代做MATHS 3023/4123/7107 Partial Differential Equations & Waves Written Assignment 2代做Python语言

MATHS 3023/4123/7107 Partial Differential Equations & Waves

Written Assignment 2

Due: 5.00 pm Monday 26 August

1. Use separation of variables to solve the partial differential equation

subject to boundary conditions

ux(0, y) = ux(π, y) = 0 for 0 < y < 1 and

u(x, 0) = 0, u(x, 1) = f(x) for 0 < x < π.

You may assume the eigenvalues are real, but should consider the three possible cases of negative, zero and positive eigenvalues.

2. Consider the regular Sturm–Liouville eigenproblem for 0 ≤ x ≤ 1 ,

u'' + λu = 0,

subject to boundary conditions

u(0) = 0 , u(1) + u ′ (1) = 0 .

(a) Verify the following properties:

i. There are an infinite number of eigenvalues, with no largest eigenvalue.

Hint: Consider the three cases λ = −k2, λ = 0 and λ = k2. Sketch or plot the graphs of y = tan k and y = −k for the case λ = k2.

ii. The nth eigenfunction has n − 1 zeros in 0 < x < 1 .

(b) Write down the orthogonality condition for the eigenfunctions.

3. Consider conduction of heat along a nonuniform. rod of length L. Let u(x, t) denote the temperature at position x along the rod and time t. The temperature satisfies the PDE

where the density ρ, specific heat c, thermal conductivity κ and coefficient α are all positive continuous functions of x. Suppose the temperature at one end is zero, while the other is insulated. Then the boundary conditions are

u(0, t) = 0   and   ux(L, t) = 0.                                         (3)

The initial temperature is

u(x, 0) = f(x),                    (4)

where f(x) is a piecewise smooth function for 0 ≤ x ≤ L.

(a) Let u(x, t) = F(x)G(t). Use separation of variables to obtain two odes, one for F(x) and the other for G(t). Show that the ode for F(x) can be written in Sturm–Liouville form.

where λ is the separation constant and p(x), q(x) and r(x) are functions that you should write in terms of ρ, c, κ and α.

(b) Write down the corresponding boundary conditions for F(x).

(c) Let λn and Fn(x) denote the eigenvalues and corresponding eigenfunctions for n = 1, 2, . . . Use the Rayleigh quotient to show that the eigenvalues λn are positive.

(d) Write down a solution of the pde (2) subject to boundary conditions (3) and initial condition (3) in terms of λn and Fn(x). Use the properties of Sturm–Liouville prob-lems to derive any formulae needed to compute any coefficients. You do not need to determine λn or Fn(x) for this question.

(e) What happens to the temperature as t → ∞? Justify your answer.