hwk8_key.pdf

7/27/2019 hwk8_key.pdf

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Homework 8

1. Solve Laplaces equation inside a circular annulus (a < r< b) subject to theboundary conditions

(a) u(a,) = f(), u(b,) = g()

(b)u

r(a,) = 0, u(b,) = g()

If there is a solvability condition, state it and explain it physically.

Answer

The shape of domain suggest us to use the polar coordinate system, in which the

Laplaces equation is expressed as

1rr

rur

+ 1

r2

2

u2

= 0

Assume u(r,) = ()G(r),

r

G(r)

d

dr

rdG

dr

=

1

d2

d2=

Using periodic boundary conditions for , we have

() = an cos(n) +bn sin(n), = n2, n = 0,1,2,

Ifn> 0,

G(r) = cnrn +dnrn

otherwise,

G(r) = c0 +d0 ln(r).

Note that we do not know whether G(0) is finite or not.

(a) Assume u = u1 +u2, where u1 and u2 satisfies

u1 = 0 u2 = 0 (1)

u1(a,) = f() u2(a,) = 0 (2)

u1(b,) = 0 u2(b,) = g() (3)

It is implied by u1(b,) = 0 that G(b) = 0. For n> 0, we get

dn = cnb2n

and we simply choose

cn = bn, dn = b

n.

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Ifn = 0, we can choose

c0 = ln(b), d0 = 1.

Now u1 could be written as

u1(r,) = a0 ln(r/b) +

n=1

(an cos(n) +bn sin(n))((r/b)n (r/b)n)

Applying u1(a,) = f(),

u1(a,) =a0 ln(a/b)+

n=1

(an cos(n)+bn sin(n))((a/b)n(a/b)n) = f()

Coefficientsan and bn could be determined by calculating the Fourier seriesof f(). Calculation ofu2(r,) is similar.

(b) There is already enough homogeneous boundary conditions we can use.

No need to separate u into two functions. It is implied byu

r(a,) = 0 that

dG

dr(a) = 0.

Thus ifn> 0cnna

n1 ndnan1 = 0

and we can choose

cn = an, dn = a

n.

When n = 0,d0/a = 0,

and the only choice we have is d0 = 0.

This result in

u(r,) = a0 +

n=1

(an cos(n) +bn sin(n))((r/a)n + (a/r)n),

and

u(b,) = a0 +

n=1

(an cos(n) +bn sin(n))((b/a)n + (a/b)n) = g().

You should know how to calculate an and bn.

2. Consider

f(x) =

0 x< x0

1/ x0 < x< x0 +

0 x> x0 +

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Assume thatL < x < L, x0 > L and x0 + < L. Determine the complex

Fourier coefficients cn. Show that cn = cnAnswer

cn =1

2L

ZLL

f(x)einx/Ldx (4)

=1

2L

Zx0+x0

einx/Ldx (5)

=1

2L

L

ineinx/L|x0+x0 (6)

=einx0/L

i2n

ein/L1

(7)

cn =einx0/L

i2n

ein/L1

= cn

3. Using the maximum principles for Laplaces equation, prove that the solution of

Poissons equation 2u = g(x,y), subject to u(x,y) = f(x,y) on the boundary, isunique.

Answer

Read section 2.5.4 in the textbook.

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