BY
S. A . S C H E L K U N O F F B ell Telephone Laboratories
Considering the practical importance of linear differential equations of the second order, or the equivalent system s of the first order equations, it is surprising th at trea
tises give little attention to effective and sufficiently general m ethods for their solu
tion. T he treatises seem to be concerned primarily w ith power series expansions, Picard’s method of successive approxim ations, numerical m ethods based on difference equations— m ethods which in theory are applicable to alm ost any differential equa In L iouville’s paper w e have found the Jeffries-W entzel-Kramers-Brillouin approxi
mation and a thorough discussion of the usual boundary value problem and associ
ated orthogonal series but very little th a t has any direct bearing on the present paper.
T he method is based on th e idea th a t solutions of linear differential equations m ay be regarded as distorted or “perturbed” sinusoidal or exponential functions— the same idea which is back of the asym ptotic approxim ation, of th e Rayleigh-Schro-
tions are needed in any special case. T he exposition is based on the second order equa
tion; the extension to higher order linear equations is sim ple enough. W hen it comes to nonlinear equations, excepting those which are only slightly nonlinear,f the virtues of the method are not quite clear at present. There is no question th a t th e results
should be better when compared to those obtained by Picard’s m ethod; but the more com plicated technique for numerical calculations m ay offset the advantages. T his is som ething to be explored.
Suppose that our problem is to find the solutions of
d V d l
— = - Z ( x ) I , — = - Y ( x) V , (1)
d x dx
subject to the initial conditions
V = V(a), I = 1(a), if x = a. (2)
Picard sim ply integrates (1) and obtains a pair of integral equations
V(x) = V(a) - f ’ z W t t W , l i * ) = 1(a) -
J
* F(£)F(f)d£. (3) T hus the stage is set for successive approximations and the solution is obtained in the form of the infinite seriesV(x) = Vo(x) + Vx(x) + V2(x) + ■■■ , T(x) = h ( x ) + h ( x ) + I2(x) + ■■■ , (4) where
V0(x) = V(a), Vn(x) = - [ ZZ ( M « - i ( m ,
J a
I0(x) = 1(a), I n(x) = -
f
Y(Z)Vn-i(Z)dZ. (5)J a
T his procedure is so simple th at it would be easy to overlook th e fact th at in sub
stance we are regarding the solutions of (1) as perturbations of the solutions of
d V d l
-— = 0, - = 0, (6)
d x dx
and th at we are dealing with a special application of a much more general perturba
tion method. Let*
Z(x) = Zo(x) + Z(x), Y( | = Y0(x) + Y(x), (7) and suppose th at the solutions of
dV o v d l0
— = - Z0(x)Io, — = - Fo(^)Fo, (8)
d x dx
subject to the initial conditions (2), are known. Then the solutions of (1) are identical with those of the following integral equations
L IN E A R A N D S L IG H T L Y N O N L I N E A R D I F F E R E N T IA L E Q U A T IO N S 349
V(x) = F 0( * ) - i ” Z ( m t ) V i ( x , m - f * Y(£)V(£)V2(x, o k ,
J a J a
K x ) = h ( x ) - f xm m i 1(x, m - f x v m ( 0 h u f e
J a J a
(9)
* In substance the theorem im plied b y equations (7), (8) and (9) is hardly new ; b u t w e have been unable to find its sta tem en t in ju st th a t form.
where V\{x, £), h ( x , £); F2(x, £), h ( x , £) satisfy (8) and are subject to the following
a satisfactory form for wave functions. John R. Carson3 em ployed Picard’s method for approxim ate solution when Z{ x) and Y { x) are slowly varying functions and suc
ceeded in summing the series and obtaining the first order correction terms in a usable form; but any attem pt to get the higher order terms by this method would seem to be out of the question. Theoretically, we should select Z0{x) and Yo(x) as near as possible to Z{ x) and F(x), subject to our ability to solve (8); but the integrations will be difficult to perform.* Thus we come back to (13) as the best compromise and it works very well.
In the more explicit form the first order correction terms are V\{x) = Vo\B{x) cosh To* — A (x) sinh To* + C(x) sinh To*]
— K0Io[A(x) cosh r 0£ — B(x) sinh r 0.r + C(x) cosh T0a;],
Vo , , (15)
Ii(x) = — l-B(*) sinh r 0x — A( x) cosh r 0£ + C(x) cosh r 0a;J K o
1946] L IN E A R A N D S L IG H T L Y N O N L IN E A R D I F F E R E N T IA L E Q U A T IO N S 351
where
+ Io[A{x) sinh Tox — B(x) cosh r 0x + C{x) sinh r®#]<
A(x) = —
f
[*— - K ô f \ cosh 2 T o ^ , 2 J o L K o JB(x) = —
f
X— - K nf \ sinh 2 r 0| ^ , (16) 2 J o L K o JIn some instances it is preferable to express the results in terms of progressive w aves; then V{x) = Fo(x)''T Fi(*) and I ( x) = I0(x) + h ( x ) become
where
F+(x) = i f 0/o+ [e-r<’1 - C{x)e~l'°x - £ (* )«r»*].
!+(*) = /o+[e-r<’* - C(x)e-r°* + E{x)er•*];
V- ( x) = - K aI ô \ e Ux + C ( x ) ^ x + £>(*)>3|*], I~(x) = I ö \ e T(,x + C{x)eT°x — D(kx)e~T,,x} ;
D{x) = A( x) + B(x) = —
f
- KoY~\ e*r °tdi;,2 J o L K o J
E(x) = A( x) - B(x) = 2 - f * X 4 r
-2 j 0 L A o J
(17)
(18)
Equations (14) and (15) express the solutions in terms of V and I at the beginning of a finite interval (0, /); one also often wants the corresponding expressions in terms of the final values. These are
’ John R. Carson, Propagation o f periodic currents over nonuniform lines, E lectrician, 86, 272-273 (1921).
* T h is objection would n ot ap p ly in strictly num erical handling o f equations.
352 S. A. S C H E L K U N O F F [Vol. I l l , N o. 4
Voix) = V(l) cosh T 0(l - x) + K0I(l) sinh r 0(Z - *),
Vi i x) = V ( l ) { [ B ( x ) - 5 ( 0 ] cosh Toil + x) - [A{x) - A( l ) ] sinh Toil + *) - [C(*) — C ( |] sinh Toil — x)}
+ Ko l il ) { [£(*) - 5 ( 0 ] sinh F0(/ + *) - [4 (* ) - ¿ ( 0 ] cosh r 0(/ + x) - [C(*) - C (0] cosh r 0(Z - * )} . (1 9 ) h ( x ) = — {[ A i x ) - 4 .(0 ] cosh r 0(/ + *) - [5(* ) - 5 ( 0 ] sinh Toil + *)
-
[C(*)
- C (0] cosh Toil - x) } + 7 ( 0 1 [Aix) - A i l ) ] sinh r 0(i + *) - [5(*) - 5 ( 0 ] cosh Toil + x)- [C{x) - C (0] sinh r 0(i - * )}.
Suppose now th at the interval is infinite and th a t Z{ x) and F(x) are slow ly vary
ing functions. In this case, there exists the Liouville-Jeffries-W entzel-Kramers- Brillouin approximation
T o the com m unication engineer these approxim ations seem natural even w ith ou t
tions. H ence the voltage F(jc) and current /( * ) associated with the progressive w aves w ill be proportional to e x p + [/]^ r(*)<£*]. If K i x ) is a slow ly varying function, we can ignore the reflections and in the first approxim ation consider the line as con
tinuously “m atched” and thus acting as a transformer. T his means th a t the voltage w ill vary directly and the current inversely as th e square root of the characteristic impedance: hence, equations (20).
There are several formal derivations;4 but the one which appeals to us m ost be
cause it corresponds closely to th e physical argum ent is also th e one which permits further im provem ents in th e approxim ation. L et us consider th e “transfer
parame-as the new independent variable. Substituting in (1), we obtain
4 John C. S later and N ath an iel H . Frank, Introduction to theoretical physics, M cG raw -H ill B ook Co., Inc., N ew Y ork, p. 148 (1933); John C. Slater, M icrowave transm ission, M cG raw -H ill B ook Co., Inc., N ew Y ork, p. 73 (1942).
(20)
where
K i x) = V Z i x ) / Y i x ) , T(x) = V Z i x ) Y i x ) . (21)
formal analysis. H e would reason as follows. If th e “characteristic im pedance” K i x ) is independent of x, a progressive w ave m oving either to the left or to the right would suffer no reflection; it is only th e sudden changes in the impedance th a t causes
reflec-ter” 0
(22)
d V d l V
— = - K(JQ)I, ---- (23)
dQ d@ K( Q)
Elim inating first I and then V w e have
K' ( Q) K'(@)
F "(0 ) “ -F T ^ r F '(0 ) - F = °- a(w; 7"(0 ) + - r z éa(W) r (0 ) - 7 =
°-If K ( Q) is constant, we have simple progressive w aves as anticipated; otherwise, w e introduce new dependent variables in conform ity w ith our idea of voltage and current transformation
V = [f^ © )]1'2? , I = [ t f O ) ] - 1/*/. (25) Incidentally, this is the transformation which should remove the first derivatives from (24). Substituting, w e obtain
1946] L IN E A R A N D S L IG H T L Y N O N L I N E A R D I F F E R E N T IA L E Q U A T IO N S 353
7 " ( 0 ) = 3 ( K ' y K " l _ _ f ( K ' y K "
1 + V, /"(© ) = 1 - +
---4 K2 2K J L 4 K2 2K I. (26)
W e now have not only equations (20) but also the q u an titative criterion of their goodness: (K ' / K) 2 and K " / 2 K should be small compared w ith unity.
T o improve on (20), we could repeat the process beginning w ith (22); but the analytical work is simpler if we turn to equations (13) and apply them to an infinite interval, assuming of course th at in the entire interval the bracketed quantities in equations (26) differ but little from unity. Thus, the solutions of
d 2y
¿ = [ i + / W ] y (27)
are also the solutions of
y(x) = y,(*) +
f
/($)?(£) sinh (x - £)d£, (28) J ooprovided the integral is convergent. T he solutions asym ptotic to are
y{x) c^. eTz + f /({ )e T* sinh ( x — £)d£, (29)
*7 oo or
y+{x) Czt e~z - \ e ~ x
f
f ( ( ) d ( + \ e zf
e ~ ^ ( Q d l^ to 00
y~(x) ~ ez + \ e z
f
/(£)<*£ - \ e ~ zf
e^f(Qd£. (30)* too J too
From these equations w e can obtain the well-known asym ptotic expansions of Bessel functions as well as expansions of other types.
T he case in which 0 = ij3 x , where is a constant, occurs so frequently th a t a repetition is justified. Equations (26) become
V"(x) = - p W + 3 (AT')2 K " - \ _ _ _ f K " ( K')21
4 K* 2KA V, /" (* ) = - P I + — - - — r r J r a" (a') H . (31)
\_2K 4K2 J
354 S. A. S C H E L K U N O F F [Vol. I l l , N o. 4 cosine integrals. Moreover, the com plete result corresponds closely to the physical picture of reflection which invariably takes place when w aves are traveling in trans
T he succeeding correction terms represent successive reflections. T he entire series re
sem bles an asym ptotic solution of the differential equation in question but it appears to be rapidly convergent.
An another exam ple, tak e the case of principal w aves on a thin cylindrical antenna when
1946] L IN E A R A N D S L IG H T L Y N O N L I N E A R D I F F E R E N T I A L E Q U A T IO N S 355
As the third exam ple we shall take R ayleigh’s equation for a nonlinear oscillator5 q + (Rtq + i?3<?3) + = 0. (36) B y (13) w e have
1 r 1
q(t) = q0( t ) I [Riÿ(r) + R3q3] sin co{t — r)dr, (37) to J o
where q0(t) is a sinusoidal function. If q = 0 up to t = 0, then q0{t) = A sin cot. S ub stitu t
ing in (37) and integrating, we obtain
q(t) = A sin cot — %(Ri + fw2i?3d 3)/ sin cot— — coi?3d 3(cos cot — cos 3cot). (38) For a periodic solution we m ust have
R1 + W R * A * = 0; (39)
then 1
q(t) = A sin oot — -—cdi?3d 3(cos ut — cos 3wl). (40) 32
Equation (39) is precisely R ayleigh’s equation for the am plitude of oscillations; equa
tion (40) differs from his equation in th at ours contains a term proportional to cos cot.
Our approximation satisfies th e initial condition g ( 0 ) = 0 w hile R ayleigh ’s does not.
Originally this work was undertaken to obtain convenient analytic approxima
tions to a number of problems in w ave theory. It has since becom e apparent, however, th at at least for a certain class of differential equations, the method would be suitable for numerical solution. T he practicability of Picard’s method for this purpose has already been explored by Thornton C. Fry;6 the present method should be quicker.
T he rapidity of convergence will be discussed in a separate paper.
‘ Ph. LeCorbeiller, The nonlinear theory of the maintenance o f oscillations, I.E .E . Journal, 79, 361 — 378 (1936).
8 T hornton C. Fry, The use of the inlegraph in the practical solution o f differential equations by P ica rd 's method o f successive approxim ations, Proc. 2d Internat. Cong. M ath. T oronto, 2 , 4 0 5 -4 2 8 (1924).
356