Powers D.L. Boundary Value Problems and Partial Differential Equations
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48 Chapter 0 Ordinary Differential Equations
has one and only one solution, unless there is a nontrivial solution of
d
2
u
dx
2
+k(x)
du
dx
+p(x)u = 0, l < x < r,
that satisfies (i) and (ii).
When a unique solution exists, it is given by Eqs. (17) and (18).
Example.
The boundary value problem
d
2
u
dx
2
+u =−1, 0 < x <π,
u(0) = 0, u(π) = 0,
does not have a unique solution, according to the theorem, because u(x) =
sin(x) is a nontrivial solution of the problem
d
2
u
dx
2
+u = 0, 0 < x <π,
u(0) =0, u(π) = 0.
Indeed, if we try to follow through the construction, we find that u
1
(x) =
sin(x) and also u
2
(x) = sin(x) (or a multiple thereof), and so all three quanti-
ties in Eq. (19) are 0.
On the other hand, suppose we try to obtain a solution by the usual method.
The general solution of the differential equation is
u(x) =−1 +c
1
cos(x) +c
2
sin(x).
However, application of the boundary conditions leads to the contradictory
requirements
−1 +c
1
=0and−1 −c
1
=0.
Thus, in this case, there simply is no solution to the problem stated.
If the differential equation (1) has a singular point at x = l or x = r (or
both), a Green’s function may still be constructed. The boundary condition (2)
or (3) would be replaced by a boundedness condition, which would also apply
to u
1
or u
2
as the case may be.
Example.
Construct Green’s function for the problem
1
x
d
dx
x
du
dx
=f (x ), 0 < x < 1,
u(0) bounded, u(1) = 0.
0.5 Green’s Functions 49
The general solution of the corresponding homogeneous equation is u(x) =
c
1
+c
2
ln(x).Thus,wewouldchoose
u
1
(x) = 1, u
2
(x) = ln(x)
so that u
1
(x) is bounded at x = 0andu
2
(x) is 0 at x = 1. The Green’s function
is thus
G(x, z) =
z ln(x), 0 < z ≤x,
z ln(z), x ≤ z < 1.
A similar procedure is followed if the interval l < x < r is infinite in length.
EXERCISES
In Exercises 1–8, find the Green’s function for the problem stated.
1.
d
2
u
dx
2
=f (x ),0< x < a,
u(0) = 0, u(a) =0.
2.
d
2
u
dx
2
=f (x ),0< x < a,
u(0) = 0,
du
dx
(a) =0.
3.
d
2
u
dx
2
−γ
2
u =f (x),0< x < a,
du
dx
(0) = 0, u(a) = 0.
4.
1
r
d
dr
r
du
dr
=f (r),0≤ r < c,
u(c) = 0, u(r) bounded at r =0.
5.
1
ρ
2
d
dρ
ρ
2
du
dρ
=f (ρ),0≤ρ<c,
u(c) = 0, u(ρ) bounded at ρ = 0.
6.
d
2
u
dx
2
+
1
x
du
dx
−
1
4x
2
u =f (x),0≤x < a ,
u(a) =0, u(x) bounded at x =0.
7.
d
2
u
dx
2
−γ
2
u =f (x),0< x,
u(0) = 0, u(x) bounded as x →∞.
50 Chapter 0 Ordinary Differential Equations
8.
d
2
u
dx
2
−γ
2
u =f (x), −∞< x < ∞,
u(x) bounded as x →±∞.
9. Use the Green’s function of Exercise 5 to solve the problem
1
ρ
2
d
dρ
ρ
2
du
dρ
=1, 0 ≤ρ<c,
u(c) = 0,
and compare with the solution found by integrating the equation directly.
10. Use the Green’s function of Exercise 8 to solve the problem
d
2
u
dx
2
−γ
2
u =−γ
2
, −∞< x < ∞,
u(x) bounded as x →±∞,
and compare with the result found directly.
11. Use the Green’s function of Exercise 1 to solve the problem stated there, if
f (x) =
0, 0 < x < a/2,
1, a/2 < x < a.
12. In confirmation of the theorem, show that the homogeneous problem a
has a nontrivial solution; problem b hasnosolution(existencefails);and
problem c has infinitely many solutions (uniqueness fails).
a. u
+u = 0, u(0) =0, u(π) = 0,
b. u
+u =−1, u(0) =0, u(π) = 0,
c. u
+u = π −2x, u(0) = 0, u(π) = 0.
13. Considering z to be a parameter (l < z < r), define the function v(x) =
G(x, z) with G asinEq.(17).Showthatv has these four properties, which
aresometimesusedtodefinetheGreen’sfunction.
(i) v satisfies the boundary conditions, Eqs. (2) and (3), at x =l and r.
(ii) v is continuous, l < x < r.(Thepointx =z needs to be checked.)
(iii) v
is discontinuous at x = z,and
lim
h→0+
v
(z +h) −v
(z −h)
=1.
(iv) v satisfies the differential equation v
+ k(x)v
+ p(x)v = 0 for l <
x < z and z < x < r.
Miscellaneous Exercises 51
14. Show that the boundary value problem
d
2
u
dx
2
+λ
2
u =f (x), 0 < x < a,
u(0) = 0, u(a) = 0,
will have no solution or infinitely many solutions if λ is an eigenvalue of
d
2
u
dx
2
+λ
2
u =0,
u(0) = 0, u(a) = 0.
Chapter Review
See the CD for review questions.
Miscellaneous Exercises
In Exercises 1–15, solve the given boundary value problem, supplying bound-
edness conditions where necessary.
1.
d
2
u
dx
2
−γ
2
u =0, 0 < x < a,
u(0) = T
0
, u(a) =T
1
.
2.
d
2
u
dx
2
−r = 0, 0 < x < a (r is constant),
u(0) = T
0
,
du
dx
(a) =0.
3.
d
2
u
dx
2
=0, 0 < x < a,
u(0) = T
0
,
du
dx
(a) =0.
4.
d
2
u
dx
2
−γ
2
u =0, 0 < x < a,
du
dx
(0) = 0, u(a) = T
1
.
5.
1
r
d
dr
r
du
dr
=−p,0< r < a,
u(a) =0.
52 Chapter 0 Ordinary Differential Equations
6.
1
r
d
dr
r
du
dr
=0, a < r < b,
u(a) =T
0
, u(b) =T
1
.
7.
1
ρ
2
d
dρ
ρ
2
du
dρ
=−H,0<ρ<a,
u(a) =T
0
.
8.
1
r
d
dr
r
du
dr
=0, a < r < ∞,
u(a) =T.
9.
d
2
u
dx
2
−γ
2
(u −T) = 0, 0 < x < a,
du
dx
(0) =0, u(a) = T
1
.
10.
d
2
u
dx
2
−γ
2
u =0, 0 < x < ∞,
u(0) =T.
11.
d
2
u
dx
2
=γ
2
(u −T
0
),0< x < ∞,
u(0) =T.
12.
d
dx
x
3
du
dx
=−k, a < x < b (k is constant),
u(a) =0, u(b) = 0(Note:0< a.)
13. In this problem, h is the groundwater level between two trenches in
which water is held at constant levels. Solve for h(x). Note that the equa-
tion is nonlinear.
d
dx
h
dh
dx
+e = 0, 0 < x < a,
h(0) =h
0
, h(a) =h
1
.
14. Solve for u(x).
d
4
u
dx
4
=w, 0 < x < a (w is constant),
u(0) = 0, u(a) = 0,
d
2
u
dx
2
(0) = 0,
d
2
u
dx
2
(a) =0.
Miscellaneous Exercises 53
15. Solve for u(x).Notetheinterval.
d
4
u
dx
4
+
k
EI
u =w, 0 < x < ∞ (w is constant),
u(0) =0,
d
2
u
dx
2
(0) = 0.
16. Show that any two of the four functions sinh(λx),sinh(λ(a − x)),
cosh(λx),cosh(λ(a − x)) are independent solutions of the differential
equation
φ
−λ
2
φ = 0.
17. In this problem, u is the temperature in a wall composed of two sub-
stances. Find u(x).
d
2
u
dx
2
=0, 0 < x <αa and αa < x < a,
u(0) = T
0
, u(a) =T
1
,
κ
1
du
dx
(αa−) =κ
2
du
dx
(αa+),
u(αa−) =u(αa+).
The last two conditions say that the heat flow rate and the temperature
are both continuous across the interface at x = αa.
18. Find the general solution of the differential equation
1
x
2
d
dx
x
2
du
dx
+ku = 0
for the cases k = λ
2
and k =−p
2
.(Hint:Letu(x) = v(x)/x and find the
equation that v(x) satisfies.)
19. Find the solution of the boundary value problem
e
x
d
dx
e
x
du
dx
=−1, 0 < x < a,
u(0) =0, u(a) = 0.
20. Solve the boundary value problem
1
r
d
dr
r
du
dr
=−r
k
, 0 < r < a,
u(0) bounded and u(a) = 0.
54 Chapter 0 Ordinary Differential Equations
21. Solve the differential equation
d
2
u
dx
2
=p
2
u, 0 < x < a,
subject to the following sets of boundary conditions.
a. u(0) =0, u(a) =1;
b. u(0) = 1, u(a) = 0;
c. u
(0) = 0, u(a) = 1;
d. u(0) = 1, u
(a) =0;
e. u
(0) =1, u
(a) =0;
f. u
(0) = 0, u
(a) =1.
22. Solve the integro-differential boundary value problem
d
2
u
dx
2
=γ
2
u −
1
0
u(x) dx
, 0 < x < 1,
du
dx
(0) = 0, u(1) = T.
Hint: Look for a solution in the form
u(x) = A cosh(γ x) +B sinh(γ x) + C.
23. Use a variation of parameters to find a second independent solution of
the following differential equation. One solution is given in parentheses.
d
2
u
dx
2
−
2x
1 −x
2
du
dx
+
2
1 −x
2
u =0 (u = x).
24. By applying the method of variation of parameters, derive this formula
for a particular solution of the differential equation
d
2
u
dx
2
−γ
2
u =f (x),
u(x) =
x
0
f (x
)
sinh γ(x − x
)
γ
dx
.
25. The absolute temperature u(x) in a cooling fin that radiates heat to a
medium at absolute temperature T obeys the differential equation u
=
γ
2
(u
4
− T
4
). Solve the special version in the boundary value problem
Miscellaneous Exercises 55
that follows, which can be done in closed form.
d
2
u
dx
2
=γ
2
u
4
, 0 < x,
u(0) =U, lim
x→∞
u(x) = 0.
26. A uniform, straight shaft exhibits violent behavior at certain frequen-
cies of rotation. Let the x-axis between 0 and a represent the undeflected
centerline of the shaft, and let u(x) be the displacement of the actual cen-
terline of the shaft measured from the x-axis. Centrifugal force provides
a transverse loading on the shaft when u is not identically equal to zero.
The equation for the displacement is
d
4
u
dx
4
−
ω
2
w
EIg
u =0, 0 < x < a,
where w is the weight per unit length of the shaft, g is the acceleration
of gravity, E is Young’s modulus, I is the second moment of the cross-
sectional area of the shaft, and ω is the angular velocity. If the shaft is
held in narrow bearings at the ends, these can be interpreted as simple
supports, leading to boundary conditions
u(0) =0, u
(0) = 0, u(a) = 0, u
(a) =0.
Find a formula for those values of angular velocity (critical values or
whirling speeds) that permit the existence of nonzero solutions to this
boundary value problem.
27. Find the lowest critical value for the angular velocity of a steel shaft with
these specifications: diameter 1.5 in.; length 48 in.; w = 0.5lb/in.;E =
30 ×10
6
lb/in.
2
, I =0.5in.
4
.
28. Sulphur dioxide (SO
2
) is a common air pollutant that reacts with water
to form sulphuric acid. If the water is airborne, the result is acid rain; if
the water is in snow, the result is acid runoff when the snow melts. Use an
analysis similar to that of Section 3 to obtain a boundary value problem
for the concentration u(x) (in units of mass per unit volume) of sulphur
dioxide in the air included in a layer of snow. Introduce q(x),theflow
rate of sulphur dioxide (in units of mass per unit time per unit of cross-
sectional area.) There are two important physical facts: (1) Diffusion is
governed by Fick’s law (similar to Fourier’s law)
q(x) =−D
du
dx
,
56 Chapter 0 Ordinary Differential Equations
where D is the diffusion constant; and (2) when the sulphur dioxide reacts
with water, it “disappears” at a rate proportional to its concentration, say
ku(x) (in units of mass per unit time per unit volume).
29. The sulphur dioxide concentration in the air in a deep layer of snow
satisfies this boundary value problem in equilibrium conditions:
d
2
u
dx
2
−a
2
u =0, 0 < x,
u(0) =C
0
.
Here, C
0
is the concentration in freely circulating air. Add an appropriate
boundedness condition and solve for u(x).
30. In “Mechanical properties of thin films from the load deflection of long
clamped plates” [V. Ziebart et al., J. of Microelectromechanical Systems, 7
(1998): 320–327] this boundary value problem is studied:
d
4
w
dx
4
−γ
2
d
2
w
dx
2
=P, −
1
2
< x <
1
2
,
w
±
1
2
=0,
dw
dx
±
1
2
=0.
The variables are w deflection, x distance measured across the short di-
mension; and the parameters are P pressure beneath the plate and γ
2
effective stress, all dimensionless. Find the general solution of the differ-
ential equation.
31. (Continuation) Solve the boundary value problem in Exercise 30.
32. (Continuation) The parameter γ
2
is related to stress, which is related to
deflection. It must satisfy the equation
γ
2
=S
0
+
1/2
0
dw
dx
2
dx.
Use your solution to find a single explicit equation that γ satisfies.
33. (Continuation) If S
0
is negative in the equation of Exercise 32, γ
2
might
be negative, say, γ
2
=−λ
2
. If this is the case, is there a value of λ for
which the solution breaks down?
34. Suppose that u(t) is a function, not identically 0, for which
u
u
=constant > 0.
Miscellaneous Exercises 57
Show that this relation is a differential equation and solve it. (Call the
constant p
2
.) Prove that exactly one of the following three possibilities
holds:
(i) u(t) = 0 for one value of t and u
(t) is never 0;
(ii) u
(t) = 0 for one value of t and u(t) is never 0;
(iii) neither u(t) nor u
(t) is ever 0.