692 CHAPTER 34. SOME PARTIAL DIFFERENTIAL EQUATIONS

19. Consider the following initial boundary value problem,

ut = uxx, ux (0, t) = 0,u(2, t)+ux (2, t) = 0,u(x,0) = f (x)

Determine the appropriate equation for the eigenfunctions and show that there existsa sequence of strictly positive eigenvalues converging to ∞. Also explain why thesolution u if it exists, must have a limit limt→∞ u(x, t) = w(x) and that this limitsatisfies w′′ (x) = w(x) = 0.

20. Consider the following initial boundary value problem,

ut = uxx, ux (0, t) = 0,ux (2, t) = 0,u(x,0) = f (x)

Determine the appropriate equation for the eigenfunctions and show that there existsa sequence of strictly positive eigenvalues converging to ∞. Also explain why thesolution u if it exists, must have a limit limt→∞ u(x, t) = 1

2∫ 2

0 f (x)dx.

21. Recall that on the circular disk of radius R centered at the origin, denoted here as DR

u(r,θ) =∫ 2π

0

1π

(12+

∞

∑n=1

rn

Rn cos(n(θ −α))

)f (α)dα

gave the solution to ∆u= 0 and f (α) a given function on the boundary where f (0) =f (2π). Show, using the divergence theorem from calculus that there is at most onesmooth solution to this problem. Then explain why

∫ 2π

0

1π

(12+

∞

∑n=1

rn

Rn cos(n(θ −α))

)dα = 1

22. Recall that on a simple computation was done which showed that

1π

(12+

∞

∑n=1

rn

Rn cos(n(θ −α))

)=

12π

R2− r2

R2−2(cos(θ −α))Rr+ r2

Therefore, ∫ 2π

0

12π

R2− r2

R2−2(cos(θ −α))Rr+ r2 dα = 1

Explain why it is also the case that

12π

R2− r2

R2−2(cos(θ −α))Rr+ r2 ≥ 0

and if |θ −α| ≥ δ > 0, then

limr→R−

12π

R2− r2

R2−2(cos(θ −α))Rr+ r2 = 0

uniformly for such α .