Reducing a singular linear two point boundary value problem to a regular one by means of Riccati transformations

to a regular one by means of Riccati transformations

Citation for published version (APA):

Loon, van, P. M. (1983). Reducing a singular linear two point boundary value problem to a regular one by means of Riccati transformations. (EUT-Report; Vol. 83-WSK-03). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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ONDERAFDELING DER WISKUNDE DEPARTMENT OF MATHEMATICS AND

EN INFORMATICA COMPUTING SCIENCE

Reducing a singular linear two point boundary value problem to a regular one by means of Riccati transformations

by

Paul van Loon

EUT-Report 83-WSK-03 May 1983

Abstract

by

Paul van Loon

In this note a method for solving a two point boundary value problem in the interval (0,1) for a linear first order system of ordinary differen-tial equations with a singularity of the first kind in t = 0 is examined. The idea is to replace it by a regular problem on some subinterval (0,1). To this end a singular initial value problem of Riccati type on (0,0] has to be solved. This initial value problem is such that the spectrum of the corresponding Jacobian at t = 0 is in the closed left halfplane ,

-Moreover, if the coefficients of the differential equation are analytic, so is the solution.

§1. Introduction

n

Let x : [0, I] + lR satisfy the differential equation (DE)

(l. I) txf(t)

=

A(t)x(t) + f(t) , t € (O,t) ,

~n n .

with A : [0, I] + lR and f: [0, I ] + lR continuous in [0, I] and ana1ytl.c at

*

t = 0 • and the boundary conditions

(1.2a) x(O)

=

lim x(t) exists t+O

and

(1.2b)

o

I sxn 0 1 s .

where B ,B € lR such that rank(B ,B ) ,. sand b € lR (s Wl.l1 be speci-fied later).

The singular boundary value-problem (BVP) (1.1) and (1.2) will be replaced by a regular BVP on a subinterval (0,1) (5)0). To this end some auxiliary singular initial value problems (IVP's) on (0,0] have to be solved. It will be seen· that these IVP's have two nice properties:

*

(i) at t = 0 the spectra of the corresponding Jacobians lie in the closed left halfplane

~-(ii) they have solutions which are analytic at t - O.

Ad (i):

Observe that for DE's of the form

(1. 3) ty'(t)

=

f(t,y(t» , (t € (0,1» ,

By analytic functions at t = Owe will mean:

real functions that. in

a:,

can be expanded around t · 0 in power series, having a_positive radius of convergence. Sometimes this radius of convergence will be given explicitly.

numerical methods with reasonable stepsize h > 0 that are numerically stable can be found if

. ( af

_{~ ay(t,y(t»}

)

_c

_a:

_{for all t € (0,1)}

(Lambert, [7J), even if the equation (1.3) is (partly) stiff.

If

~~(t,y(t»

has also eigenvalues on the imaginairy axis, then stiffly

stable predictor corrector methods can be found (Gear, [IJ).

~-In the latter case, absolute stability is required only for the fastly decaying solutions, while accuracy and relative stability are required for the other solutions. Using these methods, the real parts of the eigenvalues of the Jacobian may even be small positive. since the inter-val of integration is not a priori determined,one may choose

a

such that this condition is indeed satisfied in the whole interval

[O,oJ.

In practical applications one often finds that

a(~~(o,yo»)

c

a:-,

which by continuity implies that there exists a large scope of numerical methods and a 0 > 0 such that the solution of (1.3) can be computed numerically stable on the whole interval

(O,oJ.

Ad (ii):

The analiticity of a solution at t - 0 is an important property for starting a solution method, since it implies that all derivatives at t

=

0 exist. Moreover, most numerical integration methods are based on the assumption that the solution behaves, ' like a polynomial, and consequently perform well

1f

so. Hence, for small t, the solution can simply be computed by formal; power series expansion, and for other values of t, stable numerical~methods are available.

The fact that it gives solutions of the IVpts which are analytic at t

=

0 is not 'obvious , since for instance nhe fundamental matrices belonging to a linear IVP may contain logarithmic singularities at t

=

O.

§2. Preliminaries

For small t we have the power series

""

00

(2. 1 ) A(t)

=

l

Aktk and

k=O

f(t)

=

l

fktk. k=O

Without restriction we may assume that AO takes the form

(2.2) implies that rank Notation:

·

" .~

...

"

.

"

... .

..

. a ..

'. 0 .

A • : : pq:

·

...

.,

... .

• 0 • A • qq:

·

...

" " .. " " " k p q m

) =

q • t-p+q n

Let x E 1R. , then we will use two kind of partitions:

T

x

=

(~, x , x , T T

p q

k

and

kxp

, where xl € 1R and x2 1R€ qxm •

In accordance with this last partition let

All (t) A_I2(t)

t

k+p A(t) .,. A21 (t) A₂₂(t)

t

q+m E ) ( ) k+p q+m

First we remark that a part of x(O) can be computed directly.

Theorem 2.1. Let x(t) satisfy (1.1) and (1.2a). Then

(2.3)

where AO and fO are defined in (2.1).

Proof. Suppose AOx(O) + fO

~

O.

From the differential equation (1.1) and the continuity of the solution

we obtain (2.4) t + 0 , where £(t) = 0(1) , t +

° .

Integration of (2.4) leads to t

(2.5)

x(t) - x(t_O) = log(r-)[A x(O) + f ] + t O O

o

J

£~.)

d. , to

For t and to sufficiently small we have, since £(t)

(2.6)

Hence, lim 11 x(t) - x(t

O) II does not exist,

t+O

which is in contradiction with (1.2a).

= o( 1),

Corollary 2.2. A necessary condition for the existence of solutions

of (1.1) subject .to (1.2a) is

Observing that (2.3) can be written as

A+ ~(O) fO k AO fO pq x (0) ₊ p

= o ,

AO q fO qq x (0) q

L

A- m fO m

th t ( (O) T x (O)T, x (O)T)T • . 1 d f' d b th b d

we see a ~ ' q m 1S un1que y e 1ne y e oun ary

condition (1.2a), since

) =

k + q + m •

A

Hence, if the ODE (1.1) is homogeneous then only x (0) may be nonzero. p

1

Theorem 2.3. Let V be the subspace of C (0,1] formed by all solutions of

(2.7) tx'(t)

=

A(t)x(t) •

for which lim x(t) exists. t-+O

Then dim V

=

k + p.

t € (0,1) ,

Proof. By Theorem 2.1 and the variation of constants formula (Henrici [3], p. 199) we find that any element ~ E V must satisfy

I

t

(!)A~l A~2~2(T)

+

[A~I(T) A~Z(T)]~(T)

dT

T T (Z.8) t

I

°

k~

*

°

where ~ E:m. ,to E (0,1] fixed and A (t)

=

A(t) - A • Existence of a solution

~

of (Z.8) for any

~

E :m.k+p can be proven by the method of succes-sive approximations, starting with

~o

=

0. At the same time we obtain that

there exist constants C

I, C

z

>

°

such that, fort- sufficiently small, II<Pl (t) II ~ C₁ and II <PZ(t) II ~ C₂ ' It log tl. For other values of t the bounded-ness of <P is obvious which implies that <P E V.

Moreover, if ~ =

°

then <p = 0, which implies that any solution ~ of (Z.8) is uniquely determined by ~.Thu$, by the linearity of (2.7) and (2.8), we have

ob-tained that dim V = k + p.

o

By Theorem Z. 3 we see that the boundary condition (I. Za) imposes (q + m)

restrictions on the solution. In order to have a unique solution-of (1.1) and (I.Z) it is therefore necessary that s

=

k + p (cf. de Hoog, Weiss [4J).

To conclude this section, we investigate the behaviour of a solution of a special type of singular initial value problems in the complex plane. This complexification is necessary since the space of analytic functions at t = 0, defined on [0,1), is not complete.

Theorem 2.4. Define the set

A

by

where 01'02 >

°

are fixed.

Let F: ¢ x ¢n + ¢n satisfy

(i)

F(O,O)

=

°

(ii) F analytic in both arguments on

A,

i.e.,

aF nxn

*

Define FO(z):= F(z,O), F₁:= ax(O,O) e ¢ and F (z,x):- F(z,x) - FO(z) -FIx.

Suppose O'(F I) c:

if-Then there exists a 00 >

°

such that the IVP

(2.9a) subject to (2.9b) dx z dz

=

F(z,x)

lim

x(t)

=

0 ,

HO

telR+

has exactly one analytic solution for Izl

s

8

0,

Proof. By the variation of constants formula, we see that any solution of (2.9a) is also a solution of

1

x(z)

=

J

Using successive substitution, we shall construct a sequence of functions

i

x (i

=

0,1,2, ••• ) that converges to a solution of (2.10).

Lemma 2.4a. Define the sequence {xi} by

1

a - a

x

=

i+1

fl

x

(z):-(2.11)

o

Then there exists a

°

₃,0 < 03 ~ 01 such that for Izl ~ 03 and i

=

0,t,2, •••

(i)

(it)

xi(z) is well defined,

i

3

_ce:r

s.t. IIx (z) II ~ Clzl,

(iii) there exists a 00'

a

< 00 ~ 03' such that at

{ze:el

Izi ~ 00} the

se-quence {xi}converges to a solution x of (2.10), which is analytic at

t

=

O.

Proof. Since (ii) implies (i) it suffices to show (ii) and (iii).

(ii) (by induction): Suppose (ii) is valid for i • j (j

e:

EO) • Then

(2. 12)

I

IIxj+t(z)II

~

f

o

.

aF*

Note thatIlFO(z) II S Cllzl, 0 ~ Izl ~ 15

1, and Slnce ax (0,0) == 0,

*

lI

a

:

_{x II} S C2(lzl + II xII) , (z,x)

e:

A.

*

Moreover, F (z,O) == 0, so

*

2 IIF (z,x) II S C3(lzlllxll + II xII ), (z,x) £ A.

Together with the inequality (Wasow [11], p.93)

A of F} with Re A

=

0, we obtain (2.13) II xl • +] (z) II

s

I

z

I

I

J (

c

4 lIn v-1 (T)

I

o

2 \l where E}

=

C](C 4(v - 1): + C

s)

and E2

=

C3(C + C )(~C5 + C4(v -1)!/2 ).

Since El and E2 are independent from j and only E2 depends on C, we can

choose C € lR+ and 0 < 6

3

s

min{61,o2/C} independent from

j,

such that

j+l

IIx (z) II s cizi

This ineqti~ity is certainly satisfied for j ..

0,

which implies

(H).

(iii) is proven in a similar way. Namely,

*

i

*

i-I

IIF (z,x (z» - F (z,x (z»II ..

1

J

aF*

i-I

i

i-I

i

i-I

= II ax-(z,x (z).+ S(x (z) - x (z»)·(x (z) - x (z»d911

o

aF*

i-I

i

i-I

i

i-I

s sup lI

_ax

(z,x (z) + e(x (z) - x (z») U·Ux (z) - x (z) II

Osesl

I I

i i - I

I I

s C

2 (I + C) z II x (z) - x (z)

\I

, 0 S z s 15 3 • From this it follows that

.. II F (TZ,X (TZ» - F (TZ,X

*

i

*

i-I

(TZ» dTIl

T

s C₆

Izi

sup lIi(TZ) - i - I (TZ) II •

Hence, taking the sup-norm and choosing 0 < 00 S; 6

3 such that C6 • 00 ~ LO'<:: I,

the sequence {xi} converges uniformly in {z

~

¢I Izl S; 00} and its limit,

say x, satisfies (2.10).

Moreover, lim x(t) == 0, which implies that x is also a solution of (2.9).

UO +

t~Jl

i

To prove that x is analytic for Izi ~ 00 we show by induction that x is

analytic for Izi ~ 03 (i == 0,1, ••• ).

(a) xO is analytic for I z

I

~

°

₃,

(b) suppose xj analytic for

Izl

s;

°

3,

*

j

I I

Then Gj(z):= FO(z) + F (z,x (z» is analytic for z ~ 03'

By definition we have that G.(O) == 0, so we may write

J Then co

G.(TZ)

=

I

(Tz)kg~

J k=1 J k n (g. ~ ¢ ) • J 1 • + 1

J -

(I+F I) co k k xl (z) = T

I

(TZ) g. dT k=l J

o

==

I

k=l -I where [kI - F

1

J

exists, because a(Fl) C ¢

1 "+1

Moreover,

If

[kI - F 1]- II is bounded. uniformly in k, so xJ is analytic for

Izi :s; 03' Hence,

x

=

lim xi(t) is analytic for Izi S; 00'

i-+«> This proves the lemma.

By Lemma 2.4 it is shown that an analytic solution of (2.9) exists, so we

only have to prove the unicity of this solution.

Let

x

be another analytic solution of (2.9) and define e(z):- x(z) -

x(z).

Then, for Izi sufficiently small, 0 S e(z) S clzl. So,

1

o

S e(z) S C 2(1 + C)lzlf (c4Ilnv-I(T)I + CS) e(Tz)dT

o

S c 71z1 max e(Tz) • OS-rSI

Hence, for 0 S e S 00 sufficiently small,

o

S sup e(z) S C

7" e" sup e(z) ,

Izise

Izise

which implies that e(z) = 0, for z S e •

By unicity of the power series expansion of an analytic function we obtain

that x(z) =

i(z),

for Izl S 00'

Remarks 2.5.

o

(i) Under the given conditions we cannot maintain that (2.9) has a unique

2

solution. For example let F(z,x) • x • Then (2.9) has infinitely many

solu-tions, namely x(z)

=

0 v x(z) - (C - log z)-l.

Of course, only the first one of these solutions is analytic.

However, easy manipulations show that each of the following restrictions makes the solution unique:

restrictions to the differential equation:

(a) a(F 1) c CC- or

(b) F linear in x ,

restri~tion to the solution: (e: > 0).

(ii)

The same result could have been derived by construction of formal

power series, satisfying

(2.9).

In Hautus

(2]

it is proven that this

(unique) power series has a positive radius of convergence.

0

Corollary 2.6. If all coefficients of F in Theorem 2.4 are real, i.e.,

n

f.. . € lR , then the analytic solution of

(2.9),

restricted to the

1J 1 ••• I_n

domain [0,6

0), is real.

i n

Proof. One directly verifies that x (t) E lR , i € EO and t € [0,6

0), so n

x(t) € lR , for t € [0,6

0),

Example 2.7. Let x : [0,1] -+ lRn satisfy the initial value problem

(2. 14)

=

A(t)x(t) + f(t) { txt (t) x(o) = X o t E (0,1]

where A,f analytic at t =

°

and a(A(O» c

"l-By Theorem 2.1 we know that A(O)x

O + f(O) • 0.

Define z(t):= x(t) - xo' t € [0,1]. Then z is a solution of

{

tz'(t) = A(t)z(t) + get) ,

z(O) •

a ,

where get) = f(t) - (A(t) - tI)x

O'

t E [0,1]

By remark 2.5.(i) this solution is unique and analytic at t = 0, so (2.14)

has a unique solution, which is analytic at t - 0.

§3. Riccati transformations

To transform (\.1) and (1.2) into a regular system on (6,1) (6 >0) we use a so-called Riccati transformation

~+p

o

(3. I)

₌

t EO [0,0] ,

where R21 is chosen such that the transformed system is block upper-triangular and can be solved in a numerically stable way_

The idea to use such a transformation is not new. Russell [10] used this transformation in 1970 to identify the k-dimensional affine manifold V in En in which a solution of (1.1) and (1. 2a) is found, where he assumed that t

=

O. Nelson, Sagong and Elder [9] tried to find the solution of a homogeneous singular linear BVP by a Riccati transformation without first reducing it to a regular problem on some subinterval. However, difficult problems then arise in the course of solving the differential equations for YI' because its solution is in general not analytic at t

=

O.

After transformation (3.1) the system (1.1) is changed into

-~ All (t) A12(t)

1

fl(t) 0.2) ty' (t)

=

yet) + ~ "" AZI (t) A₂₂(t) f₂(t) ...J t EO (0,0] ,

and

'"

f₂(t)

=

-RZI(t)ft(t) + fZ(t) •

To make

(3.2)

block upper-triangular we let RZI be the solution of the

matrix Riccati differential equation:

{

tR21 (t) == AZI (t) + A_ZZ(t)R_Zl (t) - RZI _{(t)A tl (t) - RZI (t)A12(t}RZI} (t)

(3.3)

RZI (0) == 0

Theorem 3.1. There exists an e > 0 such that (3.3) has exactly one solution

on [O,e:), say R₂₁, which is analytic at t

=

O.

Proof. Consider the entries of U(t} E lR(q+m)x(k+p) as entries of a vector

u(t) E lR(q+m)(k+p) • Let F(t,U} := AZI (t) + A

22(t}U - UAll (t) - UA1Z(t)U

and F(t,u) the associated vectorfunction. Now it is evident that F(t,u)

satis-Hes the conditions (i) and (H) of Theorem 2.4 and the condition of Corollary 2.6.

Moreover,

where $ denotes the Kronecker sum.

The partitioning (3.2) is chosen such that a{F I) c

"i-

so all conditions

to guarantee the existence of exactly one solution of (3.3) on [O,e)

(e > 0), which is analytic at t • 0, are satisfied.

As direct consequence of Theorem 3.1 we obtain that the righthand side of

the DE for Y2 is analytic at t ..

o.

Since

A

22(0} .. A22(0), which has its

spectrum in

i- ,

Y2(0) s x

2(0) and by Corollary 2.6, also Y2 is analytic

at t .. O.

0)

o

Remark 3.2. Writing R

21(t)·

L

Cktk, the following relation for Ck (k=I,2, ••• )

k=l is found:

which has a unique solution for all k ~ 1 (Wasow [11], ch.2}.By remark 2.5. iii)

this also imp lies the local exis tence of R 21•

A result that will be used later is the following:

Theorem 3.3. There exists.a fundamental matrix X : [0, I

J

-+ E,nxn

correspon--I

ding to the system (I.I) such that R

21{t) .. X21 (t)X11(t), for all t € [O,eJ.

where e is defined by Theorem 3.1.

o

Proof. Let X be a fundamental matrix for which X11(t) (t € (O,eJ) is regular.

Then it follows from easy manipUlations that

X21(t)X~:(t)

satisfies the

same differential equation as R

o

If A has no eigenvalues that differ by positive integers, then there exists

a fundamental matrix X corresponding to (1.1) of the form

o

X(t)

=

P(t)tA , t € (0,1] ,

where P is analytic at t = 0 and P(O) £ I (Henrici [3J, Th.9.5.c). This matrix

X already has the desir~d property.

If AO has eigenvalues that do differ by a positive integer then the proof is

rather technical.

Elaborating Theorem 9.S.d in Henriei [3] one finds that there exists a

funda-mental matrix X ~uch that

~~ _ _ . __ .J ... where

Vet)

o

U kk

*

U ==

0

I P

o

*

A

O

_pq

A

O

qq

o

and

P

analytic at t

=

0 with

'(0) -

In •

is regular for t > 0 ,

*

0

with U kk upper-triangular +

A

and o(U kk) C ¢ u {OJ

Now

(3.4)

-I

which already implies that the last p columns of X2IX1l are of order t.

More detailed we have for the first k columns

(3.5)

...

v~J

8

a.

where V.(t)

=

t 1W.

1. 1.

(i

,

=

1, ••• ,s) with al, ••• ,a integers satisfying s

a

l ~ a2 ~ ••• <1: a s ~ 0 and regular, W __________ , ~ ____ . ...1-__ ..l. .•

With (PZ1(t»i we will mean those columns of P21{t) that correspond to

Viet) ( i · 1, •••

,8).

Then by induction one can prove that

t -+ 0 •

Since (with similar notation)

and

t ... 0 ,

the theorem is proven.

(Observe that by (3.4) and (3.5) we have a third proof of Theorem 3.1~)

Note that the function R2t defined by (3.3) can be seen as a generalization of the Riccati matrix defined by Meyer ([8J, p. 68). This follows from the fact that the boundary conditions (1.2) may also be written as

o

I

q+m

x(O) +

where x₂(O) is found by Theorem 2.1.

x(J) ""

o

o

b

t

However, in our case numerical stability of the integration method can be guaranteed.

§4. Boundary conditions

Choose

0

> 0 such that R2t exists on

[O,oJ.

Now the boundary conditions at t

=

0 will be translated in conditions at t •

O.

If we return to system (3.2) we see that the differential equation for R2t has at t "" 0 a Jacobian with spectrum in

i- ,

but also that the sets of

--'---~

bounded and unbounded~'~l~tions-ofth-e-dffferenda1-equation-( 1.1) are de-coupled for t sufficiently small.

From the resulting system'Y2 can be computed directly, since Y2(0) "" x2(0) is known, by Theorem 2. I and, by Theorem 2.4, y 2 is analytic. at t "" O. Moreover, this computation can be done numerically stable because

When Y2(0) has been compute~ the boundary condition (1.2a) can be replaced by

which indeed imposes (q+m) restrictions to the solution of (J.I).

Since Y2(t) is known for t € [0,0] we obtain for xI

=

YI the differential

equation

(4.2) t € (0,6] .•

We want to express x (0) in terms of x_{p 1}(0). To this _. en.~ let us assume there

exist functions Rpl(t) and Zp(t) such that

(4.3) x (0) c R l(t)xl(t) + z (t) ,

p p p

with

R 1(0) _p

=

(0 I ] and _p z _p (0) == 0 •

Differentiating (4.3) leads to

Hence, it suffices to take Rpl and zp such that

(4.4) and (4.5)

I

R 1(0)

tR~l(t)

== == [0 -Rp1(t)AJI(t) , I ] p p

I

tz~(t) z (0) ==

=

0 • -Rpl (t)(AI2 (t)Y2(t) p i t € (0,6] + fl(t» , t € (0,0]

Both systems of differential equations have a unique' solution. which is

analytic at t -0 and that can be computed in a nu~rically stable way.

-) Remark 4.1. Rpl is the matrix consistency of the last p rows of X

t J' wherE! X\l is defined by Theorem 3.3. Proof: Hence tXil(t)

=

A₁₁(t)X₁₁(t) + A₁₂(t)X₂₁(t) = (A11(t) + A12(t)R21(t»XII(t) == All (t)X 11 (t)

Moreover, since PI1(0) is invertible,

= lim [0 t-+O == lim [0 t-+O t I ] p -U kk

0

0

I P I ] • P - I P11(t) -I Vkk(t)

0

0

I P

Bi the boundary (4.6) =

r

Bi Bi Bi

J1

_{k+p (i=O,I),} Lk p 2

it

+-+ +-+ p q+m

conditions (1.2) can be replaced by

o

I q+m-x(o) +

o o

o

x(l) =

Now our mission is fulfilled since the singular boundary value problem (1.1) and (1.2) is replaced by a regular one, namely (1.1) and (4.6). By construction we have that the solution of (1.1) and (1.2) is also a solution of (1.1) and

(4.6). To prove that (I.]) and (4.6) have a unique solution let

BOR 1 (6) _{p p}

₀

[0 BO]X-I (6) p 11

0

6 B := = -R 21 (6) Iq+m -1 -X 21 (O)X11 (5) Iq+m

where X is defined by Theorem 3.3, and

-1 B ==

0 0 0

-Now (1.1) and (4.6) have a unique solution if and only if [B5

X(o)

+

SlX(I)J

is invertible (Keller [6J).

Theorem 4.2. (1.1) and (4.6) have a unique solution if and only if (1.1)

and (1.2) have a unique solution.

Proof. Suppose (1.1) and (1.2) have a unique solution x. Let x be that

par-ticular solution of (1.1) for which

~(O)

x(O)

=

0

x 2(O)

(4.7) _[0

is invertible.

o

-I 0 _ )

Since (4.7) is equal to [B X(o) + B X(1)]11 and ts X(o) + B X(I)]21

=

°

it suffices to show that

is invertible. This is however a direct consequence of the fact that

X(o) and XII (0) are invertible. The other way around is proved by

-re-versing the argument.

Remark 4.3. To find Y2' Rpl and zp we need R

21• However, it is not necessary to store ~omputed values of R

21(t), since all these functions can be com-puted by solving just one (p+q+m) x (k+p+l) Riccati differential equation. Namely, by writing

Wet) :: _(t E [0,0]) ,

we obtain

o

o

o

o

tW'(t)

=

+ Wet) (t E (O,oJ) (4.8) - W(t)

[~

0

I ₀

!

p p W(O) =

!

q+m

0

x2(O) +-* +-* k+p

By Theorem 2.4, (4.8) has exactly one solution, which is analytic at t

=

0, .

although the differential equation may be stiff.

o

§5. Bounded solutions

A second class of problems for which this method is applicable is obtained by replacing the condition (J.2a) by the boundedness condition

(5.1) sup IIx(t) II < 00 •

In that case we assume the following partitioning-:

!

.

.

:

.

·

"

....

" "

.

" " " " " "

.

" " "

.

: " _:

..

" "

...

.,

.

~

AO

i

pqz

o

"

.

.

.

• " • " • " ... " ... j • " ... . (5.2) D

i

AO •

~

qlqZ •

...•...

:

... .

~

AO

~

q222 '

.

" ~ "

·

" " " " " " " " " " " " " " " " " " " ~ " " " " "

.

" "

.

" "

.

" "

~

A +----+)·+c----+)·+~----++:. ,.+(----+ k p m q\xqI

where D € R is a diagonal matrix with purely imaginairy (but non-zero)

eigenvalues and q) is the sum of the geometric multiplicities of all purely imaginairy eigenvalues of AO.

Consequently we have

t

k+p+q) A(t)

=

( ) k+p+q) and x(t) partitioned x(t)

=

(x. (t) , T x (t) T , x (t) , T K P q1 where

~(t)

E Rk, etc.

_•

Theorem 5.1. Let x(t) satisfy (1.1) and (5.1). Then ~(O) ~(t) := lim t-+O xZ(O) x2(t)

exists and satisfies the relation

""~.~-.--.~~-A+ ~(O) fO u AO _fO (5.2) pq2 x (0) , p 0 q2 + _fO == 0 • A q2q2 _{x (0)} q2 m fO "A _m

Proof. The first part of the theorem may be proven by investigation of

di£-ferential equations of the form

tu'(t) Ju(t) + f(t) , t E (0,1) ,

where J is a Jordan-block, like is done in de Hoog, Weiss [5] •

The relations A+~(O) + fO _k == 0 and 0 A 22x2(0) 0 + f2 - 0

Write then t~ let)

=

a + e ( t ) , t E (0,1) , p p where £ (t) = 0(1), t ~ 0 . p Hence, = a log(..£) + p

to

to

fixed. Thus, by (5. I),

In general x2(0) is not uniquely defined by (5.3). We would like to derive a

o

o

'0

relation of the form A x (0) + f • O. However, observing that a solution

qlq2 q2 ql

of

t € (0,1) ,

where € (t)

=

0(1), t ~ 0, is always oscillatory and bounded, no extra ql

conditions for x (0) follow from this differential equation. q2

Hence, a necessary condition for x2(O) to be determined explicitly from (1.1) and (5.1) is that

AD

rank( PQ2 ) = q2 • (5.4)

AD

Q2Q2 I I r

Suppose x ₂ ==

°

q (x 2 € lR 2) • ql AO x

Since D is regular there exis ts an xl € lR such that DX 1 + q 1 q2 2

=

o

D

o

AO PQ2 AO

(:2

1

)

==

° ,

Qlq2 AO q2Q2

which implies

(:~)

=

0, because p - dim(ker(AO».

Similarly to Theorem 2.3 we have

0. So.

o

Theorem 5.3. Let V be the subspace of CleO,I] formed by all bounded solutions of

tx'(t) == A(t)x(t) , tE(O,l].

Then dim V

=

k + P + ql'

o

Hence, for the existence of a unique bounded solution (1.1), subject to (1.2b),we need that s

=

k + P + ql and

E

)

.

If this condition is satisfied,the same technique to replace the boundary conditions (5.1) and (J.2b) as in Chapters 3 and 4 can be used. Finally, we have to compute the only analytic solution (cf. Theorem 2.4) of

tW' (t) (5.5) subject to W(O) ::: -Wet)

o

I P

o

- Wet) wet) ,

o

o

o

o

o

As soon as

Wee)

is known,boundary conditions like (4.6) can simply be derived

(BO =O!).

References

[ I ] Gear, C.W.: The automatic integrations of stiff ordinary differential equations.

Information Processing 68; ed. by A.J.H. Morreli.Amaterdam,North-Holland Publishing Co., 1969; 187-193.

[2J Hautus, M.L.J.: Formal and convergent solutions of ordinary differential equations. '

Proc. Kon. Ned. Akad. Wetenschappen, Amsterdam, Serie A, 81, (1978)

,

--

~

216-229.

[3] Henrici, P.: Applied and computational complex analysis, vol. I & 2. London, etc., Wiley 1977 (Pure and Applied Mathematics.)

[4] Hoog, F.R. de and R. Weiss: On the boundary value problem for systems or ordinary differential equations with a singularity of the second kind. SIAM J. Math. Anal.

.!..!.

(1980),,41-60.

[5J Hoog, F.R. de and R. Weiss: Difference methods for boundary value problems with a singularity of the first kind.

SIAM J. Numer. Anal.

11

(1976»775-813.

[6J Keller, H.B.: Numerical solution of two point boundary value problems. Philade~hia,SIAM, 1976 (CNMS Regional Conference Series in Applied

[7] Lambert, J.D.: Computational Methods in Ordinary Differential Equations. London, etc.; Wiley, 1972.

[8J Meyer, G.H.: Initial value methods for boundary value problems. London,etc. , Academic Press, 1973.

[9J Nelson, P., S. Sagong and I.T. Elder: Invariant imbedding applied to homogeneous two point boundary value problems with a singularity of

the first kind.

Appl. Math. and Comp.,

!

(1981),93-110.

[IOJ Russell. D.L.: Numerical solution of singular initial value problems.

SlAM J. Num. Anal. ~ (1976),399-417.

[IIJ Wasow, W.: Asymptotic expansions for ordinary differential equations.