1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 4 475

b.u(x,y)=

2

π

∫∞

0

λ

1 +λ^2 sin(λx)

sinh(λ(b−y))

sinh(λb) dλ.

11.u(x,y)=

∫∞

0

2

π( 1 +λ^2 )

sinh(λx)

sinh(λa)cos(λy)dλ.

13.e−λysin(λx),λ>0.

15.e−λysin(λx),e−λycos(λx),λ>0.

17.u(x,y)=^1

π

[π

2

+tan−^1 (x/y)

]

.

19. This solution is unbounded asxtends to infinity and cannot be found
    by the method of this section.

Section 4.5

1.v(r,θ)is given by Eq. (10) withbn=0,a 0 =π/2,

an=−2(1−cos(nπ ))/πn^2 cn.

3. The solution is as in Eq. (10) withbn=0,a 0 = 1 /π,a 1 = 1 /2, and

an=

2sin((n− 1 )π/ 2 )

π(n^2 − 1 ) forn=1.

5. Convergence is uniform inθ.

7.a 0 =

1

2 π

∫π

−π

f(θ )dθ,an=

cn

π

∫π

−π

f(θ )cos(nθ)dθ,

bn=c

n

π

∫π

−π

f(θ )sin(nθ)dθ.

9. π^2

∑∞

n= 1

1 −cos(nπ)

nc^2 n r

2 nsin( 2 nθ)=v(r,θ).

11.vn(r,θ)=rn/αsin(nθ/α)has∂v/∂runbounded asr→ 0 +,ifn=1.

Section 4.6

1. Hyperbolic (a) and (e); elliptic (b) and (c); parabolic (d).


2. Only (e).


3. a.u(x,y)=

∑∞

1

ansin(nπx)e−nπy;

b.u(x,y)=

∑∞

1

ansin(nπx)cos(nπy);