Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

21.25 The terms inG(r,r 0 ) that are additional to the fundamental solution are

1

4 π

∑∞

n=2

(−1)n

{[

(x−x 0 )^2 +(y−y 0 )^2 +(z+(−1)nz 0 −nc)^2

]− 1 / 2

+

[

(x−x 0 )^2 +(y−y 0 )^2 +(z+(−1)nz 0 +nc)^2

]− 1 / 2 }

.

21.27 (a) As given in equation (21.86), but withr 0 replaced byr′.
(b) Move the origin tor′and integrate the defining Green’s equation to obtain

4 πt^2

dG

dt

−m^2

∫t

0

G(t′)4πt′^2 dt′=1,

leading toG(t)=[− 1 /(4πt)]e−mt.

(c) φ(r)=[− 1 /(4π)](p−^1 e−mp−q−^1 e−mq), wherep=|r−r 1 |andq=|r−r 2 |with

r 1 =(x 1 ,y 1 ,z 1 )andr 2 =(−x 1 ,y 1 ,z 1 ).