Computing 40, 315-328 (1988) Computing �9 by Springer-Verlag 1988

On the Stability of the One-Step Exact Collocation Method for the Second Kind Volterra Integral Equation with

Degenerate Kernel

Maria Rosaria Crisci, Napoli, Elvira Russo, Salerno, and Antonia Vecchio, Napoli

Received September 3, 1987; revised April 4, i988

Abstract - - Zusammenfassung

On the Stability of the One-Step Exact Collocation Method for the Second Kind Volterra Integral Equation with Degenerate Kernel. The local stability properties of the collocation method applied to a second kind Votterra integral equation with degenerate kernel are investigated.

A finite length recurrence relation is derived and theorems for tile Iocal stability of the methods are proved.

AMS Subject Classifications." Primary 65R20; Secondary 45D05, 45L10.

Key words and phrases: Volterra integral equations, Degenerate kernel, Stability, Collocation method.

Uber die Stabilitiit des exakten Einschritt-Kollokationsverfahrens fiir Volterrasehe Integralgleiehungen zweiter Art mit degeneriertem Kern. Es werden die lokalen Stabilitgtseigenschaften der Kollokationsme- thode, angewandt auf Volterrasche Integralgleichungen zweiter Art mit degeneriertem Kern, unter- sucht.

Eine Rekursionsrelation endlicher Ordnung wird angegeben, und es werden Sfitze fiber die lokale Stabilit~it der Methode bewiesen.

1. Introduction

Collocation methods for the numerical solution of the second kind Volterra integral equation

t

y(O=g(t)+~ k(t,s,y(s))ds t e [to, T] (1.1) tO

have been discussed by several authors (see, for example, [3], [4], [5]).

In order to introduce the one-step collocation method, let us consider a partition

H i v = t o < t l < ... < t ~ r T

of the integration range and define the subintervals:

0"o = [to, t l ] at=( t i , ti+l] i = 1 , . . . , N - 1

of size h.

316 Maria Rosafia Crisci, Elvira Russo, and Antonia Vecchio:

Let rr,, be the space of polynomials of degree less or equal to m and let

S f , ) = { u ' u ( t ) [ t ~ : = u i ( t ) ~ T r m , i = 0 .. . . , N - 1

Ul~l (tl) = u? (0 , j = 0 . . . d}

be the space of real potynomial splines of degree m and of continuity class d.

At last, let us define the set of collocation points

X i = { ~ . i , j = t i + c j h O < c l < _ c 2 . . . < c m _ l } i = 0 , . . . , N - 1

where the c j, j = 1 ... m, are the collocation parameters.

The one-step collocation method approximates y (t), the true solution of(1.1), by an element S~ )_ 1, d = - 1, which satisfies (1.1) on X~ i = 0, ..., N - 1, that is, u~ (t) satisfies the equations

t i

~ (~,~.) = g(~ , j ) + y k (~,j, s, u(s)) d~ + $o

(1.2) ~.J j = 1, ..., m

+ j" k(~i , j , s ,u~(s))ds ti i = 0 , . . . , N - 1

If c~ =0, % = 1 , u(t)e S(~ 1, that is, it is continuous in [0, T] ([6], p. 248).

Moreover, the collocation method is called "exact" if the integrals in (1.2) are calculated exactly and "discretized" if the integrals are calculated with a quadrature tbrmula.

The stability analysis of the collocation methods is still fairly developed.

In [7] we have investigated the stability properties of the exact one-step collocation method with c~ =0, % = 1, through the application to the basic test equation and to the convolution test equation.

In this paper we continue the stability analysis of the above method with respect to a second kind Volterra integral equation with degenerate kernel of rank n ([2], [9]):

y ( r ) = g ( t ) + ~ y" a3 ( t )b j ( s ) y ( s )ds , te [ to , T-J (1.3) to j = l

We observe that from the Stone-Weierstrass theorem it follows that the class of degenerate kernels is dense in the class of all continuous kernels k (t, s). Therefore (1.3) can be considered a significant test equation for the stability analysis.

Moreover, (1.3) is a generalization of the equation with convolution polynomial kernel,

m t

y it) = g (t) + Y~ ,~j ~ i t - syy (s) ds, (t.4) j = l 0

used as test equation in [1], [6] for the stability analysis of some methods. However, there are no results in the literature concerning the stability of the exact collocation methods applied either to (1.3) or to (1.4).

On the Stability of the One-Step Exact Collocation Method 317

In this paper we prove that the application of the one-step exact collocation method to (1.3) leads to a finite length recurrence relation, with coefficients depending on i, for the vector

where [Ui (~i, 1) ' " ' ' , Ui (~i, m), Z1 (ti + 1), " ' ' , Zn (ti + 1)] T,

i z j ( t ) - bj(s)u(s)ds. (1.5) to

Using this recurrence relation, local stability conditions are proved.

Unfortunately, these conditions give no information about the global behaviour of the numerical solution for large T. This is until now an open question.

The stability conditions derived are independent of the decomposition of the kernel and they require the localization of the roots of a polynomial whose degree is the minimum of the rank n of the kernel and the number m of the collocation parameters.

Moreover, we prove that, ifm is less than n, the one-step exact collocation method is not locally stable in the strong sense.

Then we consider, as a particular case, degenerate kernels of convolution type which are useful in the applications, and we prove that the above derived conditions furnish "a priori" stability conditions.

In the case of the linear convolution test equation we can deduce, from the recurrence relation here obtained, a shortened one, which coincides with that derived in [7].

At last, we report some numerical results, showing the reliability of the stability conditions derived.

2. The Recurrence Relation

In this section a finite length recurrence relation is derived for the one-step exact collocation method applied to the second kind Volterra integral equation with degenerate kernel,

t n

y(t)=9(t)+ ~ ~ aj(t)bj(s)y(s)ds, ts[to, T]. (2.1) t 0 j = l

We assume that the stability conditions for the solution y (t) of(2.1) are satisfied ([6], p. 424).

Let us make the following positions:

m

L~ (x) = [~ (x - {i,i)/( {i,k-- {;,a) (2.2) j=l j~k

318 Maria Rosaria Crisci, Elvira Russo, and Antonia Vecchio: ~ i , j

s}k= ~ aq(~i,i) f bq(s)L~(s)ds, j , k = l ... . ,m (2.3) q = l t i

:~}k = ak(~id), j = l , . . . , m k = l , . . . ,n (2.4)

i _ _ fi2k-- ~ b2(s)L~(s)ds, j = l , . . . , n k = l , . . . , m . (2.5/ t i

Let us denote with S ~, A ~, B ~ the matrices whose elements are s}k, c~}7 ,, fi}k, respectively and with I k the identity matrix of order k. Then we introduce the following vectors and matrices

ti, + 1 = Lu (~i, ,) . . . . , u ({i,,,)] r (2.6)

zi + l = [zl (ti + ,),..., z, (ti+ 1)] r (2.7)

O~ = [g (~; ~) . . . . , g (~,,,d] ~ (2.8)

�9 (lm -- S i O) (2.9) M'o = \ _ B~ I,

Ai) 9 1 (_.~o)

I,

Then the following theorem holds.

T h e o r e m 2.1 : The application of the one-step exact collocation method to a second kind Volterra integral equation with degenerate kernel leads to the finite length recurrence relation

\zi+ t / (2.11)

Proof'.

The collocation equations (1.2) in this case are

n t l

ui(~i,j)=g(~id)+ ~ aq(~id) ~ bq(s) u(s)ds+ q=t to (2.12)

n ~ i , j

+ ~', aq(~i,j) ~ bq(s)ui(s)ds, j = ! , . . . , m . q = l ti

As u~ (t) is an m - 1 degree polynomial, it coincides with the interpolating polynomial in the m collocation points, therefore u~ (t) can be written as

ui (t) = ~ L~, (t) u~ (~,,k) (2.13) k = l

where L~ (t), given by (2.2), is the k-th Lagrange canonical polynomial for~ the collocation points.

On the Stability of the One-Step Exact Collocation Method 319

Substituting (2.13) into the (2.12) and using (1.4) we have

n

ui(~i,j)=g(~i,j) + ~ aq(~i,j)Zq(ti)+ q = l

~i , j

+ ~ ~ aq(~.i,j)ui({i,k)~ bq(s)L~(s)ds, q = l k = l t i

Moreover, from (1.5) and (2.13) it follows that

j = l , ...,m.

(2.14)

m t i + 1

Zq(ti+1)=Zq(h)+ ~ uf(~,,k) I bq(S)L~(s)ds. (2.15) k = l t i

The equations (2.14), (2.15) give a finite length recurrence relation, which, using the above positions, can obviously be written in the matricial form (2.11). []

Remark 2.1 :

As it is known, the solution of (2.1) can be written as

n

y(t)=g(t)+ ~ aj(t)~j(t) (2.16) j=l

where the (j (0 are solutions of the differential system

()(t)= ~, ak(t)bj(t)(k(t)+bj(Og(t) k = l

(j (0) = 0. (2.17)

The zj which appear in the recurrence relation are the solutions of (2.17) and therefore they are significant quantities for the problem (2.1).

However, it is easy to verify that applying the collocation method to (2.17) and then using (2.16) is not equivalent to applying the same method directly to (2.1). There- fore we have to analyze the stability properties of the collocation method directly for (2.1).

3. Stability Theorems

First of all, let us observe that, as from the solvability of the system (2.14), it follows that Im-S ~ is invertible and (2.11) can be written as

ei+l/

where M z = (M~) - t M{.

As it is known, the stability of the method depends on the matrix Mz; since the elements o fM ~ are depending on the step number i, it is possible to derive conditions only for the local stability.

320 Maria Rosaria Crisci, Etvira Russo, and Antonia Vecchio:

Referring to [6] (pag. 432) we give the following

Definitions: The one-step exact collocation method is called

a) locally stable in the strong sense if all the eigenvalues of M i lie within the unit circle;

b) locally stable if all the eigenvalues of M ~ are within or on the unit circle and those on the unit circle are weakly stable, that is, they correspond to Jordan blocks of order 1 in the Jordan normal form of the matr ix M ~.

Let us define the matr ix

F (x) = (x - ~):(Im - - S ' ) - - A ~ B ~

and the polynomia~ C ( x ) = d e t F (x) ;

then the following theorem holds:

Theorem 3.1:

(3.1)

(3.2)

The one-step exact collocation method is locally stable in t i in the strong sense if att the zeros of

(x - 1 )" - m c (x) = 0

lie within the unit circle; it is locally stable if they are within or on the unit circle and i f those on the unit circle correspond to weakly stable eigenvalues of M ~.

Proof:

Obvious ly the eigenvalues of M! are the solution of

det (M~) x - M~) = O. (3.3)

F r o m (2.9) and (2.10), putt ing

/Im - S i

D(x) = ~ B i

(3.3) becomes

x" det D (x) = 0. (3.4)

Therefore, m eigenvalues of the matrix M ~ are equal to zero and the other n are equal to the zeros of det D (x).

Let us t reat separately the two cases.

I. m > n

Let us introduce the matr ix E 1 (x) defined by

i x-l, im i35

On the Stability of the One-Step Exact Collocation Method 321

where Ai and B ~ are obta ined from A i and B z by adding m - n columns and rows of zeros, respectively.

Obviously, det D (x) = (det E 1 (X))/(X - - 1) m-n

where, as it can easily be verified, det D (x) is a polynomial of degree m.

On the other hand, as - B ~ and (x - 1) I~ are commutable , by a known theorem ([-6], pag. 460),

det E 1 (x) = C (x)

and the s ta tement follows.

II. m<n Let us introduce the matr ix E 2 (x) defined by

/ In_ m 0

E 2 ( x ) = ~ ~ I"-Si_B i

So we have

~ (x - 1) I , /

det D (x) = det E 2 (x) = (x -- 1) " - " C (x)

and the theorem is proved.

F r o m the above theorem we can deduce the following observations.

(3.6)

[ ]

Remark 3.1 :

The polynomial C (x) defined in (3.2) does not depend on the kernel decomposi t ion.

In fact, let

k (t, s) = ~ aj (t) bj (s) j = l

be the kernel of the integral equat ion; the element 0, k) of the matr ix F (x) defined in (3.1) can be writ ten as

f jk=(x-1) jk - ~ k r -

~i (3.7) ti+l

- ~ k(~i,j,s)Lik(S) ds, tl

so that the stability conditions derived f rom Theorem 3.1 can be applied under the only hypothesis that the kernel is separable, also if the decomposi t ion is unknown.

Remark 3.2:

The calculation of the n + m eigenvalues o f M i is reduced, in virtue of Theorem 3.1, t o the determinat ion of the roots of a polynomial whose degree is the min imum of n and m.

322 Maria Rosaria Crisci, Etvira Russo, and Antonia Vecchio:

Now we can prove the following

Corollary 3.1 :

I f n > m the one-step exact collocation methods is not locally stable in the strong sense.

Proof'.

It is obvious, as from Theorem 3.1, that in the case n > m an eigenvalue of the matrix M i is equal to 1 with algebraic multiplicity at least n - m .

The definition of stability requires a more detailed analysis of the eigenvalue I ; this is contained in the following theorem.

Theorem 3.2:

I f n > m and det ( A ~ B ~) ~ 0 the one-step excact collocation method is locally stable in t~ if all the zeros of C (x) are within the unit circle.

Proof:

Let us denote with 0 and p, respectively, the algebraic and geometrical multiplicity of the eigenvalue 1 of M ~, and with p the rank of the matrix M ~- I.

We know that

On the other hand,

p = n + m - p . (3,8)

M ~ - ~ = (Mg) -~ ( M i - M~)

and then p is equal to the rank of M~ _ Mo .i

Because M[ - M~ contains a null block of order n, applying a known theorem (ES], p. 95 th. 108) gives

and from (3.8)

Moreover, if det (A i B i) ~ 0 we have

p N 2 m ,

p ~ n - m .

O = n - m

(3.9)

and, from the known relation O>_p, it follows that

p=O.

Therefore, the Jordan blocks corresponding to the eigenvalue 1 are of order I and the theorem follows. []

Now let us consider degenerate convolution kernels which are of considerable interest in applications. In this case from Theorem 3.1 "a priori" stability conditions can be obtained.

In fact, the following theorem holds:

Theorem 3.3: I f k(t , s) is a convolution degenerate kernel the eigenvalues of the matrix M i are constants with respec~ to i.

On the Stability of the One-Step Exact Collocation Method 323

Pro@ With the variable change s = t i + t/h in the elements of the matrix F given by (3.7), i.e.

k ( t , s ) = k ( t - s ) ,

it follows that these elements are independent of i and, from Theorem 3.1, the theorem follows. []

Finally, we report the following conclusive remarks.

Remark 3.3: The vector [ui, ~J] appearing in the reeursive relation (2.11) contains the solution u~ (t) computed in all the collocation points. So the stability conditions expressed by the above theore~ns are necessary in order that the error propagation remains bounded not only in the mesh points but also in other points in the integration range. This can be useful as the collocation method furnishes a piecewise polynomial function, which can be used as an approximate solution in the whole interval.

Remark 3.4: A shortened recursive relation for the vector [ui(t 0, z 1 (ti), . . . ,z,-l(t~)] r can be derived from (2.11). In fact, from the first m - 1 identities of (2.11) it is possible to derive, by inverting a matrix of order m - 2 , the vector [ui C.i,2) . . . . , ui (~i,m- 1), z, (~i)] r; substituting the expression so obtained in the last n+ 1 identities of (2.11) and dropping the last one, the shortened recurrence relation follows.

However, this relation is not of practical interest, since it requires laborious calculations, and the application of the stability condition on this recurrence relation requires the calculation of the eigenvalues of a matrix of order n, whereas, from Remark 3.2, with our approach it is sufficient to find the roots of a polynomial whose degree is the minimum of n and m.

Remark 3.5: In the particular case of the linear convolution test equation it can be proved, with tedious but elementary calculations, that the shortened recurrence relation is an homogeneous constant coefficients recurrence relation for the vector [u~ (t,), z 1 (t~)] T and therefore coincides with that derived in [7] for the same vector.

4. Numerical Experiments

Numerical experiments have been carried out in order to test the reliability of the stability conditions derived in the previous sections.

The integral problems choosen are the following:

A) y ( r ) = l + 3 c o s t [ e - t ( s i n t - c o s t ) + l ] - i [ 6 c o s t c o s s + l ] y ( s ) d s o

t ~ [0, 30]

with solution y (t) = e -t

324 Maria Rosaria Crisci, Elvira Russo, and Antonia Vecchio: r

B ) Y ( 0 = 1 - 480 (1 -- e - t) + 500 t + ~ [480 e (s- ') - 500] y (s) d s 0

t E [0, 30]

with solution y(t)= 1

C) y ( t ) = ~ + s i n t - 1 0 0 cos(t-s)y(s)ds 0

t E [0, 30]

sin t with solution y ( r ) - 100'

These problems have been solved with the following collocation methods :

1) m = 2 , c t = 0 , c2=1

2) m = 3, c 1 = 0, c z = 0.5, c a = 1

3) in = 4 , c 1 " ~ - 0, C 2 = , C 3 - - - - , C 4 = 1 6 6

(C 2 and c 3 are the zeros of the Legendre polynomial of degree 2 shifted in [0, 13)

4) m = 4 , c 1 =0 , c z = , c 3 - , c4 = 1 2 2

(Lobat to points).

As an example, let us describe in detail how the stability criteria apply to p rob lem A) solved with method 2).

F rom Theorem 3.1 it follows that in each step it is required to compute the roots of the polynomial of second degree

det [-(x - 1)(I - S ~) - A i B g] (4.1) x - 1

In order to determine this polynomial let us make the following abbreviat ion,

Then we have

3 2 1"~ 0 ' ( x , s ) = - 1 2 c o s x l s i n ( t ~ + s h ) ( s 2 2s-~+~-j+

+ c o s ( E ~ + s h ) ~ A - - 2 h , 3 s 2 + ~ -

L ( 2) (oi2(x,s)=24cosx sin(ti+sh) sa - s -~2 +

--h2S-l~A 4 h ( @ - @ ) + cos (t~ + s h) | +

On the Stability of the One-Step Exact Collocation Method 325

ix [ ( ) ~b3( , s ) = 1 2 c o s x sin(ti+sh ) s 2 s 2 2 h 2 +

4 s - 1 7

From (2.3)-(2.5) the eIements of the matrices S i and F i= A ~ B i a r e given by

s]j=O, j = l , ...,3

,/ti+t,+l ~),/ti+t,+l ) si2s=cks( ? , - ~ j ~ ~ ,0 , j = 1 , 2 , 3

i - - i s3s-(as(tz+l,1)-(a}(t,+~,O), j = 1 , 2 , 3 i _ _ i ylj--r O), j = 1 , 2 , 3

,~s=4)}( t'+t~+~ l),,/t,+t,+~ ) 2 ' -q~J~- 2 ,0 , j = 1 , 2 , 3

' - * 1)-qS}(t,+,,O), 1,2,3. ~3j - - ff)j(ti+ 1, J ~-

The polynomial (4.1) can be written, putting z = x - 1 , as

dio z2 + d~ z + di2

and the coefficients d} are given by

d0 = det [I - S i]

3

d~ = ~ det [Q~] k = l

3

dl = ~, det [Q2] k = l

where Q1 is the matrix obtained from I - S i by substituting the k-th column with the corresponding columrt of F ~, and Q2 is the matrix obtained from F ~ by substituting the k-th column with the corresponding one of I - S k

We observe that all the determinants required are of order three.

In the other cases, the procedure is analogous.

We wish to stress that, for problem A), we have to compute at each step the roots of polynomial (4.1), whereas for problems B) and C), which are of convolution type, in order to use the Theorem 3.3, it is required to calculate only once, io e. in the first step, the roots of the related polynomial.

At last we observe that, as for Remark 3.2, this polynomial (in the examples here considered) is always of degree 2, independent of the number of collocation parameters used.

We report in the Tables 1 - 4 the relative error in some points (t = 10, 30) applying the collocation methods 1,2, 3, 4 to the integral problems A), B), C) and where X m i m Xmax, are the minimum and maximum roots of the polynomial (4.1).

326 Maria Rosaria Crisci, Elvira Russo, and Antonia Vecchio:

F r o m the results obtained we can deduce that in these cases the col locat ion method has a stable behaviour every time the stability condit ion is satisfied.

Moreover , in some cases (problem C): methods 1) and 3) with h = 0 . 5 and problem B): method 3 with h = , 5 ) the method has been stable but the stability condit ion is not satisfied. This can be justified observing that the stability condit ions have been derived for the vector [~i,~i] and not only for u(ti+l).

Finally, in the case B, with m = 2, h = .5, m = 3, h = .5, the stability condit ion is not satisfied and the method is unstable.

I' I

[ c)

Table 1

Collocation parameters: c, = 0, c 2 = l

10.

10.

. i 25 1 ~ 1 3 1 0 0 0 ~ 4

,5 ]301 13Oo1

Relative e r r o r

.39 x 10-1

.33

.90 x 10 -2

.84 x 10 - z

.56 x 10 7

.29 x 10 +9

.10 x 10 -12

.39 x 10 la

.21 x 10-*

.2 l x 10-*

.53 x 10 -2

.53 x 10 -2

Xmln

.52 x 1 0 - 2

.68 x 10

.52 • 10 -1

.54

.40

.87

X ~ x

~99

.99

.5i x lO2

. 54 .

.22 x 10

.96

h

A) 5

B)

c)

.25

.5

.25

Table 2

Collocation parameters: c 1 = 0 , c 2 = . 5 , c3 = i

t

10.

30. 10.

30.

10. I 30. 10. 30.

10. 30. 10.

Relative e r r o r

.99 • 10 -3

.50 x 10 -2 ,52 x 10 - 4

.33 x 10 -3

.7 x 10 -6

.17 x 10 + 11

.24 x I0 -11 ~26 x 10 -11

.73 x 10 -2

.39 x 10 -2

. t l x 10 - z

.19 x 10 -3

Xmln

.60 x 10-1

.17

.14 x 10 - *

.95 • 10 -1

.88

.78

X~ax

.99

.99

.2i x 102

.69

.88

,78

On the Stabi l i ty of the One-Step Exac t Co l loca t ion M e t h o d 327

P rob lem

A)

B)

C)

Table 3

3 - V '3 3+1/~ Col loca t ion pa ramete r s : cl = 0 , c ; - , ca = ,

6 6 c4=1

Relat ive error

10. .52 x 10-1

"~ 30. .66 .2 ; 10. .11 x 10 -1

30. .20

10. .8 x 10 -11 .5

30. . 1 0 x 10 lo 10. .47 x 10-so

.25 30. .15 x 10 -1~

10. .37 x 10 4 .5 30. .41 x 10 -4

10. .87 • 10 .7 .125

30. .98 x 10 -7

.34 x 1 O- 1

.17

Xmin Xmax

.99

.99

.15 x 10 -1

.23 x 10 -1

.32 x 10

.50

.48

.99

.11 x 10

.10

Problem

A)

B)

c)

Table 4

Col loca t ion paramete rs : c 1 = 0 , c 2 2

s-l/s e 3 ~ ~ c 4 = 1

h

.5

.25

.5

.25

.5

.25

t Rela t ive error

10. .74 x 10 -5 30. .97 x 10 -4 10. .10 • 10 -6 30. .58 • 10 -4

10. .99 x 10 -11 30. .30 x 10- i2 10. .17 x 10 - to 30. .11 x 10 -1~

10. .47 x 10 _4 30. .52 x 10 -4 10. .27 x 10 -5 30. .30 x 10 _5

Xmln

.29 x 10- i

.17

.23

.56 x 10 -2

.63

.15

Xm~

.99

.99

.23

.42

.97

.99

Acknowledgment

The au thors would l ike to t h a n k Professor P. J. Van der H o u w e n for the s t imula t ing discuss ions abou t the subject o f this paper.

References

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328 M. Crisci, E. Russo, and A. Vecchio: On the Stability of the One-Step exact Collocatiorl

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integral equations of the second kind. Computing 24, 341- 347 (1980).

Maria Rosaria Crisci Dipartimento di Matematica e Applicazioni Via Mezzocannone, 8 Napoli Italy

Elvira Russo Dipartimento di Informatica e Applicazioni Universit~ di Salerno Salerno Italy

Antonia Vecchio Istituto per le Applicazioni delia Matematica - C.N.R. Via P. Castellino, 111 Napoli Italy