1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 5 479

Section 5.3

1. Ifa=b, the lowest eigenvalues are those with indices(m,n),inthisor-
   der:( 1 , 1 );( 1 , 2 )=( 2 , 1 );( 2 , 2 );( 3 , 1 )=( 1 , 3 );( 3 , 2 )=( 2 , 3 );( 1 , 4 )=
   ( 4 , 1 );( 3 , 3 ).


2. Frequencies areλmnc/ 2 π(Hz), whereλ^2 mnare the eigenvalues found in
   the text.
   5.λ^2 mn=(mπ/a)^2 +(nπ/b)^2 , form= 0 , 1 , 2 ,...,n= 1 , 2 , 3 ,....


3. a.u(x,y,t)=1.
   For b and c the solution has the form

u(x,y,t)=

∑

m,n

amncos

(mπx

a

)

cos

(nπy

b

)

exp

(

−λ^2 mnkt

)

,

whereλ^2 mn=(mπ/a)^2 +(nπ/b)^2 andmandnrun from 0 to∞.

b.a 00 =(a+b)

2

, am 0 =−^2 b(^1 −cos(mπ))

m^2 π^2

,

a 0 n=−^2 a(^1 −cos(nπ))

n^2 π^2

, amn=0otherwise;

c.a 00 =ab 4 , am 0 =−ab(^1 −mcos (^2) π 2 (mπ)), a 0 n=−ab(^1 −n 2 cosπ 2 (nπ)),
amn=
4 ab( 1 −cos(nπ ))( 1 −cos(mπ))
m^2 n^2 π^4
ifmandnare greater than zero.

9. The choice of a positive constant for eitherX′′/XorY′′/Y,underthe
   boundary conditions in Eqs. (9) and (10), will lead to the trivial solution.


10. The nodal lines form a grid:umn(x,y,t)=0atx=0,a/m,2a/m,...,a
    and aty=0,b/n,2b/n,...,b.

Section 5.4

1. The partial differential equations are the same, the boundary conditions
   become homogeneous, and in the initial conditionsg(r,θ)is replaced by
   g(r,θ)−v(r,θ).


2. In the heat problem,T′+λ^2 kT=0.Inthewaveproblem,T′′+λ^2 c^2 T=0.


3. The boundary conditions Eqs. (10) and (11) would be replaced by
   Q( 0 )= 0 , Q(π )= 0.
   Solutions areQ(θ )=sin(nθ),n= 1 , 2 ,....