Math 4263 Homework Set 1

Math 4263
Homework Set 1
1. Solve the following PDE/BVP
2ut + 3ux
=
0
u (x, 0)
=
sin (x)
2. Solve the following PDE/BVP
ux + ex uy
u (0, y)
=
0
= y2
3.
(a) Find the curves γ : t −→ (x (t) , y (t)) such that
dx
=x
,
dt
that cross the line y = 1 at t = 0.
dy
=y
dt
(b) Solve the following PDE/BVP
xφx + yφy
=
y
φ(x, 1)
=
x+1
by first finding solutions of the PDE/BVP along the curves γ determined in Part (a) and then extending
these solutions coherently to arbitrary points in R2 .
4.
(a) Find curves γ : t → [x (t) , y (t)] such that
dx
dt
dy
dt
= y
=
2y
(b) Solve the following PDE/BVP:
y
∂φ
∂φ
+ 2y
=
∂x
∂y
φ (x, 1) =
xy
x+2
5. Use the Method of Characteristics to show that the solution of
uux + uy = 0
,
u(x, 0) = f (x)
is given implicitly by
u = f (x − uy)
and verify this result by direct differentiation.
6. Use the Method of Characteristics to solve
∂φ ∂φ
+
+φ
∂x
∂y
φ (x, 0)
1
= ex+wy
=
0
2
Math 4263
Homework Set 2
1. Apply Separation of Variables to the Wave Equation φtt −c2 φxx = 0 to obtain four distinct one-parameter
families of linearly independent solutions.
2. Solve utt − c2 uxx = 0, u (x, 0) = ex , ut (x, 0) = sin (x)
3. Solve uxx − 3uxt − 4utt = 0, u (x, 0) = x2 , ut (x, 0) = ex . (Hint: find a change of variables that factors
the differential operator as we did for the wave equation.).
4. Suppose φ (x) and ψ (x) are both odd functions of x: that is, to say φ (−x) = −φ (x) for all x and
similarly for ψ (x). Show that the solution of
utt − c2 uxx = 0
−∞ ≤ x < ∞
,
u (x, 0) = φ (x)
,
−∞ ≤ x < ∞
ut (x, 0) = ψ (x)
,
−∞ ≤ x < ∞
is an odd function of x for all t; i.e. u (−x, t) = −u (x, t) for all t.
5. Solve
utt − c2 uxx = 0
0≤x<∞
,
u (x, 0) = φ (x)
,
0≤x<∞
ut (x, 0) = ψ (x)
,
0≤x<∞
ut (0, t) = 0
∀t
6. Solve
utt − c2 uxx = xt
,
−∞ ≤ x < ∞
u (x, 0) = 0
,
−∞ ≤ x < ∞
ut (x, 0) = 0
,
−∞ ≤ x < ∞
3
Math 4263
Homework Set 3
1. Use the Maximum Principle for the Heat Equation to demonstrate that there is a unique solution to
ut − k 2 uxx
= f (x, t)
0≤x≤L
,
,
t>0
(1a)
u (0, t)
= g (t)
,
t>0
(1b)
u (L, t)
= h (t)
,
t>0
(1c)
u (x, 0)
= φ (x)
,
0≤x≤L
(1d)
2. Prove the following identities
Z π
Z
sin (mx) sin (nx) dx
=
sin (mx) cos (nx) dx
=
cos (mx) cos (nx) dx
=
−π
π
−π
Z π
0
−π
π
0
if m = n
if m =
6 n
(2a)
(2b)
π
0
if m = n
if m =
6 n
(2c)
3. Consider the following Heat Equation boundary value problem:
ut − k 2 uxx
=
0
,
0≤x≤L
u (0, t)
=
0
,
t>0
(3b)
u (L, t)
=
0
,
t>0
(3c)
u (x, 0)
=
φ (x)
,
,
t>0
0≤x≤L
(3a)
(3d)
(a) Apply the method of Separation of Variables to find a family of solutions of (3a) the form u (x, t) =
X (x) T (t).
(b) Impose the boundary conditions (3b) and (3c) to find a more specialized family of solutions un (x, t) =
Xn (x) Tn (t) satisfying (1a)–(1c).
(c) Set
u (x, t) =
X
an un (x, t)
n
where the un (x, t) are the solutions found in (b), impose (3d), and then use properties of Fourier expansions
to determine the coefficients an .
4. Find the solution of the following PDE/BVP:
ut − uxx
=
0
,
0≤x≤1
u (0, t)
=
0
,
t>0
(4b)
u (1, t)
=
0
,
t>0
(4c)
1−x
2
,
u (x, 0)
=
,
0≤x≤1
t>0
(4a)
(4d)
4
Math 4263
Homework Set 4
1. Prove the Maximum Principle for solutions of the homogeneous Laplace equation: i.e., show that any
solution of
uxx + uyy = 0
,
0≤x≤a , 0≤y≤b
attains its maximal value on one of the four boundary lines
`1
`2
=
=
{(x, 0) | 0 ≤ x ≤ a}
{(a, y) | 0 ≤ y ≤ b}
`3
=
{(x, b) | 0 ≤ x ≤ a}
`4
=
{(0, y) | 0 ≤ y ≤ b}
2. Use the result of the preceding problem to prove that there exists at most one solution to
uxx + uyy
= f (x, y)
u (x, 0)
= φ1 (x)
u (a, y)
u (x, b)
= φ2 (y)
= φ3 (x)
u (0, y)
= φ4 (y)
3. Solve
kxk ≤ a
uxx + uyy
=
0 ,
u (a cos θ, a sin θ)
=
1 + 3 sin θ
4. Construct the solution to Laplace’s equation on the annular region
R = (x, y) ∈ R2 | a2 ≤ x2 + y 2 ≤ b2
subject to the boundary conditions
u (a cos θ, a sin θ)
=
g (θ)
u (b cos θ, b sin θ)
=
h (θ)
5
Math 4263
Homework Set 5
1. Determine if the following ODEs are of the Sturm-Liouville type
d
dy
p (x)
− q (x) y + λr (x) y = 0
,
p (x) > 0 , r (x) > 0
dx
dx
and if so identify the functions p (x) , q (x) and r (x).
(a) 1 + x2 y 00 − 2xy 0 + l (l + 1) y = 0
(b) x2 y 00 + xy 0 + x2 − n2 y = 0
2. (a) Find the Sturm-Liouville eigenfunctions {φn } forfor the following Sturm-Liouville system
y 00 + λ2 y = 0
,
y 0 (0) = 0 ,
y (1) + y 0 (1) = 0
(b) Suppose f (x) is a continuous function of [0, 1]. Give an integral formula for the coefficients an corresponding to the expansion
∞
X
f (x) =
an φn (x)
i=1
where the functions φn (x) are the Sturm-Liouville eigenfunctions found in part (a).
6
Math 4263
Homework Set 6
1. Show that a function f (z) = u (z) + iv (z) of a complex variable z = x + iy that satisfies the CauchyRiemann equations
∂u
∂v
∂u
∂v
=
,
=−
∂x
∂y
∂y
∂x
also has the property that both its real part u (z) and its imaginary part v (z) satisfy Laplace’s equation:
i.e.,
uxx + uyy = 0 = vxx + vyy
2. Let g (x) be any piecewise continuous function on R. Show directly from the definition, that the mapping
φg : Cc∞ → R given by
Z ∞
φg (f ) :=
f (x) g (x) dx
−∞
defines a distribution. It will be easy to show that φg defines a linear functional. The hard part will be to
demonstrate that if φg is continuous. For this purpose show that if {fn }n∈N ⊂ Cc∞ (R) converges uniformly
to a function f (x) ∈ Cc∞ (R), then
lim φg (fn ) = φg (f )
n−→∞
By the way, uniform convergence means the following
• {fn } converges uniformly to f if for every ε > 0 there exists a natural number such that |fn (x) − f (x)| <
ε for all x ∈ R and all n > N .
3. Let ψ be any distribution. Verify that the functional ψ 0 defined by
df
0
ψ (f ) := ψ
dx
is a distribution.
4. Let u (x) = u (x, y) be a harmonic function on a planar domain D. Derive the representation formula
Z
1
u (x0 ) =
[u (x) (∇ ln kx − x0 k) − (∇u (x)) ln kx − x0 k] · n dS
2π ∂D
that expresses u (x0 ) at an interior point x0 as a certain integral of u (x) and its gradient over the boundary
of D.
5. A Green’s function Gy (x) for the Laplace operator ∇2 and domain D and a point y ∈ D, is a function
defined for all x in D such that
• Gy (x) posseses continuous second derivatives and ∇2 Gy (x) = 0; except at the point x = y.
• Gy (x) = 0 for all x on the boundary ∂D of D.
• The function
1
Gy (x) +
4π kx − yk
is finite at y, has continuous second partial derivatives everywhere and is harmonic at y.
Show that such a function is unique. (You can assume such a function always exists - this is, in
fact, true.)
7
Math 4263
Homework Set 7
1.
(a) Use the Taylor expansion formula
1 ∂3f
∂f
1 ∂2f
2
3
4
(x,
y)
(∆x)
+
(x,
y)
(∆x)
+
O
(∆x)
(x, y) ∆x +
∂x
2 ∂x2
6 ∂x3
to derive the following approximations
∂u
u (x, t + ∆t) − u (x, t)
(x, t) ≈
+ O (∆t)
∂t
∆t
u (x + ∆x, t) − 2u (x, t) + u (x − ∆x, t)
∂ 2 u (x, t)
2
≈
+ O (∆x)
2
2
∂x
(∆x)
f (x + ∆x, y) = f (x, y) +
(b) Let xi ≡ i∆x, tj ≡ j∆t and ui,j ≡ u (xi , tj ). Use the approximations developed in part (a) to develop
a recursive formula yielding an approximate numerical solution for the following Heat Equation problem
ut − a2 uxx = f (x, t)
u (x, 0) = h (x)
u (0, t) = 0
u (L, t) = 0
2. Consider the problem
uxx + uyy = −4
0≤x≤1
,
0≤y≤1
u (0, y) = 0
u (1, y) = 0
u(x, 1) = 0
u (x, 0) = 0
on the unit square. Partition the square into four triangles by utilizing its diagonals and then use the Finite
Element Method to find an approximate value for u 21 , 12 .
3. Use the Laplace transform method to solve
(3a)
ut − kuxx = 0
,
0≤x≤L
(3b)
u (x, 0) = 1 + sin (πx/L)
(3c)
u (0, t) = 1
(3d)
u (L, t) = 1
(See Example 3 on page 354 of the text.)
,
t>0