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MATH 3705B

Test 4 Solutions

March 19, 2012

Questions 1-2 are multiple choice. Circle the correct answer. Only the answer will be marked.Marks]

1. The solution of Laplaces equation urr +1

r

ur +1

r

2u = 0 inside the circle r = 2 has the form5]

u(r, ) =a0

2+

n=1

rn[an cos(n) + bn sin(n)].

The solution which satisfies the boundary condition u(2, ) = 2cos(3)+2 sin(2) is u(r, ) =

(a) 2 8r3 cos(3) + 2r2 sin(2) (b) 2 r3 cos(3) + 2r2 sin(2)

(c) 2 cos(3) + 2 sin(2) (d) 21

8r3 cos(3) +

1

2r2 sin(2) (e) None of these

Answer: (d)

2. The bounded solution of Laplaces equation urr +1

rur +

1

r2u = 0 outside the circle r = 25]

has the form

u(r, ) =a0

2+

n=1

rn[an cos(n) + bn sin(n)].

The solution which satisfies the boundary condition u(2, ) = 3 cos(2) 2 sin(3) is u(r, ) =

(a) 12r2 cos(2) 16r3 sin(3) (b) 3r2 cos(2) 2r3 sin(3)

(c) 34

r2 cos(2) 14

r3 sin(3) (d) 3 cos(2) 2 sin(3) (e) None of these

Answer: (a)

3. The solution of Laplaces equation uxx + uyy = 0, 0 < x < L, 0 < y < M, satisfying the10]boundary conditions u(x, 0) = 0, u(x, M) = 0, u(0, y) = 0, u(L, y) = f(y), has the form

u(x, y) =n=1

an sinhnx

M

sin

nyM

.

Find the solution of Laplaces equation uxx + uyy = 0 within the rectangle 0 < x < 2

0 < y < 1, which satisfies the boundary conditions u(x, 0) = 0, u(x, 1) = 0, u(0, y) = 0u(2, y) = y y2. Write down the complete solution u(x, y).

Solution:

u(x, y) =n=1

an sinh(nx)sin(ny) and u(2, y) = y y2

y y2 =n=1

an sinh(2n)sin(ny)

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2

an sinh(2n) = 2

1

0

(y y2) sin(ny)dy

= 2

n(y y2) cos(ny)

10

+2

n

1

0

(1 2y) cos(ny)dy

=2

n22(1 2y)sin(ny)

1

0

2

n22

1

0

2 sin(ny)dy

= 4

n33cos(ny)

1

0

=4

n33[1 (1)n]

u(x, y) =n=1

4[1 (1)n]

n33 sinh(2n)sinh(nx) sin(ny).

4. The solution of Laplaces equation uxx + uyy = 0, 0 < x < L, 0 < y < M, satisfying the10]boundary conditions u(x, 0) = f(x), u(x, M) = 0, u(0, y) = 0, u(L, y) = 0, has the form

u(x, y) =

n=1

an sinhn(M y)

L sin

nx

L .

Find the solution of Laplaces equation uxx + uyy = 0 within the rectangle 0 < x < 20 < y < 1, which satisfies the boundary conditions u(x, 1) = 0, u(0, y) = 0, u(2, y) = 0u(x, 0) = 3 sin(x) 2 sin(3x). Write down the complete solution u(x, y).

Solution:

u(x, y) =

n=1

an sinh

n(1 y)

2

sin

nx2

and u(x, 0) = 3 sin(x) 2 sin(3x)

3sin(x) 2 sin(3x) =n=1

an sinh

n2

sin

nx

2

a2 sinh() = 3, a6 sinh(3) = 2, an = 0 for n = 2, 6. Hence,

u(x, y) =3

sinh()sinh[(1 y)] sin(x)

2

sinh(3)sinh[3(1 y)] sin(3x).