Take Home Exam due Thur Mar 13

Take Home Exam 2 due Thur May 8

Conservation laws and constitutive laws

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 = -k grad T, where 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.
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). 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/rho] Which is the correct heat capacity to use in this case?
4. 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 = C dT.

Consider the 1D heat equation u_t= D u_xx with constant D (clearly, since we took it out of the divergence) for t>0.

1. Get rid of that "annoying" D>0 by redefining time t -> Dt. Note that this new "time" , Dt, now has dimension of length²! In terms of that new time the heat equation now reads u_t= u_xx.
2. Go through the separation of variables and analysis sketched in class to deduce that the "separated" solutions of u_t= u_xx in 0 < x < L with u(0,t)=u(L,t)=0 must be e^{-(n Pi/L)2t} sin (n Pi x/L).
3. Solve the 1D heat equation in the rod 0 < x < L with u(0,t)=u(L,t)=0 and u(x,0)=f(x). Find the explicit solution when (a) f(x)=100, (b) f(x)=x(1-x). Do the coefficients A_n converge to zero as n -> infinity? How fast? Do the series converge?
4. Solve u_t= u_xx in 0 < x < L with du/dx(0,t)=u(L,t)=0 and u(x,0)=f(x). You should not have to redo the entire separation of variables procedure, you should be able to jump right away to the proper fundamental (or separated) solution.
5. Solve u_t= u_xx in 0 < x < L with u(0,t)=du/dx(L,t)=0 and u(x,0)=f(x). Again, no need to redo the entire separation of variables.
6. Use separation of variables to find solutions of the heat equation in the plane (i.e. 2D) u_t= u_xx + u_yy.
7. Find fundamental solutions of u_t= u_xx + u_yy in the rectangle 0 < x < L, 0 < y < H with u=0 on the boundary of that rectangle.
8. Use separation of variables to find axisymmetric solutions of the heat equation in the plane (i.e. 2D) u_t= u_xx + u_yy. [Use polar coordinates] What is the name of the differential equation for the radial function? [Very classical material!]
9. Use separation of variables for u_t= Div (K grad u) in a domain V with u=0 on its boundary. Assume that K is a given function of (vector) x and t>0. Is the equation separable? Is it separable if K is a function of x only? Show that the spatial operator Div (K grad) with the given boundary conditions is symmetric (self-adjoint).
10. Consider the linear algebraic operator A (i.e. a matrix!). What can you say about the eigenvectors of A if A is symmetric? What does it mean for the operator to be positive definite? what does that imply about its eigenvalues?
11. Consider the differential operator -d/dx[p(x) d/dx] + q(x) in 0 < x < L. For what class of boundary conditions is this operator symmetric? What does it mean for an operator to be symmetric? [See Sturm-Liouville theory in Haberman and elsewhere]. For those boundary conditions, show that p(x), q(x) >0 imply that the differential operator is also positive definite.
12. Show that separation of variables for rho(x) C(x) u_t= (p(x) u_x)_x -q(x) u (i.e. the heat equation with non-constant diffusivity, density and heat capacity) and source/sink term proportional to the temperature) leads to the differential equation
    -d/dx[p(x) dF/dx] + q(x) F = lambda rho C F

    Together with boundary conditions this is a Sturm-Liouville eigenvalue problem for the eigenvalue lambda.

    There are many other exercises on Fourier series and Sturm-Liouville problems in Haberman.

1. Find a solution to u_t= u_xx in x, t > 0 with u(0,t) = sin (omega t), u_x(0,t)=A sin(omega t), for some real A. Show that this problem is ill-posed.
2. Find a solution to u_t= u_xx in x, t > 0 with u(0,t) = sin (omega t) with u bounded at infinity. Show that this problem is well-posed.
3. Consider u_t= u_xxxx with u(0,t)=u(L,t)=u_xx(0,t)= u_xx(L,t)=0 and u(x,0)=f(x). Is this well-posed for t >0?
4. Consider u_t= u_xx with u->0 as x -> + infinity and u_x+h u = 0 at x=0 for all t>0, where h is a positive constant. Is this well-posed for t >0?
5. Solve u_xx + u_yy = 0 in the rectangle 0 < x < L, 0 < y < H, with u(0,y)=u(L,y)=0 and u(x,0)=f(x), u_y(x,0)=g(x) by separation of variables and superposition. Show that this problem is ill-posed [Hint: perturb f(x) by eps sin (n Pi x/L), with eps very small but n arbitrarily large].

1. Show that the functions sin(n Pi x/L)n where n is an integer, form a complete set in the interval 0 < x < L by showing that these are the eigenfunctions of a certain Sturm-Liouville problem. Expand cos x in terms of those sines in that interval. How fast does your series converge?
2. Consider the expansion of a smooth function f(x) in terms of the functions sin(n Pi x/L) in the interval [0,L]. Use integration by parts to discuss the convergence of the series. What conditions must f(x) satisfy in order to have "spectral" (i.e. faster than algebraic) convergence?
3. Same as 1 and 2 for cos(n Pi x/L). Expand sin(x) in terms of these functions.
4. Same as 1 and 2 for but for sin[(2n+1)Pi x/(2L)]
5. Expand f(x) in a Fourier series in the interval [0,L] for (a) f(x)=x, (b) f(x)=x(L-x).
6. Expand f(x)=x in a Fourier series in the interval [-L/2,L/2]. Compare to previous problem.
7. Consider the Fourier transform of a function f(x) on the real line. In practice, we like our Fourier transform to converge exponentially fast to zero as the wavenumber |k| goes to infinity [why do we like that?]. What properties of f(x) guarantee such fast convergence? [Hint: use the integration by parts argument that we used for Fourier Series].
8. Express the Fourier Transforms of f(x-a), f'(x), x f(x) in terms of that of f(x).
9. Express the Fourier transform of the convolution of f(x) and g(x) (i.e. h(x) = the integral over R of f(y) g(x-y) dy) in terms of the Fourier transforms of f(x) and g(x). Deduce Parseval's theorem.
10. Show that the Fourier transform of an odd function f(x)=-f(-x) is an odd function of the wavenumber k. Show that this implies that the Fourier transform is in fact a sine transform.
11. Show that the Fourier transform of an even function f(x)=f(-x) is an even function of the wavenumber k. Show that this implies that the Fourier transform is in fact a cosine transform.

1. Find the formal solution to the IVP u_t= u_xxx with u(x,0)=f(x) on the real line. [ Linear KdV eqn ]
2. Find the formal solution to the IVP u_t= i u_xx with u(x,0)=f(x) on the real line, where i²=-1. [ Schrodinger's eqn].
3. Find the formal solution to the IVP u_t= -u_xx +u_xxxx with u(x,0)=f(x) on the real line. Is this problem well-posed?
4. Find the formal solution to the IVP u_t= -u_xx -u_xxxx with u(x,0)=f(x) on the real line. [ Linear Kuramoto-Sivashinsky eqn]
5. Find the (formal) solution (in terms of integrals) to the IVP u_t= u_xx with u(x,0)=f(x) and u(0,t)=0 and u bounded as x -> + infinity, in 0 < x < infinity.
6. Find the formal solution to the IVP u_tt= u_xx with u(x,0)=f(x), u_t(x,o)=g(x) on the real line.
7. Find the formal solution to the IVP u_tt= u_xx - a^2 u with u(x,0)=f(x), u_t(x,o)=g(x) on the real line. [This is the Klein-Gordon eqn]
8. Find the formal solution to the IVP u_tt + 2 a ² u_t= u_xx with u(x,0)=f(x), u_t(x,o)=g(x) on the real line. [This is the telegrapher's eqn]
9. Find the formal solution to the IVP u_tt + a² u_xxxx = 0 with u(x,0)=f(x), u_t(x,o)=g(x) on the real line. [This is the beam eqn, that governs small amplitude transversal vibrations of a beam].
10. Find a (double) integral representation for the solutions of u_t= u_xx + u_yy in the plane - infty < x, y < +infty.
11. Find an integral representation for the axisymmetric solutions of u_t= u_xx + u_yy in the plane - infty < x, y < +infty. (hint: superpose exponential solutions, e^{ik_x x} e^{ik_y y} e^{-k2 t}, where k²= k_x² + k_y², then switch to polar coordinates for both x, y and the corresponding wavenumbers k_x, k_y, then determine the conditions under which the double integral is independent of the azimuthal coordinate).
12. Find an integral representation for the axisymmetric solutions of u_t= u_xx + u_yy in the plane - infty < x, y < +infty. (hint: switch to polar coordinates FIRST, then use separation of variables and write the general solution as a superposition of Bessel functions. This is called a Hankel transform).
13. Inspect the previous two formulations of the same problem and deduce integral representations for the Bessel functions.
14. Solve Laplace's equation u_xx + u_yy = 0 in the bottom half-plane (y < 0), with u bounded as y -> -infty, and (a) u(x,0)= sin kx, (b) u(x,0)=f(x). [Zauderer example 5.4]

1. Find a self-similar solution to u_t = u_xxx. Connect to Airy functions, i.e. solutions of the Airy eqn y''=xy
2. Find a self-similar solution to u_t = i u_xx.
3. Find a self-similar solution to u_t = (u²)_xx, with u >= 0 [ Porous medium eqn].
4. Find a self-similar solution to u_t + u u_x = u_xx [ Burgers' eqn].
5. Find a self-similar solution to the heat equation u_t = u_xx, such that INT u dx=0 but INT x u dx = M₁ not zero (INT is the integral over x from -infty to infty).
6. Find a self-similar solution to the heat equation u_t = u_xx, such that INT u dx=INT x u dx = 0 but INT x² u dx = M₂ not zero (INT is the integral over x from -infty to infty).
7. Another look at self-similar solutions of the heat equation: connections with Hermite functions and polynomials
   Do the change of variables u(x,t)=v(eta,t) where eta=x t^-1/2 is the similarity variable.
   • What is the equation for v(eta,t)? Note that this equation is an advection-diffusion equation for the passive scalar v . That means that in the (t,eta) plane, v is given by a balance between advection by the "velocity field" (t,-eta/2) [a stagnation-point type "flow"] and diffusion in the eta direction. Note that the flow (t,-eta/2) is `squeezing' v toward eta=0, while diffusion wants to spread v in the eta direction.
   • Separate variables and show that the v(eta,t) equation admits solutions of the form t^-mu f(eta). Show that the f(eta) equation is a Sturm-Liouville problem. The boundary conditions are that f(eta) -> 0 "sufficiently fast" as eta -> +/- infinity (how fast is that?). Deduce that the eigenvalues mu are positive.
   • Find a solution for mu=1/2. Show that the n-th derivative of that fundamental solution is a solution of the same equation but with mu=(n+1)/2. This give you all the solutions as derivatives of a Gaussian. Therefore each of these solutions has the form of a polynomial times a Gaussian.
   • So, let f(eta)=g(eta) exp(-eta²/4) and derive the equation for g(eta). Change variables: eta= 2 xi, and show that g(xi) satisfies Hermite's equation g'' - 2 xi g' + L g =0, where the primes now denote derivatives with respect to xi, not eta. Relate L to mu. Show that Hermite's equation has polynomial solutions (called Hermite polynomials of course) by looking for a Taylor series solution and showing that it terminates when L=2n (n a positive integer).
8. Find self-similar solution(s) to u_t = u_xx + u_yy.

1. Show that: x delta'(x) + delta(x) = 0
2. Find the generalized derivative of |x| H(x)
3. Discuss the meaning of the generalized function delta(g(x))
4. Expand delta(x^2-a^2)
5. Expand delta(sin(x)))
6. Express [H(x+t)-H(x-t)] delta'(t) in terms of a product of delta functions. [Hint: x and t are independent variables, so the generalized meaning of these expressions is defined in terms of double integrals over x and t multiplied by separate test functions of x and t. H(x) is the Heaviside step function].
7. What is the polar coordinate representation of delta(x) delta(y)?
8. What generalized functions f(x) satisfy x f(x) =1?
9. What generalized functions f(x) satisfy x^2 f(x) =1?
10. What are the Fourier transforms of delta, delta', delta'', 1, x, x^2, ..., H(x), sgn(x), |x|, f(x-a) in terms of the transform of f(x),...
11. Solve (x-x0) [y''-a² y] = 0 with y -> 0 as |x| -> infinity.
12. Solve u_t = u_xxx on the real line with (a) u(x,0)=delta(x), (b) u(x,0)=delta'(x). Compare to self-similar solutions.
13. Solve u_t = i u_xx on the real line with (a) u(x,0)=delta(x), (b) u(x,0)=delta'(x). Compare to self-similar solutions.

1. Derive a formal infinite series solution of the heat equation u_t = u_xx on the real line with u(x,0)=f(x) in terms of the moments of f(x). [Hint: asymptotic expansion of Fourier integral for large t, to all orders]. Note that you obtain a series expansion in terms of self-similar solutions of the heat equation.
2. Find the first few terms in the large time asymptotic expansion of the solution to the heat equation with u(x,0)= max(0,b-|x-a|), a,b >0.
3. Find the first few terms in the large time asymptotic expansion of the solution to the heat equation with u(x,0)= [H(x-2 Pi) - H(x-4 Pi)] sin(x). H(x) is the Heaviside step function.
4. What is the long time behavior of the solution to u_t = -u_xxxx on the real line with u(x,0)=f(x)?
5. What is the long time behavior of the solution to u_t = u_xx-u_xxxx on the real line with u(x,0)=f(x)?
6. What is the long time behavior of the solution to u_t = u_xx- a^2 u on the real line with u(x,0)=f(x)?
   Note that most PDE of diffusive type end up `looking like the heat eqn' for large t.

7. Evaluate the integral of sin(x)/x on the real line by contour integration.
8. Solve u_t = i u_xx on the real line with u(x,0)=cos(k0 x) exp(-x^2/a^2). What is the Fourier transform of u(x,0)? Can you get a closed form solution for u(x,t)? Sketch (or plot) your solution for various times. When |k0 a| << 1 this is a slowly varying wavepacket, its Fourier transform is localized in k-space.
   The linear Schrodinger equation u_t = i u_xx is the "generic" form for dispersive waves just like the heat equation is generic for diffusion. By this we mean that the large t asymptotics is dominated by the stationary k0's s.t. omega'(k0)=0 and omega(k) ~ omega(k0) + omega''(k0) (k-k0)²/2 and this leads to Fourier integrals similar to Schrodinger's.

1. Solve y' + a(t) y = f(t) with y in R, t > t₀ using both the integration factor/variation of constant technique and a Green's function approach.
2. Solve the boundary value problem u''=f(x) with u(0)=A, u(L)=B, using a Green's function.
3. (re-)Derive the infinite space Green's functions for the Heat equation, Poisson's equation and the wave equation in Rⁿ, n=1,2,3. For the wave equation: find the Green's function for n=3 first in two ways, by integration of the Fourier integral derived in class and by reduction to the 1D wave eqn as sketched in class. Obtain the n=2 function by suitable integration of the n=3 function (this is called the "method of descent").
4. Use the method of images to find the Green's function for Poisson's equation in the half plane. Use this Green's function to solve Laplace's equation u_xx + u_yy = 0 in the bottom half-plane (y < 0), with u bounded as y -> -infty, and u(x,0)=f(x).
5. Same question but for Neumann boundary conditions u_y(x,0)=g(x).
6. Find the Green's function for Poisson's equation in the upper quadrant x>0, y>0, with G=0 on the boundary.
7. Find the Green's function for the non-homogeneous heat equation u_t = u_xx + Q(x,t) with u(0,t)=g(t), u(L,t)=h(t) and u(x,0)=f(x). Write the solution to that problem in terms of that Green's function.
8. 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 and check out the references).
9. Find the fundamental matrix of solutions for the "0D wave equation" y''+ a² y = 0 with y and a in R.
10. 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.
11. 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.
12. 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) on the real line. Simplify the integrals as much as possible.
13. Find the solution of G_tt=G_xx in 0 < x < L, t > 0 with G(x,0)=0, G_t(x,0)=delta(x-x₀) and G(0,t)=G(L,t)=0 by the method of images as well as by a sine series expansion.
14. Solve the wave equation u_tt =u_xx 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). Use two methods: Green's function and characteristics.
15. Solve u_tt =u_xx + delta(x-Vt) on the real line, where V is a positive constant. This corresponds to a localized source traveling at constant speed. (There are several ways of approaching this problem.)
16. Find the infinite plane Green's function for the biharmonic operator del², where del is the Laplacian.
17. Can you find a solution to the wave equation u_tt =u_xx with the boundary conditions u(x,0)=f(x), u(x,x)=g(x)? (assume f(0)=g(0)). [Hint: use characteristics]

Sketch the characteristics for each problem and verify that your solution is indeed a solution of the initial value 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 characteristics 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. Sketch the solution. Compare to the inviscid Burgers' solution corresponding to a shock from u=B to u=A.
2. Let u = -2D v_x/v. Derive the equation governing v(x,t).
3. Show that if u is periodic of period L at t=0, it remains so for all time. Expand u(x,t) in a Fourier series and Write the governing equations for the Fourier amplitudes.
4. Write down the Fourier transformed Burgers' equation.