Approximate solution of integro-operator volterra equations of the first kind

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

Vo1.25,No.l,pp.199-202,1985

c041-5553/85 Pergamon

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APPROXIMATE SOLUTION OF INTEGRO-OPERATORVOLTERRA EQUATIONS OF THE FIRST KIND* B. IMOMNAZABOV An approximate regularized solution for an inteqro-operator Volterra equation of the 1st kind with an approximated right-hand side in a Banach space of abstract continuous functions is constructed. Functional Volterra equations and their systems were studied in /l/. Scalar integral Their class Volterra equations of the 1st and 2nd kind are a special case of such equations. has been enlarged with the development in present-day computational mathematics of the theory In 1970 Lavrant'ev introduced Volterra equations whose kernel of ill-posed problems (/2-a/). is a two-parametric family of linear and non-linear operators acting in a certain functional Ill-posed equations of the 1st kind with an abstract Volterra operator in space (see /6/). a separable Hilbert space were studied in /g--11/ on the basis of an abstract triangular representation with respect to a chain of orthogonal projectors. The present paper deals with constructing stable regularized solutions for integrooperator Volterra equations of the 1st kind with weaker assumptions regarding the kernel and the desired exact solution. Let Hr- Hx[O.T] be a Banach space of the continuous functions of a scalar argument t, with their values in the Hilbert space with the norm

In this space we consider the first kind

the problem

of an approximate

j A(t.~)~(?)dr=f(O. 0

solution

of the Volterra

equation

of

(1.1)

o
where a(!.T) is the two-parameter family of continuous operators mapping H into themselves, and (t.7)-Q=={(f.T) :O
the error of the right-hand side of (1). Thus, equation of the 1st kind with an approximate

A(f,r)z(?)dz-f&VI. To find a regularized solution operator equations of the 2nd kind

~,~(t) of

m,(t)

(l), we replace

j A (t, xl~,(~)d~=fo(~),

+

(2)

(2) by the family of Volterra

azo.

(3)

0 Equation

(3) has a unique

We introduce

solution

igl (0

the linear integral

in space Hr for any a>0

operator

8,:

*Zh.vychisl.Mat.mat.Piz.,25,2,302-306,1985 199

and under the condition

each

It is easy to show that E, maps the space Hr into itself, and that it is bounded for rls.l-a-'[l-erp(-T/a)). a==O, and at the same time We will use the following lemma to prove the basic result of the present communication. Lemma.

Let the abstract

Then for given z(t) and [a.l] the inequality

o>O,

function

Z(I) be continuous

for any

e>O

a number

in the segment

k%(e)>0

exists

[0,2]

and let

that in the segment (5)

IIy.(t)llrr,
Using

the identity

we obtain

z(1) is continuous

Since the function such that for 15-11~8,

in

[0,T].

for any

E>O

a number

b-i3(e)>O exists (6)

ilI(e)-z(l,e"
and [I-B. '1. durinq which f-b>0 Let us divide the segment [o. t] into two segments [O.t-p] the number B,is arbitrary, for any t. If at a certain 1=1* it occurs that f*-B'O. then because we would be able to take SC/*/Z.Then for y.(t)we have the relation

t--c

y.(r)=+

+ j

exp

5 rrp( -7) ‘

( - $)

z(e)di-exp (-;)

I(()+

[z(E)-~(OldE=

t--R

Hence, by (6) we obtain _ IlY.(t)llr<

m~rIlz(t)li j =P( - u)du+erp(-B/r)Iil(t)Ile,+ B/D ~~~-~xP(-B/a)l~2exP(-B/a)li~(l)Ilrr,+ell--erp(-~/a)].

On choosing

k=min(o,B(e)).we have inequality

This proves the lemma.

(5).

is strictly Theorem. Suppose that the operator A(t,r) with respect to t, A(t,r)=E is an identity operator, and

and continuously

differentiable

(7) Also let the exact solution ~(1) of Eq.(l) be continuous in [0,T] and let z(u)=u. Then as is chosen so that 6/a(6)-0, the regularized if the regularization parameter a-a(b)+0 GO. Z(l) of solution ~~,~,(f) converges uniformly in the segment (IJ,T] to the exact solution (1). Proof. We have the relation

arp,b(f)t

s

(8)

A(1,~)‘~.d(r)dI~--al(t)+Oa((),

II

where cp.e(~)==r,a(O -z(L), Oa(l)=ib(t) -f(l). we apply the operator ~1. to (8) :

I

1

t~exp(-~)rloa(r)dr+IS[lerp( " "

i

8

-F)A(i,T)dE]x

(9)

201

Cm subtracting

(9) from (81, we obtain

the integro-operator

Volterra

equation

of the 2nd kind

vhere

to the norm, we apply

A,(f.c) with respect

To estimate

integration

by parts to the integral

and obtain

Thus, for

A,(~,T) we have the eXpreSSiOn

Since, in accordance

with condition

(7),

we have lIA,(c,r),,
f.s(l) with

Let us estimate the element consider the expression

u.(t)--r(f)

+ -;

[ i-q

( -F)]

respect

j=P (-G)

to the r.orm of the space H.

First we

06fCT

s(E)dE,

.

For any

a>0

and

Ee[O,t] we have

llu.(~)IIHCIl~T(t)Ils+ mrx II+(E)lllIIl~(t)llH+

,’ jexp( -y)

dEC

0

04t
max Ilr(E)Ila
O
As ~(1) is a uniformly continuous function small number t,>O such that for x(0

o.zt
and

z(O)-0. for any

there is a sufficiently

a>0

04t<*

and therefore (12)

Ily.(l)llH~e. By the lemma,

for t>tC-0

we have

(5).

Thus for any f=[O,T]

the inequality (13)

lly.(t)llw<2exP (--to/a)Iz(t)ll,,+c holds on the strength

of (5) and

(12).

Then,

in view of

J(f.a(0A",~zexp-2 lIz(t)lIHr+e + $[ ( Q )

(11) and

(13) we obtain

z-exp ( -')J

‘141

202 Thus, using relations inequality llwwll+<

(10) and (14), and applying

(Per&l( -$)

llzwn"r+e+ g[

the Gronwall

lemma, we arrive

at the

2-erp (-f)])eXp(*T~)

Hence, for 6-O. a(b)-0 and b/a(6)-0 we obtain uniform convergence of the regularized solution ld.(&,U) to the exact solution r(t) of equation (1). The theorem is proved. We note in conclusion that if the exact solution exists in the segment [O.Tl, and its value at the zero is unknown, then uniform convergence occurs only in the segment [ho. TI for any ~-0. REFERENCES 1. TIKHONOV A.N., On functional Volterra equations and their application in mathematical physics, Byu. MGY, 1, No.8, 1938. 2. TIKHONOV A.N., The solution of ill-posed problems and the regularization method, Dokl. AN SSSR, 252151, No.3, 501-504, 1963. 3. TIKHONOV A.N., Regularization of ill-posed problems, Dokl. AN SSSR, 153, No.1, 49-52, 1963 (Metody resheniya 4. TIKHONOV A.N. and ARSENIN V.Ya., Methods for solving ill-posed problems nekorrektnykh zadach), Nauka, Moscow, 1974. physics (0 nekotorykh nekor5. LAVRENT'EV M.M., On some ill-posed problems of mathematical rektnykh zadachakh matematicheskoi fiziki), Izd-vo AN SSSR, Novosibirsk, 1962. 6. IAVPENT'EV M.M., Inverse problems and special operator equations of the 1st kind. In: The International Congress of Mathematicians, Nice, 1970, Nauka, Moscow, 1972. 7. LATRENT'EV M.M., ROMANOV V.G. and SHISHATSKII S.P., Ill-posed problems of mathematical physics and analysis (Nekorrektnye zadachi matematicheskoi fiziki i analiza), Nauka, MOSCOW, 1980. 8. IVANOV V.K., VASIN V.V. and TANANA V.P., Theory of linear ill-posed problems and its application (Teoriya lineinykh nekorrektnykh zadach i ee prilozhenie), Nauka, Moscow, 1978. 9. IMOMNAZAROV B., On the Volterra operator equations, Dokl. AN SSSR, 242, No.5, 997-lCO0, 1978. 10. IMOMNAZAROV B., Regularization of equations of the first kind wi'th abstract Volterra operator, Dokl. AN SSSR, 247, No.1, 25-29, 1979. 11. IMOMNAZAROV B., Regularization of dissipative operator equations of the first kind, Zh. vychisl. Mat. mat. Fiz., 22, No.4, 791-800, 1982. 12. DENISOV A.M., On the approximate solution of Volterra equations of the first kind, Zh. vychisl. Mat. mat. Fiz., 15, No.4, 1053-1056, 1975. 13. MAGNITSKII N.A., On approximate solution of some integral Volterra equations of the first kind, Vest". MGU, Vychisl. mat. i kiberntika, No.1, 91-96, 1978. 14. MAGNITSKII N.A., On approximate solution of functional Volterra equations of the first kind, Vest". MGU, Vychisl. mat. i kibernetika, N0.4, 72-78, 1977. Translated

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

Vo1.25,No.l,pp.202-206,1985

OO41-5553/85 ?ergamon

by W.C.

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ANTENNA POTENTIALS IN PROBLEMS OF DIFFRACTION BY A TRANSPARENT BODY* V.V. KRAVTSOV The method of antenna potentials parent body is substantiated.

and P.K. SENATOROV for problems

of diffraction

by a trans-

The method of antenna potentials for solving an external problem of diffraction is of this method for problems described in /l, 2/. In the present paper we give a substantiation of diffraction by a transparent body. We denote by D. Let D be a three-dimensional domain bounded by the closed surface S. ScA('s',J. the unbounded domain outside the surface Let us consider the following problem which we shall refer to as the problem of diffraction by a transparent body: find functions U, and ,,: defined in D and D. respectively which satisfy the equations (1) i\~,+k,~u,-u in D, Au2+k,fuz-0 inD., the

conditions

of conjugate

on the surface S "I- ~2ls==fl(P)Is, dU‘ Pl---Pz-

drr

dua

an

=-h(P)I~

(I

ladI

‘h! --

*Zh.vychisl.Mat.mat.Fiz.,25,2.306-311,1985