358 Chapter 5 Higher Dimensions and Other Coordinates
20.Observe that the functionφin Exercise 19 is the difference of two differ-
ent eigenfunctions of the 1×1 square (see Section 5.3) corresponding to
the same eigenvalue. Use this idea to construct other eigenfunctions for
the triangleTof Exercise 19.
21.LetTbe the equilateral triangle in thexy-plane whose base is the interval
0 <x<1ofthex-axis and whose sides are segments of the linesy=√
3 x
andy=√
3 ( 1 −x). Show that forn= 1 , 2 , 3 ,...,thefunctionφn(x,y)=sin(
4 nπy/√
3
)
+sin(
2 nπ(
x−y/√
3
))
−sin(
2 nπ(
x+y/√
3
))
is a solution of the eigenvalue problem∇^2 φ=−λ^2 φinT,φ=0onthe
boundary ofT. What are the eigenvaluesλ^2 ncorresponding to the func-
tionφnthat is given? [See “The eigenvalues of an equilateral triangle,”
SIAM Journal of Mathematical Analysis, 11 (1980): 819–827, by Mark A.
Pinsky.]
22.In Comments and References, Section 5.11, a theorem is quoted that re-
lates the least eigenvalue of a region to that of a smaller region. Confirm
the theorem by comparing the solution of Exercise 19 with the smallest
eigenvalue of one-eighth of a circular disk of radius 1:1
r∂
∂r(
r∂φ
∂r)
+^1
r^2∂^2 φ
∂θ^2=−λ^2 φ, 0 <θ<π
4, 0 <r< 1 ,φ(r, 0 )= 0 ,φ(
r,π
4)
= 0 , 0 <r< 1 ,φ( 1 ,θ)= 0 , 0 <θ<π
4.23.Same task as Exercise 22, but use the triangle of Exercise 21 and the
smallest eigenvalue of one-sixth of a circular disk of radius 1.
24.Show thatu(ρ,t)=t−^3 /^2 e−ρ^2 /^4 tis a solution of the three-dimensional
heat equation∇^2 u=∂∂ut, in spherical coordinates.
25.For what exponent b isu(r,t)=tbe−r^2 /^4 t a solution of the two-
dimensional heat equation∇^2 u=∂∂ut? (Use polar coordinates.)
26.Suppose that an estuary extends fromx=0tox=a, where it meets the
open sea. If the floor of the estuary is level but its width is proportional
tox, then the water depthu(x,t)satisfies1
x∂
∂x(
x∂u
∂x)
=
1
gU∂^2 u
∂t^2 ,^0 <x<a,^0 <t,