Antenna potentials in problems of diffraction by a transparent body

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.

$lO.OO+O.OC' Journals

Ltd.

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

203 of radiation

and the condition

at infinity for

(3)

I-.*.

and that the surface S and the functions We shall assume that k,#k,. and h Pl,PI* /I given on it are such that problem (l)-(3) has a unique classical solution. Let us describe the algorithm of the method of antenna potentials for problem (l)--_(3), and substantiate it. We shall seek the solution of the problem in the form of the antenna potentials:

where C.. n-i, 2, is the carrier of the antenna potential, which curve covering everywhere a closed non-resonant surface a., n-1, In a special case the potentials (4) can be taken on one /2/). densities p,(p) and b,(P) of the potentials are chosen so that (2) on the surface S is satisfied with the specified accuracy e that is

is a segment of an analytic 2, situated inside S (see The and the same carrier. the condition of conjugation in the norm of space Lz(S),

(5)

If 8, and c, are determined so that relation (5) holds, the functions U, and u?. constructed in accordance with formulae (4), will yield an approximate solution because the solution of the Therefore to substantiate the above method it the boundary value problem (l)-(3) is stable. is sufficient to show that functions w,and )L, exist such that condition (5) is satisfied for any s>O. Passing to the substantiation of the method of antenna potentials for problem (l)-_(3), we shall first prove some auxiliary assertions which will not only be required below, but are also of independent interest. Let us examine beforehand the question of solving approximately the inner problem for the Helmholtz equation by an antenna potential with the regular kernel

whose

carrier C belongs to the same class as before. We shall clarify the conditions under which it is possible function f(N)65(S) by an antenna potential of the form

where

n is the internal

the s~;~e&(P)),

to approximate

in the mean

unit normal to the surfaces, and g and h are the constants. beacompleteand closed system of functions in L,(C). We construct

v--i,2, ...I-,

G(I)-

( qa; +h@)

j

p.dl,

v=i.Z...

., m.

c

To explain the completeness of system to all functions of system (7),

(7) in

§ 0++(~)as-o, 8 Substituting

(7) into (8), and taking

t,(S): let the function

v-i,z,...,m.

(8)

into account that the system

IF(MldSw-0, Hence,

F(M)-Lz(S) be orthogonal

(wv(P))

is closed,

we obtain

Pd

as in /2/, we have ae

§[ qdnW.P)+WM,P) I

y

I

everywhere in the domain D (inside the surface S). Using the summation theorem for the Bessel functions form

$F(M)(q~+h)~~!ml(B,p)dS-". I

(9)

F(WQS,,-0

v=o.

1..

(see /3/),

-.

m=O.

we reduce (9) to the it..

rx

1101

204

where (r,8. up) are the spherical coordinates of the point MES (the origin of coordinates is is the Bessel function of half integral order, and l'{.m'(e. e) is a spherical inside S); J,+,~,(+) function. Let V(.lf. P)-exp (iki?ws-)lRxP. On multiplying (10) by the corresponding factors, summing over v and m, and also using the summation theorem for the Hankel function (see /3/), we have

+

hY (M, P)

B

I

F(M)dS,,rO

outside a certain sphere containing D. Being analytic, I'(P)-0 everywhere in D,. The combination qcYV/&+hV is continuous almost everywhere on passing through the surface S. Since V-0 in D,. the function V(P) in D is the solution of the third boundary value problem AV+h*V-0

in

D.

if k* is not an eigenvalue Consequently, However, if tl=h, then everywhere in I.(P)

~a; +

in D,

hVlp=0.

of the third boundary

(11) value problem

then I'(P)=0

D

sa,“:.O(P)

+

. ..“p’

(12)

(P).

are the eigenfunctions of the third boundary value problem, which where ",IL'(P),....",(.y)(P) is the rank of A,...and(il. CA correspond to the eigenvalue are certain constants. I,,. !V Hence it follows that if li*is not an eigenvalue of (ll), then system (7) is complete and closed in L,(S). Consequently any function (EL:(S) can be approximated in LZ by a linear combination of the functions of system (7), and therefore by an antenna potential (6) as well. liZ=h,. ‘ system (7) is not closed Ln L&s). Let us find this flaw. Using the properties For of the surface potentials, we seek the discontinuity of the function

h.;(P)-q.“(P) on passing

through

surface

S.

C( .a,

d”;,”

h' -

i-t

holds

almost

everywhere

on S.

dtr

-

q’u,,

(12), the inequality

)I =4x(lhl’+lqlZ)F(P)

(0

s

Let

ii= where B* is the linear

Y(P)~I in D.,and in D we have

Because h

envelope

0,

BJ..

if

k=+&.

if

k'==h,,,

1-i

, 2,...1-3

of the function

Then system (7) is complete and closed in the norm of to B. Thus we have proved the following theorem.

in the space of functions

LI

orthogonal

Theorem 1. An arbitrary function /(N)EL~(JI\U can be approximated as accurately as desired on S in the norm of Ltby the antenna potential (6) with continuous density. Let r(X) be an arbitrary metaharmonic function in D (a solution of Helmholtz's equation), which has a normal derivative from L?(S).Let us introduce the notation f=bu/dn+/tufs. We choose the quantity h so that the homogeneous problem (11) (for q=l) has only the zero solution, that is J;Z+h,. ,=I. 2...,(-. We construct the antenna potential (7):

C(M)= I Q1 W,0 P@‘Idip, c whose

density

is chosen

so that

In accordance with the stability of solution of a By Theorem 1, this can always be done. boundary vlaue problem, the potential u(M) will approximate the function u(X) in L?(0)r and in any closed subdomain Ly of domain D the approximation will be uniform. Let us look into the approximation ulg and (au/an)Is.Since a(M)-U(M)-

5 where

G,(.\f.P)is Green's

function

of the third boundary

Il~-ull‘.,s,
value problem,

IIf-(2 +fdf )I/‘,,*)-

we have

205

A,-I+Alhl.

&t+Alh(P=A&

in D by an antenna the metaharmonic function u(M) can be approximated Thus, with continuous density, the following inequalities holding:

potential

We note that this assertion, as can be seen from the proof, holds for an:' k*O. Analogous results can be obtained for an antenna potential with a singular O(M)=

fif.11)

kernel

erp(ikRup) ~(P)dl R.MP c

jY(M,P)p(P)dl,=

c

and for metaharmonic functions in D. which satisfy the condition of radiation at infinity. The study is similar to that above, and does not give rise to any difficulty; therefore we present only the final results: 1) the arbitrary function /(M)e&(S) can, with specified accuracy, be approximated iii the norm of L>(S) on S by the antenna potential (13) with a continuous density (here k is arbitrary); 2) any metaharmonic function I&(M) in D., which has a regular normal derivative on L: and satisfies the condition of radiation at infinity can be approximated by the antenna pctentlal (13) with a continuous density, and at the same time

The approximation of u(M) by the potential D(M) is uniform in any closed subdomaln domain De. Let us now substantiate the method of antenna potentials for problem (l)-_(5).

and

Theorem 2. Let Then for any e>O the continuous l*ELz(S), f+&(S). (5) holds. p,,(p) exist, defined on C, and Cz, such that the inequality

functions

D,' of

PI(P)

Proof. Since the arbitrary function from Lt can be approximated in the norm of I.* by a fairly smooth function, we will establish the inequality (5) for the fairly smooth functions 1,(M) and f,(M) only. Let 1, and h be fairly smooth functions so that the boundary value problem (l)-_(3) has a classic solution: u,(V) in D and u*(M) in D,. In accordance with what was proved earlier, the function u,(M) can be represented in the form

u,(M)= where

@(.M,P)=sin (ktR.vp)lRt,p.

s e(M,P)~,(P)dl,+Bl(N), cc

<“, F=max (IP‘I, IPZI) Ilfdl‘, c7 3 II-%‘i IIL> an and at the same time e

5

4P

and the function

u?(M) can be expressed

uz(Nw)=

as

5Y(nf,P)lrr(P)dl,+Ba(M), =I

exp(iktRd

Y(M.P)=

RYP

Because U, and u2 satisfy condition

(2) on

f,(M)= j 'X'p, dl=I

c

,

lIptIlL*
we

df’z IIZY IIL,G”.

4P

have

jVpzdl+_B,-_B~, c,

Consequently,

Inequality follows (5) _ Thus, the solution of problem (l)-_(3) can be approximated by the antenna potent1ais8. Let us now consider the algorithm for determining the functions p, and p2 sucn :i:3: inequality (5) is satisfied. We introduce a notation for the operators:

Hence

206

i,u=j

@(.M,P)p(P)dl,

?‘p=j‘+‘(M,P)fi(P)di, CI

(‘I

Let

Let us determine

the operator

K as follows:

We introduce the scalar product in the *pace of vectors I and P inthe usual way, vector spaces Lt. In this notation, inequality (5) has the form

and obtain

(14)

IIRp--fll,:,,,
By Theorem 2, the set of values of the operator K is compact everywhere in the vector space L?(S). Therefore, the system of vector functions s.(M)-:RP,. 11-i.2,....=. where &)I_ is a complete system in the vector *pace L:ICI.C=C,UCI.is also complete and closed in the vector space L:(S). In determining the vector p which satisfies condition (5), we may use <7e Tikhonov regularization method, by which the minimum of the smoothing functional

is determined, and the parameter 01 is chosen so that condition (14) is satisfied. Anumerical realization of the algorithm does not present any particular difficulty. Concluding, we note that the method of antenna potentials for problem (l)-(3) can be substantiated in different ways. The way we have chosen is the most reasonable in the transition from the study of internal or external problems to more complex problems in which the conditions of conjugation, or mixed boundary conditions are present. REFERENCES 1. KPAVTSOV V.V., Approximating functions of several variables by an antenna potential, Dokl. AN SSSR, 233, No.1, 23-26, 1977. 2. KRAVTSOV V.V., On a uniform approximation of the solution of tielmholtz's equation by an antenna potential. Vestn. MGU, Vychisl. matem. i kibernetika, No.2, 78-81, 1977. 3. GRADSHTEIN I.S. and RYZHIK I.M., Tables of integrals, sums, series and prcducts (Tablitsy ryadov i proizvedenii), Fizmatqiz, Moscow, 1963. integralov, suII1T1,

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

vo1.25,No.l,pp.206-206,1985

by W.C.

0041-5553/85 $lO.OO+O.OG Pergamon Journals Ltd.

ON THE STREAMLINESOF HELICAL FLOW* I.N. NAUMOVA Unexpectedly complex flow are described. The helical

flows

patterns

and YU.D. SHMYGLEVSKII in the behaviour

(see /l-3/) of incompressible

of streamlines

in helical

fluids obey the equaticzs

rotV=,?V. p=c-V/2, where V is the velocity vecotr, A is a constant number, p is the pressure tivided by the density, and c is an arbitrary constant. Calculating the rot of both sides of the first equation, taking into account the vector identities, we obtain the Helmholtz equation At'+h'V=O. The separation of variables in this equation defines the eiqenvalues 5, and the eiqenfunctions which are periodic with respect to all Cartesian coordinate* x, y, z. Their substitution into the first equation preserves This well-knc-wn procedure leads the eigenvalues, and somewhat narrows the class of functions. The simplicity of the calculato an inordinate amount of helical flows in an unbounded space. tions does not imply simplicity in the behaviour of the streamlines but the qeneral theory of helical flows had begun to develop before the arrival of computers, which are now essential for constructing streamline*. ~_ *Zh.vychisl.Mat.nat.Fiz.,2S,2,312-313,1985