Miscellaneous Exercises 247
More information about the Rayleigh quotient and estimation of eigenval-
ues is inBoundary and Eigenvalue Problems in Mathematical Physics, by Sagan.
The classic reference for eigenvalues, and indeed for the partial differential
equations of mathematical physics in general, is the work by Courant and
Hilbert,Methods of Mathematical Physics.
Chapter Review
See the CD for Review Questions.
Miscellaneous Exercises
Exercises1–5refertotheproblem
∂^2 u
∂x^2 =1
c^2∂^2 u
∂t^2 ,^0 <x<a,^0 <t,
u( 0 ,t)= 0 , u(a,t)= 0 , 0 <t,u(x, 0 )=f(x),∂u
∂t(x,^0 )=g(x),^0 <x<a.
1.Ta k e f(x)=1, 0<x<a,andg(x)≡0.(Thisisratherunrealis-
tic ifu(x,t)is the displacement of a vibrating string.) Find a series
(separation-of-variables) solution.
2.Sketchu(x,t)of Exercise 1 as a function ofxat various times throughout
one period.
3.The solutionu(x,t)of Exercise 1 takes on only the three values 1, 0, and
−1. Make a sketch of the region 0<x<a,0<t, and locate the places
whereutakesoneachofthevalues.
4.Ta k eg(x)=0andf(x)to be this function:f(x)=
3 hx
2 a,^0 <x<2 a
3 ,
3 h(a−x)
a,^2 a
3<x<a.The graph offis triangular, with peak atx= 2 a/3. Find a series solution
foru(x,t).
5.Sketchu(x,t)of Exercise 4 as a function ofxat times 0 toa/cin steps of
a/ 6 c.