1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

3.6 Wave Equation in Unbounded Regions 239
This method of estimating the first eigenvalue is calledRayleigh’s method,
and the ratioN(y)/D(y)is called theRayleigh quotient.Insomemechanical
systems, the Rayleigh quotient may be interpreted as the ratio between poten-
tial and kinetic energy. There are many other methods for estimating eigenval-
ues and for systematically improving the estimates.

EXERCISES

1. Using Eq. (3), show that ifq≥0, thenλ^21 ≥0also.


2. Verify the results for at least one of the trial functions used in the second
   example.


3. Estimate the first eigenvalue of the problem

φ′′+λ^2 ( 1 +x)φ= 0 , 0 <x< 1 ,

φ( 0 )=φ( 1 )= 0.

4. Verify that the general solution of the following differential equation is
   axcos(λ/x)+bxsin(λ/x), and then solve the eigenvalue problem.

φ′′+λ

2

x^4

φ= 0 , 1 <x< 2 ,

φ( 1 )= 0 ,φ( 2 )= 0.

5. Estimate the lowest eigenvalue of the problem in Exercise 4 usingy=
   (x− 1 )( 2 −x).


6. Estimate the lowest eigenvalue of the problem

φ′′+λ^2 xφ= 0 , 0 <x< 1 ,

φ( 0 )= 0 ,φ( 1 )= 0.

Use the trial functionxb( 1 −x), and minimize the Rayleigh quotient with

respect tob.

3.6 Wave Equation in Unbounded Regions

When the wave equation is to be solved for 0<x<∞or for −∞<
x<∞, we can proceed as we did for the solution of the heat equation in these
unbounded regions. That is to say, we separate variables and use a Fourier
integral to combine the product solutions.