5.3 Two-Dimensional Heat Equation: Solution 307
Thecoefficientsareeasilyfoundtobe
amn=4 ab
π^2cos(mπ)cos(nπ)
mn =4 ab
π^2(− 1 )m+n
mn ,so the solution to this problem is
u(x,y,t)=4 ab
π^2∑∞
m= 1∑∞
n= 1(− 1 )m+n
mn sin(mπx
a)
sin(nπy
b)
exp(
−λ^2 mnkt)
.
(18)
This solution is shown animated on the CD.
Thedoubleseriesthatappearherearebesthandledbyconvertingtheminto
single series. To do this, arrange the terms in order of increasing values ofλ^2 mn.
Then the first terms in the single series are the most significant, those that
decay least rapidly. For example, ifa= 2 b,sothat
λ^2 mn=(m^2 + 4 n^2 )π^2
a^2 ,then the following list gives the double index(m,n)in order of increasing
values ofλ^2 mn:
( 1 , 1 ), ( 2 , 1 ), ( 3 , 1 ), ( 1 , 2 ), ( 2 , 2 ), ( 4 , 1 ), ( 3 , 2 ),....EXERCISES
- Write out the “first few” terms of the series of Eq. (18). By “first few,”
we mean those for whichλ^2 mnis smallest. (Assumea=bin determining
relative magnitudes of theλ^2 .) - Provide the details of the separation of variables by which Eqs. (9)–(13)
are derived. - Find the frequencies of vibration of a rectangular membrane. See Sec-
tion 5.1, Exercise 1. - Ve r i f y t h a tumn(x,y,t)satisfies Eqs. (1)–(3).
- Show thatXm(x)=cos(mπx/a)(m= 0 , 1 , 2 ,...) if the boundary condi-
tions Eq. (3) are replaced by
∂u
∂x(^0 ,y,t)=^0 ,
∂u
∂x(a,y,t)=^0 ,^0 <y<b,^0 <t.
What values will theλ^2 mnhave, and of what form will the solutionu(x,y,t)
be?