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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 72, 500-511 (1979)
A Numerical Method for the Solution of Multi-Point Problems
for Ordinary Differential Equations with Integral Constraints
TAKEO OJIKA* AND WAYNE WELSH’
Department of Electrical Engineering, University of Southern California, Los Angeles, California 90007
Submitted by R. Bellman
We propose a method for the solution of a system of nonlinear ordinary differential equations with integral constraints, by transforming to multi-point boundary value problems. Examples are given.
INTRODUCTION
Numerical methods for the solution of two-point boundary value problems involving nonlinear differential equations and linear boundary conditions have been proposed by many authors (see [l, 21 for excellent surveys). Few methods have been proposed for two-point problems when the boundary conditions are nonlinear in nature and even less work has been done on multi-point problems (i.e. more than two points). For such problems, the methods of quasilinearization [3], invariant imbedding [4], collocation [5], and more recently, the initial value adjusting method [6, 71, have been devised. As early as 1922 Polya, [S], gave a sufficient condition for problems of the form
,x(n) + lplX(n-l) + a** + cpn-1X = 0, O<t<T,
x(ti) = ci , 0 < t, < ... < t, < T, (1.1)
to have a unique solution, where vi(t) were continuous functions. Bellman, [9], generalized these results to problems having boundary conditions of the form
.i
T x(t) ya(t) dt = bi , i = l,..., 71, (1.2)
0
where the yi were given functions. Observations of this type arise naturally, for example in drug concentration problems involving many compartments, [lo].
* On leave from Department of Technology, Osaka Kyoiku University, Osaka, Japan. t On leave from Department of Mathematics, University of Saskatchewan, Saskatoon,
Canada, S7N OWO.
500 0022-247x/79/120500-12$02.00/0 Copyright 0 1979 by AcademicPress, Inc. All rights of reproduction in any form reserved.
MULTI-POINT PROBLEMS WITH INTEGRAL CONSTRAINTS 501
In this paper we first consider general linear systems with integral constraints and then consider nonlinear systems of the form
2 =“f@, t), a<t<b,
s
b
/2(x, t) = c,
a
(l-3)
where x and c are n-dimensional vectors. Such problems are transformed into equivalent two-point boundary value problems of higher dimension and the initial value adjusting method is adapted to solve them numerically. This method can also be applied to problems involving a mixture of point boundary conditions and integral constraints, such as
s tj /2(x, t) = cij ) &(t1>,..., X(&J) = 0, ti (1.4)
where Q < t, < ..* < t, < b. The transformed problems then correspond to multi-point problems. The last example in section 5 is of this type, while the other examples correspond to two point linear and nonlinear systems. The computational algorithm is given in section 4, and the numerical results are provided in the last section.
2. LINEAR SYSTEMS
Consider the system
“(t) = 4) x(t) + b(t), a<t<b, (2-l)
with a constraint of the form
s b R(t) x(t) dt = c, a (2.2)
where A(t) and R(t) are continuous n x n matrices, b(t) is a continuous vector and c is a constant vector. We proceed to find an initial condition,
x(a) = x0 , (2.3)
so that the solution of (2.1) satisfying (2.3) also satisfies the constraint (2.2). Let @(t, a) be the fundamental matrix of the corresponding homogeneous
system
d(t, a) = A(t) @(t, a), @(a, a) = I, . (2.4)
502 OJIKA AND WELSH
Then, any solution of (2.1) can be written in the form
x(t) = @(t, u) x0 + jt @(t, s) b(s) ds, a
(2.5)
where x0 is an n-dimensional vector. Substituting (2.5) into (2.2), we have
j” R(t) [@(t, u) x0 + j” @(t, s) b(s) ds] dt = c. n a
(2.6)
Rearranging, we have
[ j” R(t) @(t, a) d”] x0 = c - jb jt R(t) @(t, s) b(s) ds dt. a a a
(2.7)
From the above discussion, we have the following.
THEOREM 2.1. If the matrix ui R(t) @(t, u) dt] is nonsingular, then the problem given by (2.1) und (2.2) h us a uniqw solution and the exact initial condition for the problem is given by
a”(u) = x0 = [lb R(t) @(t, a) dt]-’ [c - jab ja’ R(t) @(t, s) b(s) ds dt] . (2.8)
If (2.1) corresponds to an n-th order linear equation as in (1.1) and if (2.2) comes from conditions of the form (1.2), then the condition given in Theorem 2.1 is equivalent to the condition in [9], that
det [j” 4) y,(t)] f 0, i,j= 1 >a.*, n, a
where {ui}, i = l,..., n are linearly independent solutions of the homogeneous equation. To see this, let U denote the Wronskian of (ur ,..., u,> and let {vi}, i = l,..., n be the principal solutions of the homogeneous equation, so that @(t, a) is the Wronskian of {vi ,..., ~3. Then there is a nonsingular matrix, C, for which U = DC and
I
b
s
b
s
b
det R(t) @(t, a) dt = det RUC-1 dt = det C-r det uiqj dt. a a a
Thus, the conditions that the determinants do not vanish are equivalent.
3. NONLINEAR SYSTEMS
Let us now consider an equation
k = f(X, t), u<t<b, (3.1)
MULTI-POINT PROBLEMS WITH INTEGRAL CONSTRAINTS 503
with integral constraint
s b
h(x, t) dt = c, (3.2) n
where h and f are nonlinear n-dimensional vector functions that are twice continuously differentiable in x and continuous in t, and c is a given n-dimen- sional vector. In order to solve such a problem, we first replace it by an equivalent two-point boundary value problem. Set
3i”(t) =f(X, t), a<t<b,
s(t) = h(x, t), (3.3)
with boundary conditions
x(a) = 0, z(b) = c. (3.4)
Then we have the following theorem.
THEOREM 3.1. Suppose that (x(t), z(t)) is a solution of (3.3) and (3.4). Then
x(t) satisfies (3.2). Conversely, suppose (3.1) ha s a solution, x(t), satisfying (3.2). Then (x(t), z(t)) solves (3.3) and (3.4) where x(t) = Ji h(x, t) dt.
Proof. If (x(t), z(t)) solves (3.3) and (3.4), then
s ’ h(x, t) dt = z(b) = c. a
The converse statement is clear. According to the above theorem, if we find a solution (x, z) of the problem
(3.3) (3.4) and if x(a) = x,, , then x0 is an exact initial condition for the original problem (3.1), (3.2). Thus we can apply the initial value adjusting method to the transformed problem in order to find numerical solutions. We outline the method below.
Let (%v~, 0) be a set of initial conditions for (3.3) at the k-th iteration of the procedure. Consider the initial value problem
2(t) = f (x, t), x(a) = kx0 ,
2(t) = h(x, t), z(u) = 0, (3.5)
and denote the solution by (K~, %). In order to find k+l~O , we perturb the initial data for x in each direction and then consider the following n initial value problems
(3.6)
504 OJIKA AND WELSH
j = l,..., n, with solutions denoted (Gj , “xj) respectively, where the perturba- tion parameter, E, satisfies 0 < E < 1, and ej denotes the j-th unit vector, (0 ,..., l)..., 0).
We now define an R x n matrix “Y(t, a; E), whose j-th column has value at t = b given by
“Yj(b, a; c) = 1 [%j(b) - “z(b)], E
j = l,..., n. (3.7)
This matrix replaces the usual fundamental matrix that arises in the quasi- linearization technique as can be seen from the following theorem.
THEOREM 3.2. Let “Y, , “ul, be the n x n fundamental matrices satisfying
with
(3-g)
(3.9)
Then lim,,, “Y(b, a; l ) = kY,(b, a). If kY(b, a; E) is nonsingular, then the adjusted
initial value for (3.5) is given by
Jc+lxo = k~o + “Y-l(b, a; <) [c - “z(b)],
and the adjusting method (3.10) has the quadratic convergence property.
The proof of Theorem 3.2 is analogous to that given in [q.
(3.10)
4. COMPUTATIONAL ALGORITHM
We now summarize the preceding discussion in the form of an algorithm.
Step 0. Transform the original problem given by (3.1) and (3.2) into (3.3) and (3.4).
Step 1. Set k = 0, and prescribe the values of the perturbation parameter l , the convergence criterion C( > 0) and the initial condition Ox0 .
Step 2. Set G(a) = Go, and compute the initial value problem (3.5) and obtain the resulting terminal value ‘zz(b).
MULTI-POINT PROBLEMS WITH INTEGRAL CONSTRAINTS 505
Step 3. Compute the error G defined by
“G = 1; [c - %(b)J’ [c - k~(6)]/1’z. (4.1)
If “G < 0, then terminate the procedure. If LG > (T, then proceed to the next step.
Step 4. Set j = 1.
Step 5. Compute the perturbed initial value problem (3.6) and obtain the resulting terminal value %#). and calculate the Yj(b, a; e) given by (3.7).
Step 6. If j < z, then set j = j + 1 and return to Step 5. If j > n, then proceed to the next step.
Step 7. Determine a new initial condition from (3.10), replace K by k + 1, and return to Step 2.
5. EXAMPLES
The above work is illustrated by three examples. The first corresponds to a linear system having integral constraints with a linear kernel. The second and third examples correspond to a nonlinear system and have the same solution, but the constraints are different. The second example has a nonlinear kernel integrated over the entire interval while the third example has a mixture of point conditions and integral conditions over different intervals. This example corresponds to a five-point problem.
The tables contain a few of the calculated and actual values of the solutions, as well as the initial approximations, Ox(t), the number of grid points, p, the number of iterations, K, and the error, G. The computations were done in double precision on an IBM 370/158 at the University Computing Center of the University of Southern California.
EXAMPLE I. Consider the differential equation
f(t) + x(t) = 0,
with the constraints
I 7rl2 s n/2 x(t)& = 1, xsintdt = 1.
0 0
506 OJIKA AND WELSH
The exact solution is
x(t) = (42 - I)-l(sin t + (97/Z - 2) cos 2).
The transformed system and boundary conditions are
A comparison of the computed values, “x(t), and actual values, x(t), at the end points is given in Table 1.
EXAMPLE 2. Consider the nonlinear system
112 x1=-x1x4 )
2, = 3x, )
Le3 = xi’, + 2x3 ,
ti4 = xy2,
with constraints
I 1
xle(t+2)a/4 dt = 1, 0
s o’ (x3 + x4) t dt = $ + ;< ,
s o1 (x2 + x4)2 dt = ; + ;;,
~01(x3-x2x4)dt=-~e3+~+&. i
An exact solution is given by
q(t) = e--(f+2)2/4,
x2(t) = e3',
x3(t) = tezt,
x4(t) = (t + 2)2/4.
(5.1)
(5.2)
TAB
LE
I
Res
ults
fo
r E
xam
ple
1, p
=
500,
k
= 2,
G
= .3
x
lo-1
7
w9
x(O
)”
km
4749
“x
(742
)
x1
-1
-.751
9383
9388
411D
00
-
.751
9383
9388
961
D
00
.175
1938
3938
8410
01
.I7
5193
8393
8909
0 01
x2
2 .1
7519
3839
3884
10
01
.175
1938
3938
9180
01
.7
5193
8393
8841
10
00
.751
9383
9389
1840
00
X3
0 0
0 1
1
X4
0 0
0 1
1
. “0
.175
0 01
=
1.75
.
TAB
LE
II
Res
ults
fo
r E
xam
ple
2,
= 50
0, k
=
p 7,
G
= .5
10
x lo
-l5
ox(o
) ‘4
0)
“x(0
) 41
) W
) -_
_-
.__
-~__
___
__
-. ~~
~ -_
__
Xl
1 .3
6787
9441
1714
40
00
.367
8794
4117
0390
00
.1
0539
9224
5618
60
00
. IO
5399
2245
623
1 D
00
X2
2 1
.lOO
OO
WM
M42
30
01
.200
8553
6923
1880
02
.2
0085
5369
2339
00
02
X3
1 0
.107
7058
7356
3640
-10
.738
9056
0989
3070
01
.7
3890
5609
8963
70
01
X4
.5
1 .9
9999
9999
9868
50
00
.225
OO
OO
O00
0000
0 0
1 .2
2499
9999
9980
30
0 I
X6
0 0
.O
.I000
0000
0000
000
01
.1O
OO
OO
OO
OO
OO
O0D
01
X6
0 0
.O
.249
3097
3580
6600
01
.2
4930
9735
8066
00
01
x7
0 0
.O
.899
9345
6410
6260
01
.8
9993
4564
1062
60
01
x8
0 0
.O
-.975
0512
8271
858D
01
-.9
7505
1282
7185
80
01
TAB
LE
III
Res
ults
for
Exa
mpl
e 3,
p
= 50
0, k
=
5, G
=
.294
x
lo-l4
ox(o
) 4%
kx
(o)
x(11
4)
w
l/4)
x(1/
2)
Xl
I .3
6787
9441
1714
40 0
0 .3
6787
9441
1557
1 D
00
.282
0629
5169
3820
00
.2
8206
2951
6831
50
00
.209
6113
8715
1100
00
X$
2 1
.100
00O
OO
OO
O36
40 01
.211
7000
0166
1270
01
.2
1170
0001
6672
70
01
.448
1689
0703
3810
01
.x3
1 0
.231
6747
2269
8010
-IO
.412
1803
1767
503D
00
.412
1803
1771
298D
00
.I359
1409
1422
950
00
X4
.5
1 .9
9999
9999
9608
00
00
.126
5625
0000
0000
01
.I2
6562
4999
9559
0 01
.1
5625
-D
01
$ x5
0
0 .O
.1
4378
1300
4893
9D
01
.143
7813
0048
939D
01
- I-
x -
- 6
0 -
- -
z + F “x
( I /2
) x(
314)
‘x
(3/4
) x(
l) kx
( I)
z -._
__.
~~..
~~
____
~ ~~
~~
~~-..
_~
. .-
Xl
.209
6113
8714
4210
00
.I509
7741
8455
910
00
.150
9774
1845
1720
00
.I0
5399
2245
6186
0 00
.I0
5399
2245
5948
0 00
X2
.448
1689
0704
2890
01
.948
7735
8363
5850
01
.9
4877
3583
6474
40
01
.200
8553
6923
1880
02
.2
0085
5369
2327
10
02
Jcs
.135
9140
9142
873D
01
.336
1266
8027
5350
01
.3
3612
6680
2833
70
01
.738
9056
0989
3070
01
.7
3890
5609
9026
10 0
1
X4
.I562
4999
9995
100
01
.189
0625
0000
0000
01
.I8
9062
4999
9461
0 01
.2
2500
-00
01
.224
9999
9999
4120
01
x5
- -
- -
-
X6
.O
.O
.481
9653
2532
1680
01
.4
8196
5325
3216
80 0
1
MULTI-POINT PROBLEMS WITH INTEGRAL CONSTRAINTS 509
The transformed system consists of (5.1) together with
3i”, = X1e(t+2)z/4, x5(0) = 0, 4) = 1,
3i’, = (x3 + x4) t,
.e’ = x2 + x42,
X,(l) = ; + ;,
e3 553 X’(l) = J- + 240 T
f, = x3 - x2x4 , x3(0) = 0, 65 e2 53
fg(l)=-~e3+~+~-
The numerical results for the exact and approximate solutions are given at the end points in Table II and the values of the error criterion, G, are listed separately for each iterate in Table IV. The convergence rate appears to be quadratic, as predicted by the theory, [l 11, when the initial approximations are close to the exact values.
EXAMPLE 3. Consider again system (5.1) but with constraints
x1 (f) + x2 (4) - x3 (2) = e-(2.25)2’4 + &e1.5,
x4(O) - (xi(l)) (x2(l)) = 1 - e”.75, (5.3)
s
l/4
0
(3x, + x42) dt = e’J.75 + (2’25)i[ 11* ,
,:,(x2 + ~3) dt =
8e3 + 6e2 - 8e2.25 - 3e1.5
s 24
(5.4)
TABLE IV
Convergence Rates for Example 2 and Example 3
Iterate G, Example 2 G, Example 3
.2133041836317O.D 01
.481371663240580 00
.882735742578180-01
.180684070710000-01
.1819351543129OD-02
.293363167279670-04
.815365548691130-08
.144179629615240-14
.445866538837910 01
.61350339493972D 00
.747846627941600-01
.240842355687050-03
.837445307504540-07
.719611993662720-14
4o9/72/2-9
510 OJIKA AND WELSH
Again (5.2) provides an exact solution. The transformed system consists of (5.1) and (5.3) together with the conditions
2, = 3x, + x42, x5($) = e”.75 + (2.295 - 112
80 ’
5 = x2 + x3 3 x,(Q) = 0, x&l) = &3 + (j$ - &$*a5 _ ‘je1.5
24
From the boundary conditions, it is evident that this is a five-point problem. For this type of problem the algorithm adjusts the values at all of the points and the error criteria, G, includes contributions from the given point conditions as well as from the continuity conditions that are imposed at the interior points. The exact and computed values for all five points are given in Table III. The initial data shown in that table was given for t = 0, and the initial data for t = l/4, l/2, 314 was then obtained from the values of the first integration. The convergence rate can be seen from Table IV, where G values are listed, and again the convergence appears to be quadratic.
ACKNOWLEDGMENTS
The authors would like to thank Dr. R. Bellman for many interesting discussions on multi-point problems and also the University of Southern California for the generous use of their computing facilities.
REFERENCES
1. A. K. Azrz, “Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations,” Academic Press, New York, 1975.
2. H. B. KELLER, “Numerical Solution of two point boundary value problems,” SIAM, Philadelphia, 1976.
3. R. BELLMAN AND R. KALABA, “Quasilinearization and Nonlinear Boundary-Value Problems,” American Elsevier, New York, 1965.
4. R. BELLMAN, Invariant imbedding and multipoint boundary-value problems, /. Math. Anal. Appl. 24 (1968), 461-466.
5. R. D. RUSSELL, Collocation for systems of boundary value problems, Namer. Math. 23 (1974), 119-133.
6. T. OJIKA AND Y. KASUE, Initial-value adjusting method for the solution of nonlinear multipoint boundary-value problems, 1. Math. Anal. Appl. 69 (1979), 359-371.
7. T. OJIKA, A numerical method for the solution of nonlinear multipoint boundary value problems-Initial value adjusting method with interval decomposition, sub- mitted.
8. G. P~LYA, On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Sot. 24 (1922), 312-324.
MULTI-POINT PROBLEMS WITH INTEGRAL CONSTRAINTS 511
9. R. BELLMAN, A note on the identification of linear systems, PTOC. Amer. Math. Sot. 17 (1966), 68-71.
10. J. G. WAGNER, “Biopharmaceutics and Relevant Pharmacokinetics,” Drug Intelligence Publications, Hamilton, Ill., 1971.
11. T. OJIKA, On quadratic convergence of the initial value adjusting method for nonlinear multipoint boundary-value problems, in preparation.