Pick-up Time: Monday, May 10, 11

NAME:
Pick-up Time: Monday, May 10, 11:00am (in class)
Return Time: Wednesday, May 12, 9:30am (in class)
18.306: Take–Home Quiz # 2
You are NOT allowed to communicate with anyone,
other than the instructor, about this quiz while you are
taking it. So, work on the test by yourself!
You can use any books or notes, but you are NOT allowed
to use calculators, or any software such as Mathematica
or Maple for example, in order to solve the
problems.
Read the problems CAREFULLY, especially the
given HINT(S). Write CONCISE answers, yet JUS­
TIFY all your steps. If needed, try to use all sheets
of paper given to you with the test. Cross out what
is not meant to be part of your final answer.
Total # of points: 70.
DO ALL 3 PROBLEMS!

I. (20 pts) [This problem has parts 1, 2, 3.] Consider
the usual conservation law in one space dimension x,
− < x < +, with “density” π = π(x, t) and
“flux” q = q(x, t) = Q(π) + µ · πxx, where
∂
Q(π) = π l+1 ; l > 0, ∂ > 0, µ > 0 (l : pure number).
l + 1
1.(2 pts) Derive a PDE for π(x, t) (show all your steps
in detail).

2.(6 pts) Consider the stretching transformations (ST)
x˜ = π
x, t˜ = t, ˜ = ω π and find relations among
the real ω, κ and θ so that the PDE of part (1) is invariant
under these ST ( > 0). Construct the form of
a similarity solution for π(x, t). When would you expect
such a solution to hold? (You are NOT asked to
derive or solve any equation for the similarity solution
at this point.)

3.(a)(5 pts) Now try to solve the PDE of part (1)
by use of dimensional arguments that ONLY involve
the dimensional quantities π, x, t, ∂ and µ. Define
suitable non-dimensional quantities of the form π/A(t)
and x/B(t); give A(t) and B(t). Construct the form
of a similarity solution and compare with your result
of part (2) above. Derive an ODE for the similarity
solution for arbitrary number l. (You are NOT asked
to solve the ODE or provide any conditions at this
point.)

l .
3(b)(7 pts) In part 3(a) above, take = 2 Derive
a 2nd-order ODE for the similarity solution of 3(a).
Justify and give all conditions needed for the solution
of this ODE. (You are NOT asked to find the solution.)

II.(20 pts) [This problem has parts 1, 2.] Weakly
nonlinear shallow-water waves are governed by the PDE
ut + ω· uux + (κ 2 /3) uxxx = 0, − < x < +, where
ω and κ are positive constants.
1. (8 pts) Notice that this PDE admits constant solutions,
u = u0 = const. We seek an approximate
solution of the form u = u0 + u1(x, t) where |u1| is
“sufficiently small.” Derive a linear PDE for u1. Assume
a solution in the form u1(x, t) = g(kx−t) where
g(β) has period 2λ and cannot be a constant. (Here,
β kx − t.) Show that g is of the form e ±i(kx−t)
and find the dispersion relation = W (k). Give the
group and phase velocities.

2. (12 pts) Consider a Fourier superposition of the
waves (“wave packet”) for the g found in part (1) above
and describe it approximately for sufficiently long x and
t. Give conditions in terms of the parameters ω and κ
for the validity of your approximation. Define the caustic
and provide a graphical and analytical description
of the wavepacket near the caustic.

III.(30 pts) [This problem has parts 1, 2.]
1.(15 pts) The two-dimensional (2D) Green’s function
G(x, y; x , y ) for the Laplacian 2 in the rectangle =
{0 < x < a, 0 < y < b} is defined to be continuous
in and satisfy the PDE
Gxx+Gyy = α(x−x ) α(y−y ), 0 < x < a, 0 < y < b,
along with the homogeneous condition G = 0 on
(: boundary of ). Find G in terms of a suitable
series expansion that involves sinusoidal functions in x
via using separation of variables.

2. (15 pts) Consider the PDE
uxx + uyy + 2ux + u = µ · π(u), (x, y) in ,
where u = u(x, y) satisfies the boundary conditions
u(0, y) = u(x, 0) = 0, u(a, y) = A = const., and
u(x, b) = B = const.; π(u) is a nonlinear, bounded
function of u and µ is a positive constant.
2(a) (5 pts) Set u = δ(x, y) · w(x, y) and find a function
δ(x, y) such that the PDE for the new dependent
variable w = w(x, y) does NOT contain any wx and
wy terms.

2(b) (7 pts) Convert the PDE for w that you found
in 2(a) to an integral equation. For this purpose, define
and find a suitable Green’s function; also, define and
find a suitable solution w0( x, y) of the homogeneous
PDE for w that results by letting π 0. Hint: You
may use the result of part (1) for the Green function.
Also, write w0 = w0,1 + w0,2 where w0,1 and w0,2 each
satisfies suitable conditions on ; determine w0,1 and
w0,2 separately in terms of series expansions.

2(c) (3 pts) Solve approximately the integral equation
of part 2(b) for |µ| ≡ 1; your solution should
include terms O(µ) but no higher-order terms.

NAME:
Pick-up Time:
Return Time:
18.306 Take–Home Quiz # 1
Monday, March 15, 2004
You have 4 hours to do the test.
You are NOT allowed to communicate with anyone,
other than the instructor, about this quiz while you are
taking it. So, work on the test by yourself!
You can use any books or notes, but you are NOT allowed
to use calculators, or any software such as Mathematica
or Maple for example, in order to solve the
problems.
Read the problems CAREFULLY, especially the
given HINT(S). Justify your answers. Try to use
all sheets of paper given to you with the test. Cross
out what is not meant to be part of your final answer.
Total # of points: 30.
DO ALL 3 PROBLEMS!

I. (8 pts) In materials science, the evolution of surfaces
of materials is often studied by use of PDE. In the case
of evaporation dynamics, where material evaporates to
the environment, the surface height h = h(x, y, t) relative
to the xy-plane of reference satisfies a complicated
PDE. For simplicity, it is assumed that axisymmetry
holds, i.e., h depends only on the polar distance
r = ≤ x 2 + y 2 0 and time t > 0, h = h(r, t); h
satisfies the simplified PDE
ht = (A/r)hr,
where A is a positive constant.
Solve this PDE, and discuss why and how any initial
smooth, axisymmetric bump h(r, 0) = H(r), where
H(r) is monotonically decreasing to 0, finally shrinks
to zero as t + →.
Remark: You are asked to find
h = h(r, t) explicitly as function of r and t and discuss
implications of the solution.

II.(11 pts) Consider the nonlinear PDE
−1 1 − u2
ut = uxx + (ux) 2 , (1)
u
where u = u(x, t), x is a space variable, t is time and
is a positive constant ( > 0).
(a) (6 pts) Convert the given PDE (1) to a linear
PDE. What is that linear PDE? Show all the steps of
derivation in detail. Hint: Find a suitable transformation
w = (u). You are not asked to solve the PDE
for this question (a).

(Continuation of Prob. II.)
(b) (5 pts) Using the result of part (a), solve PDE
(1) for −→ < x < + → and t > 0 with the initial
condition
2
u(x, t = 0) = e −x , −→ < x < +→,
and the boundary condition u(x, t) 0 sufficiently
fast as |x| →. Remark: Beware. You are asked
to give u(x, t) explicitly as function of x and t.

III. (11 pts) Consider a function u = u(x, t), defined
for 0 x 1 and t 0, that satisfies the PDE
ut − (x − u) 2 ux = 0. (2)
(a) (7 pts) Suppose that u(x, t) satisfies the given
PDE (2) with the conditions
u(x, 0) = 0, and u(1, t) = 1.
Find and sketch the characteristics for this problem.
By using the characteristics, derive an explicit formula
for the solution u(x, t) as a function of x and t. Discuss
the behavior of u(x, t) at the point (x, t) = (1, 0).

(Continuation of Prob. III.)
(b) (4 pts) Now consider a u(x, t) that satisfies the
PDE (2), ut − (u − x) 2 ux = 0, but with the conditions
u(x, 0) = 0, and u(1, t) = 1
2 .
Find and sketch the characteristics for this problem.
By using the characteristics, derive an explicit formula
for the solution u(x, t) as a function of x and t. Discuss
the behavior of u(x, t) at the point (x, t) = (1, 0).

18.306: Advanced Partial Differential Equations with Applications
M.I.T. Department of Mathematics Spring 2004
Practice Set 5 Handed out: Friday, 05/07/04
Due: NEVER
You are NOT required to return solutions. None of these problems will be given in Quiz 2. The
description of the references given in parentheses can be found in the Bibliography for 18.306.
41. (a) Show that if the relation = W (k) for a wave gives a phase velocity vph(k) equal to the
group velocity vg (k), the wave cannot be called dispersive.
(b) Explain in detail why a linear wave with W (k) = ak + b, a, b →= 0: constants, cannot be
dispersive although vph →= vg . Are any discontinuities in data allowed to propagate in this case?
Why?
42. The nonlinear Schrödinger equation (NLSE), which governs the non-relativistic motion of each
atom in an interacting gas, reads
it(x, t) = −xx(x, t) + a |(x, t)| 2 (x, t) − a(x, t), −≈ < x, t < +≈; a = const. > 0.
(a) Substitute (x, t) = A(x, t)e iα(x,t) , where A and β are real functions, and derive two coupled
PDE for A and β.
(b) From (a),solve for β by assuming that A is slowly varying in both space and time (i.e., take
A const.) Derive the dispersion relation = (k) for A 1, where k βx and −βt .
Justify that your solution is a dispersive wave. What is the group velocity?
43. (Levine, Chap. 5, Prob. 5, p. 62.) The gravity-driven motion of a thin fluid film on an inclined
plane is governed approximately by the PDE: ht + (g sin ζ/λ)h 2 hx = 0. In this equation, h =
h(x, t) is the fluid thickness (height from incline) as a function of the coordinate x along the
plane and the time t (t > 0), ζ is the angle of the incline, g is the gravity acceleration, and λ
is the viscosity.
(a) Show that the general solution to the given PDE is given implicitly by the equation
h = F (x − g sin ζ h 2 t/λ), where F is an arbitrary function.
(b) Obtain a particular solution to the PDE that satisfies the initial condition h(x, 0) = κx, for
0 < x < L, and h(x, 0) = 0 otherwise. From this particular solution, derive the approximate
formula h(x, t) (g sin ζ/λ) −1/2 (x/t) 1/2 for t ≡ ≈ and x > 0, which has a similarity form.
Now re-derive the formula in (b) via a similarity solution in 2 alternative ways:
(c) Show that the given PDE is invariant under the stretching transformations x
t
= x,
= t and h = h, where > 0 is an arbitrary parameter. Hence, try a similarity solution
h = T (t) g(θ) where θ = x/t. Find T (t) and g(θ) via substitution into the PDE.
(d) Look for solutions h = c x t β . Find the constants c, α and π by substitution into the PDE.
44. Consider the usual wave equation, utt − c 2 uxx = 0.
1

x
(a) Dimensional arguments suggest a similarity solution of the form u = f(), where = ct .
Find and solve the ODE for f().
x
(b) Substitute u = xβ f() into the PDE, where = and π is real. Find an ODE for f().
ct
45. (Debnath, Prob. 8.14.7, pp. 327, 328.) The Prandtl-Blasius problem for a flat plate in a
uniformly moving fluid is described by the PDE system
uux + vuy = λuyy , ux + vy = 0,
where u and v are the fluid velocity components in the x and y directions and λ is a positive
constant.
(a) Introduce the stream function ξ by (u, v) = (ξy , −ξx) and derive the PDE satisfied by ξ.
(b) In (a) try a similarity solution of form ξ = x 1/2 g(θ) where θ = y x −1/2 . Derive an ODE
for g.
(c) Can you justify in any way, mathematical and/or physical, the form of solution suggested
in (b)?
46. (Levine, Chap. 5, Prob. 3, pp. 60, 61.) Consider the PDE ut = λ(urr + r −1 ur − r −2 u) with
variable coefficients, where r > 0, t > 0, and λ > 0 is a constant; u = u(r, t) is an axisymmetric
function in two space dimensions.
(a) Apply stretching transformations (ST) and find a relation among the exponents so that
the PDE is invariant under ST.
(b) By virtue of (a), try a similarity solution of the form u = r λ f(θ). Define the similarity
variable θ. Derive an ODE for f(θ) for any admissible value of real ψ.
(c) In (b), choose a negative ψ so that the ODE for f contains only f and f terms and no f
term. Solve the resulting ODE by applying the conditions that u(0, t): finite and lim t0 + (ru) =
k: given.
(d) Define w(r, t) ur +r −1 u. Derive a PDE for w. Try a solution of this PDE in the similarity
form w = t −1 g(θ), where θ is the similarity variable of part (b) above. Compare the results
for g with the result of part (c).
47. (a) (Drazin & Johnson, Prob. Q2.11, p. 35.) Consider the nonlinear Schrödinger equation,
iut + uxx + λ u|u| 2 = 0, where u = u(x, t) and λ > 0, and apply the ST ˜x = x, t˜ = t and
ũ = β u. Find relations among κ, α and π so that the PDE is invariant under ST. Set κ = 1:
Try a similarity solution u = t λ f(θ). Give ψ and θ and derive an ODE for f.
(b) (Drazin & Johnson, Prob. Q2.8, p. 34.) Determine µ and ω so that u(x, t) = t µ f(xt ) is a
(similarity) solution of Burgers’ equation,
ut
+ uux = uxx. Find and solve the ODE for f with the condition that f ≡ 0 as x ≡ ≈ and
f(0) = −2ν−1/2 .
48. (Carrier & Pearson, Prob. 16.5.1, p. 313.) The following problem arises in the flow of a viscous
heat-conducting fluid along a channel, and illustrates a situation in which it is useful to change
2

the scale of the dependent, as well as the independent, variable. Let u = u(x, y) satisfy the
PDE φuyy − (1 − y 2 )ux + 4y 2 = 0 in the region x > 0, −1 < y < 1, where 0 < φ ∼ 1; u obeys
the conditions u(0, y) = 0 and u(x, ±1) = 1.
(a) Find the solution u = u0 for φ = 0 that satisfies the first one of the conditions, and explain
why this is not acceptable.
− − (b) Introduce a boundary layer near y = −1 by setting ψ = φ (1 + y). Show that u0 = O(φ )
near the boundary layer. This suggests writing u = φ−ξ(x, ψ) inside the boundary layer.
Derive a PDE for ξ and determine κ. Give suitable conditions for ξ.
3

18.306: Advanced Partial Differential Equations with Applications
M.I.T. Department of Mathematics Spring 2004
Practice Set 4 Handed out: Monday, 04/05/04
Due: NEVER
You are NOT required to return solutions. Some of these problems may be given in Quiz 2. The
description of the references given in parentheses can be found in the Bibliography for 18.306.
31. (Carrier & Pearson, § 1.7, pp. 17, 18.) This problem gives a systematic way to treat a
nonhomogeneous PDE with nonhomogeneous boundary (in x) and initial (in t) conditions.
Describe how you would solve the diffusion equation with a source term, ut = λuxx + w(x, t),
where u = u(x, t), with the conditions u(x, 0) = f(x), u(0, t) = g(t), u(l, t) = h(t), where
0 < x < l and t > 0. Hint: Try to convert the given problem to one with nonhomogeneous
PDE satisfying homogeneous boundary conditions. For instance, you may introduce ξ by
ξ = u − {g(t) + (x/l)[h(t) − g(t)]}. Show that ξ satisfies homogeneous boundary conditions
and derive a new PDE for ξ. Try a Fourier series ξ(x, t) =
n an(t)αn(x). What is αn(x) and
how would you find an ODE for an(t)? What is the condition for an(t)?
32. For given > 0, derive an equation that governs the eigenvalues of the problem
2
→ u + 2 u = u; u(r, 0) = u(a, π) = 0, u(r, ν) = 0; r, π : polar coordinates
where 0 r < a and 0 < π < ν. What are the corresponding eigenfunctions?
33. This problem is an application of eigenvalue theory to statistical mechanics (Bose-Einstein
condensation). Derive an equation for the energy levels per atom in a gas of pairwise interacting
atoms enclosed in an one-dimensional (1D) box. The motion of each atom is governed by the
nonlinear Schrödinger equation, it(x, t) = −xx(x, t) + a|(x, t)| 2 (x, t) − ψ(x, t), 0 < x <
−iEt ξ(x)
L, with the boundary conditions (0, t) = (L, t) = 0 (a, ψ > 0). Hint: Set = e
and describe E so that the given PDE and conditions are satisfied.
34. (Carrier & Pearson, § 3.11, pp. 52, 53.) This problem aims to familiarize you with a uniqueness
proof. Give a proof that the one-dimensional (1D) wave equation with a space-dependent
velocity c and a forcing term f, utt − c(x) 2 uxx = f(x, t), has a unique solution u(x, t) in
0 < x < l, t > 0 if u satisfies
u(0, t) = a(t), u(l, t) = b(t), u(x, 0) = h(x), ut(x, 0) = p(x).
l
dx [π2 t /c2 + π2 1
Hint: Define the integral I(t)
= 0
x], where π is any solution of the given PDE
2
with f 0 that satisfies π(0, t) = π( l, t) = 0. Show that dI/dt = 0. Take π = u − v where u
and v satisfy the given PDE and all given conditions. Infer that π 0. What is the physical
meaning of I in the case of a stretched string?
35. Find explicitly the Green function G for outgoing waves in 3D that is specified by the solution
2
of the form w = G(r; r )e it of the PDE wtt − c 2 → w = −β(r − r ) e it , where r = (x, y, z) is
the position vector in the 3D infinite space.
1

36. Define and find the Green function G(x, t; x , t odinger equation
) for the time-dependent Schr¨
it(x, t) + xx(x, t) = V (x, t)(x, t), with the initial condition (x, 0) = a(x). Convert the
given initial-value problem into an integral equation. Show the connection with the Green
function defined in class for the diffusion equation.
37. Define the Green function G(x, y; x y
) for the nonhomogeneous PDE uxxxx − uyy = ζ(x, y) in
the strip (−≡ < x < ≡, 0 < y < 1), where u satisfies prescribed boundary conditions at y = 0
1 1
and y = 1, and is bounded as |x| ≡. Calculate G(0, ; 0, ) in terms of a (convergent) single
2 2
series. Hint: G(x, y; x , y ) should satisfy corresponding homogeneous conditions (why?). Use
Fourier Transform in x and the Residue Theorem.
38. (Carrier & Pearson, Prob. 9.3.9, p. 147.) A two-dimensional (2D) Green’s function G(x, y; x , y )
for the Laplacian → 2 in a finite region is defined by
Gxx + Gyy = β(x − x ) β(y − y ); G = 0 on boundary.
Show, by using separation of variables, that for the case of the square region 0 < x < l, 0 <
y < l, G is given by
sin(nνx
/l) sin(nνx/l)
G = {cosh[nν(l − y − y )/l] − cosh[nν(l − |y − y |)/l]}.
nν sinh(nν)
n=1
39. (Levine, Chap. 5, Prob. 4, pp. 61, 62.) Show that the first-order nonlinear PDE uy + uux = 0,
where x and y are space variables, has a general solution of the implicit form u = F (x − uy),
where F is arbitrary. On the other hand, dimensional considerations suggest a similarity form
u = H(x/y). Find H explicitly through an ODE in δ = x/y and verify its agreement with the
exact solution of the PDE when F () = and y ≤ 1.
40. (Levine, Chap. 5, Prob. 8, pp. 64–66.) The PDE Ct = k(C n Cx)x, where C = C(x, t), k > 0
and n: integer, is relevant to certain models of thermal conduction with a nonlinear heat flux,
and of filtration which concerns gas flows through porous media. Consider a solution C to this
PDE that, at t = 0, vanishes everywhere except in some neighborhood of the origin x = 0,
and also obeys the relation C(x, t) dx = Q = const., which often expresses conservation of
−
total mass.
1 1
(a) Verify that the variables (kt/Q n+2
2 ) n+2 C and = x(Qnkt) −
are dimensionless.
1
(b) Find particular solutions of the form C = [Q 2 (kt) −1 ] n+2 f (). What is the ODE and
constraints (conditions) satisfied by f ? Is f even or odd? Find unique particular solutions for
the following cases: (i) n = 1, and (ii) n = −1. Notice the difference in the nature of the two
solutions. How do these solutions behave as t ≡? Hint: Apply C 0 when |x|: suff. large
( C: continuous), and the integral constraint involving Q in order to determine the integration
constants of the ODE.
(c) Show that, if C(x, t) is a solution of the given PDE, then (αω −2 ) 1/n C(ωx, αt) is also a
solution, where ω and α are constants and α > 0.
2

18.306: Advanced Partial Differential Equations with Applications
M.I.T. Department of Mathematics Spring 2004
Practice Set 3 Handed out: Wednesday, 03/17/04
Due: NEVER
You are NOT required to return solutions. Some of these problems may be given in Quiz 2. The
description of the references given in parentheses can be found in the Bibliography for 18.306.
21. (Carrier & Pearson, Prob. 13.10.3, p. 255.) The equations of the two-dimensional isentropic
compressible flow (characterized by entropy S=const.) are
ψt + (ψu)x + (ψv)y = 0, ψ(ut + uxu + uy v) = −c 2 ψx, ψ(vt + vxu + vy v) = −c 2 ψy ,
where c = c(ψ). Discuss the characteristic surfaces in the (x, y, t) space.
22. (Carrier & Pearson, Prob. 13.4.8, p. 239.) The equations of the shallow water theory read as
ut + uux + gwx = 0, wt + [u(w + d)]x = 0,
where u is the horizontal velocity (taken as independent of vertical position), w is the wave
height, d is the depth, and g is the gravity acceleration. The second of these equations is already
in the desired form of a conservation law. To interpret these equations as conservation of mass
and momentum, obtain another desired conservation law via multiplying the first equation by
(w + d), the second by u, and adding. Can you obtain conditions on the possible dicontinuity
of u and w (called a hydraulic jump)?
23. (Levine, Chap. 24, Prob. 1, pp. 277, 278.) Show that the wave equation with a mixed derivative,
utt
− c2uxx = −2ωuxt, where ω and c are positive constants, admits the periodic in t solution
u(x, t) = A sin(t − kx). What is the relation between k and (called a dispersion relation)?
Find an approximate formula for u(x, t) for sufficiently small ωx/c and ω/.
24. (a) [Levine, Chap. 4, Prob. 10, p. 49.] Solve the PDE x 2 uxx + 1 u = utt by seeking particular
4
traveling-wave type solutions of the form u(x, t) = a(x) f(b(x)−t), which allow for x-dependent
wave speeds; f is an arbitrary function, and a and b satisfy appropriate ODE. (You may wish
to take x > 0 and t > 0.)
(b) [Carrier & Pearson, Prob. 3.4.6, p. 44.] A semi-infinite string in x > 0 has a density that
2
varies with x in such a way that the resulting PDE for its displacement u(x, t) is utt = c uxx
where c 2 = c(x) 2 = (A + Bx) 4 ; A and B are known constants. Solve the PDE via the
transformation u = (A + Bx)ξ(x, t) followed by the change of variable = (x/A)(A + Bx) −1 .
(c) [Levine, Chap. 4, Prob. 11, p. 50.] Show that a particular solution of the nonlinear PDE
ut
= [K(u)ux]x can be obtained in the form u = f(x − vt), which describes a wave form that
travels at constant speed v, where f satisfies a first-order ODE.
25. Find the (unbounded in t) temperature distribution T (x, t) for 0 < x < L and t > 0 given that
Tt
= λTxx and T (0, t) = 0, T (L, t) = ωt and T (x, 0) = 0.
1

26. (Exercise on Fourier transforms.) Find a single-integral representation for uy (0, 0) if u(x, y)
2 2 −ay 2 satisfies the PDE ≡ u − u = f(x, y) for − < x < and y > 0, with f(x, y) = e (b +
x2 ) −1 , and the conditions u(x, 0) = 0 and u(x, ) = 0; a, b and are positive constants.
27. (Carrier & Pearson, Prob. 15.4.6, p. 296.) The steady-state (Tt = 0) temperature T (x, y)
inside the strip (− < x < , 0 < y < L) satisfies the Laplace equation, Txx + Tyy = 0, with
the given conditions T (x, L) = 0 and T (x, 0) = f(x) (where f(x) 0 as |x| ). Use the
Fourier transform in x in order to find T (x, y). Invert the transform by summing residues, and
verify that the result agrees with that obtained by the direct Fourier series expansion method.
28. (Levine, Chap. 35, Prob. 11, p. 596.) In class I said that the Fourier transform (FT) is generally
applied in variables that take values in the range (−, +). However, one may also apply
FT to a variable in (0, ) when the corresponding function is taken to be zero in (−, 0).
Consider the following example.
A low-frequency electromagnetic field in a conducting medium satisfies approximately the
diffusion equation. Assume that the (scalar) component E(x, t) of a (vector) electric field in a
conducting half space x > 0 (say, sea water) satisfies the PDE
Et = (c 2 /ν)Exx, x > 0, t > 0,
along with the conditions E(0, t) = f(t), E(x, 0) = 0 and E(x , t) 0; ν is the conductivity
and c is the speed of light. By applying FT in the time t and contour integration, find
an integral representation for E(x, t) in terms of f(t ), 0 < t < t, and then derive the energy
loss formula
c t t ν
f(t ) )f(t
Q(t) dx ν|E(x, t)| 2 = dt dt
) 3/2 .
0 2 α 0 0 (2t − t − t
29. By applying FT, solve the Laplace equation uxx + uyy = 0 for − < x < and y > 0,
subject to the initial conditions (Cauchy data) of the form u(x, 0) = f(x), uy(x, 0) = g(x). Is
the solution u regular (differentiable) for y > 0? Comment on the convergence of the Fourier
integral. Argue that a small error in the data can cause a large error in the solution for y > 0.
(Therefore, you should conclude that the problem is ill-posed in the sense of Hadamard.)
30. (Levine, Chap. 24, Prob. 2, p. 278.) The pendulum-like oscillatory motions of a fluid column
inside a U-shaped tube in a vertical plane can be described approximately (in linearized form)
by the PDE
ztt − λ(zrrt + r −1 zrt) + (2g/L)z = 0, 0 < r < R, t > 0,
where z = z(r, t) is the displacement relative to the equilibrium level as a function of the
radial distance r from the tube axis and the time t > 0, and g, L and λ denote the gravity
acceleration, length of the fluid column and kinematic viscocity of the fluid. Describe the
appropriate solution z(r, t) subject to the conditions
z(r, 0) = −H, zt(r, 0) = 0, zt(R, t) = 0,
of which the last one expresses a vanishing fluid velocity at the tube wall, r = R.
2

18.306: Advanced Partial Differential Equations with Applications
M.I.T. Department of Mathematics Spring 2004
Practice Set 2 Handed out: Wednesday, 02/25/04
Due: NEVER
You are NOT required to return solutions. Some of these problems may be given in Quiz 1. The
description of the references given in parentheses can be found in the Bibliography for 18.306.
11. (Debnath, Prob. 3.6.17, p. 132:) Solve the following equations:
d(y−u) d(u−x)
(a) (y + u)ux + (x + u)uy = x + y. Hint: Use d(x+y+u) = −(y−u) = −(u−x) etc. (why?)
2(x+y+u)
4
(b) xu(u2 + xy)ux − yu(u2 + xy)uy = x .
12. (Debnath, Prob. 3.6.20, p. 132:) Find the general solution of the equation y ux − 2xy uy = 2xu.
In particular, determine the solution that satisfies u(0, y) = y 3 . [Ans. General solution:
yu = F (x 2 + y).]
2
13. [Levine, Chap. 11, Prob. 7, pp. 126, 127.] (a) Confirm that u(x, y; c1, c2) = c1x + c1y + c2
represents a “complete integral” of the PDE p = q2 (p = ux, q = uy ), where c1 and c2 are
arbitrary constants.
(b) For (a), consider the data x(s) = 0, y(s) = s, and u(s) = s 2 , where s is a real parameter.
Determine c1 and c2 in terms of s, and then give the solution u as a function of x and y only.
14. (Levine, Chap. 11, p. 126.) Describe the solution to the PDE p − q = a pq (p = ux, q = uy ),
where a is a real constant.
15. In materials science, the evolution of surfaces of materials is often studied by use of PDE. In
the case of evaporation dynamics, where material evaporates to the environment, the surface
height h = h(x, y, t) relative to the xy-plane of reference satisfies a complicated 2nd-order PDE.
For simplicity, it is assumed that axisymmetry holds, i.e., h depends only on the polar distance
r = 2
→ x2 + y 0 and time t > 0, h = h(r, t); it is also assumed that h satisfies the simplified
PDE ht = (A/r)hr , where A is a positive constant. Show that any initial smooth, axisymmetric
bump h(r, 0) = H(r), where H(r) is monotonically decreasing to 0, finally shrinks to zero as
t +≡.[Ans: By use of characteristics, h(r, t) = H( → 2At + r2 ).]
16. (Levine, Chap. 9, Prob. 11, p. 106.) The necessary and sufficient condition that µ(x, y) is an
integrating factor of the ODE
is the following PDE satisfied by µ (why?):
a(x, y) dx + b(x, y) dy = 0 (1)
(µa) (µb)
= . (2)
y x
1

Take a(x, y) = 3xy + 2y 2 , b(x, y) = 3xy + 2x 2 and show that an integral of (2) is
∂(x, y, µ) = µ → x + y = const.,
and thus obtain the general solution to (1). Note: The knowledge of only one complete
integral of (2) suffices to solve (1).
17. (Levine, Chap. 9, Prob. 12, pp. 106, 107.) Consider the quasi-linear PDE
uux + uy = 0, −≡ < x < ≡, y > 0,
with the initial data u(x, 0) = sechx, −≡ < x < ≡. Show that a differentiable (“regular”)
solution exists so long as y < y , where y is the minimum value of the function
cosh 2 s , s > 0.
sinh s
Describe (with a sketch) the solution u(x, y) for y > y . Recall: sech x = 1/ cosh x.
18. Use characteristics in order to solve the system of PDE
ut + vx = 0, vt + ux = 0
with the initial data u(x, 0) = f (x) and v(x, 0) = g(x).
19. (Debnath, Prob. 6.11.12, p. 257.) For one-dimensional anisentropic fluid flow, the Euler equations
of motion read as
pt + upx + c 2 ux = 0,
1
ut + uux + + p
x = 0,
St + uSx = 0,
where p = f (, S) is the pressure, u is the velocity, is the fluid density, and S is the entropy.
Show that this system of PDE has three families of characteristics given by
where c 2 (p/)x =const.
dx dx dx
C0 : = u, C+ : = u + c, C− : dt = u − c,
dt dt
20. A uniform solid sphere with radius a is kept initially (t = 0) at constant temperature T = T0 .
At all later times t > 0 the surface temperature of the sphere, at r = a, is maintained at
T = 0, where r is the spherical coordinate with origin at the center of the sphere. Given that
the temperature distribution inside the sphere satisfies the diffusion equation, Tt = ≥ 2 T ,
where > 0, define and calculate the average temperature Tave inside the sphere for t > 0.
Derive an approximate formula for Tave for sufficiently long times t.
2

18.306: Advanced Partial Differential Equations with Applications
M.I.T. Department of Mathematics Spring 2004
Practice Set 1 Handed out: Wednesday, 02/11/04
Due: NEVER
You are NOT required to return solutions for this set. However, these problems may be given in
Quiz 1. Problems 1-3 serve as a review of ordinary differential equations (ODE). The description of
the references given in parentheses can be found in the Bibliography for 18.306, with the exception
of the reference for Problem 3 of this set.
1. Find all solutions of the initial value problem (IVP)
2
x u (x) + xu (x) − u(x) = 0, u(1) = 0, u (1) = A; x > 1.
Hint: This ODE is the equidimensional equation (see Handout 3).
2. (a) Find the solution of the IVP
u (t) + q(t) u(t) 2 + p(t) u(t) = 0, u(0) = u0 > 0; t 0,
where p(t) and q(t) are known functions of t.
Hint: See Handout 3; you may use v = 1/u as a new dependent variable.
(b) Consider the special case q(t) = − and p(t) = µ, where µ and are positive constants
(µ > 0 and > 0). Find the relation that u0, µ and should satisfy so that u(t) develops a
singularity at finite time t = ts > 0, i.e., so that u → as t ts. Determine ts in terms of
u0, µ and .
Remark (not needed to solve the problem!): The development of singularities at finite
time is a pathological feature of many nonlinear differential equations (here an ODE). In fact,
an important “million dollar” problem in fluid dynamics is to establish whether solutions of
the celebrated three-dimensional Navier-Stokes PDE, which describe the motion of a viscous
fluid, ‘blow up’ or not in time.
3. (Prob. 1.8.3, pp. 14, 15 in G. F. Carrier & C. E. Pearson, “Ordinary Differential Equations”,
Blaisdell, 1968.) As you know, ODE are often used to model physical phenomena. This problem
provides one such example. Occasionally, a meteorite, an artificial satellite, or some other
object enters the Earth’s atmosphere on a path approximately vertical downward. The force,
F , that the atmosphere exerts on such a rapidly moving object is known to be approximated
by F = νV 2 AC/2, where ν is the atmospheric density, V is the speed of the object, A is its
projected cross-sectional area, and C is a constant that depends on the geometry; for blunt
objects C can be taken equal to 2, C = 2. Let us consider objects that only lose a negligible
amount of their mass (by melting, evaporation, etc.) and approximate the atmospheric density
ν by
ν = ν(y) = ν0 e −y ,
where y is the altitude measured in cm, = 750, 000 −1 cm −1 , and ν0 is the sea level density.
(a) Find the ODE for V 2 as a function of y and try to solve it.
1

(b) Determine the speed history of an object, i.e., its speed as a function of time, given that
its speed is V0 at y = 3 × 10
V
7 cm. Typical values of V0 might be in the range 3 × 105 cm/sec <
0 < 3 × 106 cm/sec.
4. Consider the Laplace equation, uxx + uyy = 0, in the interior of the unit circle, x 2 + y 2 < 1,
where u = u(x, y). Suppose that u satisfies the boundary condition
u = 0
ν
along the circle x 2 + y 2 = 1, where (ν, β) are the usual polar coordinates in the plane. Is this
boundary value problem (BVP) ill posed or well posed? Explain.
5. (Levine, Chap. 1, Prob. 7, p. 14.) This problem is an exercise in partial differentiation. Verify
by a direct calculation that the function u(x, t) given by
u(x, t) = 1 c sech 2
where c is a constant parameter, satisfies the PDE
2
1 ≤
c(x − ct) ,
2
ut + 6uux + uxxx = 0.
Remark (not needed to solve the problem): This solution is the celebrated solitary wave
that moves at speed c in direction x without changing its shape; t is time. The governing PDE
describes shallow water waves (KdV equation). This type of wave was first observed by John
Scott Russell on the Edinburg-Glasgow Canal in 1834.
6. (Levine, Chap. 1, Prob. 1, p. 11.) By use of the equations x = ν cos β and y = ν sin β that relate
the Cartesian coordinates (x, y) with the polar coordinates (ν, β) ( ν 0 and 0 β < 2∂),
express in polar coordinates and then solve the PDE
yux − xuy = 0.
Note: The solution of PDE is often facilitated by coordinate transformations.
7. Find the general solution in Cartesian coordinates to the 1st-order PDE
Verify your solution by direct differentiation.
2 2
yux − xuy = x + y .
8. (Levine, Chap. 8, Prob. 5, p. 90.) (a) Find the general solution to the PDE
yux + xuy = 2xy.
(b) Find the particular solution that satisfies u = y on the circle x 2 + y 2 = 1.
2

9. Determine the solution u = u(x, y) to the PDE
that satisfies u(1, y) = g(y): known.
2 x − y2 = 4.
uux − uuy = y − x
10. (Garabedian, Sec. 2.1, Prob. 2, pp. 22, 23.) Show that the solution of the nonlinear PDE
ux
+ uy = u2 that satisfies u = x on the line y = −x becomes infinite along the hyperbola
3