Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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13.9
NUMERICAL
SOLUTIONS
OF
DES
393
These formulas are more general than
needed
for a SODE, since, as mentioned
above, such a SODE is equivalent to the simpler system in which
!(x,
u, I) sa u.
Therefore, Equation (13.72) specializes to
ko = hu.. =
hin
k2 =
hin
+
~hm"
and
k, =
h(u
n
+
~mo)
=
hin
+
~hmo,
k3 =
hin
+hm-i,
Xn+l = X
n
+
hin
+
~h(mo
+
ml
+m2) +
O(h
5),
in+l
=
in
+
~(mo
+
2m,
+2m2 +m3) +
O(h
5),
(13.73)
where
mo =
hg(x
n,
in,
In),
m,
=
hg(x
n
+
~hin,
in
+
~mo,
In
+
~h),
m2 =
hg(x
n
+
~in
+
~hmo,
in
+
~m"
In
+
~h),
m3 =
hg(x
n
+
hin
+
~hm"
in
+m2,
In
+h).
13.9.4.
Example.
The IVP x+ x = 0, x(O) = 0,
x(O)
= I clearly has the analytic
solution xCt) =
sinr.
Nevertheless, let us use Equation (13.73) to illustrate the Runge-
Kutta method and compare the result withthe exact solution.
For this problem g(x,
i,
t) =
-x.
Therefore, we can easily calculate the m's:
rna = ":""hx
n,
ml
=
-h(x
n
+
!hxn},
m2 =
-h(xn
+
~hxn
-
~h2Xn),
m3 =
-h[x
n
+hin -
~h2(Xn
+
~hXn)].
These lead to the following expressions for x
n
+1 and x
n
+1:
x
n
+l = xn +
kin
-
~h2(3Xn
+
kin
-
!h2Xn),
x
n+'
= in -
gh[6Xn
+3hxn - h
2(xn
+
!hi
n)].
Starting with xo = 0
and.to
= 1, we can generate xl> X2. and so on by using the last
two equations successively. The results for 10 values
of
x with h = 0.1 are given to five
significant figures
in Table 13.2. Note that up to x5 there is complete agreement with the
exact result.
..
The
Runge-Kutta
method leuds itself readily to use in computer programs.
Because Equation (13.73) does
not
require any startups, it can be used directly to
generate solutions to any
IVP
involving a SODE.
Another,
more
direct, method
of
solving higher-order differential equations is
to substitute 0 =
-(1/
h)10(1 -
V)
for the derivative operator in the differential
equation, expandin terms
of
V,and keep an appropriatenumber
of
terms. Problem
13.33 illustrates this point for a linear SODE.
394 13.
SECOND-ORDER
LINEAR
DIFFERENTIAL
EQUATIONS
t Runge-Kutta
0.1 0.09983
0.2 0.19867
0.3 0.29552
0.4 0.38942
0.5 0.47943
0.6 0.56466
0.7 0.64425
0.8 0.71741
0.9 0.78342
1.0 0.84161
sint
0.09983
0.19867
0.29552
0.38942
0.47943
0.56464
0.64422
0.71736
0.78333
0.84147
Table
13.'2
Comparison
ofthe
Runge-Kutta
andexact
solutions
tothesecond
order
DE
of
Example
13.9.4.
13.10 Problems
13.1. Let
u(x)
be a differentiable function satisfying the differential inequal-
ity
u'(x)
~
Ku(x)
for x E [a, b], where K is a constant. Show that
u(x)
~
u(a)eK(x-a). Hint: Multiply both sides
of
the inequality by e-KX, and show that
the result can be writteu as the derivative of a nouincreasing fuuction. Then use
the fact that
a
~
x to get the final result.
13.2. Prove Proposition 13.4.2.
13.3. Let
f and g be two differentiable functions that arelinearlydependent. Show
that their Wronskian vanishes.
13.4. Show that
if
(Ji,
f{) and
(!2.
f~)
are linearly dependent at one point, then
Ji
and fz are linearly dependent at all x E [a, b]. Here
Ji
and fz are solutions of
the DE of (13.12). Hint: Derive the identity
W(fl,
f2; X2) =
W(Ji,
f2; XI) exp
{-1~2
P(t)dt}
.
13.5. Show that the solutions to the SOLDE
v"
+
q(x)y
= 0 have a constant
Wronskian.
13.6. Find (in tenus of an integral)
GI/(x),
the linearly independent "partner" of
the Hermite polynomial
HI/(x).
Specialize this to n = 0, 1. Is it possible to find
Go(x)
and GI
(x)
in tenus of elementary functions?
13.7.
LetJi,
fz,and
f3be any three solutions of y" +py'
+qy
=
O.
Show that the
(generalized 3 x 3)Wronskian of these solutions is zero. Thus, any three solutions
of the HSOLDE are linearly dependent.
13.10
PROBLEMS
395
13.8. For the HSOLDE
v"
+
py'
+
qy
= 0, show that
P=
11ft.
-
hI!'
W(fl,
h)
and
I
'f " -
f'f"
q=1221.
W(fI,f2)
Thus, knowing two solutions
of
an HSOLDE allows us to reconstruct the DE.
13.9.
Let
h,
12,
and
13
be three solutions of the third-order linear differential
equation s" + P2(X)Y" + PI
(x)y'
+
po(x)y
=
O.
Derive a FODEsatisfied by the
(generalized 3 x 3) Wronskian
of
these solutions.
13.10. Prove Corollary 13.4.13. Hint: Consider the solution u = I of the DE
u" = 0 and apply Theorem 13.4.11.
13.11. Show that the adjoint of Mgiven in Equation (13.21) is the original L.
13.12. Showthat
if
u(x)
and v
(x)
are solutionsofthe self-adjointDE
(pu')'
+qu
=
0, then
Abel's
identity,
p(uv'
-
vu')
= constant, holds.
13.13. Reduce each DE to self-adjoint form.
(a) x
2y"
+
xy'
+y =
O.
(b) s"+y'tan x =
O.
13.14. Reduce the self-adjoint DE
(py')'
+qy = 0 to
u"
+
S(x)u
= 0 by an
appropriate change of the dependent variable. What is Sex)? Apply this reduction
to the Legendre DE for
P"(x), and show that
Sex) =
1+
n(n
+ I) - n(n +
l)x
2
.
•
(l-x
2
)2
Now use this result to show that every solution of the Legendre equation has at
least
(2n +
I)/lf
zeros on
(-I,
+1).
13.15. Substitute v = y'/ y in the homogeneousSOLDE
y"
+
p(x)y'
+
q(x)y
= 0
and:
Riccati
equation
.(a) Show that it tums into v' + v
2
+
p(x)v
+
q(x)
= 0, which is a first-order
nonlinearequationcalled the Riccatiequation. Would the same substitutionwork
if
the DE were inhomogeneous?
(h) Show thatby an appropriatetransformation, the Riccati equationcan be directly
cast in the form
u' +u
2
+Sex) =
o.
13.16. For thefunction
S(x)
defined in Example 13.6.1, let
S-I
(x)
be the inverse,
i.e.,
S-I(S(x»
= x. Show that
d
-I
I
d
-
[S
(x)]
= "------"2'
x
vi-x·
396 13.
SECOND-ORDER
LINEAR
DIFFERENTIAL
EQUATIONS
and given that
s-t(O)
= 0, conclude that
fo
x
dt
S-I(X)
=
~.
o
1-
t
2
13.17. Define sinh x
andcoshx
as the solutions
of
y" = y satisfyingthe boundary
conditions y(O)
= 0, y'(O) = I and y(O) =
I,
y'(O) = 0, respectively. Using
Example 13.6.1 as a guide, show that
(a)
cosh
2
x - sinh
2
x =
I.
(c)
sinh(-x)
=
-sinhx.
(b)
cosh(-x)
=
coshx.
(d)
sinh(a
+
x)
=
sinha
coshx
+
cosha
sinhx.
13.18. (a) Derive Equation (13.30)
of
Example 13.6.4.
(b) Derive Equation (13.31)
of
Example 13.6.4 by direct substitntion.
(c)
Let)"
=
1(1
+
I)
in Example 13.6.4
and
calculate the Legendre polynomials
p/(x)
for
1=
0,
1,2,3,
subject to the condition
p/(l)
=
I.
13.19.
Use
Equation (13.33)
of
Example 13.6.5 to generate the first three Hermite
polynomials.
Use
the normalization
to determine the arbitrary constant.
13.20.
The
function defined by
00
f(x)
=
I>nxn,
n=O
where
2n
-)"
c 2 - C
n+ - (n +
I)(n
+2) n,
can be writtenas f (x) = cog
(x)
+
Cj
h(x), where g is evenandh is odd in
x.
Show
that
f
(x)
goes to infinity at least as fast as e
Xx
does, i.e.,
limx-->oo
f (x
)e-
Xx
i'
O.
Hint: Consider
g(x)
and
h(x)
separately
and
show that
00
g(x)
=
Lbnx
n,
n=O
4n
-)"
where b
1-
b
n+ - (2n +1)(2n +2) n'
Then
concentrate on the ratio
g(x)/e
xx
, where g and e
Xx
are approximated by
polynomials
of
very
high
degrees. Take the limit
of
this ratio as x
--->
00,
and use
recursion relations for
g
and
e
Xx
•
The
odd
case follows similarly.
13.21. Refer to Example 13.6.6 for this problem.
(a) Derive the commutation relation
[a,
at]
= 1.
(b) Show that the Hamiltonian can be written as given in Equation (13.34).
(c) Derive the commutation relation
[a, (at)n] =
n(at)n-l.
13.10
PROBLEMS
397
(d) Take the inner product
of
Equatiou (13.36)
with
itself
aud
use (c) to show that
len
1
2
=
nlen_11
2
•
From
this, conclude that Ic,I
2
= n!leoI2.
(e)
For
auy function f
(y),
show that
(
y
_~)
(e
Y
' /2
f)
=
_e
Y
' /2
d
f .
dy dy
Apply (y -
d/dy)
repeatedly to
both
sides
of
the above equation to obtain
d n . d/'f
(
y _
-)
(e
Y'/2
f)
=
(_I)"e
Y'/2_.
dy
dy"
(f)
Choose au appropriate
f(y)
in
part
(e) aud show that
'(
d
)n
2 2
an
2
eY /2 y _ _
e-Y
/2 =
(-I)'eY
-(e-
Y
).
dy
dy"
13.22. Solve
Airy's
DE, y" +
xy
= 0, by the power-series method. Show that the
radius
of
couvergence for
both
independent solutions is infinite. Use the compari-
sou theorem to show that for
x > 0 these solutions have infinitely mauy zeros, but
for
x < 0 they cau have at
most
one
zero.
13.23. Show thatthe functions x/'
eAX,
where r = 0, I, 2,
...
, k, are liuearly inde-
pendeut. Hint: Startingwith
(D- A)k,applypowers
of
D- Ato a linearcombination
of
x
r
e
Ax
for all possible
r's.
13.24.
Find
a basis
of
real solutions for
each
DE.
(a) s"+
Sy'
+6 = O.
d
4
y
(e) -
=y.
dx
4
(b) v"+
6y"
+ 12y' + 8y =
O.
d
4
y
(d)
-4
=-y.
dx
13.25. Solve the following initial value problems.
d
4
y
(a)
-4
=y,
dx
d
4
y d
2
y
(b)
-4
+
-2
=0,
dx dx
d
4
y
(e)
-4
=0,
dx
y(O) = y'(O) = ylll(O) = 0, y"(O) = 1.
y(O)
= y" (0) = s"(0) = 0, y' (0) = 1.
y(O)
= y'(0) = y" (0) = 0, s"(0) = 2.
13.26. Solve
v"
-2y'
+y
=
xe"
subjectto
the initialconditions y(O) = 0, y'(O) =
1.
398 13.
SECONO-ORDER
LINEAR
DIFFERENTIAL
EQUATIONS
13.27. Find the general solution of each equation.
(a) y" =
xe",
(c) y" +y =
sinx
sin 2x.
(e) y" - y
=
eX
sin2x.
(g)
v"
-4y'
+4
= eX
+xe
2x
.
(b) y" -
4y'
+4y = x
2
.
(d)
v"
- y = (1 +
e-
x
)2.
(f)
y(6)
_
y(4)
= x
2
•
(h) y" +y = e
2x
•
i =t +
sinx,
x(O) =IT/2
i =
sinxt,
x(O) = 1
13.28. Consider the Euler equation,
x
n
/
n)
+an_1Xn-1y(n-l) +...+
alxy'
+aoy =
r(x).
Substitute x = e' and show that such a substitution reduces this to a DE with
constant coefficients. In particular, solve x
2y"
-
4xy'
+6y = x.
13.29. (a) Show that the substitution (13.50) reduces the Schrodinger equation to
(13.51).
(b) From the second equation of (13.51), derive the continuity equation for prob-
ability. .
13.30. Show that the usual definition of probability current density,
J=Re[l'J*i~
VI'J],
reduces to that in Equation (13.52)
if
we use (13.50).
13.31. Write a computer program that solves the following differential equations
by
(a) Adam's method [Equation (13.62)] and
(b) the Runge-Kutta method [Equation (13.70)].
i = t - x
2
,
x(O) = 1
i
=
e-
x
t,
x(O) = 1
i=x
2t
2+1,
x(O) = 1
13.32. Solve the following IVPs numerically, with h = 0.1. Find the first ten
values of
x.
(a) x+
0.2i
2
+lOx = 20t,
(b)x+4x=t
2
,
(c) x+i +x = 0,
(d)
tx
+i
+xt
= 0,
(e) x+i +x
2
= t,
(f)
x
+xt
= 0,
(g) x+sin x =
to,
x(O) = 0,
x(O)
= I,
x(O)
= 2,
x(O)
= I,
x(O)
= I,
x(O) = 0,
IT
x(O) = 2'
i(O)
=
O.
i(O)
=
O.
i(O)
=
O.
i(O)
=
O.
i(O)
=
O.
i(O) = 1.
i(O) =
O.
13.10
PROBLEMS
399
13.33. Substitute
dldt
= D =
-(II
h) 1n(1 -
V)
in the SOLDE i +
p(t)i
+
q(t)x
=
r(t)
and expand the log terms to obtain
(V
2
+V
3
)x
n
- hp.;(V +
~
V
2
)x
n
+h2qnxn = h
2r
n
Since V is of order h, one has to keep one power fewer in the second term, Find
anexpression forX
n
in
terms
of
Xn-l,
X
n-2,
and
X
n-3.
validtoh
2
.
Additional Reading
1. Birkhoff, G. and Rota,
G.-c.
Ordinary Differential Equations, 3rd ed., Wi-
ley, 1978. The small size of this book is very deceptive.
It
is loaded with
information. Writtenby two excellentmathematicians and authors, the book
covers all the topics
of
this chapter and much more in a very clear and lucid
style.
2. DeVries, P.
A First Course in Computational Physics, Wiley, 1994. The
numerical solutions
of
differential equations are discussed in detail. The
approach is slightly different from the one used in this chapter.
3. Hildebrand,
F.Introduction toNumericalAnalysis,2nd ed.,Dover, 1987.Our
tteatmentofnumericalsolutionsof differentialequationscloselyfollows that
of this reference.
4. Mathews, J. and Walker, R. Mathematical Methods
of
Physics, 2nd ed.,
Benjamin, 1970. A good source for WKB approximation.
14
-----'----
_
Complex Analysis
of
SOLDEs
We have familiarized ourselves with some useful techniques for finding solutions
to differential equations. One powerful method that leads to formal solutions is
power series. We also stated Theorem 13.6.7 which guarantees the convergence
of
the solution
of
the power series within a circle whose size is at least as large
as the smallest
of
the circles
of
convergence
of
the coefficient functions. Thus,
the convergence
of
the solution is related to the convergence
of
the coefficient
functions.
What
about the nature
of
the convergence, or the analyticity
of
the
solution? Is it related to the analyticity
of
the coefficient functions?
If
so, how?
Are the singularpoints
of
the coefficients also singularpoints
of
the solution?Isthe
nature
of
the singularities the sarne? This chapter answers some
of
these questions.
Analyticity is best handled in the complex plane. An importantreason for this
is the property
of
analytic continuation discussed in Chapter
II.
The
differential
equation
du
[dx
= u
2
has a solution u =
-llx
for all x except i =
O.
Thus, we
have to "puncture" the real line by removing
x = 0 from it.
Then
we have two
solutions, because the domain
of
definition
of
u =
-1
Ix is not connected on the
realline (technically, the definition
of
a function includes its domainas well as the
rule for going from the domain to the range).
In
addition,
if
we confine ourselves
to the real line, there is no way that we can connect the
x > 0 region to the x < 0
region. However, in the complex plane the sarne equation,
duif
d.; = w
2
,
has
the complex solution
w =
-liz,
which is analytic everywhere except at z =
O.
Puncturing the complex plane does not destroy the connectivity
of
the region
of
definition
of
w.Thus, the solutionin the x > 0 regioncan be analyticallycontinued
to the solution in the
x < 0 region by going around the origin.
The
aim
of
this chapteris to investigate the analytic properties
of
the solutions
of
some well known SOLDEs in mathematical physics. We begin with a result
from differential equation theory (for a proof, see [Birk 78, p. 223]).
continuation
principle
14.1 ANALYTIC
PROPERTIES
OF
COMPLEX
DES
401
14.0.1. Proposition. (continuation principle) The function obtained by analytic
continuation
of
any solution
of
an analytic differential equation along any path
in the complex plane is a solution
of
the analytic continuation
of
the differential
equation along the samepath.
An analytic differential equation is one with analytic coefficient functions.
This proposition makes it possible to find a solution in one region of the complex
plane and then continue it analytically. The following example shows how the
singularities
of
the coefficient functions affect the behavior
of
the solution.
14.0.2. Example. LetusconsidertheFODE
w'-
(y
jz)w
=
Ofory
E lit. Thecoefficient
functionp(z) =
-yjz
hasasimplepoleatz =
O.
Thesolution totheFODEiseasilyfound
to be w = zY. Thus,
depending
on
whether
y is a
nonnegative
integer,
a
negative
integer
-m,
ora
noninteger,
thesolution hasa
regular
point,apoleof
order
m. ora
branch
point
at
z = 0,
respectively.
l1li
This example shows that the singularities of the solutiondependon the param-
eters of the differential equation.
14.1 Analytic Propertiesof Complex DEs
To prepare for discussing the analytic properties
of
the solutions
of
SOLDEs,
let us consider some general properties
of
differential equations from a complex
analytical point
of
view.
14.1.1 ComplexFOLDEs
In
the homogeneous FOLDE
dw
dz +
p(z)w
= 0, (14.1)
p(z) is assumed to have only isolated singular points.
It
follows that
p(z)
can be
expanded about a point
zo-which
may be a singularity
of
p(z)-as
a Laurent
series in some annular region
Yj
< [z - zo] <
r2:
00
p(z)
= L an(z-
zo)"
n=-oo
where
Yj
< [z -
zol
< rz.
The solution to Equation (14.1), as given in Theorem 13.2.1 with q = 0, is
w(z) = exp
[-
I
p(z)dz]
=
Cexp
[-a-II
Z
~zzo
-
~anl
(z - zo)ndz -
~a-nl
(z - zo)-n
d
z]
[
00 00
]
an
n+l
a-n-l
-n
= C exp
-a_IIn(z
- zo) - L
--(z
- zo) +
L--(z
- zo) .
n=O n + 1 n=l n
402 14.
COMPLEX
ANALYSIS
OF
SOLOES
We can write this solution as
w(z)
=
C(z
- zo)"
g(z),
(14.2)
where
a sa
-a-l
and
g(z)
is an analytic single-valued function io the annular
region
rl < [z-
zn]
<
rz
because
g(z)
is the exponential
of
an analytic function.
For
the special case io which p has a simple pole, i.e., when
a-
n
= 0 for all
n 2:2, the second sum io the exponent will be absent, and g will be analytic even
at
zoo
In fact, g(zo) = I, and choosing C = 1, we can write
(14.3)
The
singularity
of
the
coefficient
functions
of
an
FOLDE
determines
the
singularity
ofthe
solution.
Depending on the nature
of
the siognlarity
of
p(z)
at
zo,
the solutions given by
Equation(14.2) have differentclassifications. Foriostance,if
p(z)
has a removable
siogularity (if
a-
n
= 0 V n 2: I), the solution is
Cg(z),
which is analytic. In this
case, we say that the
FOLDE
[Equation (14.1)] has a removable singularity at
zoo
If
p(z)
has a simple pole at zo
(if
a-I
f=
0 and
a-
n
= 0 V n 2: 2), thenio general,
the solution has a branch poiot at
zoo
In this case we say that the
FOLDE
has a
regular singularpoint.
Fioally,
if
p(z)
has a pole
of
orderm >
I,
then the solution
will have an essential singulatity (see Problem 14.1). In this case the
FOLDE
is
said to have an
irregular singularpoint.
To arrive at the solutiongiven by Equation (14.2), we
had
to solve the FOLDE.
Sincehigher-orderdifferential equations are not as easily solved, it is desirable to
obtaio such a solution through other considerations.
The
followiog example sets
the stage for this endeavor.
14.1.1.
Example.
A FOLDEhas a uniquesolution,to withina multiplicative constant,
givenby
Theorem
13.2.1.Thus,given a solution
w(z),
any
other
solutionmustbe of the
form
Cw(z).
Let zube a singulatityof p(z), andlet z - zo = re
le
.
Start at a pointz and
circleZQ so
that
f) --+e+21f.Even
though
p(z)
mayhavea
simple
poleatzo,the
solution
mayhavea
branch
point
there.
Thisis
clear
from
the
general
solution,
where
a maybe
a
noninteger.
Thus,
w(z)
ea w(zQ+ re
i
(8+ 2
:n
) may be
different
from
w(z). To
discover
this
branch
point-without solvingthe DE-invoke
Proposition
14.0.1 and
conclude
that
w(z) is also a solutionto the FOLDE.Thus, w(z) can be differentfrom w(z) by at most
a multiplicative constant:
w(z) =
Cw(z).
Definethe complex number
ex
by C = e'h<ia.
Thenthefunctiong(z)
'"
(z - ZO)-aw(z) is single-valued around
zo0
In fact,
g(zo +rei(B+'h<) =
[ri(B+2"lr
a
w(zo
+ re
i(8+Z,,)
=(z -
zo)-ae-Z"iae'h<iaw(z)
= (z - ZO)-aw(z) = g(z).
This argumentshowsthat a solution w(z) of the FOLDEof Equation(14.1)can be
writtenas
w(z) = (z - zo)ag(z), whereg(z) is single-valued.
III