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. When we write that the heat flux q is proportional to minus the gradient of temperature (Fourier's law of heat conduction), 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). In practice, they are determined by controlled, macroscopic experiments, although molecular dynamics computer simulations are starting to be competitive.

1. 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). Compare the results.

2. Using Gibbs' equation for the entropy: T dS = dE + p dV, where T is absolute temperature, S is specific entropy (entropy per unit mass), E is specific energy, p is pressure and V is specific volume (volume per unit mass, i.e. the inverse of the mass density), derive the entropy evolution equation from the equations for the mass density and the energy density. Assuming that the flux of energy consists of both a macroscopic flux and a microscopic flux (but do not assume a Fourier-type constitutive law! keep the microscopic heat flux undetermined). Your equation should have the usual conservation form. Identify the entropy flux and the entropy production term. You could first assume that mass density and heat capacity are constant so Gibbs' eqn is simply T dS = dE.

For diffusion in an infinite 1D domain (real axis), dimensional analysis and conservation suggest solutions of the form u(x,t)=M0 (D t)^(-1/2) f(eta) where eta=x (D t)^(-1/2) is non-dimensional and M0 is the conserved integral of u over x from -infinity to +infinity.

1. Assuming that u and its derivatives go to zero sufficiently fast as |x| -> infinity, show that the integral of (x^n u) over the real axis (call the value of that integral M_n) is conserved for any integer n >0 provided M_i=0 for i=1,...,n-2.
2. Find a self-similar solution u(x,t)= t^a v(xt^b) to the heat equation when M₀=0 but M₁ is not zero. What is a?
3. Find a self-similar solution for u(x,t) such that M₀=M₁=0 but M₂ does not vanish. What is a?
4. Find solutions of u_t= u_xxx in the form of Fourier modes. What would be the general form of solution on the real axis? Are there self-similar solutions of that equation? What can you find out about them?
5. For the heat equation u_t= Del u in Rⁿ, where Del is the Laplacian, find a self similar solution u(x,t)= t^a v(xt^b) with x in Rⁿ, t > 0. Find b from similarity. Find a from conservation, assuming that the integral of u(x,t) over Rⁿ is finite and non-zero. Look for a radial solution v(eta) = w(|eta|) where eta is in Rⁿ and |eta| is its Euclidean norm.
6. Is the equation u_t= D u_xxxx well-posed? D is a constant. Discuss as a function of the sign of D.
7. New : Consider the heat equation u_t= u_xx in x > 0, t > 0, with boundary conditions u_x = - |h| u at x=0, u bounded as x goes to infinity and some initial condition u(x,0) = f(x). Is this problem well-posed? What can you find out about its solution?

Green's function for the heat equation (Integration factors - Variation of parameters - Duhamel's principle)

• What is the general solution to the initial value problem y' + a(t) y = f(t)? with y in R, t > t₀

1. What is the "fundamental matrix of solutions" for a linear system of ODEs y'=A(t) y, where y is in R^N and A is N by N?
2. What is the general solution to the initial value problem y'' + b(t) y' + c(t) y = f(t)?
3. What is the connection between the fundamental matrix of solution for linear ODEs and the fundamental solution of the heat equation?
4. Calculate the integral of exp(-x²) over the entire real axis (R). What is the integral over R of exp(-x²/a) where a > 0? What is the integral over R of exp(-x²/a) exp(b x)? From the latter, deduce the integral over R of xⁿ exp(-x²/a) for any integer n (Hint: differentiate with respect to b or, equivalently, Taylor expand in b).
5. The solution of u_t= u_xx for x in R with u(x,0)=g(x) is given by a certain integral which corresponds to a weighted average of g(x). Assume that g(x) is analytic so you can expand it in a Taylor series. Evaluate the integral term by term by expanding g(x₀) in a Taylor series about x.
6. The solution of y'=ay with y(0) given is y=e ^{a t} y(0). Use this formal approach to find the solution of u_t= u_xx for x in R with u(x,0)=g(x) with a=d²/dx². Use Taylor series to make sense of e ^{a t}. Compare to the previous problem.
7. Write the solution to the inhomogeneous heat equation u_t= u_xx + f(x,t) in terms of the known forcing f(x,t) and initial and boundary conditions u(x,0), u(a,t), u(b,t) using a certain adjoint Green's function.
8. Rewrite the previous solution in terms of the standard Green's function.
9. Find the Green's function for a=0, b=infinity using the method of images.
10. Find the Green's function if u_x (0,t) is known instead of u(0,t) (and b is still infinity).
11. Find the Green's function as an infinite series if u is prescribed at x=a and x=b, both finite.

1. Give some physical contexts for these equations.
2. Solve the 1D Laplace equation in 0 < x < L with boundary conditions u(0)=A, u_x(0) = 0
3. Solve the 2D Laplace equation on 0 < x < L, 0 < y < H, with u(0,y)=A sin (pi y/H) , u_x(0,y) = 0, u(x,0)=u(x,H)=0. What is the solution if u(0,y) is perturbed by very small but highly oscillatory noise (e.g. eps sin (n pi y/H) for a small eps but any large n)? What do you conclude about this problem? Compare to the previous 1D version.
4. Same problem as above but with u(L,y)=0 instead of u_x(0,y) = 0.
5. Back to the heat equation: Solve u_t= u_xx in x > 0 with u(0,t)=eps exp(i omega t), u_x(0,t) = 0 with omega real. (hint: look for u(x,t) = eps exp(i omega t) exp( s x)). Note that this is an initial value problem in the x variable. Is this problem well-posed?
6. Find the Green's function for Poisson's equation in R.
7. Find the Green's function for Poisson's equation in R with G=0 at x=0 and x=L.
8. Solve u_xx = f(x) with u(0)=A and u(L)=B in terms of the Green's function.
9. Find the Green's function for Poisson's equation in the plane.
10. Find the Green's function for Poisson's equation in R³.
11. Find the Green's function for the half plane (e.g. y > 0) with (a) G=0 at y=0, (b) dG/dy=0 at y=0.
12. Find the solution to Poisson's equation in the half plane with u(x,0)=f(x) in term's of the Green's function.
13. Find the Green's function for Poisson in the quarter plane (x > 0, y > 0) with G=0 on x=0, y=0.
14. Find the Green's function for Poisson in a ball of radius R with G=0 on the surface of the ball (Hint: use an image at x'₀ = R² x₀ /|x₀|²).

1. Solve the wave equation in x > 0, t > 0 with u(x,0)=f(x), u_t(x,0)=g(x) and (a) u(0,t)=0, (b) u_x(0,t)=0. Sketch u(x,t) for the case where g(x)=0 and f(x) has compact support in x > 0 (i.e. f(x) is a localized bump of some kind).
2. Find the fundamental matrix of solutions for the "0D wave equation" y''+ a² y = 0 with y and a in R.
3. Find the solution to G''+ a² G = delta(t-t₀), where delta is the Dirac delta function. Compare this solution to the fundamental matrix of solutions found in the previous problem.
4. Write the solution to y''+ a² y = f(t) with y(0)=A, y'(0)=B in terms of (a) the fundamental matrix of solutions, (b) in terms of the function G(t;t₀) found in the preceeding problem.
5. Write the explicit solution (i.e. not in terms of an unspecified "G") to u_tt =u_xx + h(x,t) with u(x,0)=f(x), u_t(x,0)=g(x). Simplify the integrals as much as possible.
6. Use d'Alembert's formula to find the solution of u_tt=u_xx with u(x,0)=0, u_t(x,0)=d'(x-x0) where d'(x) is the derivative of the delta function d(x).
7. Solve u_tt=u_xx + d(x-Vt) with u(x,t)=0, u_t(x,t)=0 as t -> -infinity. (i.e. forget initial conditions and just find the forced response. You can use the Green's function or look for the forced response rightaway.)
8. Solve u_tt=u_xx in x > 0, t > 0 with u(x,0)=0, u_t(x,0)=0 and u(0,t)= h(t) (semi-infinite problem) using Green's functions.
9. Find the solution of G_tt=G_xx in 0 < x < L, t > 0 with G(x,0)=0, G_t(x,0)=d(x-x₀) and G(0,t)=G(L,t)=0 by the method of images as well as by a sine series expansion.

1. We have made sense of the derivative of the Heaviside function H(x). Find the generalized derivative of a function f(x) that is piecewise C¹ (i.e. a function that has jumps at some points but is continuously differentiable elsewhere).
2. There's two good ones in Problem Set 1

Linear and Quasilinear 1st order PDEs

Sketch the characteristics for each problem.

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,o)=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 characterisitcs do not intersect.
8. Solve u_t + u u_x = 0 for t > 0 with u(x < 0 ,0) = u₁, u(x > a, 0) = u₂ and u(0 < x < a,0) linear between u₁ and u₂, the latter being constants. Discuss u₁ < u₂ and u₁ > u₂.
9. Show that the equation r_t + q_x = 0 with q=q(r), which expresses conservation of the x-integral of r(x,t), implies an infinite number of conservation laws. (e.g. Let u = dq/dr, show that (uⁿ)_t + F_x=0 for some F(u) and for all n.) Show that each conservation law leads to a different shock velocity.
10. For r_t + q_x = 0 with q=q(r) (e.g. simple traffic flow model) what is the most general form of q(r) for which the shock velocity is the average of the wave velocities before and after the shock? (i.e. V=(u₁+u₂)/2).
11. Triangular wave: Solve r_t + r r_x = 0 with r(x,0)= bx for 0 < x < a, r(x,0) =0 otherwise.
12. Solve r_t + r r_x = 0 with r(x,0)= A (constant) for 0 < x < a, r(x,0) =0 otherwise.
13. N wave: Solve u_t + u u_x = 0 with u(x,0)= u₀+ c x for a < x < b, u(x,0) = u₀ otherwise, with a < 0 and b, c > 0.
14. Consider u_t + u u_x + a u = 0 where a is a positive constant and u(x,0)=f(x). Under what specific conditions will shocks form?

1. Burgers' equation is the nonlinear advection-diffusion equation u_t + u u_x = D u_xx where D is a positive constant. Find a traveling wave solution of the form u(x,t)=U(x-Vt) for some constant V to be determined with u -> A (constant) as x -> -infty, u -> B as x -> + infty with A > B. Skecth the solution. Compare to the inviscid Burgers' solution corresponding to a shock from B to A.
2. Find a self-similar solution to Burgers' equation for x in R and t > 0.
3. Let u = -2D v_x/v. Derive the equation governing v(x,t).

1. What is the function that is the signed distance to the curve x=0?
2. What is the function that is the signed distance to the curve x² + y²=1 ?
3. Use Huygens' principle to find the signed distance function to the parabola y = x ² . Sketch level curves of that distance function. matlab m-file
4. More on Envelopes: Find the envelope of the normals to the parabola y = x ² . (Such a curve is called the evolute ).
5. Find the distance to the parabola y = x ² but now using the general "method of rays" to solve the problem.
6. Find the distance function to the curve y = cos x using both methods.
7. Geometrical acoustics : Here's a harder problem from Thomas and Finney 5th edition, example 25 on p. 25 (for those of you teaching calculus): The speed of sound in an ideal gas is c = sqrt( g R T) where g and R are constant (the ratio of specific heat and the gas constant, respectively) and T is (absolute) temperature. In a standard atmosphere, T = T_S - a y, where T_S is the surface temperature, a is a positive constant (2 degree Kelvin/1000 ft or 6 Kelvin/km) and y is height above the surface. So temperature and the speed of sound decrease with altitude. This is similar to waves impacting on a beach where the velocity decreases like sqrt(depth), here velocity decreases like sqrt(Temperature), however velocity decreases as waves approach the beach while here velocity increases as sound approaches the surface. If you have a point sound source at a certain height, say y_o at x=0 , find the sound "rays" (or beams) from the eikonal equation. You should find that the sound beams curve upwards and therefore there is a maximum distance at which the sound will reach the surface. Beyond that distance, the sound beams have turned back toward higher elevations away from the surface. The Thomas and Finney problem refers to an asymptotic formula for that maximum horizontal distance.

1. Derive the equations for the 1D motion of a gas down a tube from conservation of mass and momentum. Assume that the pressure p is a function of the density rho only p = C rho ^gamma where C and gamma (ratio of specific heats) are constants (this is the case for isentropic flow of a polytropic gas, i.e. a perfect gas with constant specific heats). Compare the resulting equations to the shallow water equations. Linearize the equations about a uniform state of rest.
2. Find and sketch the "+" characteristics in the dam break problem.
3. [moved to 8 below]
4. Piston withdrawal: solve the shallow water equations for the case where water of uniform depth H0 for x < 0 is held by a gate at x=0. At time t=0, the gate is moved back at constant speed V > 0.
5. What are the shock conditions for the shallow water equations?
6. Deduce the equation of conservation of (kinetic + potential) energy from the shallow water equations. Should this energy be conserved across a shock?
7. Piston pushing: At t=0 at piston starts moving at constant velocity V in water of inital constant depth H0 [Creates a shock].
8. Re-investigate the dam break problem with initial conditions u=0, H=H1 for x < 0 H=H2 for x > 0 with H2 < H1, both constants. [This problem requires a shock].

1. Consider the heat equation with initial condition u(x,0)=A (constant) for a < x < b, zero otherwise. Find an asymptotic expansion for u(x,t) that describes the long time behavior of the solution.
2. Express the solution to u_t+ 2 u_xx+ u_xxxx+u=0 for u(x,0)=f(x), x in R in terms of a Fourier integral. What is the long time asymptotic form of the solution?
3. Calculate the integral over R of exp(i x²) where i²=-1 using contour integration. Justify carefully.
4. Find the exact Fourier integral representation of the solution to Schrodinger's equation: i u_t = - u_xx, with u(x,0)=f(x), on the real line. Find the asymptotic form of the solution for long times assuming that the integral of |f(x)| over the real line is finite.
5. Find the exact Fourier integral representation of the solution to the linear KdV equation: u_t = u_xxx, with u(x,0)=f(x), on the real line. Find the asymptotic form of the solution for long times assuming that the integral of |f(x)| over the real line is finite.
6. What is the Riemann-Lebesgue lemma?
7. Let F(k) be the fourier transform of f(x). Assume that f(x) is differentiable and that the integrals of |f(x)| and |df/dx| are bounded. What is the asympotic behavior of F(k) as k -> infinity?
8. The wave equation in R² admits plane wave solutions of the form u(x,y,t)= exp( i (k x + l y - w t)) for any k, l real. Write the equation in polar coordinates: x =r cos theta, y=r sin theta. Show that it admits cylindrical wave solutions of the form F_n(r) exp(i (n theta - w t)) (same "w" as for the plane waves). Find an integral representation for F_n(r) using Fourier superposition of plane waves [Hint: write the wavevector in polar coordinates as well: k = K cos a, l=K sin a]. Find the asymptotic form of F_n(r) for large r.