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MATH704 Linear Partial Differential Equations. Assignment 2

Due on Monday 30 March 2020, 4 pm

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MATH704. Linear Partial Differential Equations. 2020.

Assignment 2

Question 1. The distribution of heat in a metal rod is modelled by the following nonhomogeneous

heat equation:

ut = 2uxx + 3 + 4 cos x

Solve the equation if the initial temperature u(x, 0) = 5 + 6 cos x + 7 cos 2x, and the ends of

the rod are insulated:

ux(0, t) = ux(π, t) = 0, t > 0.

Question 2. Solve the following initial-boundary value problem modelling the vibration of a

string with length L = 1 and fixed ends.

utt = uxx + xu(x, 0) = 3π3sin(πx)ut(x, 0) = x

u(0, t) = u(1, t) = 0, t > 0.

Question 3. Solve the following initial-boundary value problem modelling the vibration of a

string with length L = 2π and fixed ends.

utt = 4uxx + 2π − xu(x, 0) = 7 sinut(x, 0) = 0

u(0, t) = u(2π, t) = 0, t > 0.

Question 4. The following non-homogeneous Laplace equation (Poisson equation) models

the distribution of electrical potential when an outside charge is present:

uxx + uyy = π − x

Solve the equation subject to the following boundary conditions:

u(x, 0) = π − x, u(x, π) = 0,

u(0, y) = u(π, y) = 0.

Question 5. The temperature u(x, t) of a narrow metal rod of length L = π with a heat

source is modelled by the following non-homogeneous heat equation:

ut = uxx + x.

Solve the equation if the initial temperature u(x, 0) = 2π + 4x, and the ends are kept at

constant temperatures as follows:

u(0, t) = 2π, u(π, t) = 5π, t > 0.

Hint: Transform the non-homogeneous boundary conditions into homogeneous ones for

w = u − c1x − c2.2

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