APM3701 Assignment 2 - DUE 3 September 2024

APM3701
ASSIGNMENT 2 2024
UNIQUE NO.
DUE DATE: 3 SEPTEMBER 2024

, APM3701

Assignment 2 2024

Due Date: 3 September 2024

Partial Differential Equations

QUESTION 1

To demonstrate the uniqueness of the solution u(x,t)u(x, t)u(x,t) for the temperature
distribution in a 2-dimensional domain Ω\OmegaΩ using the Maximum-Minimum
Principle for the heat equation, we can proceed as follows:

Step-by-Step Solution:

1. Understanding the Heat Equation and Boundary Conditions:
o The heat equation in a 2-dimensional domain Ω\OmegaΩ is given by:
∂u∂t=ΔuinΩ,\frac{\partial u}{\partial t} = \Delta u \quad \text{in} \quad
\Omega,∂t∂u=ΔuinΩ, where Δu=∂2u∂x2+∂2u∂y2\Delta u = \frac{\partial^2
u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}Δu=∂x2∂2u+∂y2∂2u is the
Laplacian of uuu.
o The boundary condition is given as u(x,t)=f(x)u(x, t) = f(x)u(x,t)=f(x) for all
x∈∂Ωx \in \partial\Omegax∈∂Ω, where f(x)f(x)f(x) is a prescribed function
on the boundary ∂Ω\partial\Omega∂Ω.
2. Initial Condition:
o Initially, the temperature in the domain Ω\OmegaΩ is given by
u(x,0)=u0(x)u(x, 0) = u_0(x)u(x,0)=u0(x).
3. Maximum-Minimum Principle:
o The Maximum-Minimum Principle states that if u(x,t)u(x, t)u(x,t) is a
solution to the heat equation in a domain Ω\OmegaΩ, then the maximum
and minimum values of u(x,t)u(x, t)u(x,t) in the closure of Ω×[0,T]\Omega
\times [0, T]Ω×[0,T] occur on the parabolic boundary (either at t=0t = 0t=0,
or on ∂Ω\partial\Omega∂Ω).
4. Uniqueness Proof: