Assignment 1 Final

Assignment 1
Anthony Cummins
September 2020
Contents
1 Question 1 1
2 Questions 2 and 3 3
i
Question 1
We consider an object occupying the simply-connected domain Ω ⊂ R3 .An

exothermic chemical reaction inside the object produces heat at the steady in
time and uniform in space rate s ≥ 0. The thermal conductivity of the object,
k > 0 is also uniform in space. The surface of the object, S, is kept at a con-
stant temperature, T0 . The object has a unit outward normal vector n, defined
everywhere.
We take the object to be in thermal equilibrium and in steady-state. Figure 1.1

shows this.
Figure 1.1: A sketch of Ω
We will designate the flow of heat through the boundary φ, the heat Q and the
rate of creation of heat R. We will also designate the temperature function of
the object T (x, t). Then, the global conservation law is given by
1
dQ
= −φ + R = 0 (1.1)
dt
with φ = −k∇T (x, t) and R = s
dQ
= k∇T (x, t) + s = 0 (1.2)
dT
We model the flow of heat with Fourier’s law, as given in the slides of Lecture
1, slide 30. It is clear that in order for thermal equilibrium to be achieved,
there must be a flow of heat across the boundary ∂Ω. If there were not, the
temperature in the domain would increase without bound. We here remark that
∇T (x, t) should be positive for the problem to be physically correct. Integrating
over the domain and applying the divergence theorem,
ZZ ZZZ
(−k∇T (x, t))dS − sdV = 0 (1.3)
Ω Ω
ZZZ ZZZ
∇ · (−k∇T (x, t))dV − sdV = 0 (1.4)
Ω Ω
yielding
−k∆T (x, t) = s (1.5)
2
Questions 2 and 3
Question 2
Equation 1.5 is essentially Poisson’s equation. The boundary-value problem is
−k∆T (x, t) = s, x∈Ω (2.1)

T (x) = T0 , x ∈ ∂Ω (2.2)
Question 3
3a
Equations 2.1 represent the BVP with Dirichlet boundary conditions. Since the
sign is the same in front of all derivatives (this can be seen if we unpack the
vector notation), the equation is elliptic, and linear with constant coefficients.
3b
We can show that there exists at most one solution to this BVP by the Dirichlet
Uniqueness Theorem. This is Theorem 2.3.4 in the book ”Numerical Methods
for Partial Differential Equations”, to which all further references to theorems
shall refer.
Dirichlet Uniqueness Theorem

Let Ω ⊂ R3 be a bounded region with boundary ∂Ω, and suppose that u ∈
C 2 (Ω) ∩ C 1 (∂Ω) satisfies
−∆u = f (x), x∈Ω

u = g(x), x ∈ ∂Ω
Then there exists at most one such solution. This theorem is applicable to the
BVP as derived in equation 2.1.
3
3c
Non-Negativity Theorem (Theorem 2.3.6)
We can determine that the solution to the BVP is non-negative by use of the
non-negativity theorem. Let Ω ⊂ R3 be a bounded region with boundary ∂Ω,
and suppose that u ∈ C 3 (Ω) ∩ C 2 (∂Ω) satisfies
−∆u ≥ 0, x∈Ω
u = 0, x ∈ ∂Ω
Then u(x) ≥ 0 on Ω. This is also true when u = f, x ∈ ∂Ω and f ≥ 0. If

it were less than 0, the minimum principle would be violated. Therefore, it is
applicable to equation 2.1, and its solution is not negative.
Minimum Principle Theorem (Theorem 2.3.2)

Let Ω be an open bounded domain with boundary ∂Ω and closure Ω̄ = Ω ∪ ∂Ω
Suppose that u ∈ C 3 (Ω) ∩ C 2 (Ω) is superharmonic on Ω. Then, u(x) takes its
minimum at the boundary ∂Ω.
We can therefore say that this BVP given in equation 2.1 is positive and has its
minimum on the boundary of the object.
3d
Stability of the Solution
If the boundary temperature changed from T0 to T1 , the temperature in the
object would change by not more than |T0 − T1 |. This was shown in Lecture 2,
slide 35. It is also a consequence of the conservation of energy.
Let Ω ⊂ R3 be a bounded region with boundary ∂Ω, and suppose that u1 , u2 ∈

C 2 (Ω) ∩ C 1 (∂Ω) satisfy
−∆u1 = f,x ∈ Ω
u1 = g1 ,x ∈ ∂Ω
−∆u2 = f,x ∈ Ω
u2 = g2 ,x ∈ ∂Ω
Then, |u1 (x) − u2 (x)| ≤ maxx∈∂Ω | g1 (x) − g2 (x) for all x ∈ Ω
4
3e
Formulating the BVP with a Neumann Boundary Condition
Suppose that, instead of prescribing the temperature on the boundary of the
object, we prescribe the density of heat flux instead. Our BVP then becomes:
−k∆T (x, t) = s, x∈Ω (2.3)

−k(∇T (x)) · n = j, x ∈ ∂Ω (2.4)
We must satisfy a certain compatibility relation, given in Theorem 2.2.1. Let

us denote the boundary ∂Ω as Γ. Then, the equation to be obeyed is:
Z Z
− jdΓ = sdΩ (2.5)
Γ Ω
We also remark that the solution is no longer unique, and only determined up
to an additive constant.
If the flow of heat were set to 0 at the boundary, then the object would, in
essence, become a perfect insulator. No heat would be able to escape the object.
Mathematically, equation 1.1 would become:
dQ
=R (2.6)
dt
with R = s, and the internal temperature would increase without bound.

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Assignment 1

We consider an object occupying the simply-connected domain Ω ⊂ R3 .An

We take the object to be in thermal equilibrium and in steady-state. Figure 1.1

Figure 1.1: A sketch of Ω

−k∆T (x, t) = s (1.5)

−k∆T (x, t) = s, x∈Ω (2.1)

Dirichlet Uniqueness Theorem

−∆u = f (x), x∈Ω

Then u(x) ≥ 0 on Ω. This is also true when u = f, x ∈ ∂Ω and f ≥ 0. If

Minimum Principle Theorem (Theorem 2.3.2)

Let Ω ⊂ R3 be a bounded region with boundary ∂Ω, and suppose that u1 , u2 ∈

Then, |u1 (x) − u2 (x)| ≤ maxx∈∂Ω | g1 (x) − g2 (x) for all x ∈ Ω

−k∆T (x, t) = s, x∈Ω (2.3)

We must satisfy a certain compatibility relation, given in Theorem 2.2.1. Let

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