AN ABSTRACT OF THE THESIS OP
.S.
Harold Lien
Mathematics
------------------ for the -------------------------
(Name)
(Degree)
(Maj'r)
May 12, 198
Date Thesis presented ---------------'U
ICL SOLUTION OF !L S!.00ND ORDER
Title
iI L ì»UATION
Abstract Approved:
(Major Professor)
ABSTRACT
1ie
integration of differential equations
the engineer,
nd the seiproblems, the differential equation is
xiwiierieal
is impDrtant to the computer,
Entist. In many
that the solution
cannot be obtained in terms of
functions. In such cases methods hich yield cri
approximation to the solution are reauire.
8uch
k:rio'wn
(x,o(, 1), of the differential econtain8 two arbitrary cons tcnts,
are determined uniquely when
These
oontant
and
the solution is required to pass tIuouh a given point
and to have a given slope at that point. The point rxd
The solution y
quat.1.on f -: f(x,y,yt
)
'
slope must be such
thet £(x,y,y') has no singularity for
Those constants are also determined,
en there are two given pointa
but not always uniquely,
through which the solution is to pasa. The numerieal processeg available for the sciutlon of the second order differential equation are those for which the boundary conditions relate to one point.
the required wilues.
This thesis is a discussion and comparison of the
methods which can be ap1ied to obtain an approximation
to the solution of the second order differential equation,
yti
f(x,y,y') when the boundary conditions relate to two
The first part is a compriaon of three iuethods
points.
in.ttih the solution is obtained by successive approximatiefl_ :fIc second part is e discussion of a step-by-step
methoa by which the approximation is obtained by assuming
comlitions relating to one point and then correcting the
assumption so thet the approximation
satisfies
third
the requir-
part is a
e
condition at the second point. The
discussion of the convergence of the approximation to the
soluLion,
SOLUTION OF
SECOND ORDER
DIFFERENTIAL ET5ATION
NIThIERICAL
TI
b
HAROLD LIEN
A TEESIS
submitted to the
OREGON STATE COLLEGE
in partial fulfillment of
the requirements for the
degree of
blASTER OF SCIENCE
June 1938
ÄPPRO'vED:
epartrnent 01 iviatriernacics
In Charge of Major
Chairman of School
Acknow1egement
The writer is indebted to Dr. 7. E.
lume
for the
many valuable suggestions made during the writing of
this thesis.
TABLE OF CONTENTS
Chapter
I.
Page
Introduction
i
Successive Approximations
3
i.
Bridger's Method
3
2.
A Modification of Bridger's Method
9
3.
A Third Method of Successive
Approximation
II.
III.
IV.
Step-by-Step Integration
Method
i.
Iviilne's
2.
Method of Variations
13
25
25
28
Convergence
32
Conclusion
39
i!
SECOND ORDER
NIJIVIERICAL SOLUTION OF TIB
DIFFERENTIAL EQUATION
INTRODUCTION
The numerical integration of differential equations
the computer,
is important to
entist.
In many problems,
the engineer,
and the sci-
the differential equation is
that the solution cannot be obtained in terms of
such
In such cases methods vthich yield an
knonn functions.
approximation to the solution are required.
The solution, y
f(x, y,
equation y''
stsnts,
ly when
point,
and
tliie
.
(x,
,(), of the differential
yt) contains two arbitrary con-
These constants are determined unique-
solution is required to pass through a given
and to have a given slope at that point.
The
point and slope must be such that f(x, y, yt) has no sin-
gularity for the required values.
These constents are al-
so determined, but not always uniquely, when there are
given two poinbs throuh which the solution is to pass.
The numerical processes available for the solution of the
second order differentel equation are those for w7ich
the boundary conditions relate to one point.
This thesis is a discussion and comparison of the
methods which can be applied to obtain an approximation
to the solution of
the second order differential equation,
2
-tt
=
f(x, y, y')
two points.
when the boundary conditions relate to
The first part is a comparison of
three meth-
ods in which the solution is obtained by successive ap-
proximations.
The second part is a discussion of a step-
by-step method by which the approximation is obtained by
assuming conditions relating to one point and then correcting the assumption so that the approximation satisfies
the required condition at
part is
a
the second point.
The third
discussion of the convergence of the appro:dma-
tion to the solution.
3
SUCCESSIVE APPROXIMATIONS
I.
i.
RIDGERS METhOD
C.. Bridger*
has devised
method by which one can
s.
obtain the approximate solution to
(1)
y".
f(x,
y'),
y,
when the boundary conditions relate to both ends of the
range, viz.,
x.
(2)
x0,
xx,
y0;
y
yy,.
The method consists in dividing the range into 2
subintervals.
r
equal
The approximation to the second derivative
p
values of x in the range; by two
is found for these 2
successive integratïons by Simpson's Rule,
tion to the solution is found.
an approxima-
To illustrate the method,
let us use the second order differential equation
y"_
1+
-
y
y'2
subject to the conditions
x
O,
y
1;
x
1,
For the first approximatïon,
y
2.
an ordinate y,, is as-
sumed at the midpoint of the range and also three slopes
y, y,
y
at the first end point,
the midpoint,
and the
second end point of the range respectively.
*C.A. Bridger, On the Numerical Integration of the
Second Order Differential Equation with Assigned End
ate College, August, 1936.
Points. A thesis, Oregon
ri
The usual assumption Is to take the slopes as equal to the
joining the two end points and the ord-
slope of the line
inate y
For the example this Is
as lying on this line.
put into the table
X
y
y1
o
i
i
1
i
0.5
1.0
1.5
2.0
are substituted into the dif-
These values for y and y
ferential equation, obtaining three second derivatives
1.33
y"
1.00
y
The assumed slopes and ordinate are refined by the f ol-
lowing set of formulas:
(i
(il)
y'
(iii)
V'
(ïv)
at this stage
(y+
(Y'7:-i!»
Y,
_____
-
-:
-
- (
y
?
-?
+
(
j'*(
y'
h=
2y'i-
y'.
2y)
--
)
y'
),
where,
For actual computation the work
2.
is arranged in the table
x
y1'
y'
y
X4,
y"0
y,
y.
X,
y",
y
X
y
y'
a
l0?-i-
yL
5
Since this is the first approximation, two decimal accuracy will suffice.
In the example the table is
X
o
0.5
1.0
y"
y'
2.00
1.33
1.00
0.22
1.04
1.83
Using these values for y and
y
1.00
1.32
2.00
the
y'
second derivative is
again calculated.
For the second approximation the range is now divided into four equal subintervals.
There are three second
derivstives calculated; for the end points and the midpoint of the range.
To obtain two more second deriva-
tives, we interpolate,
using two formulas derived from
Lagrange's Interpolation formula in which interpo1aLd
values are to be midway between two consecutive given
second derivatives:
(y)
3y
+ 6y
(-y"
bo
6y
(
1
=
(vi)
Five second derivatives
a five
-
y
3y'
)
)
are now available.
At this point,
column table is used:
II
I
y'y
ytt
X
III
IV
y
y'
The entries in column II sre calculated by
(vii)
y'
-
(viii)
'sr'
Ïi.z
y'
O
I
-
yI 2
i
9y
2
(
i-
y".
-
t.L
l9ç"
-
4y"
--
1+1
5y
"
4-
)
y'
)
and
),
the quantity
(y'
-y'
(vi.
Adding
is Simpson's rule.
ow (viii)
)
-
J
)
to
(?J-
By advancing
is obtained.
one step at a time the entries in column II 8re
(viii)
c.e.
The initial slope
calculated.
the first entry in
column III is calculated from the equation
(Y-
(a-xe)
(ix)
+
where
-
y
)
-
y
)
il,3,5 ----- a -
subscript
a
[L -
-----]2
-----j + (y
+
y'
2,4,6 ----- a
and k
1,
-
L
)J
-
2.
The
denotes the values at the second end of the
The remaining entries in column III ere obtained
range.
by
(y
)+------
y0)
adding algebraically
The entries in column
y'0
to the entries in column II.
or the ordirates are calculated
IV,
by formulae similar to (vii)
-
(
y
(x)
y
(xi)
y.¿*2 _
.
-3
9y
C
+ l9y
y'.
¡4&
and (viii),
-
y
5y'
i-4y¿ii
--
r!
)
The work for the example appears in the table:
x
O
0.25
0.50
0.75
1.00
y"
1.0484
1.2981
l..5679
1.8579
2.1680
y' 0.2929
0.6507
1.0785
1.5713
y'
0.3035
0.5964
0.9542
1.3820
1.8748
1.0000
1.1112
1.3036
1.5941
1.9999
7
From these values for y', y" is calculated; and the process froni (vii) to (xi) inclusive is used to refine our
values.
X
y'-
y11
1.1094
1.1728
1.4504
1.7809
2.2465
o
0.25
0.50
0.75
1.000
y'
y
y'
0.3465
0.6156
0.9507
1.3492
1.8514
0.2791
0.6042
1.0027
1.5059
To obtain further atproximaticns,
1.0000
1.1189
1.3133
1.5994
1.9997
subintervals
the
The necessary second derivatives
are divided in half,
needed are calculsted by interpolation formulae
5y'
':
(xii)
-i-.
y
t,
-
5y
+ 9y
9y
(-g::
y3
15y
y',
= i--
- y",
)
)
1
16
(
y
-
5-y"
-+-
l5y
-
5y
)
The last interpolation formula is advanced one
time until the end of the interval is reached.
ration is carried out using formulae (vii)
sive.
step at a
The integ-
to (xi)
inclu-
After every interpolation a refinement is made to
correct for variation due to the interpolation.
The fol-
lowing table carries the work out to 16 ordinates.
roi
X
y
?J'YJ
o
i.oie
.125
.25
.375
.50
.625
.75
.825
1.1979
1.2689
1.3455
1.4552
1.6077
1.7994
2.0108
2.2074
0.1437
0.2980
0.5522
0.6357
0.9178
1.0390
1.3685
1.5411
1.0753
1.0689
1.3291
1.4253
1.4105
1.7107
1.7163
2.1090
2.1480
0.1290
0.2783
0.4634
0.6300
0.8292
1.0454
1.2744
1.000
o
.125
.25
.375
.50
.625
.75
.825
1.000
o
.0625
.1250
.1875
.2500
.3125
.3750
.4375
.5000
.5625
.6250
.6875
.7500
.8125
.8750
.9375
1.000
1.1043
1.1279
1.1487
1.1782
1.2223
1.2810
1.3484
1.4129
1.4569
1.5308
1.6165
1.7063
1.7971
1.8903
1.9921
2.1131
2.2687
Ik.5579
.0680
.1409
.2118
.2885
.3649
.4488
.5339
.6250
.7166
.8166
.9187
1.0299
1.1434
1.2664
1.3928
1.5312
y'
y
0.2744
0.3181
0.6724
0.8266
0.9101
1.1922
1.3134
1.6429
1.8155
1.0000
1.0312
1.0925
1.1810
1.2962
1.4168
1.5877
1.7539
1.9918
0.3229
0.4519
0.6012
0.7863
0.9529
1.1521
1.3683
1.5973
1.8808
1.0000
1.0483
1.1138
1.2001
1.3096
1.4397
1.5983
1.7827
1.9999
0.3299
0.3979
0.4708
0.5417
0.6184
0.6948
0.7787
0.8638
0.9549
1.0465
1.1465
1.2486
1.3598
1.4733
1.5963
1.7227
1.8611
1.0000
1.0227
1.0498
1.0815
1.1176
1.1588
1.2046
1.2561
1.3127
1.3754
1.4436
1.5187
1.5998
1.6887
1.7841
1.8873
1.9997
2.
A i1ODIFICATIOi
OF FRIDGEi-'S
ThOD
A modification of the above process is
to divide the
range into an even number of equal subintervals.
of
these subintervals is to be sufficiently small.
The range
The ap-
proxirnation to the solution is obtsined by repeated. use of
formulae (vii) to (xi).
This method does away with the
interpolation formulae and the formulae used in making the
This process is repeated until re-
first approximation.
peated use produces no change to the required number. of
decimal points in the result.
In this process the first
approximation for calculations of the second derivative is
obtained by passing
a
straight line through the two end
points.
In applying this method to y"=
condition
xO, yl;
x.l, y.2
with boundary
and with ten subinter-
vals, we have the results listed in the table.
X
ytt
o
yt_
r?
.
.1
.2
.3
.4
.5.
.6
.7
.8
.9
1.0
o
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
o
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
2.00
1.82
1.66
1.54
1.43
1.33
1.25
1.18
1.11
1.05
1.00
1.e526
1.1388
1.2483
1.3577
1.4706
1.5586
1.6424
1.6979
1.7511
1.7768
1.8044
.1909
.3647
.5212
.6730
.8075
.9396
1.0578
1.1756
1.2801
1.3859
.1093
.2285
.3590
.5002
.6523
.8118
.9798
1.1513
1.3291
1.5067
i
1
1
1
1
1
1
1
1
1
1
.2293
.4202
.5940
.7505
.9023
1.0368
1.1689
1.2871
1.4049
1.5094
1.6152
.3131
.4224
.5416
.6721
.8133
.9654
1.1349
1.2929
1.4644:
1.6422
1.8198
y
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
1.0000
i.03$2
1.0837
1.1514
1.2336
1.3313
1.4408
1.5646
1.6982
1.8451
2.0001
1.0000
1.0367
1.0848
1.1754
1.2196
1.3384
1.4133
1.5650
1.6723
1.8581
2.0007
X
o
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
o
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
o
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
y'
1.1088
1.1520
1.2064
1.2746
1.3587
1.4586
1.5713
1.6990
1.8537
2.0183
2.2231
1.0980
1.1367
1.1922
1.2351
1.3623
1.4435
1.6189
1.7071
1.8803
1.9896
2.1558
.1130
.2308
.3547
.4863
.6269
.7784
..9417
1.1201
1.3128
1.5251
.1115
..2279
.3495
.4777
.6204
.7695
.9413
1.1138
1.3152
1.5136
i.ino
1.1520
1.2063
1.2724
1.3509
1.4521
1.5628
1.7084
1.8494
2.0353
2.2029
1.1105
1.1474
1.2025
1.2797
1.3526
1.4603
1.5621
1.7057
1.8443
2.0464
2.2083
.1130
.2308
.3622
.4857
.6331
.7764
.9468
1.1179
1.3282
1.5244
.1128
.2299
.3530
.4857
.6247
.7776
.9385
1.1186
1.3095
1.5265
.3282
.4412
.5590
.6829
.8145
.9551
1.1066
1.2699
1.4483
1.6410
1.8533
.3319
.4434
.5598
.6814
.8096
.9523
1.1014
1.2732
1.4457
1.6471.
1.8455
.3240
.4370
.5548
.6862
.8097
.9571
1.1004
1.2708
1.4419
1.6522
1.8484
.3299
.4427
.5598
.6829
.8156
.9546
1.1075
1.2684
1.4485
1.6384
1.8564
y
1.0000
1.0384
1.0884
1.1504
1.2252
1.3136
1.4165
1.5353
1.6710
1.8254
1.9999
1.0000
1.0387
1.0888
1.1508
1.2254
1.3132
1.4161
1.5342
1.6708
1.8243
2.0001
1.0000
1.0380
1.0876
1.1494
1.2240
1.3121
1.4153
1.5331
1.6695
1.8226
1.9995
1.0000
1.0382
1.0887
1.1504
1.2256
1.3137
1.4170
1.5355
1.6713
1.8255
1.9999
12
This modification of Bridger's Process seemed to be
much faster.
In the example we know at the second approx-
imation the solution to one decimal place, for all the
required values of x.
The speed of this modification is
due to two things perhaps,
the choice of the length of
the subinterval and also the differential equation itself.
The chief advantage of this modification is that fewer formulas need to be remembered.
is
The disadvantacre here
the same as in Bridger's Process, namely that so far
the error term of the rth approximation has not been de-
termined.
The only check is that after a few approxima-
tions the successive values obtained for y, y', and y"
do not change to the required number of significant fig-
ur es.
L3
3.
A
ThIRD METHOD OF SUCCESSI
APPROXIMATION
The method developed in this section is exceedingly
easy to use when a calculating machine is available, and
cn
be used to good advantage even when a calculating
machine is not available.
Of the three methods tested
using successive approximations,
this seemed to be the
fastest.
If the function y = f(x) has a continuous fifth der-.
ivative
in the neighborhood of x
x0,
then f(x) can be
expanded by the extended theorem of the rican
(1)
f(x)
(Xjj,
f(x0)+ f'(x) (x-x0)
J
f(x)
(x-x0)
f (s)
4
X0
------
The integral
/'x
c)
f (s)
is the remainder term.
If we make a chanr,e
by transforming the origin to
independent variable,
f(h)
(2)
Iì
Equation
(3)
(2)
hf'(h)
xx0
of variable
and let h be the ne
the (i) becomes
hZ.
hf'(0) i---f! (Q
f (o)j
f(0)-s-
ds
4)
(h-s)
h3
__.flI
(°)
(s)
4
f (0) ds.
yields the further equations
hf'(0)+
h*ft(0)+f1(Q)+f
(o)
14
3
h (h-s)
+
hftI(h)
f(s) ds
J
(4)
h3f"(0) +
hf"(0)
«,
f (0)
+
f
h2(h-s) i
(s)ds
Multiplying (2) by 12 and (3) by -6 and adding to (4) we
have
+
12 f(h)-6 hf'(h)
(5)
+ h
hLffl(h)
= 12 f(0)-6 hÍ'(0)
11-i
Z
f1t(0)
4
f
-
(s)
J
£
(hs)
-
ds
2
If f (h) exists and is continuous in the interval from
to
hh,
(h
r)
h=O
the integral in (5)
(h-si\2 s Z
Jf (s)
2
can be written
(5)
f
ds
(.)) fh
h
&
s
ds
o
o
as
s2'(h-sY
z
f
L)
is positive for all values
(h-s)
5
-
of
s
h,
and h.
Equa-
tion (5) may be rewritten in the following form
12 r
f(h)f(0)[f'(h)1f0)J
6
72O
(7)
y.
¡#,
hr
z
If equation
('7)
L
12
(9.
If 4- (h) is replaced by
becomes
-Lr,'-r"(o)J
y, and
ii-s-t-yll
-12
f(0) by y.,
't
H
-y
equation (6)
___y
(Ç).
is used as a formula of integration,
its
15
error is found to he about 1/8 tue error of Simpson's
rule.
For use in integrating second order differential e-
-
(g)
,
and
r
-z
L'
¿
f':
--
:]
-ii-
-y'-2
y'.
the two formulas
we have
quations,
--(y'!
_y!]
ith these formulas is that
The only difficulL
may
times
the third derivative is very difficult to calculate.
If
the third derivative is easy to compute arid if h is taken
small enough, these equations (8) give a very rapid method for computing the approximate solution of
To use
f(x,y,y').
in the calculation of the approximation
(8)
f(x,y,y') passing through the end
to the solution of y"
points, x
f
y
x
;
x.,
y=
ye,,
the range of integ-
ration is divided into a number of equal subintervals of
The integration is carried out step-by-step
length h.
over each subinterval.
iven by
The initial slope is
(9)
0
_______
y'
where _e
tian.
r
i
L
n- e
(1-
t)
y"(t)
The formula is derived in
.
The integral in (9)
evaluated
a
dt,
later sec-
by the following
considerations, is
r(x)
f
o
(
¿
-t)
y"
(
t dt.
Then
it
follows
(E_x) y"(x)
r'(x)
rtt
and
(x)
nl
- (d-x)
/
y 'x)-y"(x)
r(x) for each subinterval, then
If we apply (7) to
h
r1,1
)
-
rfl
12
Hence
1.I
dt=
r()J-t)
o
But
rO,
r,=
ç
'L2'
ì=l
z
L y"
-h
rt
,
rLir)
2
--
r
_yII
r-
(
r'
y'-y"
Therefore
,
2.
(li)
f(L_t)"dt
= h
(2-1)y"_
-y
h
't
_
o
.
12
"'c
Substituting (li) In (9) we find that
4a
'.
(12)
y'=
o
'
--:i/(
-x
i
)
y"
¡
-
-h
2
h2
-(
-y'
)
=1
h
I.
,
r
For the first approximation of y" and y" we take
-xo
(x-x0)4y0
which is the straight line passing through the two points,
and calculate y" and y".
we now by the use of (8)
With these values of y" and y"
and (12)
proximation to the solution.
calculate the first ap-
With this first approxima-
tion, we again calculate the second and third derivatives
and integrate using (s) and (12).
The process is repeat-
ed until further integrations produce no change In the
approximations.
Let us apply this to the familiar example,
tion is required for
(1, 2)
.
Now
y'L
l+L2passin
the solu-
through (0,1) and
yII_
The work is arranged in the following manner.
TABLE
1
X
y
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
.0
.1
.2
.3
.4
.5
.6
.7
.9
1.0
6
5
IT
2
y'
3
4
tV
y11
2.00
1.82
1.66
1.54
1.43
1.33
1.25
1.18
1.11
1.05
1.00
1
1
1
1
1
1
1
1
1
1
1
2.00
1.65
1.38
1.18
1.02
0.89
0.78
0.69
0.61
0.55
0.50
8
7
9
(1-x)
.0
.1
.2
2.000
1.638
1.328
1.078
1.000
.910
.830
.770
.715
.665
.625
.590
.555
.525
.500
-0.35
-0.27
-0.20
-0.16
-0.13
-0.11
-.18
-.16
-.12
-.11
-.10
-.08
-.07
-.07
-.06
1.0 -.05
.4
.5
.6
.7
.8
.9
-0.08
-0.06
-0.05
Using equations (8) and (12) an
late the entries in column
Table Ii
i.s
calculateJ.
0
Table
Table II.
2 of
calculated from column
using equation (8) and Table
.858
.665
.500
.354
.222
.105
,
2
I,
we calcu-
Column 7 in
in Table II.
column
1
Again
in Table II is
These operations are repeated to obtain the
successive tables which are the approximations to the
solution.
TABLE II
1
2
4
3
yTT
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.0000
1.0324
1.0830
1.1504
1.2331
1.3301
1.4405
1.5634
1.6981
1.8439
2.0003
5
.2276
.4189
.5931
.7533
.9019
1.0400
1.1691
1.2906
1.4051
1.5131
1.6156
6
.6
.7
.8
.9
1.0
7
.O9
.110
.120
.100
.095
.075
.060
.050
.033
.022
.133
.323
.210
.180
.141
.118
.077
.02
.023
-.005
.113
.209
.296
.376
.450
.520
.584
.645
.702
.756
.807
.2280
.3610
.6846
.8940
1.0740
1.2150
1.3330
1.4100
1.4420
1.4650
1.4600
8
9
(lx)y"
y'/2
x
.1
.2
.3
.4
.5
1.0518
1.140
1.250
1.370
1.47Ö
1.565
1.640
1.700
1.750
1.783
1.805
.525
.570
.625
.685
.735
.782
.620
.850
.875
.891
.902
1.051
1.026
1.000
.959
.882
.782
.656
.510
.350
.178
0
20
For the second approximation we have:
yt
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
:
1.0000
1.0367
1.0849
111456
1.2199
1.3089
1.4135
1.5340
1.6715
1.8268
2.0000
5
Ay"
.0
.0
.1
.2
.3
.4
.5
.6
.0398
.0550
.0770
.0950
.1050
.1350
.1370
.1430
.1370
.1320
.8
.9
1.0
tt
.3136
.423].
.5426
.6736
.8156
.96Th
1.1275
1.2945
1.4670
1.6436
1.8229
6
4
3
2
i
x
,
1.0982
1.1380
1.1930
1.2700
1.3650
1.4700
1.6050
1.7420
1.8850
2.0220
2.1540
7
.3438
.4548
.5960
.7950
.9170
1.0870
1.2780
1.4710
1.6550
1.8250
1.9590
8
9
Ay"
y/2
y"/2
Li-x)?
.1110
.1412
.1990
.1620
.1700
.1910
.1930
.1840
.1700
.1340
.156
.211
.271
.336
.407
.483
.563
.642
.733
.821
.911
.549
.569
.596
.635
.682
.735
.802
.871
.942
1.0982
1.0242
.9544
.8890
1.011
1.077
.8190
.7350
.6420
.5226
.3770
.2022
0
21
For the
third approximation
T
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
i
2
Y
Yt
1.0000
1.0388
1.0885
1.1501
1.2244
1.3124
1.4152
1.534Q
1.6702
1.8253
2.0006
5
.3280
.4398
.5563
.6794
.8111
.9528
1.1065
1.2738
1.4551
1.6504
1.8592
6
x
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
.04159
.05416
.06774
.08329
.09966
.11806
.13805
.15681
.17368
.18820
.12321
.12843
.13565
.14631
.15840
.17353
.19073
.20642
.21857
.22681
we
have:
4
3
yn
yt11
1.10725
1.14884
1.20300
1.27074
1.35403
1.45369
1.57175
1.70962
1.86643
2.04011
2.22831
7
y'/2
.169
.219
.278
.405
.476
.553
.636
.727
.826
.929
.36317
.48638
.61481
.75066
.89697
1.05537
1.22890
1.41963
1.62605
1.84462
2.07143
ytt/28
.5536
.744
.6015
.6353
.6770
.7268
.7858
.8548
.9332
1.0200
1.1141
9
(1-x)y"
1.1072
1.0339
0.624
0.8895
0.8124
0.7268
0.6287
0.5128
0.3732
0.2040
0
22
For the fourth approximation we have:
2
i
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
4y
7
6
ti
It)
.
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
.0434
.0540
.0620
.0880
.0990
.1140
.1299
.1481
.1730
.2020
y
1.1086
1.1520
1.2070
1.2690
1.3570
1.4560
1.5700
1.6999
1.8480
2.0210
2.2230
y
.3296
.4423
.5597
.6833
.8144
.9547
1.1058
1.2607
1.4483
1.6435
1.8568
1.00000
1.0385
1.0885
1.1505
1.2252
1.3135
1.4168
1.5358
1.6715
1.8259
2.0007
5
x
4
.3
t,
x
.1059
.1523
.1258
.1550
.1580
.1670
.1830
.1870
.2270
.2440
/
.164
.221
.279
.341
.407
.4'77
.557
.634
.724
.821
.928
.3640
.4699
.6222
.7480
.9030
1.0410
1.2240
1.4080
1.5950
1.8220
2.0660
8
yi2y2
t
''I
ti
/
.5543
.5760
.6035
.6345
.6785
.7280
.7850
9
(1-x)y
1.1086
1.0368
.9656
.8883
.8142
.7280
.6280
.5097
.3696
.2021
.8499.
.9240
1.0105
1.1115
0
ti
23
v:
For the fifth approximation we
1
2
yT/2
x
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.0000
1.0385
1.0886
1.1508
1.2256
1.3139
1.4168
1.5355
1.6712
1.8254
1.9999
.3297
.4427
.5605
.6841
.8152
.9557
1.1069
1.2702
1.4474
1.6407
1.8527
.164
.221
.280
.342
.407
.477
.553
.635
.723
.820
.926
2.
If the calculated
a
y,
does not check with the given
mistake has been made in the calculation.
has been made it is well to check first,
as
if
is
the most important
ordinates.
y,,,
If a m.isake
the value for
for the determination of the
25
II. STEP-BY-SThP INTEGRATION
MILNE METHOD
1.
The most convenient method for the numerical solu-
tion of differential equations is the step-by-step method
This method requires two calcu-
devised by W.E. Milne.*
lations.
First, predicted values are calculated.
This
is followed by a calculation which checks the predicted
values.
The method may be applied to
the solution of second
Suppose that four values of
order differential equations.
the solution to
(i)
f(x,y,y')
y"
are known:
(2)
X
y
x,
y6
y
X,
y,
y
X
y
y'
;
The predicted value for y
(5)
y
y2
i-
is
2y"-y'
-
y
i3T
y
:
The predicted value for y'
(4)
y"
y'
calculated from
2y')
is calculated from
-14- 4iY)
Now y" is computed; with this value of y",
b
4y
y
y'
is checked
'J'
*W.E. Milne, On the Numerical Integration of Ordinary
Differential
pp. 455-460.
EiatTn,
Am. Math
Monthly, vol.33, 1926
e between the predicted and
If there is a discrepancy
y,
checked values of
value of y
and if e /29 will not effect the
to the number of decimals required,
it is as-
sumed that the predicted value is correct to the reauired
accuracy.
If
too large the celculation is repeated
is
until no corrections are necessary.
To obtain the starting values to the solution of the
second order differential equation y"
to x = x0,
y'=
y = y0,
approximation is used.
f(x,y,y')
subject
A process of successive
y,.
Tho additional slopes and two ad-
ditional ordinates are calculated by
-
V
h3
j
shy'
T
y.-i--
,,
y
h
ii
h
y-hy'--y
a
6
2o ---y
n
y
s
(7)
20
s
y-hy
nj
h.
y+hy"
y'
n
h
.s----y
n
D
With these values of y,,
vatives
f,and
y'
ya,,
y,
and y,,
are calculated.
the second deri-
The values for the
lopes are refined by the following formulas:
t
y'.-(7y"
y'
(8)
-
- l6y'
-1
y'
y", )
4
..
h
y'
i nl
hy
(7y",+ l6y'
*
F
)-.-
a n'
hy
a-
4
With these newly calculated slopes, the ordinates are
checked by
2'7
h
7-I
Y1
-I-
)
ç
+
(9)
h
Y
¡
11T'
(
7y,
-t-
l6y
-F-
y
)
4-
At this stage the solution is known for one backward
point and one forward point.
To approximate the second
forward point, the formulas
2h
vi
--
Y
(
5y"
-
y"
-y',
)
-2h
"
(lo)
+-
are used.
(
t- 4Y,'
-4-
From these values
Now the
is calculated.
second forward solution is checked by
Y'
'+--
y
h
+ --
(11)
o
4y
(
(r'
-t-
J
4y'
y
and
yt
g
If the checks have a large discrepancy,
(11) is repeated
until further repitition produces no change to the required number of decimals.
Let us now solve y"
L
h'lO
J4Y
with x=O, y
.
1,
't
,
-_l
l.O5OO
o
.3
.4
i.00000
1.00500
1.02006
1.04533
1.08109
.r
l.12762
.6
.7
1.18547
1.25516
1.33744
1.43308
1.54309
.1
.2
.8
.9
1.0
O,
y'
l.O5Ol
-O.lOi7
o
o
0.10017
0.20134
0.30453
0.41076
0.52112
0.63664
0.75861
0.88810
1.00501
1.02007
1.04535
1.08108
1.12765
l.Ï8545
1.25521
1.33742
1.43312
1.54309
1.026M
1.17520
15
-1
-9
2
12
.3
-2
0
with
2.
METHOD OF VARIATIOITS
in-
The method described above is excellent when the
itial conditions all relate to one point.
tial conditions relate to two points,
Then the ini-
the initial slope
must be chosen so that the solution will pass through the
two points.
ie&i
-r
(i)
y"
- f(:;,:;í,;'
x, y.y0be
be the given equation; let x
be a solution obatined by a step-by-step
Let
point.
process,
the initial
using initial conditions
XX, yy,
- (2)
the
Thus we have
.
M'= Y
table
X
:i
)?
H
t
y
X,
xi
(3)
:
y1
:
x
A solution is desired for
where
Jyt
is small.
from (i) we have as
(4)
above.
Then
the
y= y0, y
y+cÇy,
y
cÇy
yt+yt,
y
and
principal infinitesimal
y,
is
substituLiri
xx0,
where
found by differentiating (i) and then
tue known values oP
,
'
foi
the
t.ble
29
is a linear horr!ogeneous
NOvi (4)
the second
order in ¿y.
differenti1 equation of
A solution is desired for the
initial condition
é1T.::
cfrt= i
O,
at
x=x0.
since it is linear and homogeneous,
CJy
ution of (4),
we
is
(3)
y is a sol-
is a solution.
The desired end point is
the table
then if
x, y.
(x,1,b).
To determine
The end point of
the correction,
set
r
b,
whence
(1
Since the initial slope for
for Cd'y will be C.
t
the initial slope
I
o
Example.
was 1,
Hence the corrected slope is
Y-Y+
(r)
¿y
f
ApDly the method of variations to
yfl
subject to
x=O,
yl
and xrnl,
y=2.
From the solution in the preceding section, the table for
j
lY'and
f
is
constructed.
30
ØY
X
-1.0000
-1.0000
-1.0000
-1.0000
-1.0000
-1.0000
-1.0000
-1.0000
-1.0000
-1.0000
-1.0000
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
0
0.19934
0.39476
0.58265
0.75991
0.92428
1.07407
1.20876
1.32806
1.43263
1.52318
By the use of tJis table, the solution to
(
\
7
\
JO y
+1
J
'
2
/'
(
\'
is constructed by
J
Mime's Method.
x
-.10016
-.1
o
o
.1
.2
.3
.4
.10016
.20133
.30453
.41075
.521111
.63688
.75860
.88831
.5
.6
.7
.8
.9
1.0
,
1.02648
1.17536
1.00500
1.00000
1.00500
1.02015
1.04534
1.08115
1.12758
1.18554
1.25510
1.33710
1.43282
1.54268
-.10018
0
0
-8
7
0
4
.10018
.20138
.30454
.41083
.52110
.63647
.75856
.88745
1.02622
1.17514
1.17536
Jij,
From
¡y"
E
5
the corrected slope is y'
.
.38874.
31
Mime's Method
The solution by
x
y
0.96681
1.00000
1.04469
1.10144
1.17086
1.25405
1.35113
1.46433
1.59384
1.74232
1.91042
2.10093
-.1
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
is
y'
y"
.27566
1.11292
1.5112
1.20246
1.26790
1.34768
1.44281
1.55520
1.68477
1.83448
2.00722
2.19930
2.42003
0.38874
0.50616
0.62970
0.76023
0.89964
1.04943
1.21122
1.30704
1.58026
1.78930
2.02097
The slope here is evidently too large.
A second applica-
tion of the process of variation gives the corrected
slope to be y
.33027.
x
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
y
1.0000
1.0386
1.0888
1.1510
1.2261
1.3147
1.4180
1.5369
1.6731
1.8277
2.0028
The solution is tabulated.
y'
0.3303
0.4433
0.5612
0.6853
0.8170
0.9577
1.1092
1.2730
1.4517
1.6451
1.8578
y"
1.1091
1.1520
1.2077
1.2768
1.3599
1.4583
1.5729
1.7082
1.8573
2.0278
2.2226
he step-by-step process is perhaps the most accurate,
at least the approximate error can be determined.
disadvantage is that it is time consuming.
The
32
iI:r.
CONVERGENCE
The numerical integration of second order differen-
tial equations with assigned end points is a process of
successive approximations.
Consider the equation
(1)
f(x,y,yt
Y"
let the solution be required to satisfy the conditions
xi,
ya;
xO,
(2)
y=-b.
These boundary conditions may be considered as quite general.
If there are any other boundary conditions, say
xx,
y
=
x=x,,
y0;
=;
by means of the transformations
x-x
I
Y=a
J4_
+(y-Y0\
)b
0,/
we have the boundary condition (i).
(y,-y0\
L
dx
dX
Lx-x)
b
dx
dX
Lx-x
and
L
''
Now
are to be substituted
o
in equation (1).
The differential equation (i) can be transformed into
the integral equation
(3)
Bx
Y
j(xt)
f(t,
y(t), y'(t)
dt.
Now A and B must be determined so that (3) satisfies (2).
Lo(4g,and
b=
a
ottJ
(Z-t) f(t), y(t), y'(t)
dt
3:r
whence
1í
b-a
-J
B
(2-t) f(t), y(t), y'(t) dt
o
Thus equation (3) becomes
(4)
y= a+
b-a
x
x-
L-t) f(t, y(t), y(t) dt
f.
(x-t) f(t, y(t), yt(t)
dt
which satisfies both (i) and (2).
Let
yy0
(x)
be a continuous function which satisfies
Now substitute into equation
condition (2)
(4)
and we
have
b-a
ai-1
(5)
x-
(x
-
x
J
-
(2-t) fedtl-J (x-t) f0dt,
o
o
where
f(t,
and
y(t),
y'
(t)
b-a
j
odttf
(2-t)
?0dt.
From (5) and (6) we determine
f
wefind
and
f(x, y,'(x),
+ ò-
y,(x)
i
x-
f
Now substituting f in (4)
;
-
(1-t) f,dt
(x-t) f,dt
PX
b-a
f
y'..
z
.1-t)
at+
By means of similar substitutions
functions
J
Jo
,dt.
we find a sequence of
ç,
y
(7)
I
'Jo
rg.i
a
-
J o (2t)
fr(3.t
(x-t) frdt
(t) frdJ
(X
b-a
T
f
where
f(t,
let
Now
Yr.,
rdt.
(t))
(t),
us examine the sequence
(8)
r
e,
Ç1-
r
:-'
2
f2Lt)
(-L) dtJx
(x-t)
(-)
dt
(?rr..,) dtt.JX (r-,) dt
If
1'
over
Is continuous
for each value of
9f
and,
in the interval
x
exist
and are
to
to y.
2, and 1f
and y',....,
=
U
finite in this
region, then from the law of the
(io)
r,
the range
x, y
and
mean
y,),
Ç-Ç
ere
and
denote the values of the partiaÏs taken at
y=3r
+
4-
I
T'=
e(y
o
_:rT:._i)
=
e -
i.
Let Mr denote the maximun absolute value of ? and N,.
when x varies in
denote the maximiwi absolute value of
the range
equality
O
x
j.
,
The
equality
(le)
becomes the
in-
(ii)
I
ç
-
- (et-t)
- £ Í
Thus
I]
£
IrJEI
(Îvi,(6r1+
o
J(x-t)
r'r1
(
o
Let
er
r
II)
)
dt
dt
E.
denote the maximum absolute value of
terval
L
x
'
of
rI-I
r
the
in
(11-a) lE
<
I
,'.
and
in the in-
denote the maximum absolute value
same interval.
f
x
-
Jx
(ß t) ("rer
(x-t)
Tr
(irerl
i
o
Q;
r'
r
)
)
dt
dt
If we integrate the inequality (11-a), we have
-
Ier+il
(xz
rr
-)
rer
"re
'
t
(Mrr
Hence
e,<. (ii.
(12)
. *f
In a similar manner we find that
(Llrer
.
Hr
)
(Z -t)
(I'/Irer
IE',I
4(Li1.
)--
f19
N
x(ie-i Nr
e(e+et
(13)
-.
e+ìe
+
dt
)
dt,
).
bothjL
c'3T
Now let
(12)
stant, whence
(14)
er.,.,
<
Q,
be less than Q a positive con-
cy
and (13) become
Z
£
LS_
e'f+I
(15)
and
2
4er'
)
and
(r+
)
Now adding (14) and (15) we have
e,e,<L Q(Z+ 3/2)
(i6)
ere
C
er-te'r
o
)
<Q(Z+3/2)
From (14) and (16) we see that
£Q2( 2(+
er+,
eri
(r)
<
3)
j..(
2t-t- 3)
-(
2g-f- s)
(
eri#e..,
),
e,..e,-f-e
(
In a similar manner, it can be shown that
(18)
3L
<
'
Q
C
2
Now
-0)
T
o
Since Iyr,,yrk
2j
er.,.,,
creases indefinitely,
Yr+,
1e
3)
(
-
)
1
etj
------- (r
-\T
).
converges to a limit as r ±fl
provided that the series of positive
cons tants
e,+eLe.i. ---------------- +
(19)
converges.
than
1,
If
the
ratio
R.=
for r greater thanj'
ere,
becomes and remains
, however
less
1arge( may be,
37
the series
(is)
Since
converges.
r
r-.
(
-
2
(
22+
r-,
21+
:3)
the ratio R will be less than 1,
(20)
4-e'
(
'
according as
3)
Now (20) is satisfied when
3L
QL2
c
<.
or when
1,
(21)
<
.2
Thus the first sequence of (J?) will converge u.nifoxm1y to
a limit,
and this limit will be a continuous function.
Let us consider the second sequence of (7)
case
As
yt11:
\;.1
(22
-?J:i
e;
(y3T-y) --------
e,
e
,
e
+ e;
e'
(Ò)
From (1G) we see that
(
y
3)
).
series of positive constants js
the
This serles converges if e.4,
than 1.
In tiaia
becomes and remains less
e±
remains less thanl
<1.
This is the same condition which insures the convergence
of the first sequence.
We have shown that the sequences
(7)
do converge.
The problem now is to show that these sequences actually
do satisty the differential equation (i).
s i nc e
a +
y
(7)
X
,
L
ri-e
¡
rdt
-t)
(
(x-t) frdt,
Jo
Jo
then upOn differentiating twice with respect to x,
(23)
yt'
r+I = f
r
Subtracting the quantity f"
from both sides of (20),
we have
"r+
whence from
(
24)
(
"
f-f r',t
=
ri-i
I
11),
1y"rs
s
-fr-,
From (17) and (18),
'
Q
(
e.
e,f+
(21) becomes
lQ(I
(25)
t?ip,
-f
).
r.(
1<
2L+
¿)
2''
s
(ee1
I
).
I
The right hand side of (22) can be made less
O
is
=
1,
however small
may be, provided that r
taken large enough and furthermore,
(20),
£(2Li-
(20)
must satisfy
3J
1.
2
Hence the sequences (7) satisfy the differential equation
ir
£
satisfies condition (20).
In many practical cases
ried out over
a
the integration can be
car-
much larger range than that given by (20).
The integration can be carried out, however,
at least as great as
that given by (20).
over a range
39
i«
*
CONCLUSION
Of the methods investigated
f(x,y,y') passing through
imation to the solution of y"
two end points,
in obtaining the approx-
the speediest and simplest was
method developed in partI.
the
third
The disadvantage is that al-
though the results obtained do apparently converge, there
is no way at the present time to check the limit of the
error.
i.E.
The most accurate method is that developed by
Mime, along with
the process of variations.
In using a step-by-step method,
to
solve the second
order differential equation with assigned end points,
firsL guess of the initial slope
a
is important.
the
To make
good first guess, we use the process of successive ap-
proximations to obtain an initial slope.
With this in-
itial slope, the differential equation is solved by means
of
a
step-by-step process, and incidentally the results
obtained by successive aprroximations are checked.
the
If
solution obtained by a step-by-step process and this
initial slope misses the second end point, lt can be corrected by the process of variations.
In studying any second order differentiel equation,
lt would be well to make a rough graphicel solution and
then proceed to the numerical processes.
The graph would
give some indication as to the total range of integration
permissible, and also indicate the best length of sub-
interval to take.
41
BIBLIOGRAPHY
Bridger, C.A.,'On the Numerical Integration of the Second Order Differential Equation with Assigned End
Points." A thesis, Oregon State College, August 1936.
Committee on Numerical Integration,"Nujnerical Integration
of Differential Equations."
Bulletin of the National
Research Council, Number 92
Levy, h,, and Baggott, E.A.,"Numerical Studies in Differential Equations.' London, Watts and Co., 1934
Mime, W.E., "On
the Numerical Integration of Ordinary
Differential Equations, Am. Math. Monthly, vol.33,
1926, pr.455-460.