If you have never seen these derivations, Chap 1 in Haberman is pretty good. Any book on continuum mechanics will have some of that (Fluid mechanics, solid mechanics, transport theory...). The extra little bit that we did was to use conservation to derive that Q(x,t,n)=q(x,t). n. This is a nice result due to Cauchy (yes, that Cauchy again). It's often plopped down without justification.

When we write that the heat flux q = -κ grad T, where κ > 0 is a diffusivity and T is temperature (Fourier's law of heat conduction, Fick's law of diffusion), we use what's called a constitutive law. Such laws can, in theory, be deduced from a lower level (i.e. microscopic) description of matter (i.e. kinetic theory). The second law of thermodynamics (entropy of a closed system increases) also provides constraints (see (4) below). In practice, they are determined by controlled, macroscopic experiments, although molecular dynamics computer simulations are starting to be competitive.

1. Describe in words the meaning of Q(x,t,n) and specify its units in the case of (1) conservation of mass, (2) conservation of energy. We derived the form of Q(x,t,n) in 2D using a small triangle, can you do it in 3D using a small tetrahedron?
2. Derive conservation of mass for a gas flowing down a tube of cross-sectional area A(x) where x is the direction along the axis of the tube. Do this in two ways (1) from first principles assuming that the variables (mass density and gas velocity) are uniform over a given cross-section and (2) by integration of the general 3D equations over the cross-sectional area A(x), this time with no uniformity assumptions. Compare the results.
3. Derive the heat equation, i.e. conservation of (internal) energy in a medium at rest (macroscopically speaking, at the microscopic level the molecules are bouncing around very fast even when it seems that the fluid, gas or solid is not moving). Energy density e(x,t) is equal to the mass density &rho times the heat capacity C times the temperature T . In a gas, there is an important distinction between C_V and C_P, the heat capacity for constant specific volume V and constant pressure P, respectively. [specific volume = volume of unit mass = 1/ρ] Which is the correct heat capacity to use in this case?

1. What are the units of v and κ ?
2. What is the general solution of u_t + V u_x=0? (where u(x,t) is the "unknown" function, V is a constant velocity and -∞ < x < + ∞ and t > 0).
3. What is the general solution of u_t + A u_x + B u_y =0? (where u(x,y,t) is the "unknown" function, A and B are constant velocity components and -∞ < x, y < + ∞ and t > 0).

• We encountered successives derivatives of a Gaussian in doing so: (∂/∂x)ⁿ (π a)^-1/2 e^-x2/a (with a = 4t). Show that these derivatives have the form of a polynomial of degree n times a Gaussian. What are those polynomials?
• Give an asymptotic form for the solution of u_t = u_xx on the real line, when u(x,0)= U (constant) for a < x < b, and 0 otherwise.
• Give an asymptotic form for the solution of u_t = u_xx on the real line, when u(x,0)= sin x for 3 &pi < x < 5π, and 0 otherwise.

Many of the following exercises were essentially solved/sketched in class, but maybe a bit too fast/sloppy for you? Slow down, digest, fill in the details, lay it out for yourself in a satisfactory way.

IVP = Initial Value Problem, BVP = Boundary Value Problem

1. What is the basic Fourier solution of u_t = u_xx+u_yy in the real plane?
2. What is the fundamental solution of u_t = u_xx+u_yy in the real plane?
3. What is the general solution of u_t = u_xx+u_yy with initial conditions u(x,y,0) = f(x,y) (known) in the real plane?
4. What is the solution of u_t = u_xx+u_yy with initial conditions u(x,y,0) = δ(x-x₀)δ(x-y₀), with x₀, y₀ some arbitrary real constants, in the real plane?
5. Pure BVP: What is the ("steady") solution to u_t = κ u_xx, with κ > 0, on x > 0, when the boundary condition is u(0,t) = 2A cos ω t ? (A, ω real and "known"). SKETCH the solution. You may want to solve the simpler complex :-) problem first: u(0,t)= e ^{-i ω t} and show that the solution to that is e ^{-i ω t} exp[ (i sgn(&omega) -1)(|ω|/(2&kappa))^1/2 x ]
   How deep do the boundary fluctuations penetrate into the medium?
6. What is the ("steady") solution to u_t = κ u_xx, with κ > 0, on x > 0, when the boundary condition is u(0,t) = 2A cos ω t, u_x(0,t)= 0 ?
7. Show that the previous problem is ill-posed. What does that mean? Conclude that we need to have "bounded at infinity" as a boundary condition (actually any limit on exponential rate of growth as x → ∞ would be OK mathematically speaking, e.g. something like |u| < exp(4 x) would be mathematically acceptable).
8. What is the ("steady") solution to u_t = κ u_xx, with κ > 0, on x > 0, when the boundary condition is u(0,t) = f(t) ? (real and "known" but NOT necessarily simple harmonic like 2A cos ω t ).
9. What is the thermal diffusivity (κ) of soil? of water (∼ 1.5 x 10^-7 m²/sec)? How deep do the daily fluctuations of temperatures propagate into the soil/water?
10. If you have studied fluid dynamics: what is a "Stokes layer"?
    [Answer: it's the solution to problem 5 but for the fluid velocity above an oscillating plane, oscillating temperature or oscillating velocity, same math!].

Some other `pure Boundary Value Problems'
(i.e. boundary forcing has been applied for a long time, transient response to initial conditions is long gone and we're just looking for the forced response, often called the `steady response' although it is not necessarily `steady'):

1. Find the forced response for u_t = κ u_xx in x > 0 with u_x(0,t) = B cos ω t. (Neumann boundary condition).
2. Find the forced response for u_t = κ u_xx in x > 0 with u = α u_x at (0,t) where α is some positive real constant (Fourier-Robin boundary condition a.k.a. `of the 3rd kind').
3. Find the forced response for u_t = κ u_xx in x > 0 with u = -α u_x at (0,t) where α is some positive real constant.
   Is this problem well-posed? What is the physical interpretation of that boundary condition? (think of u as temperature, so (- κ u_x) is the heat flux in the x direction)
   [Hint: this is kind of like `the rich get richer and the poor get poorer' economic model. Previous problem is: `the rich get poorer and the poor get richer'].

Et voilà!

These are exact results, but the ideas (deforming contours to go through saddle points in steepest descent directions) apply to many more problems as asymptotic methods. That's the method of steepest descents (see e.g. Bender and Orszag Chapter 6). The method of stationary phase is a special case of steepest descent.

REALITY CHECK: prepare the following problems to present to the class. You can talk to each other, read any book you would like but YOU have to be ready to present and explain and show that you understand the solutions, not just that you are able to regurtitate someone else's solution. You should not talk to other faculty or postdocs. YOU have to do the work/research on your own or with your peer group. You must acknowledge your sources (e.g. say what you found in what book if you use results from that book).

1. Consider u_t = κ u_xx in x > 0 with u = +/- α u_x at (0,t) where α > 0, and u(x,0)=f(x) known, and u(x,t) well-behaved as x → ∞. Discuss the boundary condition physically for both + and -. Are there simple Fourier solutions? Is this problem well-posed? What is the general solution?
2. Consider u_t = κ u_xx in x > 0 with u(x,0)= &delta(x-x₀) and u(x,t) → 0 as x → ∞. What is the long time asymptotics of u(x,t) for (i) u(0,t)= 0, (ii) u_x(0,t)=0. How long is long? Same question for u(x,0)=cos(x-x₀) when |x-x₀| < π, 0 otherwise. (Some thinking/discussion of relation between x₀ and π is needed).
3. Consider u_xx+u_yy = λ u in R², where λ is a real constant. What are simple Fourier/exponential solutions of this equation? Discuss λ < 0 and > 0. What is the general solution to this equation when λ < 0? Superpose Fourier solutions to form an axisymmetric solution u(x,y) ≡ u₀(r) where r=(x²+y²)^1/2. Do you know what that solution is? What is its behavior for small r? How small is small? What is its behavior for large r? How large is large? [Extra: How should you superpose Fourier modes to obtain a solution of the form u(x,y) ≡ u_m(r) e ^{i m θ} where θ is the usual polar angle and m is an integer?]

• Find the forced response (∼ "particular solution") to u_t = a u + &delta(t-t₀) where u=u(t) only (ODE) with a and t₀ real constants.
• Find the general solution of u_t = a u + &delta(t-t₀) with u(0)= u₀ (known). Discuss t₀ < 0 and > 0.
• Find the general solution u(t) of u_t = a u + F(t) with u(0)= u₀, with the forcing F(t) and the initial condition u₀ "known".
• Find the forced response to u_t = κ u_xx + A cos(5x) + B cos(6x) in -∞ < x < ∞ where A, B are real constants.
• Find u(x,t) if u_t = κ u_xx + A cos(5x) + B cos(6x) with u(x,0) = 0, in -∞ < x < ∞ , t > 0, where A, B are real constants.
• Find the forced response to u_t = κ u_xx + A sin(ω t) cos(k x) in -∞ < x < ∞ where A, ω and k are real constants.
• Find the forced response to u_t = κ u_xx + A δ'(x-x₀) in -∞ < x < ∞ where A, ω and x₀ are real constants.

1. Find one or more self-similar solution for u_t = u_xxx ( - ∞ < x < ∞, t ≥ 0). Try to use both a direct approach and a Fourier approach. Note: you may not find the final solution in closed form but you should process the integrals as much as possible.
2. Find a self-similar solution for u_t + u u_x= ν u_xx ( - ∞ < x < ∞, t ≥ 0)

• Derive the sound speed for an ideal gas. How does it depend on temperature?

1. complete the calculation of the 2D Green's function.
2. Solve the wave equation in 1D and 3D with a moving point source: δ(x-Vt) [δ(y)δ(z)] with V constant. Discuss.

1. Solve x u_x + y u_y = u with u(x,1)=f(x) known. Plot the characteristics and give the solution in parametric and explicit form.
2. Solve y u_x - sin(x) u_y = 0 with (a) u(x,0)=f(x) and (b) u(0,y)=g(y). Discuss both cases carefully.
3. Solve u_t + x u_x = 0 for x > 0, t > 0, with u(x,0)=f(x). What is u(1,10)?
4. Solve u_t + x u_x = 0 for x > 0, t > 0, with u(0,t)=g(t). What is u(1,10)?
5. Solve u_t +sqrt(x) u_x = 0 for x > 0, t > 0,with u(x,0)=f(x). What is u(1,10)?
6. Solve u_t + (1+x) u_x = 0 for x > 0, t > 0, with u(x,0)=f(x) and u(0,t)=g(t).
7. Consider u_t + u u_x = 0 for t > 0 with u(x,0)= sin(x). Find the x,t domain where characteristics do not intersect.
8. Derive the traffic flow equation. What is a simple but realistic model for the car flux q(ρ) where ρ is the car density?
9. Show that the traffic flow equation with q=q(ρ) can be reduced to the inviscid Burgers equation.
10. Derive the shock speed condition. Where does it come from?
11. Show that the shock speed for the full traffic flow equation and for the inviscid Burgers eqn are the same iff q=q(ρ) is quadratic in ρ.
12. Jake has proposed the following traffic flow model: ρ_t + C ρ_x = 0 with C=C(ρ)=U (1-ρ/ρ_M) and studies the IVP: ρ(x,0)= ρ_M for x < 0, ρ(x,0)= ρ_M/2 for 0 < x < a, ρ(x,0)= 0 for x > a. Interpret the model and the initial conditions. Solve the problem by the method of characteristics. Analyze and discuss carefully.
13. Solve the same problem for a simple but realistic q=q(ρ).
14. Solve the traffic flow problem for initial conditions: ρ(x,0)=0 for x < 0, ρ(x,0)= x ρ_M/2 for 0< x < 1, ρ(x,0)= ρ_M/2 for x > 1,

• Burgers' equation is the nonlinear advection-diffusion equation u_t + u u_x = ν u_xx where ν is a positive constant. Find a traveling wave solution of the form u(x,t)=F(x-Vt) for some constant V to be determined with u → u₁ (constant) as x → -∞, u → u₂ as x → ∞. Sketch the solution. Show that there is no such solution when u₁ < u₂. Compare to the inviscid Burgers' solution corresponding to a shock from u=u₁ to u=u₂. What is the width of the transition (i.e. shock) region as a function of ν (assumed small)?
• Cole-Hopf transformation: Let u = -2ν φ_x/φ in Burgers' equation. Derive the equation governing φ(x,t).