TI
TISIS OP
for the
M.S.
AN ABSTRACT OF
-
Laurence Alfred Harvey
Date Thesis presented
Title
Ma
The Operational
in Mathematics
,_l5O___
olutionof
Abstract Aproved
A Laplace vector transform is defined in a
to the usual s calar definition and it
analogous
manner
application to vector derivatives is illustrated in Section 2,
In Section 3, the linear vector differential
equation with constant dyadic coefficients is solved in the
The solution depends upon obtainirg the
operational form.
which is a function of the
reciprocal 'Ir- o± a d,iadic
transform operator p. The reciprocal is found for a special
form of this diadic which applied to a large class of linear
differential equations with constant coefficients.
Application is made in Sectïon 4 of the relationships obtained in Section 3 to solve a first order linear
differential equation which contains a cross product in the
first derivative. A simple second order differential equation of the motion of a projectile subject to various initial
conditions is then treated in a similar manner and a complete
solution obtained in terms of the initial vector velocities
llore difficult equations of
and the gravitational vector.
product with one or more
vector
second order that involve the
solutions
operational
of its terms is then treated and
en
combinaalthough
expressed. It is noted in closirg that
tion of one or two terms in R( t) and its time derivatives
with the cross product may possess a solution, a combination
of all three terms with the cross product does not since
fails to exist and no solution can be found.
the dyadic 1i
TEE OBRATIONAL SOLUTION
DIPPENTIAL EQUATIONS
OP VECTOR
by
LAURENCE ALFB
A
hARVEY
THESIS
submitted to
OREGON STATE COLL3GS
in artia1 fulfillment of
the requirements for the
degree of
AT
OP SCIENCE
june 1950
PROV1D:
professor of Mathematics
In charge of Major
Head of Department of Mathematio8
chairman of School Graduate Committee
Dean of Graduate School
Dale thesis is presented
Typed by Prances Harvey
//
A QKNOWLDGT
The writer wi8he$ to exre8s hi8 deep
appreciation to Dr. I. M. Hotetter for hie
able direction, constant encouragement, and
infinite patience.
TABLE
OB'
Section
CONTENTS
Title
1
INTRQDuCON
2.
THE VECTOR TRANSFORM
3.
TH
4,
APPIIQATIONS
6
TABLES
.
.
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page
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.
LINEAR VECTOR BQUATION
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3
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5
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10
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16
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.
TEE OPERATIONAL SOLUTION OF VECTOR
DIFFERENTIAL
UATI ONS
1.
INTRODUCTION
Physicists and mathematicians have long found
the vector notation particularly advantageous in ex-
pressing differential equationß governing natural
phenomena.
Very little has been done on the solution
of vector equations completely in vector form without
recourse to systems of coordinates.
The facility of
vector, or more generally, polyadic algebra has not
been fully exploited in the solution of such problems.
It is the purpose of this paper to apply oper-
ational methods directly to solving a limited clase of
vector differential equations.
For simplicity of
nota..
tion we will let capital Latin letters represent vectors, capital Greek lettere represent dyadics, and
lower-case Latin letters represent scalars.
Section
2 defines
a vector transform analogous
to the Laplace Transform and obtains the transform of
derivatives of a vector function in terms of the trans-
2
form of the function.
Sectiona 3 and 4 illustrate the
use of the Laplace vector transformation on linear vector differential equations and its applications to sev-
eral problems.
A short table of related transforms
appear in section 5.
2.
TEE VECTOR TRANSPOBM
A vector function P(
t)
defined for all positive
values of the parameter t, maltiplied bi
integrated with respect to
t
peP
and
over the range from zero
to infinity describes a vector function G(p), i.e.
(2.1)
1 P(
P( t)
t)
dt
G(p)
The new function, G(p) is the Laplace Vector
transform of p( t).
The transformation is designated
by the bold-faced L.
L1.
cated by
The inverse transform in indi
The transform pairs, P(t) and G(p) are
tabulated in a manner similar to that of tables of integrals.
Rules and definitions which apply to the ecalar
traneform also apply to the vector transform.
treatment by Churchill (4, pp. 2-60)
110-155)
The
or Pipee (b, pp.
and associated theorems and definitions may be
applied directly by the substitution of the vector transform pair
pair
f(t)
P( t)
and
and
g(p).
G( p)
for
the s calar transform
From the definition of the vector transform
-
L F'(t)
p/e
"(t) dt.
Integrating by parts,
(2.2)
ePtÏJo
pP(t)
L F'(t)
p2Je_PtI(t) dt.
+
o
Hence
(2.3)
F'(t)
L
- pP(0)
p L F
p G(p)
- pF(0).
Similarly,
(2.4)
1
P"(t)
= L H'
p 1
p2G(p)
H
- pH(0),
-pP(0) - pF'(0).
epeated applicatiom of the foregoing procedure
yields the formula
(2,5)
We
note
(2.6)
L
n-1
F()(t)
L F
-
P(o)
p
that
p()
n-1
A
=fk
L.
-
2k=O
i
kj.
A
also,
(2.7)
LP(h1)xÁ[P11G(P)
_P(1c)(o)ph1ia
5
3.
UATION
THE LINEA.R VECTOR
It is well known that any linear vector function
may be written in the notation of Gibbs (1,
where
is a dyadic.
266) as
H
F.
(3.1)
p.
All the possible forme linear in
a vector P, namely
aP, PxA,
(3.2)
FA
and
B,
.P
Consequently, it is
may be expressed in this form.
obvious that any linear vector differential equation
with constant coefficients may be expressed in the
RL+
(3.3)
where
(3.4)
o'
l'
R''!1+
2
R"2+
.....
forni
= P(t)
+
are constant dyadics, and
R' = dR/dt, R" = d2R/dt2, etc.
We seek the solution of (3.3) subject to the initial
conditions
(3.5)
I«O) = R0, R'(o) = R
T
,
,,
B)(0)
R(n)
Enrploying the vector transform as defined in
Section 2, to equation (3.3) we obtain
o+[(p)_PRJ
(3.6)
+
...
+[P¼()_pRo_pRo7]
.
'2
R(k)(0) pn_k1.n
+
L P
_
Solving for the
operational
(3.7)
solution
G( p)
=fRo
transformed variable ((p) the
may be expressed as
.1+[2R0+10]
R
+
.2+[p3ito+p2R6+PRcr]
(o)n_k]
3
.
+
where
(3.8)
=
o
+
P]
P22
+
is the reciprocal of
and
,.
+
provided such a
',
reciprocal exists.
Let us find the
Y=uI
(3.9)
Let
reciprocal of a dyadio of form
be a
(3.10)
+
vix
set of unit orthogonal
11.j
¿,
i&, and I
where repeated indices indicates
indices.
(
3 . 11)
We have (1, pp.
Y =
uij
+
vectors, then
summation on those
288-297)
vjxA
The reciprocal of Y is
(il,2,3),
=
then
= 31
(
Uj
given by
-F
vjxA) =
(1, p.
290)
7
N1j,
(3.12)
and
1Ç1
where
N1
is the reciproca]. set
= u
to Nj.
vxA
+
such a
set always
exists provided the vectors Nj are independent.
shall so assume. The reciproca]. set N1 18 given
(3, 13)
N2xN3
N2XN3.N1
N1 =
N3xN1 '
N3XN1.N2
N2
We
by
N1XN2
N3
N1xN2.N3
or from (3.12)
(3.14)
(2+vE2x) x( u+vxA)
N-
u2+v2xA) x( u3+vxA)
with similar expressions for
form of the numerator
N2
.
u+vmxA)
The expanded
and N3.
is
(3.15)
=
u22x3+uvAx(33x2) -v22xA]Â
u2E1 +
.(uY
=
Applying
+
+
+
uv2(1.A)2
u3
+
uv2A.A
=
Ej.(UY
+ v2AA)
as
v2).(u
u3
the
v2.i&
is expanded
The denominator
(3.16)
uvxA
+
+
vxA)
u(xA).(xA)
same expansions for N2
and
we find
that
j.(uY
N1
(3.17)
+ v2AA)
(1123).
u3 + uv2A.A
With the aid of (3,12) and
uY + v2
u3 + uv2A"A
yl
(3.18)
have
or applying (3.9) we
U2I + uvlx& + v2AA
u(u2 + v2A'A)
(3.19)
as
(3.10)
the reciprooal of
y =
(3.20)
uX + vIxA.
Let us now consider
the
differential equation
of the form
(3.21)
a0R+a11'+a2R"+. . .+a
+aq1R
ql)+
1
.
.+a
+aqR
n)
= F( t).
This may be written as
(3.22)
a0R.I+a1Rh.I+a2Rh1.I+...+aq_1R(1),I+aq),IxAq
q+1)
It is of the form (3.3)
(3.23)
= a,I,
8qIXAq,
R(n) 'IXA
1q+i
P( t).
where
= a11,
..,
'
q-i
q+1 = aqiIxA.qi,...,
=
aIxL.
(3.8)
Prom
write
we
(3.24)
(ap)I +Ix
=
This is the form of the dyadie
Y
of (3.6) and (3.20)
where
1-'
(3.25)
u
'7
IZa1p1, y
(O
1,
a1A1
A
Consequently, by (3.19)
where
tion
i)21
ap)1xA
+
(ajpi)f(jajPi)2
A
we
(3.27)
G(p)
with
(8
1
(3.26)
+ AA
+ ASA]
(3.25). For the o:perational soluobtain from (3.7)
is defined
f
1i bj/
J/ ko
by
fl
i-I
EajiR11) +2Eajpi_1B)xAj
defined
by
(3.26).
+
10
4.
AII1I0ATIONS
eetion 3 to develop
We now apply the method of
the complete solutions of certain problems.
Consider
the solution of the linear differential equation of first
order.
+ mR = A ein t,
R'xA
(4.1)
subject to the initial conditions
We write
R(0) =
this
R'.IxA
(4.2)
+ mR.I = A sin t.
With the aid of (3.25) and (3.26)
(4.3)
I
pIx,
= ml +
Applying (3,2?)
to
i1
=
m21 + pmlxA +
+ mp2A.A
.
equation (4.3) and carrying out the
vector multiplication, the operational solution is
(4.4)
&(p)
A
pn3ItoxA
=
(p2a2
m2)
(p+
+
1)
From the tables of transforms we now have the complete
s
olution,
(4.5)
i(t)
R0xA
sin mt/a_
+
A sin
t
11
As an example of a simple second order differential equation let it be required to find a solution
to
the vector equation of the motion of a projectile
fired with a muzzle velocity
having a velocity
U.
V0 from a moving platform
The forces on the projectile are
mG
the gravitatiozm]. force
and the air resistance W,
which we will assume proportional to the velocity,
W
Consequently, if the posïtion vector of the
-kR'.
projectile referred to its initia], position at the time
of firing is denoted by
R,, the equation of motion is
(4.6)
= G
R1'
+ hR'
With Rc' = U + V0, i«O) =
R,
and h = k/rn.
Then
by specializing (3.26) we have
(4.7)
a0
O, a1 = h,
ph, y
I
-.-1
(4.8)
1, u
a2
p2 + ph
Prom (3,27)
(49)
G'(p)
=
+
p
Prom the tablee of
+ pR0'
+ G
+ph
eotion 5, we have at once the
O,
12
complete solution
(4, 10)
R( t)
1_t/'m) (
R0
jm
mt
k
nk2
Consider a complete solution for an equation of
the type
R" + R'XA. + m2R
(4.11)
B,
subject to the initial oonditiom
Referring to equations (3.25) and
a1
1, a2
1.
R(0)
(3.26),
R0, R'(0)
a0- in2,
we find
Hence
(p2 + m2)i +
+ (2
rn2)pIxA + p2AA
(p2+ m2)21
(p2+ rn2)[(p2 rn2)2 + p2aJ
The transform solution is obtained from (3.27) as
(4.12)
(p2R0+pR0xA + pR0' + IB)
G(p)
(p2R, + pR0xP. + pRo' + LB)(p2+ in2)
p+ m2) + p2
+
p3R0xA + p2R0'x
-
[(p2+
m)
p4R0.AJ4. +
(p2
in2)
+ pLBx&
+
p3R0'.AÁ + LB.AA
p+ m)
+
To evaluate the Inverse transform
G(p)
it
13
may be
necessary, in
as
tables such
oases, to
many
refer
to extensive
In
those of Doetech (5, pp. 3-184).
the
absence of such tables it is possible to evaluate many
forms that are similar to (4.12) by utilizing the
134).
Heaviside Expansion Formula (3, p.
If by proper
choice of the point of reference we may let
null
be a
R0
vector, six terms will be dropped from the opera-
tional solution.
(4.13)
The solution, R(
t)
sin vt-u
B(t)
,
is written
sin ut
]
cos ut - cos vt
+ (R0TxL + LB) [ _
V-U
L
+ (m2R0' + LBxA)
+
+
m2lB
Í2
sin vt_v
u
cos vt
-
utj
y2 ces ut
u2v2(v2_u2)
ut)/u2
B.AÂ1°°
[m2_u2)(u2_v2)
in ut
+
sin
[(m2_u2)(u2_v2)
(cos mt)1m2
(cos vt)/v2
(m2_v2)(u_v2)
V
(m2_u2)(m_v2J
sin vt
m
sin
(m2_v2)(u2_v2) - (m2_u2)(m2_v2
where
U2
2m2+a2+a14m2+a2
2
mt
V2
2m2+a2_aj!+a2
2
14
Consider next the case where the second ordered
derivative in (3.21) occurs in a vector :product.
R"
The
ER' +
+
operational solution
and (3.26)
F (t)
mR
be found by applying (3.24)
may
in the usual manner,
(4.14)
(pu
=
+
p2Iï
m)I +
(pn+
+ p2( pn + m) I xA + pAA
'
(pn+m)f3pn+m)+p4A'A]
write the operational solution
From which, by (3.27) we
(4.15)
G(p) =
p2R0xA. + pR0Tx
+ puR0 +
For the equation
(4.16)
wo
R" + mR' + nRxà. =
P(t)
have
-=
(p2
(p2
+
(p2
mp)I
+ nIxA.
mp)21 + n(p2 + mp)IxA.
+ mp)
and by the
usual procedure
(4.17)
G(p) =
[(p2
+
[(p2
+ nrp )
+ n2AA
+ n2A 'A]
+ mp)R0 + pR0' + Ls']
Consider the case where ail terms of the left
member of (3.21) Involve a vector product.
(4.18)
+ mR'xA1 +
Prom equation (3,20)
lt is
'o
evident
P(t).
that
u
u occurs as a factor In the denominator of
reciprocal is not defined and
Por example,
Since
0.
the
no solution existe.
F(t)
G(p)
t-
ap
- n)
1 - e
i
(p+a)
i
p(p + a)
a
i-e -at
a
(p + b)2 + a2
P
2
Cos5
(p2 + a?) -
Cosh at
(p+a)
pa
Sinh at
(p2-a2)
e -at
p
Sinat
(p2 + a?)
ap
a
t
F(t)
G(p)
J-4
(p2-a2)
ebt
sin at
a2>Q
p2
{P2-f(b-1-a)2jf+(b-a)2J
1
2ab
sin at sin bt
f-1
F(t)
G(p)
d(r)
pfl
(pb2)2+p2a
pfl
(p2+c2)[(p2+b2)2+ p2a23
dt() [
Cos
Cos
F
dt(
[(c2_u2)(u2_v2)u2
0, 1, 2, 3,
n
I
J
vt
Cos
-
ct
c2_u2)(c2_v2)c2j
2b2+a2-ablli.b2+a2
-
2
'
ti..
+-ut [u sin wt
+ w
a
v+e1t lu sin vt
cos vrJ
w(u2+w2)(v2+w2)
+
-
Libc
IL
n = 0, 1, 2, 3,
ut
cos
(c2_v2)(u2_v2)v2
-
2b2+a24aIZ?
2
dt'
- y2
u2v2(v2-u2)
ut
d(')
u2
(p2±ap)22c2
vt
2
a?
+ y
cos
vt
v(u2+v2)(v2+w)
1ibc + a2
14
14.
I-J
I!j
LITATURB
1.
CIT'
L
B., Vector analysis oundd upon the
Wilson,
Now Haven,
lectures oÍ J. Williard Gibbs.
436p.
1901.
ress,
University
Conn. Yale
2.
Page, Leigh. Introduction to theoretical physicS.
New York, D. Van Nostrand, 1928. 66l.
3.
Pipes, Louis A. Applied mathematics or eiineers
New York, McGraw-Hill, 1946.
and. physicists.
6l8p.
4.
Churchill, uel V. Modern operational mathematics
in engineering. New York, McGraW-Hill, 1944.
306p.
5.
Doetech, G. Tabellen zur Laplace transformation
pringer-Ver1ag,
und anleitung zum gebrauch.
Wien, Austria, 194?. l85p.