Heaviside Operational Calculus by J R Carson
Heaviside Operational Calculus by J R Carson
Heaviside Operational Calculus by J R Carson
T H E H E A V I S I D E OPERATIONAL CALCULUS*
BY J. R. CARSON
The Heaviside operational calculus is a systematic method,
originated and developed by Oliver Heaviside, for the solu-
tion of systems of linear differential equations with constant
coefficients, and linear partial differential equations of the
type of the wave equation. Its important applications in
mathematical physics are to the dynamics of small oscilla-
tions, the Fourier theory of heat flow and to electrical trans-
mission theory. I t was, in fact, in connection with problems
of the latter class that the operational calculus was developed,
and in the solution of such problems it is an instrument of
remarkable directness, simplicity and power.f
The operational calculus may be described with sufficient
generality for our present purposes, in connection with the
solution of the system of linear differential equations
#nlAl+#n2^2 + +annhn = 0 ,
in the auxiliary variables hi, • • • , hn. As in (1) and (2), both
sides of the equations are supposed to be identically zero for
/<0.
For the equilibrium boundary conditions the solution of (4)
supplies the solution of (3) by virtue of the formula
d r*
(5) Ki
= 77 f(t-T)hi(r)dT, (*=1, 2, - . . , » ) .
at Jo
This formula is easily established in a number of ways;
perhaps the simplest, and at the same time the most general,
method of proof is to regard f(t) as the limit of a summation
of the form
51<Kt-fM)tfn,
where </>(/) is defined as a function which is zero for / < 0
and unity for t^O. The solution of (4) applies to this case
and passing to the limit (ô/—»0) formula (5) results by usual
processes.
1926.] HEAVISIDE OPERATIONAL CALCULUS 45
(8) "Wp)+ym'
where y(t) is the complementary solution, so constructed as
to satisfy the equilibrium boundary condition. But by (5),
we have also
d r'
(9) % = — I e*^h{r)dr ,
dt J0
whence, by direct equation,
ept d f*
-y(t)=—l eW-^h(j)dT .
dt J0
Carrying out the indicated differentiation and simplifying,
we get
1
f y{t)e~pt = h(t)e~pt+p I h{j)e~TpdT .
H(p) Jo
Finally, setting / = <x>,
1 f00
= I h(t)e-ptdt ,
1926.] HEAVISIDE OPERATIONAL CALCULUS 47
i r00
pH(p) Jo
as a means of deriving the rules of the operational calculus,
and as an instrument for the direct solution of problems. My
own work on this problem has been directed as follows :
(1) guided by the inductive work of Heaviside on the opera-
tional equation, to develop general rules for the expansion
and transformation of the integral equation, leading to general
types of expansion solutions. (2) The accumulation of a table
of definite integrals of the type
J o
4>(t)er»'dt.
(10) l/H(p)~tlon/p*.
0
p(-—-ao)=hu(0)+ f e-*WKt)dt.
\H{p) / Jo
(11) *(0=Sa»*V»l
o
We thus arrive at the Heaviside rule :
In the operational equation,
h= l/H(p),
expand \/H(p) in inverse powers of p :
(10) l/H(p)~a0+a1/p+a2/p*+
The explicit solution f or h{t) results by replacing \/pn by tn/n\
in the asymptotic expansion, so that
(11) h(t)=a0+a1t/ll+a2t*/2\+ • ••
is the required power series solution*
It may be remarked in passing that this type of solution,
while extremely direct and always possible in the case of
systems having a finite number of degrees of freedom, is of minor
practical importance unless the power series can be recognized
and summed. Furthermore this form of solution does not
exist in many important technical problems.
The Solution in Terms of Normal Vibrations. It is known
from the usual elementary theory of linear differential equa-
tions that the solution of equations (4) may be written in the
form
m
(12) A*=Qo+X) d^i\ (i= 1 , 2 , . . . , » )
o
where pu p2, • • • , pm are the roots of the equation
Dip)
(13)
B^-ink-*'
Mi(p)
* It is interesting to note that the series (11) is Borel's associated
function of the series (10), and that the infinite integral is the Borel
sum of the series (10),
50 j . R. CARSON [Jan.-Feb.,
(14) h = Co+J^C3eV,
(16) -J^=±Co+£-ii~,
f H{p) p p-Pt
where, i t will be recalled, pj is a root* of H(p).
Multiplying through by pH(p) we get
^ pH(p)
(17) c,H(p)+j:——C,-I.
P—Pi
We now introduce restrictions which obtain in physical prob-
lems : H(p) has no zero root, no repeated roots, and 1/H(p)
is a proper fraction of the form M(p)/D(p) where the
numerator and denominator are prime to each other, and
the numerator of lower order in p than the denominator.
Now if we set £ = 0, the summation vanishes, and we get
C0=l/ff(0).
Lap Jp=pj
We thus arrive at the solution
then
d
g(/) «—*(/) ,
at
provided A(0) = 0 .
3. If h and g are defined by the operational equations
h=l/H(p) , g=l/H(p+\) ,
then
g ( / ) = ( l + x f dt\e-^h{t) .
( f{t)e~^dt
Jo
1 1
x =
T{p) -H{p)~x~{i>)y
and by the corresponding integral equation
i r00
X(p) Jo
ox
(Cp+G)V= / ,
dx
thus formally eliminating the variable time t. Successive
elimination of V and of I give the equations
d2
2
7 /= / ,
dx2
d2
y2V = V ,
dx2
2
where y = (Lp+R)(Cp+G). These equations are satisfied
by solutions of the form
Cp+G,
/=— (Ae-**-Bé**) ,
7
where ^4 and 5 are constants of integration.
We now assume that the line is infinitely long so that the
reflected wave vanishes. We also assume a voltage Vo(t) = Vo
applied at x = 0. The equations then become
F = F 0 e-.73
Cp+G
ƒ=— V<>e-yx
Cp+G
I=— e->x .
1926.] HEAVISIDE OPERATIONAL CALCULUS 55
Let us write
V2
where
R G R G
v*=l/LC , p=— + — , 0-= ,
2L 2C 2L 2C
and let us consider the operational equation defining a new
variable F:
(20) F = —e^x = , ^ e •
r'dF
(22) V=-v dt.
Jo dx
Our problem is thus reduced to evaluating the function F,
as defined by the operational equation (20) and the cor-
responding integral equation
(23) := F{t)e-^dt .
V(p+p)*-o* J0
r™ e~*VFH
Jx Vp*+l
56 J. R. CARSON [Jan.-Feb.,
(25) , I e-ptJoiWP-Wt .
2
e-^V(p+p) -^ /»°°
(28) =\ <r**e->*UWt%-*/*)M ,
v
( ^ + p ) 2 ~ e2 J*h
1926.] HEAVISIDE OPERATIONAL CALCULUS 57
(30)
7 = 0 for t<x/v
(31) ax r* e-ptlxiaV^-xt/v2)
= e-px/v_\ I fa, t^x/v.
» Jxh VT2 — x2/v2
The foregoing is believed to be an excellent example of the
value of the operational calculus and in particular of the
advantage attaching to the recognition of the integral equa-
tion identity. The directness and simplicity of the solution,
as derived above, as compared with its derivation by classical
methods, is noteworthy. At the same time it must be admitted
that a direct solution from the operational equation, without
recognizing the integral equation identity, presents formidable
difficulties. Heaviside's own attack on this problem from the
operational equation, while distinguished by extraordinary
ingenuity and almost uncanny intuition, can hardly be
regarded as entirely satisfactory.*
However, without a knowledge of the integral identity (24)
on which the preceding solution is based, it is possible to de-
rive a series solution as follows. The method will be sketched
for the operational equation
V = e~y*
for the voltage. The procedure for the current is the same and
will require no explanation. The method of solution depends
on Theorem 4, given above. Since
x
y X sa y/(p + p)2 — a2
V
7 = 0 for t<x/v
p f /-*/» (t-x/v)2
(t-x/vY \
+A +
>—jr- --T
It is easily verified that this series solution in the "retarded"
timet—x/v is absolutely convergent and identical with the
expansion of the solution (21).
The Asymptotic Solution of Operational Equations. An
extremely interesting and important part of the operational
calculus relates to the derivation of asymptotic expansions
directly from the operational equation. A study of the many
problems for which Heaviside obtains asymptotic expansions
1926.] HEAVISIDE OPERATIONAL CALCULUS 59
shows that they may be divided into two classes : (1) those
of which the operational equation is of the form
(I) I=VpF(p) ,
(II) h=4>Wp) .
Heaviside himself gives no justification or proof of his ex-
pansion solutions. He does not formally distinguish between
the two classes, and he gives no information regarding the
asymptotic character of the series. While a completely satis-
factory theory of these expansions has not as yet been worked
out, the application of the Laplace integral equation to their
investigation throws a great deal of light on the problem,
and at least reduces it to a form to which the orthodox
processes of analysis are applicable.
We start with the type of problem (class 1), which is
symbolically formulated by operational equations of the type
(32) h = VpF(p) ,
h=(aQ+a1p+a2p2+a*pz+ • - • Wp ,
Now since
1 f00 dt
--= e-?'—— ,
Vp
/p Jo V irt
h = F(p)V~P ,
is that the definite integral (37) shall be asymptotically represent-
able by the series
( i f r00 i r 0 0
/
i-3 r" t*
m +
W I nmd'
1-3-5 r00 t* \
s yW
I ^ (20 Jo 3 ! /
This theorem is an immediate consequence of the preceding
analysis, for by (38) the series (39) is simply
— - (a0-ai/2t+h3a2/(2t)2- • • •) ,
Vwt
whence, by (37),
h(t)~—— (ao-a1/2t+l-3a2/(2t)2- • • •) ,
Hp) = f mentit,
Jo
If we write the definite integral in the form
(41) h = <t>(Vp) ,
1 _ r 00
(42) —<I>WP) = h(i)<rpiàt .
P Jo
1926.] HEAVISIDE OPERATIONAL CALCULUS 63
1-3 b,
g3(i) = *&(<) - —2
2 VTt'
1-3-5 b7
g*(t) = Ht(t) +
~V~ Va'
64 J. R. CARSON [Jan.-Feb.,
X g(t)e-ptdt~—z
Vp
as p-+0 ,
X t'gi{t)e~ptdt^
2Vp
as p->0 ,
1 • 3 65
X' t>g2(t)e-ptdt~
2
z
Vp
as p->0 ,
1-3-5 h
X t'gS)e-vldt~
23 V^
as p->0 .
N o w since
o v^i v^
it follows from the preceding that the functions gi, g2, g3, • • •
all converge to zero as l/t\/wt as /—><*>, provided g(/) contains
no term which converges in an oscillatory manner. It is this
latter restriction, namely that g{t) must contain no oscil-
latory term, say of the form
1
cos / ,
VTT/
(49) h = e-vZp ,
where a is a positive real parameter. This equation, it may
be remarked, formulates the propagated voltage in a non-
inductive cable.
The corresponding integral equation is
1 _ r00
(SO)
P Jo
of which the solution is known. It is
1 cT e"1,T
(51) Afl).-— -—dr9
VT JO TVT
where r = 4//a.
Equation (49), it will be observed, falls in class 2; that is,
it is of the form h^^Ç^/p). In accordance with the Heavi-
side rule for this class, we expand (49) as a power series in
Vp. It is
y/ap ap apy/ap
1! 2! 3!
1926.] HEAVTSIDE OPERATIONAL CALCULUS 67
Rearranging,
\ 3! S! / ' \2! 4! /
Now apply the Heaviside rule; that is, discard the series in
integral powers of p, replace y/p by l/y/ict and pn by dn/dtn
in the first bracketed series. We get
(53) A—
V
P/P+1
(54) — — « f k(t)e-p'di,
J
pVp/p+1 °
of which the solution is known. I t is
/T r™ ere*
(55) A(/) = l - ^ 4 / — I —— it.
£°f(t)<r»dt
to sum divergent series. The foregoing suggests that they
may be profitably employed to obtain asymptotic expansions
of f{i), when such an expansion in inverse fractional powers
of t exists.
D E P A R T M E N T OF D E V E L O P M E N T AND RESEARCH,
AMERICAN T E L E P H O N E AND T E L E G R A P H COMPANY