Math2065 Partial Differential Equations

MATH 2065 — Introduction to Partial Differential Equations —

Assignment 2

This assignment is due by Thursday, October 24, 4 pm. It should be posted in the locked collection boxes opposite the lift in the Carslaw building on Level 6. Your assignment, with a cover sheet, should be stapled to a manilla folder, on the cover of which you should write the initial of your family name as a LARGE letter.

• Assignments that are turned in late or in the wrong box will not be accepted!

• This Assignment has 4 problems and a total of 50 marks.

1. [10 marks] – Heat Equation

The temperature variation u = u(x,t) in a rod of length π, initially at temperature given by f(x), then positioned with one end in ice and the other end insulated, can be modeled by the heat equation

∂tu = ∂ 2 xu, 0 < x < π , t > 0 ,

with boundary conditions (BCs) u(0, t) = 0 , ∂xu(π,t) = 0 ,

and initial condition (IC) u(x,0) = f(x) .

(a) Show that the general solution satisfying the heat equation and the BCs is given by

u(x,t) =

∞∑ n=0

An sin

( 2n + 1

2 x

) exp

[ − ( 2n + 1

2

)2 t

]

(b) Show that, when fitting the IC, the unknown coefficients An can be determined from f(x) via

An = 2

π

∫ π 0 f(x) sin

( 2n + 1

2 x

) dx.

You may use the result∫ π 0

sin

( 2n + 1

2 x

) sin

( 2m + 1

2 x

) dx =

{ 0 , n 6= m π/2 , n = m

n,m ∈ N .

2. [10 marks] – Laplace’s Equation for an annulus

Find a solution v = v(r,θ) to the following Dirichlet problem for Laplace’s equation in polar coordinates.

∂2rv + 1

r ∂rv +

1

r2 ∂2θv = 0 , 1 < r < 2 ,−π ≤ θ ≤ π ,

v(1,θ) = sin 4θ −cos θ , v(2,θ) = −sin3 θ + 3(cos2 θ)(sin θ) .

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3. [20 marks] – Fourier Series

Consider the function f(x) = x for 0 < x < π.

(a) [5 marks] Extend f(x) to an even function F(x) of period 2π.

i. Sketch F(x) in the range −3π < x < 3π.

ii. Find the Fourier Series of F(x) .

(b) [5 marks] Extend f(x) to an odd function G(x) of period 2π.

i. Sketch G(x) in the range −3π < x < 3π.

ii. Find the Fourier Series of G(x) .

(c) (Gibbs Phenomenon) The American mathematician J. W. Gibbs observed that near points of discon- tinuity of f, the partial sums of the Fourier Series for f may overshoot by approximately 9% of the jump, regardless of the number of terms. Consider

f(x) =

{ −1 , −π < x < 0 , +1 , 0 < x < π .

i. [2 marks] Show that the partial sums are given by

f2n−1(x) = 4

π

[ sin(x) +

1

3 sin(3x) + . . . +

sin ((2n−1)x) (2n−1)

] ii. [2 marks] Sketch the partial sums f11,f51 and the original function f.

iii. [6 marks] Assume that for each partial sum the maximum occurs at x = π/(2n). Show that

lim n→∞

f2n−1(π/(2n)) ≈ 1.18 .

4. [10 marks] – Fourier Transform

Let u(x,t) solve the partial differential equation

∂tu = ∂ 2 xu−u, x ∈ R, t > 0 ,

with initial condition u(x,0) = v(x) for x ∈ R. Furthermore, denote by û = F{u} the Fourier transform of u with respect to x. Show that û satisfies the equation

∂tû = −(k2 + 1)û

with initial condition û(k,0) = v̂(k) = F{v}. Find the solution û of the equation in Fourier space and thereby the solution u of the original equation.

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