Iterative method for systems of two-point boundary value differential equations based on contraction mapping principl

Manicke, R. L.

English

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MANNED SPACECRAFT CENTER.
HOUSTON,TEXAS
June 1969
S
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.L.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
MSC 00122
INTERNAL NOTE
AN ITERATIVE METHOD DERIVED FROM EXISTENCE
AND UNIQUENESS THEOREMS FOR SYSTEMS OF
SECOND-ORDER, NONLINEAR, TWO-POINT-
BOUNDARY-VALUE DIFFERENTIAL EQUATIONS
AUG 1970
^rCfii VE9QSA
	
!.	 STI FIC,11^
	
^	 IBPUT 8+1A1.'Cy
°,o	 (ACCES	 R ►.	 5	 (THRU)
^_	 vO	 (PAGES) (COID)
QNASA Ck OR TMX OR AD NUMBER)
	 (CATERY)L&
IMSC 00122
INTERNAL NOTE
AN ITERATIVE METHOD DERIVED FROM EXISTENCE AND UNIQUENESS
THEOREMS FOR SYSTEMS OF SECOND-ORDER, NONLINEAR,
TWO-POINT-BOUNDARY-VALJE DIFFERENTIAL EQUATIONS
ic. C
PREPARED BY
R. L. Manic e
Theory and Analysis Office
Computation and Anal y sis Division
APPROVED BY
, k J. M. Lewallen
Chief, Theory and Analysis Office
Computation and Analysis Division
JM
ugene H. Brock
Chie , C	 utation and Analysisaivision
NATIONAL AERONAUTICS AND SPACE ADMINISTRATICN
MANNED SPACECRAFT CENTER
HOUSTON, TEXAS
June 1969
Section
	 P
.^.
SUMMARY . . . . . . .	 1
INTRODUCTION	 . . . . .	 . .	 . . . . . . . . . . .
	 1
SYSTEM WITH IMPOSED CONDITIONS. . . . . . . . . . . .
	 2
GREEN'S FUNCTION FOR THE SYSTEM . . . . . . . . . . .
	 3
CONTRACTION MAPPING THEOREM . . . . . . . . . . . .
	 7
THEOREMS FOR UNIQUE EXISTENCE . . . . . 	 . . . . .	 7
ITERATIVE PROCEDURE . . . . . 	 . . . . . . . . . . .
	 12
EXAMPLE . . . .	 . . . . . . . . . . . . . . . .
	 13
CONCLUSION . . . . . . . . . .	 . . . . . . . .
	 . .	 16
BIBLIOGRAPHY . . 	 . .	 . . . . . .	 . . . . . . . .
	 17
YS
k sri:; rY
^d :•
°^ o
P_
AN ITERATIVE METHOD DERIVED FROM EXISTENCE AND UNIQUENESS
THEOREMS FOR SYSTEMS OF SECOND-ORDER, NONLINEAR,
TWO-POINT-BOUNDARY-VALUE DIFFERENTIAL EQUATIONS
By
R. L. Manicke
Manned Spacecraft Center
SUMMARY
;:- This paper develops some applications of the contrac-
tion mapping principle. 	 It shows sufficient conditions to
guarantee existence of a unique solution of a two-point-
boundary-value system of differential equations.	 Moreover,	 4
4 ;	 k.
it exposes an iterative method that must converge from any
initial guess to the unique solution, provided certain
criteria are satisfied. 	 An example is given to clarify the
r
procedure.
4 4	 `4t	 •
.s.	 N
N	 •^'
INTRODUCTION
r
A boundary-value problem in ordinary differential
equations differs from an initial-value problem in that it
must satisfy conditions at two or more points. 	 An nth
order differential equation can have 	 n	 conditions at one
point or one condition at 	 n	 points.	 In fluid mechanics
and trajectory mechanics, systems of second-order equations
with conditions specified at two points are often encoun-
tered.	 This paper will consider only systems of this form.
x
^v
u
SYSTEM WITH IMPOSED CONDITIONS
Consider
I' Y(t) + fl (t ► 	 Y 1 ( t ),	 Y2(t)'	 ..., Yn (t) ► Y1(t) ► Y2,(t), ...,	 yn(t ))	 0
R
Y2 (t) + fz(t,	 Y 1 (t) ► 	 Y2 (t),	 ..., Yn(t) ►
 y(t) 	 Y2(t) ► ...,	 y;,(t))	 0
CI) '•
yn(t) + fn (t ► 	 Yi (t)^	 Y2 (t),	 ..., Yn(t), Yi(t) ►
 yz(t) ► ...,	 Y;(t ))	 0
Yi(a) Ai	 Yj(b)	 -	 Hi
or in vector notation
t	
° Y ►► + T( t ^ Y , Y')	 _	 0	 with y	 CYl, Y2
..8	 yn^T
s
"
When	 f	 satisfies a Lipschitz condition t, it means
a wlt5 a 	 g.
:	 y=1 that each component satisfies the following
'fi(t^ y l ^ y2 ^	 ...	 yi,	 ..., y	 P	 Y1V	 ... V	Yif "'	 Yn)
-	 f i ( t t Y 1 t ...	 ...Y2'	 '	 Xi'	 9 .,.	 X!Y n t	 y^1'	 ,	 i s .•.	 y^n)
Ki IYi	 - xi ,	 + Li ly !	 -	 X!j with	 K1 , L 1 0
}1
h1^	 1ri
Y
6E^;
it
Ar
and (t, Y 1 9 Y2 0 ..., yip ..., yn' ylP ..., yi, ...0 Yn),
Ct! ylf y2, ...^ xi' ...' ynt Y 1
19... ' xi• ...^ yn ) elements
of the domain of f i	 [a,b] x R2n , 1 ;6 i S n	 Define
the minimum Ki , L i that will preserve the inequality as the
Lipschitz conetanta of fi . Notice that this condition
does not presuppose the existence of a solution y(t) of
system (I) . ,,
GREEN'S FUNCTION FOR THE SYSTEM
Now, in order to construct an iterative procedure, it
is sufficient to use a Green's function to transform each
equation of a homogeneous boundary-value system into an
equivalent integral equation system. (For the definition
and properties of Green's functions, see No. 2 in Biblio-
graphy.) Such a Green's function is:
(b - t) (s- a
	 a	 s	 t	 ba
G(t,$)
-(b s) (t ,	 a t s i ba
(This Green's function is for the operator -y".) Since
zero-boundary conditions are rarely encountered, it is
usually necessary to transform the system into one with
zero-boundary conditions and satisfy the original Lipschitz
condition. Such a transformation is
T t	 aB bA + A- 1"W tC)	 a- b
3
Clearly, T i (a) = A i and T i (b) = B i 	Therefore, if-T(t)
is a solution of problem (I) , then z'(t) _ -7(t)  - T(t) is
a solution of
(IT)
T(a)	 0	 z'( b)
where
F (t, Z Z') n f (t, Z + -"O( t) , 'Z' + 7' (t) )
Since
^F (t•..	 z	 •..	 z	 ...	 zr	 ..^	 z	 (t	 ....	 ...	 ...	 1	 ...	
n)
	
n	 Ii	 r	 r	 i t 	 r	 r	 ► 	 i t 	 r	 n
r )	 F i	 r	 r qt r 	 r z n r	 r q r	 r Z
I fi(t, ...,
 
z 	 T i (t), ..., zn + Tn (t)r ..., zi + T (t),	 .' zn' + Th(t)
...
	 +	 ...	 1	 "
	
+ ." t
	
egg, z r + T tf i (tr ...r qi + Ti(t),	
r zn	
T n (tJr	 r q i 	 i( )r	 r n	 n( )I
S KL I( z i + T i (t )) - (qi + r j (t))I + L i l(zi + Ti(t))	 (qi + Ti(t))I
Kilzi - 
qi ) + Lilzi - qil
this is exactly the same Lipschitz condition that each
f , 1	 i ;S n , must satisfy.
4
rw
Theorem:
Problem (II) is equivalent to
b
-..	
-*.^
TW 	 G(t, s)F(s, Z(s) z ( s )) d s	 C 1)
a
(Two problems are equivalent if and only if they have
exactly the same solutions.)
Proof:
i t	 G(t, s)F(s, i(s), 'z' (s)) ds() ifa
b at fa
(s _ a)F(s, Z( s), w ( s )) ds
bt - a	 (b - s)F'(s, (s) , '' (s)) dsb a t
Clearly, Z(a)	 0	 (b)
By the product rule for differentiation, it may be
'	 found that
i' (tla
	
(s - a)F (s, Z (s) , Z' (s)) dsfa
b	 t (t - a)F (t , z ( t) , z + (t))
1	 b
F—"^'a
	
Ch - s) F (s z ( s ), Z ' ( s )) ds
t
_	 s
(b	 t) F (t, z(t)	 ' (t)
5
t
AZ' Lt)--a Ct - a)-F" ( t : Ct) Z' Ct)
+ (-2t + a + b)F Ct, z ( t) , Z' ( t ) )
'b = a (t - a) F' (t , (t) , Z' (t)
+	 b	 a F (t, z(t) , Z' (t)
+' (2t - a - b) F
 ( t , Z (t) , z' tt)
t)(a
	 t)- ' (t, z ( t ) , Z' M )
5 ----	 (t	 b) F(t, ZCt), xCt))
- a
-F (t , Z (t), Z' (t)
or
The fundamental theorem of contraction mappings will
be applied to this crass of systems. (A function
T : S i—n—t—°.8 where 5 is a normed linear space is said
to be a contraction mapping if and only if there is a real
number a,0 < a < 1 , such that for all x, y e S
II TX	 Ty II ; all x - y II )
• il k.	 i
CONTRACTION MAPPING THEOREM
For every contraction mapping defined on a complete
nonmed linear space S , there exists a unique fixed point
x e S such that Tx x	 Moreover, for any x  e S
the iterates
w	 Tnx	 Tn -1x	 = Tn-2x 	 . : " Tx
	
x
Q	 1	 2	 n-1	 n
-converge to the fixed point x , and
n
II x
n
 - X1  S	 a=--a II x1 x0 II
!	 (See Bibliography No. 3, pp. 213-216.)
THEOREMS FOR UNIQUE EXISTENCE
	
First consider the system TV + f(t 'y)	 0, 7(a) A,
and y(b) B	 Let S be the space of continuous functions
on [a,b]. This .linear space is complete with norm
IIvII
	
	
max j maxj v, (t)j j
1^iSn [a,b]
In the following theorems it will be necessary to obtain
upper bounds for J 
b	 f b aGG(t,$) ds and 	^tss), ds
Ja	 a
Lemma 1:
a
bG i s ds	 b a 2 and
f b l 
aG i s ds	 b a
a	 a
7
^I
r.
Proof
bG (t, s) ds = b	 t	 t (s - a) ds + t	 a	 bF^a (b - s) dsa	 a	 b a ft
^b	 t t- a _
2 
+ t- a b- t 2
- a	 a)
_ -t t__,a,) S b- a.28*
and
J
b I aG t s	 =	 1 1 t (s - a) ds +	 1 ft b (b	 s) ds-a a a
_ (-t=- a Z +(b t) 2	 b a
2 (b
-- a)	 2
Theorem 1;
If the system y" + T(t,'y) = 0, T(a) = A, and y'(b)
satisfies the Lipschitz condition t and b - a < (8/K)1/2
where K = max K-. 	 then a unique solution exists.
I^i.n 1
Proof:
First, the system must be transformed into a homogeneous
system; it then becomes	 " + T(t,-z) = 0 , z'(a) = 0 = i(b)
With the right side of equation (1) as the operator in the
following equation, an attempt is made to find sufficient
conditions when it is contracting. Notice that the operator
T defined by
ti
v^F,_
3b
T(F)	 G(t,$)F(s, z(s), z' (s)) ds is a linear mapping
a
from S into S
b
Tu i (t) - Tvi'(t)	 f G (t v s) [Fi (s, ul (s) , u 2 (s) , ... 1 ui (s) pa
.. un (s)) - F i (s p ... 0 v i (s) ,
... , vn (s)) ds
Now, the Lipschitz condition is used to find
b
ITui ( t) - Tvi M	 S	 G(t,$)Ki,ui(s) - v i (s) I dsa
_ _ b
ITu i (t) - Tv i (t)l	 Ki +u - vII J G(t,$) dsa
Since by Lemma 1
fb G(t,$) ds	 b 8 a)2
a
7
rM1''
	
S	 '
d.
	
J`	 ^tAq^{i
i
r
or if
b - a < (8/K)1/2
a unique solution exists, namely: T(t) = i(t) + `T (t)
Now, consider the system y'" + f( to 'y, y') _ 0,
T(a) = A, and 7(b) = B	 This time, let S be the space
of continuously differentiable functions. The space is
complete with norm jjvjj	 max	 max (K,lv,(t)l + L,jvi (t)f
1jign [a,b]
Theorem 2:
If the system T + f(t, y, y'') = 0, y'(a) = A, and
O) ='F satisfies the Lipschitz condition t, and
with K = max K.K b	 a 2 + L b- a	 < 1	 tin i
and L = max L.
1 ;Sifln 1
then a unique solution exists.
Proof:
The same transformation as Theorem 1 shall be used;
again using T(t), the system is transformed into a new one
with homogeneous boundary conditions:
71 + F( t, It X,)	 0
T(a) _ 0 = X(b)
10
Ai
rNow form,
fbG(t.$) I 	(s,	 (s ).	 (s),	 , u (s).	 (s ).	 I u (s))ITu i (tj	 Tv i (t)I s	 fi	 1	 i	 n	 1	 n
a
f i (s, v i ( 's ), •••, vi(s), ..., v(n), vi(s), ..., vn(s))l ds
b
	
S	 G(t,$)(Kilui(s) - v i (s)I + Li lu,'(s) - vi(s)l) ds
a
*
 S 	 bI I3 - 'tTI I f G(t,$) ds S	 b- a	 1 1'u - 'v i I
a
and
b
d Tu	 i(t) - d Tv (t)	 aG t s	 f (s, ... u O ( s ))l ut i
	 at	 fa I	 at 11 
y:	
- f i (s, ..., v ' (s))	 ds
r	
u' f b	 t s ' dsI aG ata
Hence by Lemma 1
b	 a 
II u - vll2
which together imply
2
( I Tu' TvII 5 xb+ L b _ a Liu - 'vfI
where K	 max K., L _ max L
s	 l^ign 
1	
l^ n 1
:a
s	 Y.
.•F Vii•
S
F't
	
Therefore, if
K (b 	 + L h - a	 < 1
8
the mapping 'T is contracting. Hence, a unique solution
exists, namely: y(t) ='x(t) +'(t)
ITERATIVE PROCEDURE
If the system is of the form 711 + F(t, z') = 0 ,
'z(a) = 0 = T(b), the second part of the contraction mapping
(0
theorem states that for any starting value T(t) E S
the iterates of the ith component converge to z i (t), that
is
b	 (n-1)	 (n-1)	 (n-l)
(z^(t)	 =	 G(t,$ ) F. s, z (s), •••, z.(s), •••, z (s)	 dsi	
a	
i	 1	 1	 n
(0)	 (0)	 (0)	 (0)	 T
converges . to z  (t) for any Ti (t)	 z1 (t) , z 2 (t) , • • • , z n (t)
or
-w (	 (n-1)
+ F t, 'z
	
0
(n )	 (n)
T(a)	 0 = T(b)
converges to a solution.
0
Moreover, convergence to the solution is uniform, and
4
t bound for error on each iterate is given by
n	 ,
	
(0)
T z ^
	
a	 •
Phe method is as follows
(0)
(1) Select any appropriate 'T(t).
(1)	 (1)	 W.(2) Calculate f(t); select this 'T(t) as a near z (t) ;
proceed like this using each iterate as a new
EXAMPLE
Consider the system
``	 (1) Y1( t ) + Y 1( t) + Y 2( t )	 0
(2) y2(t) + y 1 (t) + sin y 2 (t) - 0
Y1(o) = 1	 yl (l) = 1
Y2 (0)	 0	 y2 (1)	 0
mss;'. s
13a
• t
	 so that
I f i ( t , Y l , y2 9 ..., yi, ..., Yn' yl, ... , Yn)
- f i (t , yl, Y2, ... , Xi, ... , Yn , Yi , '' • , Yn) I
K
	 ;S Ki l yi 
- Xil
is for (1)
IY 1 + Y2 - xi - Y 2 1 = ( 1 )I Y 1 - X1 1
and for (2)
IY l + sin Y2 - y l	 sin x2 1 = ( 1)IY 2 - X2 1
2
a = max K  b - a	 = 1/ 8
T l (t) = 0 - 1 1 - 1 
^1 1)t = 1
Y_
r
T2 ( t )	 0 ,
The system is transfor
Z 111 (t) + z l (t)
V2 (t) + z l (t) +
since (t,, z') - 7(t, z
so T (t) 	 (1,0)T
ned into the homogeneous system
+ Z 2 ( t) + 1	 0
1 + sin z 2 (t)	 0
14
	'	 (o)	 TLet	 z = (1, 7/2) . Now, using the Green's function
for this system, it is found that
(1)	 1
z 1 Ct) s f G(t,$) (2 + n/2) ds
0
	
t	 1
(1 - t)	 s(2 + n/2) ds + (-t)	 (s - 1) (2 + n/2) ds
fo 	 ft
	
t 2 1- W4 	 n _ t s - 1 2 4+ -ff
R
'— t
= t (i	 t_) 4 +	 + t t _ 1) 2- 4 + 7r2 	—^-	 2	 —2
'	 - 
4n
It
o	 t 3 + t3 - 2t 2 + t]	 4--^- t (1 - t)
:4
Similarly,
	
C1)
	 3t c 1 _ t)
.	 z (t)x	 2
so
	
-.	 4	 3t(1 t
	
z	
— t (1 _ t) '	 2
	
°ta	 1S
a. b
r
f
-^
By the contraction mapping theorem,
1/8	 4 + ^r
	
3t 1 -	 t
11 z	 -	 z	 I 1	 --	 (1,	 ir/ 2)z -	 "" _ t (1	 t	 ,	 ---	 -
(0)
	 (1)1	
max	 max	 z	 -	 z,	 where	 i	 102fi	 i[0911	 1
.385
Since this must converge uniformly, the next iterate obtained
( 1 )	 (0)
from using	 'z	 as a new	 z	 must be even "closer," (which
is made precise by the definition of the norm).
^I
CONCLUSION
This investigation has shown how an iterative method
to solve two-point-boundary-value systems of differential
equations Js obtained from the principle of contraction
AF
4 mappings.	 Moreover, it states some conditions that will
guarantee this solution to be unique, and that the method
A: will uniformly converge to this unique solution for any
starting value.	 Possible logical extensions of these
- results could be found by assuming a variable but bounded
terminal value	 b	 In cases where the length b-a is too'
large to guarantee unique exis tence, find conditions that
can determine specific intervals on the real axis for which
unique solutions exist.
:., 16
r'
=' '	 17
BIBLIOGRAPHY
1. Courant, R.; Hilbert, D: Method of Mathematical Physics.
Wiley and Sans,	 Inc.,	 1953, Chap V.
2. Yosida,	 K.: Lectures on Differential and Integral
Equations. Interscience Publishers, Vol. X, 19609
pp.	 64-76.
3. Collatz, L.: Functional Analysis and Numerical Math-
ematics. Academic Press, 	 Inc., 1966.
4. Bailey, P.; Shampine. L. F.; and Waltman P.:	 Nonlinear
Two-Point Boundary Value Problems.	 Academic .Press,
Vol.	 44,	 1968.
S. Keller,	 H.P.: Numerical Methods for Two-Point Boundary
Value Problems.	 Blaisdell Publishing Co., 1968.
e
w
^. n
4
r


An iterative method derived from existence and uniqueness theorems for systems of second-order, nonlinear, two-point-boundary- value differential equations

An iterative method derived from existence and uniqueness theorems for systems of second-order, nonlinear, two-point-boundary- value differential equations

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