MATH 314: Fourier Methods & PDEs Spring 2023 Homework Set 2
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MATH 314: Fourier Methods & PDEs
Spring 2023
Homework Set 2
Due: Thursday, January 19 (11:59pm)
Continuity; Heat and Diffusion Equations, Equilibria
· Course information is on Canvas: http://canvas.sfu.ca
· Textbook references are to:
[G] Mark S. Gockenbach
“Partial Differential Equations: Analytical and Numerical Methods” (2nd edition).
Reading:
· [G] Chapter 1: Introduction — Classification of DEs, review of terminology
· [G] Chapter 2 — S2.1: Heat flow in a bar, Fourier’s law, ICs & BCs, steady-state solutions, diffusion; S2.4: Conservation laws, advection
I. Textbook Questions:
。[G] Section 2.1: (pp.18—20)
Physical interpretation and equilibria for the diffusion equation
II. Additional Questions:
Also submit solutions to the following problems:
1. Continuous function:
Let f : [a, b] →e R be a continuous function on some interval [a, b] c R (where a < b).
(a) For x0 ∈ [a, b], give the ε-δ definition of the statement “f is continuous at x0”. Use this to prove that if f (x0 ) > 0, then there exists h > 0 so that f (x) > 0 for all x ∈ (x0 _ h, x0 + h).
(b) Assume that for every subinterval [x1 , x2] c [a, b] (with a < x1 < x2 < b) we have
生2 f (x) dx = 0.
生1
Prove that f (x) = 0 on [a, b].
Write down an extension of this result to functions on R3 .
2. Heat equation with circular symmetry:
In this problem we will derive the heat equation with circular symmetry.
(This will be equivalent to working in 2d polar coordinates (r, θ) with no θ-dependence; or in 3d cylindrical coordinates (r, θ, z) in which quantities are independent of θ and z .)
Assume that the temperature is circularly symmetric: u = u(r, t), where r = ^x2 + y2 .
[Similarly, the thermal energy density (energy per unit volume) is e = e(r, t), the heat flux density (heat per unit area per unit time) is q = q(r, t), and any material properties depend at most on r .]
Consider a thin lamina: a circular annulus a < r < b, with constant height h (in the z-direction).
Tˆ
(a) The heat energy density is related to the temperature by e(r, t) = ρ(r)cp(r) ╱u(r, t) _T0、, where T0 is some reference temperature at which the total internal energy is E0 .
Show that the total heat energy in the annular lamina is
E (t) = E0 + 2πh b e(r, t)r dr = E + 2πh b ρ(r)cp(r)u(r, t)r dr
a a
(for some constant E), and hence find an expression for dE/dt.
(b) Any heat flow is assumed to be purely in the radial direction, so the heat flux density is q = q (where = cos θ i + sin θ j is the radial unit vector).
Show that the flow of heat energy per unit time out of the region at r = b is 2πh b q(b, t), and obtain a similar result at r = a.
Hence find the total heat flow per unit time into the annular region (as an integral over the region a < r < b).
(c) We saw in class that in one dimension (i.e. x-dependence only), the continuity equation — the differential equation expressing conservation of heat — in the absence of external sources and sinks, is ∂e/∂t + ∂q/∂x = 0.
Find the analogous expression in the circularly symmetric case (dependence only on r). (d) Fourier’s law of heat conduction, in general (for isotropic materials), is that the heat
flux density is proportional to the temperature gradient:
q = _K1u,
where K is the thermal conductivity. In particular, if u depends only on r, then heat flows are only in the radial direction, and
q = q . = _K . 1u = _K ∂u
Assuming that all material properties (ρ, cp and K) are spatially homogeneous, use the results from parts (a)– (c) and Fourier’s law to derive the circularly symmetric heat equation without sources or sinks,
= ╱r 、,
where κ is the thermal diffusivity.
(e) Find the equilibrium temperature distribution u(r) inside the circular annulus a < r < b if the outer radius is at temperature T2 and the inner radius is at temperature T1 .
3. Equilibrium solutions:
Determine which of the following PDEs have equilibrium (steady-state) solutions, that is, time-independent solutions (satisfying ut = 0):
In each case, either find the equilibrium solution and sketch its graph (by hand or with the help of Matlab or another software environment), or explain why it doesn’t exist.
(a) = ╱x 、 + 1
(b) utt + 3ut = u北北 + t
(c) =
with u(1, t) = 0 and u(2, t) = 0.
with u(_1, t) = 0 and u(1, t) = 0.
u(0, t) = 0 and u(2, t) = cos t.
(d) ut = u北北 + u北 with u北 (0, t) = 1 and u(1, t) = 0.
4. Non-existence of equilibria:
Consider the heat equation ut = κu北北 on the interval 0 < x < é, with Neumann boundary conditions u北 (0, t) = _1 and u北 (é, t) = 0.
Give a physical interpretation of the PDE and these BCs; and use it to explain why this system will not have an equilibrium. Explain why, instead, you expect the temperature u(x, t) to increase as time t increases.
[You do not need to do any calculations for this problem! Just give a clear explanation based on the physical interpretation; we will be studying in some depth what is going on here mathematically.]
5. Inhomogeneous PDE and BCs:
Consider the heat equation with a source
ut = u北北 + x(1 _ x) with u(0, t) = 0 and u北 (1, t) = _1.
(a) Give a physical interpretation of the boundary conditions.
(b) Find the equilibrium solution, denoted by p(x), say; and plot it using Matlab. [You may wish to use the given code hw2code .m as an example; this code plots the equilibrium solutions from Examples 2.2 and 2.3 of the textbook, as in Figure 2.3.]
(c) Let u(x, t) = p(x) + v(x, t), where v(x, t) is the deviation from the equilibrium. Show that the function v(x, t) satisfies the associated homogeneous PDE and BCs
vt = v北北 with v(0, t) = v北 (1, t) = 0.
2023-01-30