 277-289 ◽  
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
General Solutions ◽  
Particular Solutions ◽  
Image Position

In the first paper of this series [1] which will be designated I, particular solutions of various kinds have been found for the iterated equation of generalized axially symmetric potential theory (GASPT) which, in the notation defined in I, is (1) where the operator is defined by

scholarly journals The iterated equation of generalized axially symmetric potential theory. I. Particular solutions

1967 ◽  
Vol 7 (3) ◽  
pp. 263-276 ◽  
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Stream Function ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Symmetric Flow ◽  
Symmetric Potential ◽  
Dimensional Flow ◽  
Physical Problems ◽  
Two Dimensional Flow ◽  
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The iterated equation of generalized axially symmetric potential theory (GASPT) [1] is defined by the relations (1) where (2) and Particular cases of this equation occur in many physical problems. In classical hydrodynamics, for example, the case n = 1 appears in the study of the irrotational motion of an incompressible fluid where, in two-dimensional flow, both the velocity potential φ and the stream function Ψ satisfy Laplace's equation, L0(f) = 0; and, in axially symmetric flow, φ and satisfy the equations L1 (φ) = 0, L-1 (ψ) = 0. The case n = 2 occurs in the study of the Stokes flow of a viscous fluid where the stream function satisfies the equation L2k(ψ) = 0 with k = 0 in two-dimensional flow and k = −1 in axially symmetric flow.

The iterated equation of generalized axially symmetric potential theory. V. Generalized weinstein correspondence principle

1970 ◽  
Vol 11 (2) ◽  
pp. 129-141 ◽  
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Correspondence Principle ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
Image Position

Solutions of the iterated equation of generalized axially symmetric potential theory [1]where the operator Lk is defined bywill be denoted by except that when n = 1, fk will be written instead of . It is easily shown [2, 3] thatby which is meant that any function is a solution of (1).

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