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JITL II^TELnLJ^-riFFo^iV^TI^L E\U GIC^L ^LF FC. 1J_."
( ïo be T re rente c^ l'oi’ o.n F.ec. la^ree by ..orir i... J e.Die e. ;
The theory ci uiifereno^-^ii'i'elentir 1 eçuntiens is er-
with svecisl reieience to the vi’cbleh; oi finding o solution ly
of transforms, one the discussion is confinée, nr inly to the 1:
equation, only brief references being rnace to the non-linesr i
The question of simule exponential solutions is ccnsii
f ir s t. Folio .'inn this the rublishac. ma, te ri 9.1 is dealt ui th
chronologically, beginning -.vith a ra ie r by .ch.mic.t in 1911, r
proceeding to discussion of the main contributions irom rnui
present day. re tween 1911 and 1921 a number of German m-.them s
studied these equations in considerable detail, --nd some of ti
are shown to have used methods based on transforms. They ri-e
by Eochner end Titchmarsh, both of whom made nefinite use of -
Fourier transform in their work on the subject, enn lin^’lly sc
papers by Wright are considered, which clearly exhibit the voi
the Laplace transform method. ,,right avoids certain 'ssumptic
in the e a rlie r papers, and his results are seen to be by far ■
important and far-reaching.
The dissertation concludes with a. reference to the wc]
these equations have been trea.tea in practical problems of va;
type s .
D issertation submitted for the Degree of M.Sc.
in the University of London.
THE USE OF TRAÎTSPORMS IN CONNECTION WITH
DIFFERENCE -DIFFERENT IAL EQUATIONS and RELATED TOPICS.
By
Doris Maude James
Royal Holloway College.
March, 1950,
ProQuest Number: 10096352
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CONTSÏTTS
Section Page
I. Introduction ... ... ... 1
II. The Transcendental Equation 4
III. Schmidt .................. 11
IV. Schürer .................. 26
V. Hilh .................. 34
VI. Hoheisel ... 41
VII. Boohner .................. 55
VIII. Titchmarsh ... 67
IX. Wright 78
X. Applications 102
XI. Conclusion ... 113
1.
THE USE 0¥ TRAlISPORIi/IS IÎT CONKECTIOU WITH
DIFtmSHCE^DIPFEREFJIAL EQ.UATIQHS and RELATED TOPICS.
I. IHTRODUCTIOH.
The study of differen ce-d ifferen tial equations has
teen pursued in considerable d etail during the present
century, and much information about these equations has
been obtained by the use of transforms and sim ilar
operators. The f ir s t paper of importance was published
by Schmidt (29)* in 1911. His method of finding a
solution involves the use of a formula which is seen to
be equivalent to the inversion formula of a transform.
From 1911 onwards the study of the subject has developed
continuously, culminating in the rigorous discussions by
Wright (42-4-7), published in the la st few years. His
work is based almost en tirely on the use of transforms.
By a d ifferen ce-differen tial equation is meant here
an equation of the form
* References of the form (l), (2), ... are to the
Bibliography, those of the form (1 *1 ), (1-2), ...
(2*1), ... are to the equations.
2.
■ ■ ', ■ '■ ■> ■ ■ -, ^^”'t> n-K )j - o
where the 4r^ are independent of X , and ^ (xj
is the unknown function. Of such equations, the type
f ir s t discussed was the lin ear equation,
VW -#1
j^Z.O 'V - *
where each term contains only one function ^
and the functions and i^Cpc) are known. I
shall be mainly concerned here with the linear equation,
making only b rief comments on the non-linear equation,
since the theory of that type is s t i l l being developed.
Some of the methods used for the solution of
lin ear d iffere n tia l equations may be adapted to the
solution of lin ear d ifferen ce-d ifferen tial equations,
although the analysis is usually more complicated.
For example, when simple exponential solutions are
considered, the usual auxiliary equation is found to be
a transcendental equation. With the development
of the Operational Calculus, however, a method of
solving d ifferen tial equations by Laplace transforms
was evolved, and th is may be applied to difference-
d ifferen tial equations with considerable success.
3.
In considering the transform method a number of
problems are found to arise and, in p articu lar, it is
seen that the order at in fin ity of a solution is of
great importance to the v alid ity of the method. One
of the f ir s t steps in a rigorous approach is the proof
of an existence theorem stating conditions under which
the equation has solutions of a certain type. The
asymptotic behaviour of solutions under certain conditions
is also of in tere st, together with the question of
obtaining an actual solution in certain simple cases.
It seems convenient in discussing linear
differen ce-d ifferen tial equations to follow a
chronological scheme, beginning with Schmidt*s work
in 1911, but f ir s t the transcendental equation, already
mentioned, is considered.
I should like to acknowledge my indebtedness to
Miss B.U-. Yates for the valuable help which I have
received from her in frequent discussions.
4.
II. THE TRAFSCmOENTAl EQ.ÜATIOU.
Certain points of notation which are used throughout
the dissertatio n w ill be stated here so that repetition
may be avoided.
The number C is a positive constantw hich is not
always the same at each occurrence, while
... are positive constants each of which has the same
value at each occurrence. The numbers Ayi, f^ &«2/ " V
represent arbitrary constants, and S is any small
positive number.
The general lin ear equation is taken int he form
(1.2 ) where (x) e ^ W and 0 z 4 -Ir, ' ' ' 4 .
It is also supposed that >/ \, -n y/ I , The
linear equation which is considered in greatest d etail
is that with constant coefficients, namely an equation
of the form
m A
where the numbers d are real or complex constants.
This is referred to as the non-homogeneous equation, and
the equation
^ (x-hlr ) - - O iji-Aj
f^z 0 -y Z.0
5.
as the homogeneous equation.
As in the case of lin ear d ifferen tial equations
it is clear that the most general solution of (2. 1 )
is given hy adding any particu lar solution of (2«1 )
to the general solution of (2.2). By analogy, the
general solution of (2.2 ) is sometimes called the
complementary function.
Occasionally it is convenient to use the
operator A defined by
r - o /V * 0
in which case equation (2*2 ) may he w ritten in the
form
The number J is a complex quantity given by
 r (^-4- y t" where o" and f are re a l, unless it is
otherwise stated. It is seen immediately that
where
<4H 'V ^
J„x
Thus u ix)i d is a solution of (2-2) if An.
represents a root of the equation
ra -fyl
y ii) - o .
