Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
The left hand side of Eq. (10.7) is a function of x and y, while the right hand side is
a function of z alone. If Eq. (10.7) is to have a solution at all, each side of the
equation must be equal to the same constant, say k 2 3. Then Eq. (10.7) leads to
d 2 Z
dz 2 ‡ k2 3Z ˆ 0;
1 d 2 X
X dx 2 ˆ1 d 2 Y
Y dy 2 ‡ k2 3:
…10:8†
…10:9†
The left hand side of Eq. (10.9) is a function of x only, while the right hand side is
a function of y only. Thus, each side of the equation must be equal to a constant,
say k 2 1. Therefore
d 2 X
dx 2 ‡ k2 1X ˆ 0;
d 2 Y
dy 2 ‡ k2 2Y ˆ 0;
…10:10†
…10:11†
where
k 2 2 ˆ k 2 1 k 2 3:
The solution of Eq. (10.10) is of the form
X…x† ˆa…k 1 †e k 1x ; k 1 6ˆ 0; 1< k 1 < 1
or
X…x† ˆa…k 1 †e k 1x ‡ a 0 …k 1 †e k 1x ; k 1 6ˆ 0; 0 < k 1 < 1: …10:12†
Similarly, the solutions of Eqs. (10.11) and (10.8) are of the forms
Hence
Y…y† ˆb…k 2 †e k 2y ‡ b 0 …k 2 †e k 2y ; k 2 6ˆ 0; 0 < k 2 < 1; …10:13†
Z…z† ˆc…k 3 †e k 3z ‡ c 0 …k 3 †e k 3z ; k 3 6ˆ 0; 0 < k 3 < 1: …10:14†
ˆ‰a…k 1 †e k 1x ‡ a 0 …k 1 †e k 1x Š‰b…k 2 †e k 2y ‡ b 0 …k 2 †e k 2y Š‰c…k 3 †e k 3z ‡ c 0 …k 3 †e k 3z Š;
and the general solution of Eq. (10.6) is obtained by integrating the above equation
over all the permissible values of the k i …i ˆ 1; 2; 3†.
In the special case when k i ˆ 0 …i ˆ 1; 2; 3†, Eqs. (10.8), (10.10), and (10.11)
have solutions of the form
where x 1 ˆ x, and X 1 ˆ X etc.
X i …x i †ˆa i x i ‡ b i ;
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