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Vrije Universiteit Amsterdam Code 400163

Partial Differential Equations Assignment 2

Hand in your answers to the class on Monday 15 March.Please note: late submissions will not be marked.

Good luck!

Please work tidy & justify your answers.

1. With reference to the picture below study the heat conduction on a thin circular ring.

Figure 1: As x runs from (L , L], the black point traces the circle of radius 1. x represents thearclength along the ring. Thus despite the two dimensional shape, we deal with a one dimensionalproblem (see also page 152 of Asmar).

The temperature on the ring is governed by the heat equation ut = c2

uxx, L < x L, withinitial condition

u(x, 0) = f(x) , L < x L

and periodic boundary conditions

u(L, t) = u(L, t) , ux(L, t) = ux(L, t) .

(a) Set L = ; using separation of variables derive the differential equations

T kc2T = 0X kX = 0 , X() = X() ; X() = X()

(b) Argue that a positive choice of k leads only to trivial solutions for X. Show that fork = 2 nontrivial solutions arise only for = n = 0, 1, 2, 3... and find them.

(c) Conclude that

u(x, t) = a0 +n=1

(an cos(nx) + bn sin(nx))en2c2t .

Write explicitly the formulas for a0, an, bn.

2. Solve the heat problem in a square plate with dimensions a = b = 1 and given the initialtemperature distribution u(x,y, 0) = 100 Celsius. Assume that the edges are kept at zero

temperature and that the constant c in the equation is equal to 1.

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Vrije Universiteit Amsterdam Code 400163

(a) Approximate how long it will take for the maximum temperature in the plate to drop to50 Celsius?

(b) What is the answer if c = 2? What is your conclusion about the speed of heat transferas a function of c?

3. Consider a thin disk of radius R and use polar coordinates with origin at the centre of the disk.Then the temperature u(r, ) satisfies Laplaces equation

2u(r, ) =1

r

r

r

u

r

+

1

r22u

2= 0

Assume the temperature is given on the boundary of the disk by u(R, ) = f(). We imposetwo additional conditions in order to obtain a unique solution: (i) we assume that the solutionis bounded; (ii) we assume periodicity conditions:

u(r,) = u(r, ) andu

(r,) =

u

(r, )

motivated by the fact that (r,) and (r, ) are different polar coordinates of the same point.Show that a solution by separation of variables, in which u(r, ) = F(r)G(), leads to theordinary differential equations

r2F(r) + rF(r) F(r) = 0 and G() + G() = 0

subject to the conditions that G() = G(), and G() = G(). By considering the cases = 0, < 0 and > 0 separately for the problem for G(), show that the eigenvalues aren = n

2 with corresponding eigenfunctions

Gn

() = an

cos n + bn

sin n

for n = 0, 1, 2, . . ., where an and bn are constants.Show that the corresponding solutions for F(r) take the form Fn(r) = Anr

n, for n = 0, 1, 2, . . .By applying the boundary conditions, show that

a0 =1

2

f() d, an =1

Rn

f()cos(n) d

and

bn =1

Rn

f()sin(n) d

Suppose now that the disk has radius equal to 3 and that f() = 10+2

3

for .Given that

x sin nx dx = x

ncos nx +

1

n2sin nx

and x3 sin nx dx =

x3

n+

6x

n3

cos nx +

3x2

n2

6

n4

sin nx

determine the steady state temperature distribution for the disk.

4. The Laplacian of a function u(x, y) is defined as 2u

x2+

2uy2

. Let x = r cos and y = r sin and

compute the solution of the equation 2u = 0 defined in the region {(x, y) R2 , x2+y2 1}

and such that u(x, y) = y on the circle x2

+ y2

= 1.

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Vrije Universiteit Amsterdam Code 400163

5. For each of the following differential equations, apply the indicated change of variables toobtain a Bessel equation of appropriate order and hence determine the solution of the originalequation:

(a) 9x2y + 9xy +

4x2

3 16

y = 0; z = 2x1

3 .

(b) 36x2y 12xy +

36x2 + 7

y = 0; u = yx23

.

END

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