Estimation of the error of the numerical solution of a non-linear integral equation

B. A. Bel’tyukov and G. Sh. Nutfullina

228

REFERENCES 1.

SERGEEV, V. O., The regularization of Volterra equations of the fllst kind. Dokl. Akad. Nauk SSSR, 197, 3,531-534,197l.

2.

PETROVSKII, I. G., Lectures on the theory of ordinary differential equations (Lektsii po teorii obyknovennykh differentsial’nykb iravnenii), “Nauka”, Moscow, 1970.

3.

GOURSAT, E., Course of mathematical analysis(Kurs matematicbeskogo ~o~ow-~n~~ad, 1934.

analiza), Vol. 3, part 2, Gostekbizdat,

ESTIMATION OF THE ERROR OF THE NUMERICAL SOLUTION OF A NON-LINEAR INTEGRAL EQUATION* and G. Sh. N~FULL~A

B. A. BEL~OV

Irkutsk (Received 3 January 1974) THE PROBLEM of obt~~g an efficient estimate of the error of the n~e~c~ solution, obtained by any method whatever, is solved for a non-linear integral equation of Hammerstein’s type. The method of approximating degenerate kernels is used. A fundamental theorem on the estimation of the error indicated above and an auxiliary lemma on the estimation of the norm of the inverse matrix of the degenerate kernel method in terms of the norm of the resolvent of an alrnost degenerate kernel are proved. An ilhrstrative example is given.

The important problems connected with the problem of the approximate solution of nonlinear integral equations include the problem of the bound of the error of a numerical solution found by any method. This problem was first considered by Mysovskii [I] , and then in [Z] , where the method of squared sums was used to solve it. In this paper this problem is solved by the method of approximating degenerate kernels. In some cases this approach is more efficient. We

consider a non-linear integral equation of Hammerstein type b s(t)=

g(t)+

s

G(t,s)fEs,z(s)lds,

t =[a, bl.

Suppose we know the approximate numerical solution of this equation: f z

Ll . * . _ xl * . .

f, :“n

where

a
1. We first prove a lemma, We consider some kernel K(C s) and a degenerate kernel close to it m &I (4 3) =

ZJ

i-1

*Zh. vj%hisl.Mat. mat. Fir., l&S,

1323-1329,1975.

ai(t)z(s)B‘(s?>

nzz=rt.

(1)

229

Short communications

where the functions a#) and pi(s) are linearly independent. The solution of the linear integral equation of the second kind with the degenerate kernel K,(t, s) leads, in the usual way, to an algebraic system with the matrix E-A, where E is the unit matrix, A=! ($ir Fj) 11,i, j =1, 2, . . . , m, (Tfh, aj) is the scalar product of z(s) fii(s) by aj(s) in [a, bf . We indicate the bound of the inverse matrix (E--JI)-~ in terms of the bound of the resolvent R(t, s, 1) of the kemelK(t, s). Lemma Let the kernels K(t, s) and Km@, s) satisfy the conditions:

1) h= i is not a characteristic number of the kernel K(t, s), and

i JaII?@,s, 1) fds G r;

sup

2)

t J IR(t,s)-K,(t;s)Idsg6;

sup

3)

(l+lYtl,

4)

Then the matrix E-A is not singular and the following bound is satisfied II(E-A)-‘III

Q 1+

aw+r)’

(2)

I-(l+lyS

proof: We take two arbitrary vectors II= (yt, . . . , Y,) and p= (p,, . . . , p,,,), s&@&g equation y=p+Ay.

We construct the functions

and

For q(t) we have

&e

(3)

230

3. A. Bef’tyukov and G. Sk. ~u~~l~i~ Then from Eq. (3) we obtain

since

From Eq. (4) we fmd

from which

I< (l+r)suPIIl(t) I.

3UPlf4m t

1

(7)

Substituting the original cp(t) in the first term of Eq. (4) and taking Eq. (5) into account, we obtain

and consequently, supIll,(t)]~allpll1

+6supIqmI.

i

C-9

I

From (7) and (8) we fmd a(l+I-) SUPIW) I G I-(1+Iy t

and ~bstitut~

IlPllI

this into (6), then lldi I<

l-lE

Mw+r) 1- (l+lq

3

IlPllI,

from which, in particular, it follows that the matrix E-A is non-singular, since, taking asp the zero vector, and asy any solution of the system (E-A) y-0, we obtain thaty is a zero vector. But from Eq. (3) we have Ityll~
(9)

231

Short communications

since in any case we can take r=r’+r”. Thus the problem posed above is reduced to the replacement in (9) of the number r by a smaller one. We construct a degenerate kernel m

c

Ui(t)fii(s)t

G”, (4 5) =

man,

i-1

close to the kernel G(r, s), and represent Eq. (1) in the form b z(t)=

g(f)+

p(t)+

J a

(10)

Gn(4~)f[~,z(~)lds,

where 0 p(t) =

[G(t, s) - G,@, ~1IfIs, z(s) Ids.

J

0

From Eq. (10) we obtain

g(t) + PW +

r(t)=

z

CiUi(t)

(11)

9

i-1

where b

Ci=

J (1

i=l, 2,.

bi(S)f[S, X(S)]&,

. ., m.

Substituting JZq.(10) into this, we obtain a system of non-linear equations for ci, i= I, 2,. . . , m:

(12)

P(c) -0,

where c=(c,, . . . , cm); the vector function

Suppose we know the constants Ci, i=l, 2,. . . , m, determined by the linear system m Zpj(ti)=

Ei

-

, 2r...rn,

i=i

g(ti),

c j-l

man.

(13)

We notice that in fact finding the i?i means the interpolation of the numerical solution f= (z!,. . . , s,) by means of the function m

.F(/)=g(t)+

c

EjUj(t).

(14)

J-i

We write ?= (El,...,

Gil),

p= (Pl, . . * pm), 1

Pi=P(ti).

B. A. Bel’tyukov and G. Sh. Nutfullina

232 ThenfromEqs.(ll)and(l4)wehave

We consider some kemelK(t, s), close to the kernel L(t,

S)=G,,,(t, s)~~‘[s, z(~) I, where T(S) is

(14).

Let the following conditions be satisfied: 1) h=l

is not a characteristic number of the kernel K(t, s), and sup *,R(f,s,i),ds4r, t J II

where R(t, s, I) is the resolvent of the kernel K(f, s); 2) the following inequalities are satisfied:

mar *IC(ti,s)-Gn(ti,s)IdS<6r, f

~~pjIG(t,~)_-G.(f,r)Id.~8. t a

I cl

sup * pc(t, s) - a*
and also m

nl

max iZ

lUj(ti)

1G a’,

sup

t z i-i

J-1

max(bi,

Ia&) l G a”,

t(s) = fx’is, f(s) I;

IPil)c 6,

x=D,‘;

3)

4) the system (13) is consistent, and for the constants Ei, 1=1, 2,. . . , m, defined by this system we have . 0

max

Q f

pi(s)f[s,ZI(s)]ds

ti -

il

J a

I

(in fact this is the bound of the error of the system, similar to (12), but without p(s), with the constants 6);

Ifxz”(s,

5) in the domain Dr,=(uQsGb, =j I~(s),thereby

Ir-z”(s) I
where rt>&M, we have

Ifx’(s,

m

max(N,

1pil)GH,

sup

I

c

i-1

l%(t) I(@, IsA) s T;

5) I
233

Short communications Iz-Z(S)

6) in the domain D,-,={a
8) h=LBqG’h,

I Gr?), where r,~zcl”f+M(2@&+62),

we have

where rl=~(f+6&~), a”~(l+r) B=l+

(16)

I- (I+ r) (63+&MT) *

Then the following inequality holds: i- (I-2h)‘” Ils-ii%

G

cc’

h

q

+

(17)

Mb,.

Proof: We will show that for the system (12) in the space fm,, of the vectors C= (ci, . . . , c,) the conditions of a theorem due to Kantorovich [S] on the Newton method are satisfied, if as the initial approximation we take the vector E= (c,, . . . , L). We estimate the norm II[P’(c’)l-‘111, where P’(F)=E--4, b

A=

fx’[S,z(S)+

i, j=l,

p(S)]&(S)aj(S)dS,

2,.

. ., m.

I

(1

We use the preceding lemma for the kernel K(t, s) and Km(t,S)= +

CCi(t)fx’[sq2”(s)+ P(S)lpi(S).

We first determine 6 for condition 2) of this lemma. We can write K-K,=K-KmFrom this, considering that t(s) =Drl, and E(s), Z(S) +I)($) ED,,, we fid

‘,K(t,s)-K,(t,s),ds~*s+6*MT=6.

sup

t s

Then, by the lemma, we have IIIP’(c^)]-lllr=II(E-A)-‘IIL
where B is (16).

We estimate the discrepancy P(Z). We have b

P(F)= E, -

s

p,(s)f[s,f(s)]ds

b

-

s

(1

Bi(S){f[S,f(s)+ P(J)l-ft~~~(S)l)d~~

i=l,2,.

. ., m.

(Km-K,).

234

8. A. 3el’~ukov and G. Sk. ~u?~~~~na

From this we obtain IIP@)fll<~+Mh& and consequently, ll[P’(t)]-*P(ir)ll,~~.

Then we estimate the norm P”(C) in the domain &={llc--Cllr92Tj},

(18)

P”(C) is a bilinear operation, that is, (P”fC)QC” $2

i ~z$~[S,Z*(S)+P(sf]~i(~)~j(S)~~(s)dsCA’Cj”~ j=f

i=l,2,.

&mf

0

. ., m, 111

x*(s)= g(s)+

II4cm (4, I-1

where if we take (18) into account, then z*(s) -I-~(S)ED,,. Then we obtain that the norm IIP”(c)II~L in the domain SO. FinaIly, for A-LB~ condition 8) of this theorem is satisfied. Therefore, all four conditions of the theorem of [5] are satisfied. By this theorem we have 1- (I-2h)‘h (Ic41

g

h

rl.

Substituting this in (15) we arrive at the estimate (17). The theorem is proved. We consider an ihustrative example. it is required to estimate the error of the numerical solution t

0.25

0.5

t”

0.9989

0.9954

(19)

of the equation z(t)=

00s y

t

‘18

+

J

t sin tsei-x(8) ds,

t =

[O,

Vz].

(20)

0

From Eq. (20) it is easy to obtain Oar 4.053417. We use this region as L&r.We choose m=n=2, we fmd c’t==O. 107659 and c2= -0.040960. G,(t, s) d*~-t%~/6*at sin ts. From this system (13), We ob~~~(t, s) =-Gz(t, s) exp [I-iT(s where Z(S)= cos (s/2) +0.~0?~9 ~~-0.~27 s&.We consider lir(t, s) =Rz(~, s),6~0. We calculate the remaining quantities occurring in the conditions of the theorem, and then by formula (17) we find the required estimate Us-~111~0.00679. The exact solution of Eq. (20), and accordingly, the actual error of the numerical solution (19) is x (1) = 1 and Ils-z”l+=O.O046.

In conclusion we mention the following. The smallness of the term MS 1 and in part of the quantity q in (17) can be arranged by constructing a degenerate kernel G, (t, s), man, sufficiently close to G(f, s). The system (13) may then contain free unknowns which must be so chosen that the quantity f (see condition 4) of the theorem) is as small as possible. For example, it is possible to supplement Eqs. (13) by the equations

Short e~mmunicat~~s

235

k

6 -

s

pi(s)f[s,z(s)]ds

= 0,

4

the ordinal numbers of which are the same as those of the free unknowns, and solve the combined system. This paper as a whole is complementary to [ I] instated

by J, Berry

REFERENCES

non-l&ax integralequation.

1.

MYSOVSKIKH, I. P., Estimation of the error of the numerical solution of a Dokl. Akod. Nauk SSSR, 153,1,30-33,1963.

2.

DOAN KIM LUAN, Approximate solution of integral equations, Candidate Dissertation, Leningrad State University, 1972.

3.

MYSOVSKlKH, 1. P., The method of mechanical quadrature5 for the solution of integraI equations. Vestn Leningr. un-ta. Ser. mekhan. iastron., 7,2,18-88,1962.

4.

LOZINSKII, S. M., Estimation of the error of the numericaI integration of ordii izv. uuzov. M~t@~tika, 5(6), 52-90,195s.

5.

KANTOROVI~, L. V. and AKILOV, G. P., Functional analysis in normed spaces (Fu~t~on~nyi normirovannykh prostranstvakh), Firma@, Moscow, 1959.

different&I equations. I.

analiz v

STRICT ESTIMATES OF THE RATE OF CONVERGENCE OF AN ITERATIVE METHOD OF COMPUTING EIGENVALUES* v. c. PRiKAzcHIKov Kiev (Received 1 I Februqv 1974) STRICT estimates of the rate of convergence of an iterative process for calculating the minimum and maximum eigenvalues of the generalized eigenvahre problem with linear self-conjugate and positive-definite operators are established. 1. We consider the following eigenvalue problem: Ay=b&

0)

where A and Q are linear operators defined in a real fmite~men~onal the conditions A==1?‘>0,

Hilbert space H, and satisfying

Q=Q’>O.

(2)

Problem (l), (2) is equivalent [l] to the variational problem on the minimum of the functional I[yl=(Ay,

Y)/(QY,

Y),

it has a finite set of eigenvalues hr. I= 1. 2, . . . . N, and a complete system of corresponding eigenvectors, orthonormed in the following sense: *2h. vychisf. Mat. mat. Fiz., l&5,

133@-1333,1975.