MEMOIRS OF
NUMERICAL
MATHEMATICSNUMBER 3

1976

EDITORIAL BOARD
SIN HITOTUMATU (Executive Editor)
Research Institute forMathematical Sciences
Kyoto University
HIROSHI FUJITA
Department of Mathematics
Faculty of Science
University of Tokyo
SHIGERU FURUYA
Department of Mathematics
Faculty of Science
University of Tokyo
MASAYA YAMAGUTI
Department of Mathematics
Faculty of Science
Kyoto University

The MEMOIRS OF NUMERICAL MATHEMATICS will publish
original papers concerned with numerical mathematics, and henceforth will be published once a year.
All correspondence should be addressed to
Sin Hitotumatu, Research Institute for Mathematical
Sciences, Kyoto University, Sakyo-ku, Kyoto,
Postal 606, Japan.

We are sorry to inform that our ex-Chairman,
Prof. Minoru Urabe passed away on September 5,

1975.

nontents

Hirofumi MORIMOTO

Kunio TSURUTA
and Katsushi OHMORI

Mitsunobu KURIHARA

On the stability Condition ofDifference Schemes for Initial
Value Problems

1

A posteriori enor estimation for
Voltena integro-different1al
equations33

On the existence of an Approximate
Solution in Chebyshev Series of a
Nonlinear Integral Equation ofFredholm Type49

On the Stability Condition of Difference
Schemes for Initial Value Problems

Hirofumi(

MORIMOTO

Received February 14, 1975)
RevisedJanuary

9, 1976

1. Introduction
The stability condition for the scheme of difference equations
with constant coefficients with periodic boundary condition isproved by P.D. Lax and R.D. Richtmyer(5), to be equivalent to theuniform boundedness of the norm (see §2) IGn(L\t,k) I
1~n~T/6t,k}.

in D={O<6t<T,

Here k=(k1,···,k d ), k i =2wr i /L i , ri=O,±l,±Z,···, d isthe number of spatial variables andLi are the periods.

G(6t,k)is the amplification matrix introduced by the Fourier transformationof the difference equation, and its dimension

1m

is equal to thenumber of dependent variables.

J. von Neumann's necessary condition on the eigenvaluesof

G is IAI~l+K~t, whereA

K is a constant in Do

T. Kato (3) showed that, roughly speaking, if the larger
eigenvalues of(1)

G have simple elementary divisors andif either

the eigenvalues are separated from each other by a fixed amount,or (2) the associated normalized eigenvectors have the Gram determinant (see §2) larger than a fixed number, then the von Neumanncondition is sufficient.

H.O. Kreiss [4] obtained four necessary and sufficient conditions, but his conditions are often difficult to apply to thepractical matrix calculation.

M.L. Buchanan [1] found the necessary and sufficient condition for the upper triangular matrix form, to which G(~t,k) canbe transformed by a unitary matrix ~nto
a12(~t,k)----al(1.1)

A(~t,k), m(~t,k)oA2 (~t ' "k)-----a 2
meAt
k)
,'

0----0A=
m- 1(~t,k) a m-l , m(~t,k}0-----------0

A (~t,k)m

with the diagonal elements nested with a nesting constant R(i.e.
IAi-Ajl~RIAr-Asl for all ranges with l~r~i~j~s~m).

Her condition for the stability is that there exist constants
K and L such that in D<-<
1 =1=m,IAil ~ 1 + K~t,2

atid*~(1. 2)

L, l~i<j~m.

The lemma (1) of T.Kato is deduced from her result, but the
1emma'(2) is not.
The equivalence theorem of P.D. Lax and R.D Richtmyer is forthe limit 6.t+O.
6.t.

The actual calculation is practised for a t.ini te

Richtmrer and Morton showed the examples in which the stability

condition is satisfied and the difference calculation runs practically unstable even in early time steps.

Hence, they imposed a

severer condi tion on the maximum eigenvalue I AImax in D for a finite
6.t for practical stability as a modified von Neumann condition [8].
In this paper we take the eigenspace intQ consideration.
§ 2 ,weIn

show that if ! GI is bounded and its spectral sets are separat-

ed each other by a fixed amount, then the Gram determinant of thematrix of the normalized basis of the subspaces for these sets canbe made larger than a positive constant.

Hence Kato'~ condition(2) is deduced from his condition (1).

And Buchanan's criterion(1.2) is necessary only for the set of such eigenvalues Ai (6.t,k(6.t))in D, that converge to a multiple eigenvalue for k=k(6.t)

in thelimit 6.t+Q.
In §3 we show that occasionally the norm increases with n forsome eigenvalues A, i.e. !Gnl_nhIAln-1IGI, where h~l.

This would*Her result is misquoted by Richtmyer and Morton [7] and V.

,

Thomee [9].
3

cause the practical instability in the early time steps, rather
than I'Gnl~IAlnmax' in the case IAlmax-l.
If IAI<l, it decreases to 0 for sufficiently large n, then thepractical instability will disappear,
eigenvalue

unles~

there is a large

IAlmax>l in D.

In §4 two examples of the .practical instabil i ty are discussed.
One of them indicated by H.Takami (Fig.2) is that, IAI=l in D andthe stability condition

is not satisfied, but the modified von
We show in this example that for

Neumann condition is satisfied.

nthe eigenvalue with index 2, GN4 and IG 1-4n, which causes the"practical instability, where IAlmax=l.
In another example, one in Richitmyer and Morton's book[8],the modified von Neumann conditlon is satisfied and yet the condition
for the eigenvalue to be of index 2 is satisfied.

Therefore, thepractical instability occurs in the early time steps .

..

Before stating our results, we recall some relevant notionsof the 'spectral theory and matrix representation ·of linear op&t'ators (see Dunford and Schwartz [2], T.Kato [3] and van der Waerden[10 ]') .

Let G be a linear operator in unitary space X of dimen-sion m(dim X=m<oo), AI, " ' , As.
G and the set o={A(s~m)

be the distinct eigenvalues of

,

,

1 '

... ''As } be the spectrum of G.
n

The set of.

all Uf.,X such that (G-A i ) u=O for some integer n forms an algebric
eigenspace X(A i ) of. X.

The smallest n for which this is true forall u£X(A i ) is called the i"ndex of Ai 'and is denoted .by 11 i .
vector u~O is an eigenvector of G for Ai if (G-Ai)U=O.

4

A

Any subset

a'~ ais called a spectral set, and the set X(a') of all ue X suchthat A.~a,(G-Ai)nui= 0 for some integer n is an invariant subspace
1.

for G.

For two separated sets a' and a" (a 'Il a"=O), the spectral

distance of a' and a" is defined by dist(a' ,a")= min IA'-A"I.
1 'E a'X"E-O"

Let {x 1 ' ... , xm} be an orthonormal basis in the unitary sp~ce

X.

m

.

Then a vector u = raix.EX 1S represented by the column matrix
1

1.

u =

G can be represented by an mxm matrix as a linear transformation of

the column matrix u into Gu.
column vectors.

In the following, u, v, •.. indicate

The inner product of u and v(= !1 b.x.)
is (u:~)=
l. 1.

~~ -a. b ., an d t h e norm 0 f vector u 15.
lul=(u,u)~/~.
1
1. l.

F urt h er t h e

IGul/lul and the norm ininvariant subspace X'=X(a') is defined by IGl x '= max /Gui/iul.

norm of matrix G is defined by IGI= max

O,,"UEXLet {u'1 ' ... , u'},m

P;\utrX'
be a basis of any invariant subspace X'=X(a')

and U'=[u1',···,u' m ,] be an mxm' matrix of which
the columns are the'basis, then G is reduced to an m'x~' matrix G' such that GU'=U'G'.
Further let U=[U', U", ... ] be the union of such matrices U', UtI,
then G is reduced to a direct sum such as GU=U.(G'~G'~···).
The Gram determin~nt.o£ U' is a determinant of the productof conjugate transposed matrix U'* and U', i.e. det[U'*.U'], andis a square of the volume of parallel polyhedron spanned by thebasis{u~'··"

u'm'}'

5

We consider the initial value problem with constant coefficients and with simple boundary condition, so that the Fourier
integral representation of the solution can be used .

If" there.

are p dependent variables in d-dimensional space, then the differequation ~s given as~ntial_d_.u

at(x, t)= P'( _d_) u (x, t) ,a,x

where u is a vector with p components and x is a vector with dcomponents.
ofP( ;x) is a pxp matrix whose elements are polynomials

a ... , --.
a
--,
dX 1

If the solution of the initial value problem. 'is expressed asu(x,t) = (27T)-d/2

Jdk ~(k,t)ei(k,x),'then (2.1) becomes

a . .u(k,t)..
""= P(ik)u(k,t).

~
Hen~e

the solution is(2.2)
Now let us start our statements.

Let G be a linear operator

in a unitary 'space i, and a' and a" b~ separated

spectral setsof G,' the~ X'=X(a') and 'X" =X(a") are invariant sub~paces of G.

.

'

Let dim X'=m', dim X" =m", and n'= E11i' 11" = E11i
, 'A. f a'A. .'&- a"ibe' the sums of1th~ .ind~ces .

. -.The fir,st lemma,. reads:
Lemma 1 ~.Let,:u.'eX' atld u"e)C' be vectqrs,such tha,t lu'I=lu"I=I,jthen the following(2. 3)

i~eq~aljty

holds for.the,Gram determinant~det{ [u' ,u" ] 'It. [u' ,u" ]}~ [dist(a' '~")) ~Ml
IZGI
~

6

whereM1 =m i n (n ' m' , , n" m') .
Proof.

Let u"r:X" be decomposed such thatu" = cu' + w,

(2.4)

(u' ,w)=O,

where the coefficient c is given by(u' ,utI )=c(u' ,u')=c,. Sinceu 'E X', we haveII
(G-A) n·l.u'=O.
A ~ a'
ii

Therefore

I II ( G- Ai) ni u "I = I IT ( G- Ai) niw I ~
AiEa'

AiEa'

(I G I + I Ai I ) nil wI

IT
AifO"

(2.5)

IT (G-A )ni is regular in the invariant subspace

The operator\

1\

X" =X(a" ).

i E
a'It follows thatlu"I=1 IT

i

-ni
(G-A)

Ai EO"(2.6)

On the other hand, Kato's lemma (see[3]) says for any linear
operator T in a unitary space X with the inverse T- 1 that

For an invariant subspace

X" E X, the lemma is thatm" = dim X" ,

(2.7)

7

where IT I
on X"

X"I

i~

.

is the restriction of T•

the norm in X"(G-A )ni.

for T = TI

Ai Ea 'i

Then (2.7) becomes
I (G- Ai)-1I x" ~ 12GIm"-l/dist(a' ,a" )

m"

Since

we have for A.l. E a'

I =

TI

A;(:a'>
IA -A I =dist(a' ,a") m" .
jJisubstituting the above relation into (2.6), we have

(2.8)

Iwl

~

( dist(a',a")-) n'm" IutI I .
12GIThen by (2.3) the Gram determinant becomes** .

det{[u',u"] '[u',u"]} = det{[u',cu'+w] .[u',cu'+w]}, * ·[u',w)'}= det{[u',w]

(2.9)

lu'12

=

o

o.
Iwl 2

dist(aV,a") 2n'm"~
12GI

lu'I'= 1.
In (2.9), X' and X" can be exchanged and hence the lemma 1 is proved.
The second lemma reads:
Lemma 2.
{v"l ' '"Let normalized basis of X'', X" be {v'1 ' ~ .. , VI m' } andV'=[v'l ' ... ' 'v'] V"=[v"··· v"
]
mt '
1"
m"
then the following inequality holds for the Gram determipant:,v" m'" }their matrices be8t ") 2M
det{[V',V"1 * '[V',V"]}~( d'IS t (a,a
) 2 . det{V'*.V'}'dettVil*.V"},

12GI

where

Proof.

We choose a normalized basis{utI (! X",ii=l,··· ,m"}so that the following decomposition holds:(2.10)

utI

=u'

+

w

iii'

U'E X'ifor i:\:j.

'

Let the matrix of their vectors be
U"=[u"····u"1mil ] ,

U'=[u""'u' mil ] and1

W=[w 1 ····wm'" ]then
UtI = U' + W.

Let the matrix of any normalized basis {vIEX', i=l,"',m"}beThen the Gram determinant is given as follows:det{l'P,U"] * '[V',U"]} = det{[V',U'+W] * '[V',U'+W]}
V' * • V'.j= det{[V',W] * '[V',W]}= a(2.11)m"= det{V' * 'V'}' TI

Iwil2pi=1

by making use of (2.10).

Let the matrix of any normalized basis{v" '~" X", 1'=1,"" ,m"} be VIi -- [Vii1 " ' "iV lmil'l'jSince UtI is also (he matrix of normalized basis in X", there
existsa regular matrix S" such that9V" = U" S".

(2.12)
Then

I ,
. [ m

[V' ,V"] = rV', Utt ]owhere Im.'is aunit matrix of dimension m' .

Therefore, itfollows for the Gram determinant that
det{[V',V"] * • [V',VtI]}=det{[V',U"] * '[V',U"]}'det{S" * .S"}

(2.13)by (2.11) and (2.12), and for the Gram determinant of (2.12) thatdet{V" * ·V"}=det{U" * ·U"} ·det{S" * ·S"}(2.14)~det{S" * .S"},

since U" is a matrix of normalized basis so that det{U"*·U"}~l.
Substitution of (2.14) into (Z.13) yields(2.15)***
mil
det { [V I , V"] . [V I , V"] }~det {V' . V' } . det {V" . V"} i~1 1 wi 1 2,

where
mil
II

i=1

(a' 'O)2M
tI)'
Iw 12 ~ (d 1St
mil
IZG 1

i

from lemma 1.

Since

1

X', X" can be exchanged in (Z.15),

Lemma 2 is proved.
Hence Kato's condition (1) in §1 is deduced from his condition (2).
Lemma 3.

Let T and U be m-dimensional square matrices, rows orcolumns of U being normalized, and let
det{U * ·U} ~ 6 2 > O.

10

Then

I [U- 1 TI [l12U] ij I < -JIlllJ.
I

T U-1]ijl ~ -max ITijl,i,jm2
maxlTijl.
lJ. i,jThis lemma is easily proved by Cramers rule.
following lemma
Lemma 4.

canFurther, thebe deduced from lemma 2 and lemma 3.

Let the spectral set 0 of a linear operator G in a uni tary

space X be decomposed into two subsets= {a ' , a" } ,

(J

with
dist {a' ,a"}~ p > 0,

aad let X' and X" be subspaces for a' and a", thenm3( 12pGIr

M2 max(IGnl "x

IGnlx,,)~IGn'l~ m3 ( llil.)+M2 • max ( IGn I ,., I Gn I " ) •pxx

In the difference scheme of an initial value problem the dimension
m of the amplification matrix G is nearly equal to the number of
dependent variables.

Elements of G are rational expressions oftime step lJ.t, space differences lJ.x j and exp[ikjlJ.x j ], where j=l, 2,d, and d is the number of space variables [5] .
For the difference scheme,afunctional relation lJ.xj=lJ.Xj(lJ.t)

is assumed such that lJ.Xj~O as lJ.t~O.

Hence we may take the ampli--fication matrix. fbr G(lJ.t,k) in D.
Take any functional relation k=k(lJ.t) in D.

If a(k(lJ.t)) is

the maximum spectral set of eigenvalues which converge

to a multi-

ple eigenvalue and if the invariant space X'=X(a'(k(6t])is definedin O<lJ.t<T, then the following theorem is derived from lemma 4.

11

The necessary and sufficient condition for IGn(At,k) I

Theorem.

to be uniformly bounded in D is that there exists a constant M
U

IG (k(lIt)) I x ,< M,
in D and for all k(At), where X'=X(cr'(k(At))) is such a subspacethat cr'(k(lIt)) is a spectral set of which elements converge to a
multipleeigenvalu~

as

At~O.

Particularly when all eigenvalues have index 1 in the limit

I GU(k(lIt)) I x , becomes I AU(k(At)) I and this condition coin-

At~O,cides with the von Neumann condition.

In this case the von Ne~manncondition is sufficient as well as necessary.

3. Practical stability condition for the finite lit.
The stability condition is a condition in the limit oflIt~O.

Richtmyer and Morton [8] shqwed the various examples which satisfy
the stability condition, but their calculations were practicallyunstable fora: fini te At.

They take as a modified von NeumanB

condition the restriction(3.1)

where r=max

Re~,

~

is the eigenvalue of P(ik) in (2.2) for all k

and all eigenvalues for each·k.

·Although the practical instabili-

ty often depends on the initial·condition, we consider only thepractical instability caused ~rom the amplification of the vector
by GO.

The power of the eigenvalue IAnl max is not always adequateU
to use for the norm IG I.
In fact the square norm IGI2 equals

themaximum e1genvalueof G* G, but IGI2 is not suitable to deal with the iterated norm
Sometimes it happensthat IAI max is near 1.
.
12

In this case,·IAl n max grows slowly for small n, but the calculation becomes pracn

tically unstable in the early time steps, although IAI max varies
slowly for small n.
In the following we estimate I Gn I in terms of its eigenspace,hand we show that IGnl grows sometimes as n IAI
where h~l.

n-1

IGI for small n,

We shall let the elements of G remain constant andincrease n.
(1) Let A', A" be distinct eigenvalues of G, and u' and u" benormalized eigenvectors for respective eigenvalues.

For the eigen-

vectors lemma 1 is simplified as
A"I
I A'- 2(det{ [u' ,u"] * . [u' ,u'~]}) 1/2

I GI ~

(3.2)

Then

where

(3.3 J~For IA'I<l, IG

n

11,-1

IIA'I-IA"II'IA"I'nfor I A' I> I AII I.

I decreases to 0 for sufficiently large n.

Concerning the limiting case A"-+A' in (3.2) we deal with the
eigenvalue of which the index is larger than unity.
(2) Let the restriction of G on the irreducible eigenspace
X'=X(A')be

13

. . . •. a

'a 12

A'

o A'

a

1 m'

2 3

A' =

(3.4)

AI

a m, - 1 ,

o

A'

m'

for an orthonormal basis of X' .
We decompose (3.4) into two parts such that
(3.5)

A'

= A'I m'+ A",

where I m , is an unit matrix of dimension m' and A" is the remainder,
rna trix for which we can show (A") m =0.
Using (3.4), we have
(3.6J

n
n
n- 1
n
n- h hnn - ~ + 1 m' -1
A' =AI I m' +n A'
A" + . .. . + ( h ) A'
A' (. + • • • + ~m'-1
(
) AI
A"

Since (i,i+l) elements are zero in A" 2 , AII 3 , . " ' , we havelaI
IA,n l x ,~nIA'ln-,1.max
ii'i+1'(3. 7)

i+1~0

where ai'for i=l,"', m'-2, as X' is irreducible.

Next letA"hmax 1---hhi . A I•

then from (3.6) we havem'<IAlln(nB)- 1 < 2(nB)m'-1IA ' l n ,
'x'
nB-l

IA,n lfor nB~2.

In the case of I AI I =l-c, where c is a" small positive number, theabove power attains to its maximum at n=(m'-l)/c, and then decreases
to 0 as n increases.

14

Thus even if the stability condition for~t+O

is satisfied,some practical instability may occur, when the di~tance between
the normalized eigenvectors is small, or when there is a largecoefficient in the upper triangular matrix for orthonormal basis oftheirreducibl~

eigenspace.

If the absolute value of theeigenvalue in question is 'smaller than 1, this practical instability may disappear for sufficiently large n.

4. Examples of the practical instability.

We apply the above considerations to the following examples.
In the first example of the numericalso~utionf0r the wave

equation by H. Takami (private cornrnumication. Fig.2) the absolutevalue of the eigenvalue is 1, and therefore the stability condition
is not satisfied, but the modified von Neumann condition is satisfied.

The eigenvalue with index 2 is -1 and the upper triangularelement of (3.6) is 4, and hence

IGI 4.
N

In the second example by Richtmyer and Morton about the wave
equation coupled with heat flow (Fig.3.see [8]), an instability
occur~

from thefi~ststep and grows more and more with n.

In

this case the stability condition is satisfied but the modified
The maximum absolute eigenvalue

von Neumann condition is not.
is 1.7.

We use the same example to see the effect of existenceof the eigenvalue with ipdex 2 of the amplification matrix.

.

The

modified von Neumann condition is necessary for the condition thatnu

eigenvalue of index 2 should appear, although both c~nditions

almost coincide numerically (Fig.l).

15

The wave equation is a 2 u/at 2 =a 2 u/ax 2 , or equivalently

Example 1.

(4.1)

J

l

av
at

=

aw
ax

aw
at

=

av
ax

where v(x,t)=au/at, w(x,t)=au/ax.

According to the scheme of

Courant-Friedrichs-Lewy, we set
11

(4.2)

V.

= V(j f::.x, nf::.t)nw
j= w((j++)f::.X, (n+-12-) f::.t).

J

J

l

The finite difference equation corresponding to (4.1) isn+1= Vjn + v(w nj - W~_l) ,
Vj
(4.3)

1

vn+1)
Wn+l = wn + v (n+!.
Vj +1 j

t
where

j

j

= W

v= f::.t/llx.

G(k,f::.t)

.1

n
n
n
2( n
2 n
v(v'+ l V j ) + v wj + 1 - Wj + wj - 1) ,J

The amplification matrix isiI(4-.4')

n +

1

= !
l2ive iW / 2 sin(w/Z)

where2ive- iW / 2sin(W/Z)J'
l-4v 2 sin2(w/Z)

w=kf::.x, kc 2nr/L, r being an integer.
The eigenvalues are
A=(1-2v 2 sin 2 (w/Z» ± 2vsin(w/2)' v 2 sin 2 (w/2)-1

Therefore, IAI max >1
for Ivsin(w/2)1;1.
Let v=l, then the matrix (4.4) is diagonalized as16

(4.5)= [ e-J\ (00)

iw

0

o

J

e iw

by the matrix U of the normalized eigenvectors:

(4.6)

U(w)

1

=

1

e- iW / 2-1

e iW / 2[

12

J

Eigenvalues of (4.5) tend to multiple eigenvalues -1 as 00 tends
to

7T,'

and the Gram determinant of U tends to O.

According to the statements mentioned in §3, the matrix ofan orthonormal basis for (4.4) is~]

v = ._1_(4. 7)n:and G is transformed by the unitary matrix (4.7) into
A(wl

(4.8)

=

(:-iW

4sin 2 (00/2)

J.

e iw

The stability condition is not satisfied due to Buchanan'S
criterion. nevertheless the modified von Neumann condition (3.1)
is satisfied.

Further
2 tan ( 00 / 2) . sin ( nw) ] •

(4.9)

e inw

For

W=7T "(4.10)

the above matrices become
A(7T)

=

r

-1

L 0

and
(4.11)n
A (7T)

(_:1"
=
17

-: 1
-

..I

(-11n_1 4n ](_l)nThe multiple eigenvalue A(n)=-l has index 2, and by (4.11)

n

IA I

n

grows approximately as 4n, while IAI =1.
In this case (v=l), the finite difference equation (4.3) has
the general solution:(4.12)JV(X,.t) =f(~+t) -g(x-t)

JV! = F (j + n) - G(j .- n) ,nr..,

~W(x,t)=f(x+t)+g(X-t)\W j =F(j+n+1)+G(j-n) ,with arbitrary functions F(j)=f(j6t) and G(j)=g(j~t).

Note thatthe solution (4.12) are the same as that of the differential
equation (4.1) restricted on the mesh points.
Now we give some examples to see the behavior of the solutionof the practically unstable finite difference equation (4.3) in thecase of v=l.

Let us take an initial condition:x < 0 ,

(4.13)

x > 0
0,then we obtain the following solution:
1v(x,t) =x ~ -t)

1/2 ,-t<xSt,,t < x •

0

(4.14)(

x ,-t ,

0w(x,t) = ~. -1- t<x~t t

~

~0 ,
SO

t < x

,

that the initial condition for the finite difference equation(4.3) is
18

(

0

=
J

v.

t

1

j

0

j < 0teli\J(4.15)

0

w. =
J

2 f

o,

0.

$

j

= 0,

j

!;. 0 1and its solution is the restriction of (4.14) on the mesh point's.
In stead of (4.15), we take an initial cond'i tion:
0

Vj(4.16)

=

\

,

1

j

~O

0

j

> o,

This is equivalent to take additional functions v and w, the initialcondition of which is

j

= 0

j

~

0

Its solution isn

Vj= E(n+j)-E(n-j)=else',

f

n

Wj

n+j
,
(-1)
/2

= E(n+j+l)+E(n-j)=

lo

'
else

where
E(·J.I"to,j-l• (-1)
/2

j

,

,

~

I

0 ,

j > 0 •

To introduce periodic condition, we take ,the convolutions of

19

the above solution and a periodic function of which the mean value
is zero : e.g.
P(j) =-1

j =0 ,

+1

j=No

o

else

O<N o <N ,where N is the period of the functions.

Since the equation islinear, the convolutions
00

Pvj -

r

N-l

r

R,,..-oo k=OnP(k+R,N)v j _ k ,etc.
nare also the solution.

It can be shown that

PY jetc. are welldefined and periodic with the period N.
Consider, for example, the error terms
If both N and No

aregrow infinitely as n~oo.

n

PY j

of these convolutions

even,ampli~udesThis corresponds to the infinite growthof the iterated amplification matrix IGnl when w=~.
the infinitely growing component of

n

PV j

In fact,

is the one proportional

to (-I)j+n.When either No ~r N-N o is odd, no. solution grows
indefinite~ybecause N is fixed and no·Fourier component grows inn

definitely. Anyway, the amplitude ofma,gni vude as .tha tnand P(v j

0 f

PVj

is the same order ofitself, and hence the profiles ofn+Vj ) are quite different.

Aside from the wave equation, it is difficult to find an
initial condition with no loss of accuracy and with no growing
components.

In practical usage,

components for largek(=Nw) are not important, and practical instabilities occur atsuch large k's, since the components for small k behave approximate-20

ly like solutions of the differential equation by virtue of the
Therefore,

consistency property of finite difference equation.

we can avoid the instability at the expense of accuracy; thatis, suitably mollifying the initial conditions we apply the samedifference equation to get rather smooth solution which contains
small components with large k's.
For example, we start by the initial conditions which containthe convolution of the given initial values and the smoothing
function ~(j)=(~)2-J, with an integer J~O.

Spatial shift by

J, 2 is brought about, and its correction is necessary.
Fourier TransformY' (j) = 2- J (i)
J

-J
*~= 2

0

PV j

r

(cos (k'IT/N))J

Ikl~N/2 1

J ............ oJ0(s) PV j -ss=O>(cos(k'IT/N)) Pv (k).

)

n

n

Let us apply this scheme to the solution PVj, PWj (J=l).
n
PVj *

:r =

1/4

j=-n+1,

_1/4

j=n+1

0

else

The following figure will be useful to understand the above result.........

~.

.

..

.

. .. . '. .. ....... .
....~.--"'.1---:

L.~.~.
_ _....~•. -...~•. -....~•. -....~.t-JCl\the exact solution of differential equation~

the solution of finite difference equation with the initialcondition(4~16)

21•

the smoothing solution of A with correction shift by -1/ .
2

Note that, in the case of wave equation, the convolutions with~(~) give the exact solution for v;=o, and they give the solutioll
shifted by 6.t/ 2 on the space axis and by 6.t/2 on the time axis for
These facts are deduced from the general solution (4.12).
Next let us take the following initial condition:(4.18)

j\vo= 2f(x) ={l-x 2 ,Ix I ~1 ,0

Ix I > 1 ,wo= 0

Its solution is

j v = f(x+t)+f(x-t)( v; = F(j+n)+F(j-n),\w = f(x+t)-f(x-t)~ 1Wnj = F(j+n+1)-F(j-n)

Let us take the initial condition for the finite difference equation
corresponding to (4.18) such as

to,1-(j6.t)2,= 2F(j)=otherwise ,(4.19)- (j+1/2)6.t 2
w~
J,= F(j+1)-F(j) ={o ,where 6t=1/M (M:a positive integer).

otherwise,
We obtain the exact solutionfor (4.1) and (4.18) as the solution of the finite difference eqs.
(4 . 3) and (4.19).
If, instead, we start with the initial condition22vj = 2F (j),
{
WO

j

= 0

J

we get the solutionnnVj = F(j+n)+F(j-n)+oj
n

Wj

n

= F(j+n+1)-F(j-n)+E j

with additional error terms:
9>(j +n) - P(j -n)~ (j +n +1 ) + ~(j - n)

,

where

o ,

jf1t~-l

,

j+M

)f1t,

<'p(j) =-lsj f1t~+l ,

j-M

(-1)

the profile of
n <

8~
J

jf1t~l6t),

is given as follows-M-'YI. . ~0<At)2

M ---41.t--4.t--4I.~y Y-M+'r1. 0

M-1'I.

n >-At

In this case also, the mollification by thefunctiongi:tres

the exact solution,

Example 2.

The wave equation coupled with heat flow.

This stability condition is discussed by Richtmyer and Morton.
The amplification matrix for their scheme is23

r(4.20)

G =

1

ivs

-i(y-l)vsivs

1-v 2 s 2(y-1)v 2 s 2-ivs
11S2+12-(y-l)v 2s2+l
11S2+l

2

V S
----

11S2+1

where y=cp/c,,~ a=thermal concluctivity/c v ' s=2 sin(-~- 6x), k=21Tr/L,2

r is an integer, v=c6t/6x and 1l=a6t/(6x)2.
The stability condition [6] isv = c6t/6x < 1.

(4.21)

Though the above stability condition is satisfied, the calculationof .the propagation of a simple jump shows sometimes the unstableprofile from the first step.
is

IAI~l,The modified von Neumann conditionand leads to the practical stability conditionv < 1(1+211)/(y+211) .

(4.22)

Fig.3 shows the unstable profile in an early time step for(4.23)

y=3.0,

v=(13+2)/s,and V/11=S/3,

where the stability condition (4.21) is satisfied but the modified
von Neumann condition is not, and the instability grows more andmore with n.
Now, we shall show that even if the stability condition c6t/6x<land the modified von Neumann condition
satisfied at the same time, the solution can be practicallyunstable.

First we shall look for the condition for the matrix

G of (4.20) is reduced to the diagonal form

24

o

o

o

o

G

o

o

Then the rank of G-Al is 1, the necessary and sufficient condition
for this is obtained from (4.20) as
Al=l

and

s=O

or

For this case the rank is 0 and G isv=O.
a unit matrix.

Hence, itis found that the double eigenvalue has the index 2 when G has two
equal and one distinct eigenvalues.

We shall" look for the condition

for G to have double eigenvalues.
The characteristic equation for (4.20) is

and since

~ = 0
dA

for the double eigenvalue, we get(4.25)

f(A,K)=(YK+(A-l)(A+1))(K-l(A-1)+Y(A+1))-(A~1)(A+2)2=0,

where
and

(4.26)",2 S .2 =-(A-I) ((I+YK)A-(l-YK))
A(A+2)The double root of the cubic equation (4.24) with real coefficients is real, and from (4.25) we see that the double root of(4.13) is in the interval25

2-1 < A ~It is not difficult to see that the eigenvalue of the greatest
modulus is near -1 in the complex plane, and that it moves monotonically to the double root of the modulus less than 1 when we vary theparameters(~,v, s) continuously, and if s=2

sin(k~/N)~O,

corresponding components become stable in any sense.
that the modified von Neumann condition

theTheis means

11..1<1 is necessary for G tohave no double eigenvalue of index 2.
Therefore, we investigate the case of v/~=a, where a is thelower bound of v/~ to have the double eigenvalue A for each fixed v.
In this case the component corresponding to the double eigenvalue
A is one with s=-2 (i.e.

k~x=~),

that is, the component is propor-

tional to (_l)n-jIA/n, where A is noted to be real.

This suggeststhat the mollification used in the first example is also effective.
Since the general solution or any other elementary expression forthe solution is not found, some of our numerical results are given
in Fig 4a) and b).

Acknowledgement.
The author wishes toexpre~shis cordial thanks to Professor

ISao Imai for his encouragement, Professor Hiroshi Fujita for his
critical remarks, Professor Hideo Takami for indicating an interesting example and to Professor Yoshiaki Chikahisa for revising themanuscript of this paper.

He is also gr.ateful to Professor Satiomi

Kaneko for his valuable discussions and constant friendship throughout this work.
26

References(1)

M.L.Buchanan :"A necessary and sufficient condition for stability of difference schemes for initial value proh1ems~ ..
J.Soc. Ind,ust. App1. Math.,(2)

11(1963) pp919':'9,35.

N. Dunford and J. J . Schwartz. : "Linear Operat~r:?': Part 1.
New york 1958.Chap.7.

(3)

T.Kato :'Estimationof iterated matrices, with application to
the von Neumann condition~ Numer. Math,. 2(1960),pp.22-29.

(4)

"
"
H.O.Kreis :"Uberdie approximative Loseng
von linearen partiellen Differential gleichungen mit Hi1fe von Differenz
gleichungen~(5)

Trans.Roy.

In~t.Stockho1m,

128(1958).

P.D. Lax and R.D. Richtmyer :"Survey of the stability oflinear finite defference equations~' Com.Pure. App1.Math.,
9(1956)pp 267-293.

(6)

H.Morimoto : "Stability in the wave equation coupled with heatf1ow~'(7)

Num. Ma'th., 4(1962) pp136-145.

R.D.Richtmyer and K.W.Morton : "Difference Methods fot Initial-Value Prob1ems~' ,2nd ed. (1967), Interscience, New York.p81.

(8)

R.D.Richtmyer and K.W.Morton : Loc. cit. sec. 10.4.

(9)

V. Thomee:" Stabi1 i ty the0fY rpt" p~rtial difference operaters'!
SIAM Review,11(1969)p167.

(10)

B.L.v.d.Waerd.en :"Moderne Algebra'! 2nd ed. (1940), Springer,
Kap.15.

27

v=c6t/6x
Stabilityfor----~-6t4Q.
______~condition__________

1.Or~----~~

this line, IAlmax~1
Y=3.0
v=( 13+2) /5
x v/lJ.=5/3

1

n

o

Above this broken line an eigenvalue with

1//3
0.5

6t-I/8index 2 exists.
2

4

v/lJ.=c6x/a

o

1.0

2.0

Fig.l. Stability diagram for coupled sound andheat flow for y=3, between ,,=c6t/6x and
ljJ=c6x/a. The stability condition is satis':" 6fied below the curve. Two lower c~rves . ~~--~------------------------~--~
approach still more as y tends to 1.

6t=1/32nq-----------------~---

y=3. 0\)=(13+2)/5
11=3,,/5
n=5

3.0

2.0 -

1.0""---"""1.

16
o

5

10

-1.0 ..

Fig.2. Profiles of the solution of wave
Fig.3. Calculated profile after 5 cycles,
equation in example 1, in §4, forin a run in which the modified von,,=6t/6x=I(By H.Takami)
Neumann condition was violated, although
c6t/6x<I(From Richtmyer'arld Morton[8]).

28Fig.4 a)29

;. !' ,. !' Ii "'r I 'Ii A. ····T. r: i ·.'···r·'·· ,.. . it! :., ! .\. i "! II..~!.'I :.I111,..': ~III.
~J·)ll-~I
WJ~:r--~·I.~: .L\...~JJ.!. ·:.J. .\-·l.·.l· I
,'" .
t\
!

I'

t J'" "1'I I I,,,. i· .;"' I :'":-1\':' ::'; ,,'1.:.,:./: :";', -.....
,::,:~ ., .,' . 'I' Ifl' '1' ,:l ,J" ,.:.,':. :' .,. '1 ~'. .:. :. .'" .: . . . :,. . .'.:':, . :_;';"1'-1::.L_..... - .. -.... . ,. --,::T
:
:
'
:
'
:
.
.
:
;
.
:
.
/
.
:
:il:'-"':'~""':I"''''''; 'l' ":,' ' ':"':i.Y"'·:'j:·' . '. "I: "L,. ,I'r~'Lt"i ".:. '::1:' ,.\ '-;' ::... ; N,.

I

I

,'i=-

j, ~~.- ,.,: - , - -: .- i·--i...... i'j~~- •. - .. -1.:, ..

-~""I'",;"C1:' '.
=f""-1:1:"' I': "i I !
.,-.,--'1........
. .,." i:....Ij\''::f=r~~-+-!----+--+---+--+-+-l~'':~I''''''- t·· . I.... ,..,... -- ..- ., ,.. . +."-j'" ...........
~........
'. U' l ' ,I " ,., ! '·l', ' : ' .,j " " ,j"!-1\~,·r:1 1" 'r-' ;;(·t'.I" -1~ _. ~"'J: "'1'-')" "'," .!, ::'," j:' '~" J.; 'J"
I' I I ,':';",:. . TiT ":1:; I:'1" IT "j' "rl-r(f'( ,. [' ~!~~: ::'[:> (:"'T- '-f~ -, '::"f"'RJJ
I f--.t' !. .:.. : .-:--L;) :' .r ,.. 'rr ~ r t.!.T !' :" !·:~:"i·:, :~r, ,:L·..; .! ' '~.I Lt·...~" ' .~. 1'\" '.,.. ,1, -."" ,..; ... _, .... '''!~I'''I .'I '-I", . ;·-.1·.. ·".. "·r.:· 'I', + --, ::., . ':'1'T'"
I ~:! ' ! ,I,,! ': '!',"!',': I' .\. !.;: ,1,· ,".: . . ··.1· .... "',I ,",.....L

I'

I.
-!-

""I:\.

L. ,.

- ....

... ... ......

"

"'1'

.,

!/r-

l..,

"

!

,:

'1 .. ' ..

1
I" "',,.,1 ..... : I 'j. "'" '''1'':'!'! . ,' ,!,. I ' I ' !"i' "--I"i" ,,:

I~-t',:;' ittil+I:'tti~··i~i i:;=(1~tl:f1' ~!"'ilji··I:!:1":~·I ~ T ! .~ 1,.1.· j-"i- I· ~' '"'':1''1''' "-I-'! :. ! I':" .,. i· :";"r iT' ,. .!. i .j.:" "'! I' ,
t- I!
:~ ' :,''I II. I,,:i +'I 1 ,'.,;,I ' I "'I illi"'
j '. ,' ,J':r'I' ' 'II! '.;, 'V'\.I.!
':
il"1

j"!'

'ji'-:'", ".
.
'--..... 1
I'
,I
:
!,'
i
'~j'
-""'"il''' i"I:"1' i"~"':I"':,. '-1
;..
, 1
"1
i. .
I' I
t' i
,. . : 'I' .' i I "... ,- -I-- .. : ... -1-, :1·"; ,'"I''''! ,. ",1,· ,+ I " .. , .. -';'
I' .•. !- i '1 .,. ·1..
.I,!
r-, >-' : .I... _I. 'n' I : l..L.. i_I . . .;... -"-~'- 1 " --~.-J--L--·I
I ... L' i j.
f.-i. :'j . 1
f·. f,J· f· "i·j .. !.:[ ... -yi· 1 ;.. ":",
. I', .'' I1 ,,1,
I ' I I· . . I
f\.'
.
. ..
! : I!"
I ! ',.,''''1'.... ~ .... ".;.- "', .. 1. . ·1.·· .".:" :': "'1' .~ ,- .".j ... ii. 'l'r ;""'1'''
i . 'r '1"L~i- .
.
.. ~. f'-' '~--'r-"-----r;,,1 . :., .• ':'!" .\:.: . .1:. ':'1·...
I·..
,.,!:
: , , .,
, . : I· "I "
"1"
i
I .,
j .H.!
: l'r-+-..L_L~-'I

I"'l" ''':',11'','

.....

-

',"-' .

l'I.,--" ........./.":/.....j:!. ',' ....,.. :,-- ,-,. -!,···,·.
"":'f I" I ! .' ".....
:, i

.

·1

j.- ,..

!. : , . ''';'' . ",:'1 . ',,-·t·~... , ",......, ,'-,-~
1"·'-'I~,L,

H· ··r·I·"

vi ' -~-,'r' !. .:,.,:.- i .:-t·... ,).........,.-.. -.~ .... '--I
' - ' - ..

I·-r··:r.. ,._...... I ' ! ': l-i : ',-!
I· ... " ':.. ; ',',---

j'\ ': ,', ,.

-;, '1 ,:::::1,:; :'If·.i~ ;T~ :~. :J" '.' : : ::~I:':: I::·
I~ h.lr :::1 . ,:'.(' .:"!... ~ ... :.. -I.. ." . .
.:.:1., '......:.
r",'! ' r ,.;. '.1". '~ .
If.
. .
i··.;-·!.·~· .+- .- I" ',....... ,,,·f .".:1 .. l·"k~+" . . ',... - " ....I·.. .i ~ :.1 .. i:
'
'0"
'I· I
.,' "'I '1· " .' ..vi .! ! , .

'. i -. ""1 I'

1 ... ".

. ' ""!"1"I
1 .. 1
1'i ' T .. .... j ... '~r-- ...:--. ·.. ·1........ j.... , ....... :"'1-" ,-, ... : .,I .... I ""r'.. ",'"I ... -:"1'"I "I . ', .... ' j ' , 'i :'1 '" .' .:""
"'1'
'!',
1,:. " , '. _....... '''1' . :.: ,""'I" ~..: '!"!.., ,.

I,,!.

_._'..
'-r,I, ..
11'-,+I I.... I . I +, .1"" . (. .!., +1
I"i'L"I, II~" I"}'!" I-"i'-I' +, I.
,'I,
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+:1"'['"j'I!... .:t ,",,,! ..... ,.

.. ':-

,

+

t

i f !

!

'.

_.. _

'''!.':L_.~,J ....:- -"""I'~ ...l: "1' ·t'.... r 'V T "I .. ,· ....'1"··' ..1-·
r. "I.·-;--'r--!'-'j- '~1~ -r :t ~I'~' ·T.j .... ·1' i.. ' I-:"r-·r '~T' ....1·: .._·[· -,:I I '.... I ", . !·.... j'. .. :'..:'·ii ~ t. :r ':. ! I ", --N,!.. . ..... . ,'..
I )-+~-1:"l- ~ -t~ 1- I~~, -~. I· ··i.. .. I·'\· ··1·, ... ,'" . i· ....,/......,:. J~ YI'·~ :-1 ~ :'r ":r:'~j, .1 , 1I'"! ....... "....
!,
r : 1,\ : ..1...: . ~~:.. .~, r. ~-, . :-. '.~~ ·n' i , ._.' v:.-. ~l .....~, ~.~...~~ J'l
I ii,
i I' j':"1 ~ .j ..
j , :.. ;1... ·I-j.. : ..

l.

·'·1'

1

I

II"

1.. 1

I

.1

r

'i

'"

i·

·1·

',"

I

,

" .

:.

t

·1i ..

. , o·

I,

'

.

j "-

!

,

!

I

;

1

'_L:

1

;

:

,

iii

I

:

I· I I.!

:.. [: ..•

I l '

!

:1

f· !:',.;:·'I.1 ~I I ;l~i:·!: +tj· : ;'i' (" ~·Iill ~·I-jtj[
f

I· ;

II

j

.... -j..... 1 .. !.
I.!. -;- ~·-~+:'·l'· ... -,--~.. _:- "!"1'rw':' j.. ~.! ....;.____, ·;"'I·-·~-!-I. ",',!.,
I
. " .
1
._ _ ••

Fig.4 b)30

Fig.4.

Evolution of the weak shock wave coupled with heat flow.

Governing

equations are the same as those in Richtmyer and Morton's book [8] , where~/v=0.04938l

and v=0.975933.

index 2 at k6x=n.
a)There is a double root, which is -0.953112 of

No absolute eigenvalue is greater than 1.

The shock wave structure is completely hidden behind the error in theinitial values from the second time step.
b)

On the other hand, the initial values are smoothed.

The shock wave

structure can be observed in this figure.

Faculty of General Education
Tokyo University of Agriculture& Technology

Fuchu-shi, TokyottNote by the Editor:

The author of the present paper suddenly passed
away on December 27, 1975 by an accident.

31

A posteriori error estimation
for Volterra integro-differential equations
Kunio Tsuruta and Katsushi Ohmori(Received January 5, 1976)~.

Introduction
In this paper we consider a posteriori error estimation of

the approximate solution of the following nonlinear Volterra
integro-differential equation

(1.1)

(1.2)

dx = f(t,x) +dt

1og(t,s,x(s))ds,

O..{t..{T<+co.

Recently, for Volterra integro-differential equations aswell as in the case of ordinary differentia! equations, various
approximate methods, for example, the Chebyshev series m'ethod(1], the Linear multistep method [2J, the Euler method [3J,'
the Runge-Kutta method [4J and the Spline function method [5J,
have been reported. In them, though a priori error estimatesto approximate solutions have been given, it seems that a posteriori error estimates have no~ been given to approximate
solutions.
In the sequel, re~ardine equations (1.1)-(1.2) as an operator equation and applying the theorem on the Newton method forfunctional equations by Urabe [6J, wetprove the existence of

3,1

an exact solution from an approximate solution and obtain aposteriori error estimate to the approximate solution. In thiscase this error bound is calculated by the uniform norm, andso larger than the exact one for each point. In the main
Theorem 2 we obtain a pointwise error estimate under the weakerconditions than those in Theorem 1. To obtain the first approximate
Wolfe

solution~r11.

we use the Picard iteration method discussed by

The merit of using the Chebyshev series is that its

derivative orinte~ralalso can be easily.

can be computed easily and the residualThou~hthe Wolfe's method is essentiallythe application of the method of Clenshaw and Norton[7J, theproper error estimate of the computational Chebyshev series toan exact solution was not discussed. On the other hand, insection3 our main theorem guarantees the possibility of gett-ing the proper error boun to an approximate solution obtainedby Wolfe's method. In general, the speed of the convergence by
the Picard iteration method is easier than the Newton method
to get the computational solution. In our case, if we could
not get the adjoint kernel X(t,s) explicitly, it is not easy
to do the Newton iteration, but it is possible to .do the theoretical Newton method and80

get the better estimates thanthat of the Picard iteration method, provided that a starting
approximate solution were obtained by the Chebyshev series
iterated appropriately.

34

2. Preliminaries.
In this section we consider the nonlinear Volterra integrodifferential system (1.1)-(1.2) under the following conditions:x(t): an unknown n-dimensional vector-valued function,f(t,x): a given n-dimensional continuous vector-valued
function on I and continuously differentiable
in xED,g(t,s,x): a given n-dimensional vector-valued function,(H)

which is continuous on (t,s)(o~s ~ t ~ T),continuously differentiable with respect to x in
D and g(t,s,x) =° (if t < s),where D is a given domain of n-dimensional
Euclidean space Rn and I= [O,TJ.

Moreover we suppose

Lipschitz continuous in xED.

Lete( I ;.D) be a space of all continuous bounded functions
from I into D with the norm IIx!, = ~~..plx(t)1 for any x E: e(IiD),..-~!
1
where I-I denotes the Euclidean norm. Let e (I;D) be a spaceof all continuously differentiable bounded functions from I
into D. Furthermore we define a product space V = e( I ;Rn) X Rn.
For any y = [v(t),} ] E V, if we define the norm lIyll" = IIvn + I~I'
then it can be easily shown that V is a Banach space.

35

Definition.

For any x( t) E: C1 CI;D), we define the' residual

operator pcx):,dx
f(x) (t) = dt - f(t,x(t))

(2.2)

t

-10 g(t,s,x(s))ds.

Under the above preparations, we consider the following operator]' ( x) ( t )= [P( x) ( t) , x ( 0 ) -7l .

We also consider a linear operator from C1 (I;D) into V such
that

(2.3)

L(x)(t) = [~~ - A(i;)x(t)t-1o a(t,s)x(s)ds, x(O)],where A(t) is an nx n continuous matrix on I and aCt,s) ann)(n continuous matrix on (t,s) (O.{. S.s.. t~'r), aCt,s):: 0 if t< s.,Lemma 1.

The resolvent solution X(t,s), which is the uniquesolution of the equationtaX(t ,~ = -X(t,s)A(s) - J
X(t,u)a(u,s)du,dS
so .s.. s ~ t .s.. '11,X(t,t) = I,

is differentiable with respect to t and satisfies the equationt- ACt)X(t,s) +La(t,u)X(u,s)du,36

o .s.. s ~ t,~ '1'.

And there exists an inverse operator L- l for (2.3) such that
L-l(y)(t) = X(t,O)~ +Remark 1.

1o X(t,s)v(s)ds,

y =[v, ~J.

It is well-known that if A(t) and aCt,s) are continu-

ous functions, the adjoint kernel X(t.s) exists ([8]).

'rhus under the sui table conditions we can establish the modi fied Newton method for the operator equation F(x) = 0 as follows:(n= 0,1,2, ••• ).

Applying the theorem on the Newton method for the functional
equation by Urabe [6J to the system (1.1)-(1.2), we have afollowing existence theorem.

Theorem 1.

Assume that the system (1.1)-(1.2) has an approxi-

mate solution x = x( t) E D, x( 0)= '1, for which there are apositive number &and a nonnegative number ~~l such that(i )DO" = t'iI {x /1 x - x(t) I ~ 0 } CD,(i i )~~ f { Id ( t , x ( t » - ,A ( t) I +t£I.~( t , s, x ( s» - a ( t , s) Ids} ~for any x E C(I;D o)'
(iii)

Mr

1
_ -

K

< cf

-,

where r 2 0 and M > 0 are constants such that
and

37

*

Then the system (1.1)-(1.2) has one and only one solutionx = x(t) E Cl(I;Do) on I, and for this solution we haveMr

Ilx - x II~--1 - K

3.

A pointwise error estimation

In Theorem 1 assumptions (2.1) and o~ X < 1 are rather strong.
In this section we obtain a theorem relating to the poftEwise
error estimation under the weaker assumptions than those in
Theorem 1. In order to prove Theorem 2, we use the following
lemma which could be easily proved.

Lemma 2.
t~

For a given nonnegative continuous function u(t),I, and a constant c 20, we define a sequence {rn(t;c)}(n = 0,1,2, ••• ) as follows:r +1(t,s) = cl u(t-s)r (s;c)ds.

nOn

Then N~Q)
lim .ft=D~ r n (t; c) = R( t; c) exists uniformly on I, ·andR(t;c) is nonnegative, nondecreaing, continuous and satisfies
the following linear integral equation:

R(t;c) = rO(t;c) + c

Furthermore if c

ht u(t-s)H(s;c)ds,

t 6: I.

~

L u(s)ds < 1, then we have R(t;c)~()

By using Lemmas 1 and 2, we get the following38ro(t;c)1. -th~orpm.

c foU(S)dS·Theorem 2.

Under the hypothesis (H) we consider the Volterra

integro-differentia1 system (1.1)-(1.2). Suppose that the system(1.1)-(1.2) has an approximate solution x = x(t) E D, x(O)=1,

for which there are positive numbers 0 and M, and a nonnegative
number K such that= V \ xlix - x( t) I s.. 6 } c. D,

(i)

Do

(ii)

sup {I¢(t,x(t»

tEl

tE: I

- ACt) I + llf<t,s,x(s»0for any x ~ C(IjDo>,(iii)

IX(t,s)1 ~ u(t-s)(iv)"P( x)" ~ r.

- aCt,s) Ids}~ )<.

M'r£u(s)ds ~ M,

and. Then if rR(Tj~)< r, the system (1.1)-(1.2) has one andM - aonly one solution x = x(t) ~ C1 (I;Do), and for this solution wehave
Ix(t) - x(t)/ ~ rH(tjX),

tEl,/P(xn)(t) I ~ r-f r n_1 (t;-f")

if ~~ 0 (n = 0,1,2, ..• ),p< xn) (t) = 0

if)( = 0 (n. = 1,2, ••• ),

M

and

11

where r -1 ( t; ~ ) = V

if)( I. 0 and {xnl (n = 0,1,2, ••• ) isdefined in (2.5) with Xo =Proof.

x.

At first from the definition of {xn } we havet

= -{ X(t,s)f(xn)(s)ds (n = 0,1, ••• ).
o39The proof for K = 0 is essentially the same in the case ofKIo. So we prove the theorem only for K ~ O. Suppose that

K I O. Then we obtain by mathematical induction·
(3.2)
and

In fact the results are obvious for n = O. Assume that x 1 ,x2, ••• ,xN are all in D6 and (3.2), (3.3) hold for n = 0,1,2, ••• ,N. From (3.3) it follows that IxN+1 (t) - xO(t) I ~ &, thatis, x N+ 1 (t) ~ Dd' t~

I. Then the definition of f(x), Lemma 1,(3.1), the induction hypothsis and the mean value theorem
imply that

I

£

dXN+1
t
IP(xN+1 )(t) I = dt
- f(t,x N+1 (t)) g(t,s,xN+1 (s))ds I• I~~- P(xN)(t) - £oX~i'S2p<XN)(S)dS - f(t'XN+1(t»-£g(t,s,xN+1 (s»ds I

~

1

t

1o {/A(t) - 0(t,x,/t) + 8(x N+1 Ct) - xN(t)) / + fla(t,s)
0-l:"Ct,s,xN(s) +8(x N+ 1 (s) - x N(s))/ds}d8 rr N(t;-1t)
40

and

~ r..l:L } u ( t - s) rN ( s ; $.. ) d s
Iv! -bM= rrN +1 ( t ;+ )·Therefore (3.2) and (3.3) are proved for all n. As the remaindersof the proof are similar to those in Theorem 1 in [6J, they areomitted here.

Corollary 2.1.

Under the same assumptions as in Theorem 2, we

have

Ixn ( t) - x ( t) I ~ IP( xn )11 R( t ; X)
M

for K ¢

° and n = 0,1,2, ••••The proof of Corollary 2.1 follows ~ediately from the proofof Theorem 2 if we regard

Corollary 2.2.

f(x n )

as r in fheorem 2.

In Theorem 2, if K satisfies the inequalitY~<l,

we haveMr
1 - K

I rR(t;..1S-) Is-..--M

IIx -

Mr

xll~--1 - k.

':('hie is just the result of Theorem 1. 'rhe proof of Corollary2.2

41

follows from Lemma 2 and Theorem 2.

Corollary 2.3.

Under the same assumptions as in Corollary 2.2,

we havefor n = 0,1,2, ••••The proof of Corollary 2.3 follows easily from our hypothesis
and the definition of r n •Corollary 2.4.

If the adjoint kernel X(t,s) satisfies theinequality

I X ( t , s) I .$. c e xp ( b ( t - s ) ) ,

t €

I,

where band c are some constants, then we obtain= {R ( t ., ..k...)'
f'i

b +

c
K

c

(exp(b+~c)t) - 1)
M

if b+~cIO,
fv1

fVI

if b+~c = o.

ctMIf we solve the integral equationtt

I exp(b(t-s))R(s)ds ,M 0R(t) = (c£eXP(bs)dS) + CK

then we find the result of Corollary 2.4.

42

4.

Computational procedure and numerical examples
In his paper [1], Wolfe established the Picard iteration

method with a Chebyshev interpolation series to computenumerically the solutions of (1.1)-(1.2). In this paper weused his procedure to provide a first approximate solution
in the following examples. Though, in general, it is difficult
to find the residual for x exactly, it is easy to compute it
approximately, provided that a first approximate solution xwer~

obtained by Wolfe's procedure. In fact, from the defini-tion of the residual function we find P(x)(t) so that neglecting truncation errors,

P(x)(t)dYN =where -dt

dYN
=

dt

dYN+1
dt

t={, AN'jTj(t) is the N-times computationallyn=O
iterated function by means of Wolfe's procedure and x = YN.

-

Wolfe's procedure is easily extended to solve the adjoint
kernel X(t,s) numerically with a double Chebyshev representation. Thus the Newton iteration (3.1) could be carried out
computationally. This procedure has not been executed in thispaper because it is primarily our purpose to obtain an aposteriori error estimate for Volterra integro-differential
equations.
We now give two examples. Especially the first has beenused oy many authors 19J.
43

Example 1.

~~ ~ 1 + 2t - x(t) + £t(1+2t)e S (t-S)X(S)dS,
x(o) == 1,

0 So. t ~. 1.

Chebyshev coefficients of x
a O 3.118449253672

as

0.000006752824

a10.800895417852

a90.000000S46036

a20.287218928325

a 100.000000105245

a3

0.056192726040

a 110.000000011974

a4

0.012351784489

a 120.000000001343

a50.002004280830

a 130.000000000141

a6

0.000338715218

a 140.000000000015

a7

0.000047631421

a 150.000000000001

a 160.000000000000

Starting order of the Chebyshev series : 4 •The number of the Picard iteration : 12 •(1) Error estimates by Theorem 1:-11 , K = 0, II x- - /\x /I So 1.068 x 10 -8•{Ii == 1 78, r = 5 • 2 x 1 0(2) Error estimates by Theorem 2:Ix(t) - 2(t) I ~ O.?r(e 5t - 1),
xn(t)
cf.

=

x(t) = exp(t 2 )x(1)=0.(t..{1,(n = 1,2, ••• ).

2.718281828590 ,~(1) == 2.718281828459 ,(Theorem 2) Ix(1) - 2(1)1 '1.540 x 10- 9 •44

Example 2.
dx
dt
x(O)

0.02,

=

0 ~ t

~

2.

Chebyshev coefficients ofx

aO

0.1360872

a4-

0.0002507

a1

0.0602175

a50.0000205

a2

0.014-1604-

a6

0.00000.10

a3

0.0022177

ai

0.0(7 ~ i< .29)Starting order of the Chebyshev series

10.

The number of iterations : 19 •(1) Error estimates by Theorem 1:-6-4 ,M = 15.5731, r = 1.660><10 ", 'K = 3.483X10
IIx - )illS. 2.5861 x10- 5 •(2) Error estimates by Theorem 2:
M = 7.032, r=

1.660 x 10-6 , r( = 7.098 x' 10- 5 ,Ix(t) - ~(t)1 ~ 1.55x10-6(e1.0725t_ 1), O~ t~ 2.
cf.

Ix(2) - :i(2) I ~ 1.17 x 10- 5 •All the computations were carried out on the digital computerIB}'I 370 NODEL/135 at the Computation Center, Waseda University.

'llhe authors would like to express their gratitude to Professors. Sugiyama and Professor K. Nakashima. for their valuableadvices and constant encouragemerit.

45

[lJ f\1.A.VJolfe, The numerical solution of non-singular integral
alld integro-differential equations by iteration with the
Chebyshev series, Compt.J.,12(1969),193-196.

[2J P.Linz, Linear multistep methods for Volterra integrodifferential equations, J.Assoc.Comp.Mach.,16(1969),295301.

[3] C.Cryer and L.Tavanini, The numerical solution of Volterra
functional differential equations by Euler's method, SIAM
J .Numer.Anal. ,9(19?2) ,105-129.

[4J N.A.Feldstein and J.R.Sopka,Numerical method for nonlinear Volterra integro-differential equations, SIAM J •. Numer.Anal.,11(1974),826-846.

[51 G.Micula, The numerical solution 'of Volterra integrodifferential equations by spline functions, Rev.Roum.Math.
Pures Et Appl.,Tom XX,No.3(1975),349-358.
[61 M.Urabe, The Newton method and its application to boundary
value problems with nonlinear boundary conditions, UnitedStates-Japan seminar on differential and functional equations, Benjamin,New York,1967,384-410.

[7J C.vl.Clenshaw and H.J .Norton, The solution of nonlinear
ordinary differential equations in Chebyshev series, Comt.
J.,6(1963),88-9~.

[8] B.l.Grossman and H.h..I"liller, I'urturbation theory for
Volterra integro-differential systems,1 J. Differential
Bquations 8(19'lO),45'l-474.

[9J C.VJ.Cryer, Numerical methods for functional differential
equations, in Proceedings of a Conference Held at Pa~k46

City, Utah, Delay and Functional Differential Equations
and Their Applications, K.8chmitt,ed.,Academic Press,
New York,1972,17-101.

Kunio Tsuruta
anti
Katsushi Ohmori
Department of Mathematics}'acul ty qf ocience and Engineering
Waseda University
Illokyo, Japan

47

On

~he

Existence of an Approximate Solution

in Chebyshev Series of a Nonlinear Integral
Equation of Fredholm Type

Mitsunobu Kurihara(Received January 10, 1976)

Abstract
In the present paper we study a nonlinear integral equationof Fredholm type and prove the existence and the uniqueness ofan approximate solution in the form of finite Chebyshev series
accurately as it is desired for an isolated solution of theintegral equation. We use Galerkin's procedure based on Chebyshev
polynomials and determine the coefficients of the series by toe
method of Newton-Raphson's to obtain the desired approximate
solution. The proof we give is analoguous to that by M. Urabe [3].

O.

Introduction

M. Urabe [3J studied multi-pointbo~ndaryvalue problems fornonlinear ordinary differential equations and proved three basic
theorems (Theorem 1, Theorem, ,2 and Theorem 3) on approximate
solutions in Chebyshev series of the boundary value problems. He
used Galerkin's procedure based on Chebyshev polynomials anddetermined the coefficients of the finite Chebyshev series bv themethod of Newton-Raphson's to obtain the approximate solution.

49

Theorem 1 says thatfo~

any isolated solution there exists anapproximate solution ~ccurately as it -is desired by Qomputing
finite Chebyshev polynomialseri~s.

TheorSm 2 says that theobtained Chebyshev approximation corresponds one to one to theisolated solution. Theorem 3, which plays an important role inpractical applications, says that one can always assure theexistence of an exact isolated solution by checking several
conditions on the obtained Chebyshev approximation and further
it gives a method to obtain an·-error bound of the obtained
approximate solution.
It is expected that' the analbgllous. conclusion· will beobtained for

nonlinea~fact, ShimasakiM. andintegral equations of Fredholm type. In
T~

Kl~ono[IJ obtained the numericalsolutions of the nonlinear integral equations using the methodof Chebyshev series analoguous to that by M. Urabe [3J. Theygave some numerical examples and stated a fundamental theorem
analoguous to Theorem 3 to obtain the error bounds of thesenumerical solutions.
In the present paper we prove two theorems analoguous to
Theorem 1 and Theorem 2 by M. Urabe [3J on the existence and theuniqueness of an approximate solution in the form of finite
Chebyshev polynomial series for an isolated solution. We consider
a nonlinear integral equation of Fredholm type of the form:
1(0. 1)

uCt)=f(t)+J_IK(t,s,U(S»dS

on the interval [-I,lJ. Denote by J the closed interval [-l,l}.

50

Let D be an open interval. Here we assume on the equation (0. 1)
the conditions that f(t) is continuously differentiable functionof t on the interval J and that K(t,s,u) is continuous functionof t, sand u on the region JXJXD and also twice continuously
differentiable function of the arguments t and u on the same
region.
In order to obtain an approximate solution of the equation(0. 1), we take the finite Chebyshev polynomial series withunknown coefficients an(n=O,I, ~- ... ,m) such thatmreau (t)=T (tj.
mn=O n n n

(0. 2)

Here we denote by Tn(t) the Chebyshev polynomial of degree ndefined in the following form:(0. 3)

Tn(t)=cos(ncosand also denote by e
( 0 • 4)

n

-1

t)

for tEJ and n=O,l,the constant number of the form:

eO=l, e n =/2

for n=1,2,·····.

It \'1ill be reasonable to determine these m+l coefficients an(n=O,l,····· ,m) so thatu (t)=(P f)(t)+P JI K(t,s,u (s»ds
m
m
m -1
m

(0. 5)

may hold. Here P

is the operator which expresses the truncationmof the Chebyshev series of the operand discarding the terms of
the order higher than m. In what follows, the finite Chebyshev
series u (t) satisfying the equation (0. 5) will be calledm

Chebyshev approximation of order m.

51

According to the definition by Shimasaki M. and T. Kiyono[lJ, we call u=O(t) isolated solution of the equation (0. 1) when
it is a solution of the equation (0. 1) the first variational
equation of which(0. 6)

V(t)-J~l~~(t,s,n(S))V(S)dS=O

has no continuous solution except v(t)=O. The nomenclature comesfrom the fact that for any isolated solution u=O(t) of theequation (0. 1) there is no other solution of the equation (0. 1)
in a sufficiently small neighborhood of U=Q(t).

(see [lJ) . .

The conclusion of the present paper is the following twotheorems.
Theorem 1.

If the equation (0. 1) has an isolated

solution u=O(t) lying in an open interval D for any tEJ, thenfor sufficiently large mO there exists a Chebyshev approximation
u=um(t) of any order m>mO such that the sequence um(t) converges
uniformly to the solution O(t) on the interval J.
The Chebyshev approximation u=u (t) stated

Theorem 2.

min Theorem 1 is determined uniquely in a sufficiently smallneighborhood of u=O(t) provided the order m of the Chebyshev
approximation u=u (t) is ,sufficiently high.
m

In order to prove Theorem 1, we use the following lemma provedby M. Urabe based on the Newton-Raphson's procedure to determine
the coefficients of desired Chebyshev approximation.

52

Lemma. (M.Urabe [2] and [3])(0. 7)

Let

F(a)=Obe a given real system of equations, where a and F{a) are vectorsof the same dimension and F(a) is a continuously differentiable
function of a defined in some region n of the a-space. Assume that
(0. 7) has an approximate solution a=B for which the determinantof the Jacobian matrix J(a) of F(a) with respect to a does notvanish and that there are positive constant~and a nonnegative

constant K<l such that(1)n8={aIRa-an<8}Cn(2)nJ(a)-J(a)U~K/M'(3) .M'r/(1-K)<8,
where rand M' are numbers such that
IIF(B)I\ ~randThen the system (0. 7) has one and only one solution a=n in noand for a=nit holds thatdetJ(n)~O

andHere we denote by the symbol"

II Euclidean norms for vectors andmatrices.

531.

Some Properties of Chebyshev Series

Denote by C(J) the family of all continuous functions of ton the interval J. For any function f=f(t)€C(J) we use the twokinds of norms \I fll nand 1\ fllq' which are defined as follows: .
(1. 1)\\f\\ =sup! f(t)!
n tEJ

and
(1. 2)

When we use the notations (0. 3) and (0. 4), it is known
that any function f=f(t)EC(J) is expanded in Chebyshev series ofthe form:00(1. 3)f(t)~Lea T (t),n=O n n n

wherea ==€'l lT fl f(t)T (t)(1_t 2 )-1/2 dt .
nn -1nApplying Parseval's equality to the expansion (1. 3) and using
the definition (1. 2), we have
(1. 4)

In particular for finite Chebyshev series of the form:mf(t)= Lea T (t),mn=O n n n

the equality (1. 4) implies that
(1. 5)

where a is the vector a=(a ,a , ••••• ,a ).
o 1
m
54

Moreover using the equality (1. 5) and Schwarz's inequality forthe same finite Chebyshev series, we have
(1. 6)

By the definition of the operator

Pm~C(J)+C(J)for anyf=f(t)EC(J) we havem(P f)(t)= Lea T (t).
mn=O n n n
Consequently it follows that
00

«I-P )f)(t)~ L e a T (t),mn=m+l n n n
where I denotes the identity operator. If f=f(t) is continuously
differentiable function of t on the interval J, it is proved that
(1. 7)~ ( I - Pm ) fl\ n~ a ( m) \I ( I - Pm- 1 ) f \I q~ a ( m ) \1 f U
q

and
(1.

8)

for m=O,l,·····, where P_l=O, f=df/dt and the functions a(m) andal(m) are monotone deceasing of m satisfying respectively
(1.

9)

and

These properties of Chebyshev series was proved in detail in thepaper by M. Urabe [3].

55

2.

Fundamental Inequalities

If u=O(t) is an isolated solution of the equation (0. 1)
lying in the interval D for any tEJ, there exists a positive
number 0 such thatDo~{ul lu~O(t)l<o for some tEJ}CD.

Denote that 0 m=P mO. It follows from the inequalities (1. 7) and(1. 8) that
(2. 1)

and(2. 2)

where

Hence from the inequalities (1. 9) and (2. 1) there exists a
number ml sufficiently large such that for any m>m=m
for any tEJ.
The coefficients a=(aO,a1,·····,a m) of our desired Chebyshev
approximation of order mmu (t)= Lea T (t)mn=O n n n .

will be determined from the equation (0. 5), that is equivalent
to the system of nonlinear algebraic equations(2. 3)

F(m)(a)=(Fo(a),Fl(a),~ •••• ,Fm(a»=o,

where

(2.

4)

I
m
K(t,s,u (s»ds= L e F (a)Tn(t).
u (t)-(P f)(t)-P
m
m
m -1
m
n--0 n nJ

56

The equation (2. 3) is called determining equation of Chebyshev
approximation.
In order to determine a domain where the function F(m)(a) is
well defined, we take a numbe,r m2 >m 1 such that for any m~m2

sincelu-a(t)I<lu-am(t)I+lam(t)-a(t)I<~-Mla(m)+"am-a"nfor anyU€Vm~

Let us put thatex>a(t)= Lea T (t)n=O n n n

a=(~

and

,a ,·····,am).01

and define the domain of the form

n ={a Il\a-all~m

For any a=(a

a

0'1'

1 (~-Mla(m»}.
{2m+l••••• a )En'mm'

we obtainmu (t)= Lea T (t)E:V CDmn=O n n nm

for any te:J

sincefrom the inequality (1. 6). Therefore it is concluded that thefunction F(m)(a) is defiried on the domain nmand continuously

differentiable function of a on the same domain from the definition(2. 4).

Let Jm(a) be the Jacobian matrix of the function F(m)(a).
To investigate the properties of the matrix Jm(a), let us consider

57

a linear equation of the form
(2. 5)
wh~r'ea=(aa••••• a )eO
0' l ''mm'
and

If we putmum(t)= Lea T (t),n=O n n n

m

m

v(t)=

Lex T (t)n=O n n nandw(t)= Ley T (t),n=O n n nthen by the definition of the Jacobian matrix J(2. 6)m(a) we have

Jl aKv(t)-P~(t,s,u (s»v(s)ds=w(t).
m _loUmcorresponding to the linear equation (2.

5).

Substituting Qm(t) for um(t), we rewrite the equation (2. 6)in the form
(2. 7)

V(t)-J~l~(t,S,Q(.»V(S)dS-W(t)+R(t),

where

Jl'aK

R(t)=-(I-P)~(t,s,Q(s»v(s)dsm _loU
-P

J

l
aK
aK
[~(t,s,Q(s»-~(t,s,Q (s»]v(s)ds.
m _loUaUm

It is easy to prove from the inequality (1. 8) and the Parseval's
equality (1. 4) that
(2. 8)

58

+\1

Cl [;~(t,s ,O( s))-;~(t, s ,Om(s)) ]v( S)dSll q<M 2 o 1 (m)1\ V\\q +M3 \\a-a m\\ qll V\\q~ (M 2 +M 3Ml ) 0 1 (m)1I V\\q ,

where

and

From the definition of the isolated solution u=a(t) and wellknown theory for linear integral equations of Fredholm type it
follows that there exists a constant number M such thatfor the equation (2.

7).

(see [lJ or [4J) Therefore from theinequality (2. 8) we have

If we takem3~m2

sufficiently large, we obtain for any ~3

By the equality (1. 5) this is equivalent to the inequality(2 •. 9)It readily follows from the equation (2. 5) that for any m~3(2.10)

detJ mUO;lO

and(2.11)

59

where M' is a constant number. In fact, put n=O in (2. 5), then~=O,by (2. 9) we have

which implies (2.10). By (2.10), from (2. 5)

we have

Then by (2. 9) we have (2.11). This inequality (2.11) plays animportant role in ~he proof of Theorem 1.
Let"""=(a"a' ••••• , a')and....
0' a"l ' ••••• , a")mm
be arbitrary vectors belonging to the domain Qm . Then both~'=(a'

0' l 'mm

u'(t)= I e a'T (t)mn=O n n nandu"(t)= I e a"T (t)mn=O n n nlie in VmCD for any t&J. For any vector ;=(x O,x 1 '·····,x m) let us
put(2.12)

Jm(~');=n'and.Jm(~");=n",

where
and

n "=(y"0' y"l ' ••••• , y")m •Corresponding to the equations (2.12), we have
(2.13)

v(t)-P Jl ~K(t,s,u'(s»v(s)ds=W'(t)m _lOUmandv(t)-Pl aK-;;;-(t,s,u"(s»v(s)ds=w"(t),fm _lOUmwheremv(t)= Lex T (t),n=O n n n60

mw'(t)=

L e y'T (t)n=O n n nandmw"(t)= L e y"T (t).
n=O n n n

By the equation (2.13) and (2.14) we havew'(t)-w"(t)=-P Jl [~K(t,s,u'(s»_~K(t,s,u"(s»Jv(S)dS.
m -1 aUm
aU
m,
Then it is easy to prove from the Parseval's equality (1. 4) and
Schwarz's inequality thatnw'-w"1I q=<I\Jl-1 [~K(t,s,u'(s»-~K(t,s,u"(s»Jv(s)dsllq
aU
m
aU
m

On the other hand, it follows from the equality (1. 5) and theequa~ions(2.12) that

andHence we have

which implies(2.15)
for all vectors a' and a" belonging to the domain nm. Thisinequality (2.15) also plays an important role in the proof of
Theorem 1.

61

3.

Proof of Theorem 1

Let u=O(t) be an isolated solution of the equation (0. 1)
lying in the interval D for any tEJ. Denote that 0 =P 11 and put
m mmorn. (t)= n=Or e n ~ n Tn (t)andA1 , ••••• ,am.
A )a O ,a~=(At\

It is concluded from the previous section that there exists a
number m3 such that for any

m~m3 the function F(m)(a) defined in(2. 3) and (2. 4) is continuously differentiable of a on thedomain st m and the Jacobian matrix J (a) of the function F(m)(a)m

has the inverse J-l(a) at a=& satisfying the inequality (2.11)m
and satisfies (2.15).
Now we will apply the Lemma in the section 0 to the equation(2. 3), the roots of which are the coefficients of our desired
Chebyshev approximation. For any m~m3 let us put

am(t)-(Pmf)(t)-PmJ~lK(t,s,am(S))dS=Rm(t).
This is rewritten in the form
R (t)=P Jl [K(t,s,O(s»-K(t,s,O (s»]dsm
m -1mand hence from the Parseval's equality (1. 4) and Schwarz's
inequality it follows that

II RmII q < IIJ 1 [ K( t , s , 0 ( s ) ) - K( t , s , Om ( s ) ) ] d s It q
-1

<M 4 11 0 m- O IIq <M4 Ml 0 1 (m),

where

62

If we putFCm)C&)=p,then we have from the equality (1. 5)

(3.

1)

Since 0l(m)=(m+l) -1 ,the inequality (3. 1) expresses that a=& isan approximate solution of the determining equation (2. 3) for
any sufficiently large m.
In order to check the conditions (1), (2) and (3) in Lemmain the section 0, we take an arbitrary nOnnegative number K<l andput

There exists a number m4>m 3 so that
[M'M 1M4 /(1-K)]ol(m)<ol//2m+l- (m)=O(m -1/2 ) as m~oo. If wemay hold for any m>m4 since / 2m+loltake om such that

(3. 2)(3. 3)

QO :;:{aIJla-&I\<omKQ ",.
m

In fact, for any a~Qo

and any m~m4
mlIa-&\I<om~Ol//2m+l~[o-Mlo(m3)J/12m+l~[o-Mlo(m)J//2m+l,

which implies a~Qm. Moreover for any aEQ om(3. 4)

63

and any m~m4 we have

Finally by the inequalities (3. 1) and (3. 2) we have

(3. 5)

M'Hp"<[M'M M /(l-K)]o (m)<~.
l-K =
1 4
1
= m

The expressions (3. 3), (3. 4) and (3. 5) show that the conditions(1), (2) and (3) are fulfilled.
Thus by the Lemma in the section 0 we see that the determining
equation (2. 3) has one and only one solution a=a in the domainnosatisfyingmand(3. 6)

If we puta=(aa ..... , am'\
0' l '

andmu (t)= Lea T (t),mn=O n n nthen um(t) is a Chebyshev approximation and satisfies for m~m4
<llu m-a m1\ n +Ilo m-all n=<y'2m+llla-&II+Ml o (m)I\u m-ol\ n=<[MIM I M4/(1-K)]y'2m+lo l (m)+M l o(m)from the inequalities (1. 6) and (3. 6). The functions y'2m+lo l (m)and oem) are equal to O(ro- l / 2 ) as m~oo. This proves the existenceof a Chebyshev approximation u (t) being convergent uniformly tomthe solution o(t) as m~oo.

64

4.

Proof of Theorem 2

Let u=u(t) be an isolated solution of the equation (0. 1).
For any positive number E we choose a number mO such that for any
m>mO,ol(m)=l/(m+l)<E and suppose that for any m>mO there are two
Chebyshev approximations(4. 1)

u=u (t)mandu=u' (t)msatisfying(4. 2)

andHence the two Chebyshev approximations (4. 1) lie in the domain
DE={U\ \u-U(t)\<E for some t~J}CD
for any tEJ. Let us put v (t)=u (t)-u' (t). By the de.finition ofmmm
the Chebyshev approximations (4. 1) we have
(4. 3)

vm(t)=P Jl [K(t,s,u (s»-K(t,s,u'(s»]ds
.
m -1
m
m
l ClK
=J _laU~(t,s,u(s»v (s)ds+R(t),
m

where
ClK
8 (s»-~(t,s,u(s»]v
ClK
R(t)=P J l Jl [~(t,s,u(s)d8ds
m -1 0 aUmaUm-(I-P )Jl ~K(t,s,u(s»v (s)dsm _l.aUmand
8 (t)=u'(t)+8[U (t)-u'(t)].
Ummmm
Noting the fact that ue(t)ED for any 8E[0,1] and any tEJ, it ismEeasy to see from the Parseval's equality (1. 4) and Schwarz's

65inequality that

where M2 and M3 are the constant numbers defined in the section 2.
Using the theory for linear integral equations of Fredholm type
for the equation (4. 3) and the definition of the isolated solution
u=Q(t), we obtain

where M is the constant number used in the section 2. Since E isarbitrary, the inequality above implies that

From the Parseval's equality (1. 5) for the finite Chebyshev seriesvm(t) it follows thatv m (t)=O

for any tfJ,

that is,um(t) =u'm(t)for any tEJ.

This proves the uniqueness of Chebyshev approximations and hencecompletes the proof of Theorem 2.

66

References[lJ

Shimasaki M. and T.Kiyono:

Numerical solution of integral

equations in Chebyshev series. Numer.

Math~21, 373-380(1973).
[2J

Urabe M.:

Galerkin's procedure for nonlinear periodic

systems. Arch. Rat. Mech. Anal. 20, 120-152 (1965).
[3J

Urabe M.:

Numerical solution of multi-point boundary

value problems in Chebyshev series, Theory of the method.
Numer. Math. 9, 341-366 (1967).
[4J

Hukuhara M.:

Integral equations (Japanese). Kisosugaku-

koza 18, Kyoritsu Shuppan (1955).

Mitsunobu Kurihara
Depertment of Computer Science
University of Yamanashi
Kofu, Japan

67