5.3 Two-Dimensional Heat Equation: Solution 305
The sum of a function ofxand a function ofycan be constant only if those
two functions are individually constant:
X′′
X=constant, Y′′
Y=constant.Before naming the constants, let us look at the boundary conditions on
φ=XY:
X(x)Y( 0 )= 0 , X(x)Y(b)= 0 , 0 <x<a,
X( 0 )Y(y)= 0 , X(a)Y(y)= 0 , 0 <y<b.If either of the functionsXorYis zero throughout the whole interval of its
variable, the conditions are certainly satisfied, butφis identically zero. We
therefore require each of the functionsXandYto be zero at the endpoints of
its interval:
Y( 0 )= 0 , Y(b)= 0 , (9)
X( 0 )= 0 , X(a)= 0. (10)
Now it is clear that each of the ratiosX′′/XandY′′/Yshould be a negative
constant, designated by−μ^2 and−ν^2 , respectively. The separate equations for
XandYare
X′′+μ^2 X= 0 , 0 <x<a, (11)
Y′′+ν^2 Y= 0 , 0 <y<b. (12)Finally, the original separation constant−λ^2 is determined by
λ^2 =μ^2 +v^2. (13)
Now we see two independent eigenvalue problems: Eqs. (9) and (12) form
one problem and Eqs. (10) and (11) the other. Each is of a very familiar form;
the solutions are
Xm(x)=sin(mπx
a)
,μ^2 m=(mπ
a) 2
, m= 1 , 2 ,...,Yn(y)=sin(
nπy
b)
,ν^2 n=(
nπ
b) 2
, n= 1 , 2 ,....Notice that the indicesnandmare independent. This means thatφwill have
a double index. Specifically, the solutions of the two-dimensional eigenvalue
problem Eqs. (6)–(8) are
φmn(x,y)=Xm(x)Yn(y),
λ^2 mn=μ^2 m+νn^2 ,