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2. If ri — ri is not an integer, then the smaller root ri of the indicial equation generates a second solution of the form
X2 (t) = (t - to )r2^2 bn (t - to )n
which is linearly independent of the first solution xi ( t).
3. When ri - ri is an integer, a second solution of the form
X2 (t) = Cxi (t) ln( t - to ) + Y2 bn (t - to )n+r2
exists, where the values of the coefficients bn are determined by finding a recurrence formula, and C is a constant. The solution X2 ( t) is linearly independent of xi (t).
? Bessel Functions
If t is very large, Bessel s equation looks like the harmonic oscillator equation, XX' + x = 0.
The roots of the indicial equation are p and — p.
Consult the references for the derivation of the formula for
For any nonnegative constant p, the differential equation
t2X'(t) + tX(t) + (t2 — p2)x(t) = 0
is known as Bessel’s equation of order p, and its solutions are the Bessel functions of order p. In normalized form, Bessel’s equation becomes
a"(0 + -^(t) +
x(t) = 0
From this we can see that tp(t) = 1 and t2q(t) = t2 — p2, so that tp(t) and t2q(t) are analytic at to = 0. Therefore zero is a regular singular point and, using equation (8), we find that the indicial equation (with P0 = 1, Qo = — p2)
r(r — 1) + r — p2 = r2 — p2 = 0 Application of Frobenius’ Theorem yields a solution Jp given by the formula
Jp (t) = tpJ2
^ 22nn!(p + 1)(p + 2) ???(p + n)
The function Jp(t) is called the Bessel function of order p of the first kind. The series converges and is bounded for all t. If p is not an integer, it can be shown that a second solution of Bessel’s equation is J— p(t) and that the general solution of Bessel’s equation is a linear combination of Jp(t) and
J— p (t).
For the special case p = 0, we get the function J0(t) used in the aging spring model in the second submodule of Module 11:
? ( _ 1 )n
2n t2 t4 t6
= 1 1---------------------
4 64 2304
Note that even though t = 0 is a singular point of the Bessel equation of order zero, the value of J0(0) is finite [J0(0) = 1]. See Figure 11.2.
? Check that J0(t) is a solution of Bessel’s equation of order 0.
When p is an integer we have to work much harder to get a second solution that is linearly independent of Jp(t). The result is a function Yp(t) called the Bessel function of order p of the second kind. The general formula for Yp ( t) is extremely complicated. We show only the special case Y0 ( t), used in the aging spring model:
Actually y is an unending decimal and non-repeating (or so most mathematicians believe), and 0.5772 gives the first four digits.
Y0 (t) =
/ t \ ?
(y + ln 2 )Jo(t) + Y2'
(-1 )n+1Hn (n\)2
where Hn = 1 + (1/2) + (1/3) + limn^?( Hn — lnn) ^ 0.5772.
+ ( 1 / n) and y is Euler’s constant, y =
210 Chapter 11
Figure 11.2: The graph of J0(t) [dark] looks like the graph of the decaying sinusoid *J2/jttcos(t— jr/4) [light].
The general solution of Bessel’s equation of integer order p is
x(t) = Cl Jp(t) + C2 Yp(t) (10)
for arbitrary constants ci and C2. An important thing to note here is that the value of Yp( t) at t = 0 does reflect the singularity at t = 0; in fact, Yp( t) ^ —<Xi as t ^ 0+, so that a solution having the form given in equation (10) is bounded only if C2 = 0.
Bessel functions appear frequently in applications involving cylindrical geometry and have been extensively studied. In fact, except for the functions you studied in calculus, Bessel functions are the most widely used functions in science and engineering.
? Transforming Bessel's Equation to the Aging Spring Equation
See “Aging Springs” in Bessel’s equation of order zero can be transformed into the aging spring equa-M°dule n. tion x" + e-atx = 0. To do this, we take
t = (2/a) ln(2/as) (11)
where the new independent variable s is assumed to be positive. Then we can use the chain rule to find the first two derivatives of the displacement x of the
Transforming Bessel's Equation to the Aging Spring Equation
We use w in place of x in the aging spring section of Module 11.
aging spring with respect to s:
dx dx dt dx
ds dt ds dt
dt2 ds [~Vs ) + Ttl?
__2\ (2_\ + d_x^
as as dt as2
_ d2x 4 dx 2 dt2 (as)2 ^ dt as2 Bessel’s equation of order p = 0 is given by:
2 d2 x dx 2 s —tj + s— + s2x= 0 ds2 ds
and when we substitute in the derivatives we just found, we obtain
' d2 x 4
dx 2 dx 2 2
Using the fact that
s = (2/a)e—at/2
(found by solving equation (11) for s) in the last term, when we simplify this monster equation it collapses down to a nice simple one:
+ -Jxe~atx= 0
d2x 4 dt2 a2
Finally, if we divide through by 4/a2, we get the aging spring equation,
x" + e-atx = 0.
The other way around works as well, that is, a change of variables will convert the aging spring equation to Bessel’s equation of order zero. That means that solutions of the aging spring equation can be expressed in terms of Bessel functions. This can be accomplished by using x = ci Jo(s) + C2Yo(s) as the general solution of Bessel's equation of order zero, and then using formula (12) to replace s. Take another look at Experiments 3 and 4 on Screens 2.5 and 2.6 of Module 11. That will give you a graphical sense about the connection between aging springs and a Bessel's equation.