a numerical method for the solution of multi-point problemsboundary value problems. examples are...

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)


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-dimensional 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 perturbation 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 quasilinearization 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).


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


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, submitted.

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.


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.

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