Laplace,s Equation--Bispherical Coordinates
المؤلف:
Morse, P. M. and Feshbach, H
المصدر:
Methods of Theoretical Physics, Part I. New York: McGraw-Hill
الجزء والصفحة:
...
21-7-2018
1699
Laplace's Equation--Bispherical Coordinates
In bispherical coordinates, Laplace's equation becomes
 |
(1)
|
Attempt separation of variables by plugging in the trial solution
 |
(2)
|
then divide the result by 
to obtain
 |
(3)
|
The function 
then separates with
 |
(4)
|
giving solution
 |
(5)
|
Plugging 
back in and dividing by 
gives
 |
(6)
|
The function 
then separates with
 |
(7)
|
giving solution
 |
(8)
|
Plugging 
back in and multiplying by 
gives
 |
(9)
|
so Laplace's equation is partially separable in bispherical coordinates. However, the Helmholtz differential equationcannot be separated in this manner.
REFERENCES:
Arfken, G. "Bispherical Coordinates 
." §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.
الاكثر قراءة في المعادلات التفاضلية الجزئية
اخر الاخبار
اخبار العتبة العباسية المقدسة
