A method of solving exterior boundary value problems for second-order elliptic equations

Short communications

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4.

MIRANDA, K., Partialdifferential equations of elliptic type (Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa), Izd-vo, in. lit., Moscow, 1957.

5.

KRASNOSEL’SKII, M. A., ZABREIKO, P. P., PUSTYL’NIK, E. I. and SOBOLEVSKII, P. E., Integral operators in the spaces of summable functions (Integral’nye operatory v prostranstvakh summiruemykh funktsii), Nauka, Moscow, 1966.

6.

KOLESOV, Yu. S., Periodic solutions of quasilinear second-order parabolic equations. D-. Mask. matem. ob-va, 21, 114-145, 1970.

7.

YANOVSKII, L. P., Periodic solutions of the quasilinear parabolic problem in the case of a second-order increase of the non-linearity with respect to the derivative of the solution, in: Collection of papers of post-graduatestudents of Voronezh State University(Sb. rabot aspirantov VGU), Vol. 2,82-85, Izd-vo VGU, Voronezh, 1972.

A METHOD OF SOLVING EXTERIOR BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER ELLIPTIC EQUATIONS* I. I. KOCHETOV Moscow (Received 13 December 1973)

A METHOD of solving exterior boundary value problems for second-order elliptic equations by reduction to first-order integral equations is presented. Subject to certain conditions, the existence of an approximate solution of the integral equation is proved. The method is applicable for domains of arbitrary shape. Examples of the numerical solution of some boundary value problems are given, It is known [l] that the numerical solution of exterior boundary value problems for elliptic equations presents considerable difficulties. The net methods developed are applicable only for interior boundary value problems. For this reason attempts have been made to reduce the exterior boundary value problem to an interior problem, either by specifying a known asymptotic solution on a sufficiently distant exterior boundary [2], or by using, for example, a conformal transformation of the exterior domain into the interior one, but thereby the shape of the domain was often made complicated, which necessitated the application of complex algorithmic and logical methods [3]. In the case of a known fundamental solution for the differential operator of the problem an efficient method of solving interior boundary value problems for a domain with an arbitrary boundary was justified in [4]. The boundary value problem for the differential equation was thereby reduced to the solution of an integral equation of the first kind. A similar method used in mechanics is called the “method of sources” [5]. We will consider the use of this method to solve exterior two-dimensional boundary value problems. Let the domain G, with the continuous boundary S, be the complement of some bounded domain Go to a complete space. We consider the exterior boundary value problem Lu=O

*Zh. v_?chisl.Mat. mat. Fiz., 15,3,779-781,

1975.

m GI,

Buls,=f,

(1)

228

I. I. Kochetov

where L is an elliptic operator of the second order, and B is a differential operator of not more than the first order. We will assume that the problem is correctly posed in the following sense: a solution exists which uniquely and continuously depends on the initial data in any bounded domain. In GO we construct a smooth closed curve S,, situated at a distance not less than some a>O. from SI . Then the integral

s

k(P,P’)rp(P’)dP’,

sz

where k(P, P’) is the fundamental solution of the operator L, satisfies the equation Lu = 0 in G, for any function rp(P’) EL (S,). If the solution of problem (1) is representable in the form (2) then the following equation for the function cp(IJ’) must be satisfied:

s

Bk (P, P’) cp(P’) d” 1PES,

=

f.

(3)

S1

If a solution of Eq. (3) exists, then because of the correctness of the original problem the integral (2) will be a solution of problem (1). So that the basic problem is the question of the existence of even an approximate solution of Eq. (3). Remark. An exact solution of Eq. (3) does not exist, for example, if the functionfis specified as continuous, since the representation (3) imposes on f the requirement of infinite differentiability.

It is not possible to prove the existence of an approximate solution by the direct application of the method of proof of the similar result for interior problems [4] because of the impossibility of spacing the boundaries S, and S, at an arbitrary distance on account of the boundedness of the domain Go. We make an additional representation. In the domain G1 at an arbitrary distance from S1 we place a smooth contour S, surrounding S1. Then the integral

(4)

wilI also satisfy the equation Lu = 0 in the neighbourhood of S, . Lemma

If the function u(P) satisfies the equation Lu=O in the whole plane and the curves S, and S, are sufficiently smooth (&, Ss=P), then in G there exists a representation by means of the integrals (2), (4), arbitrarily accurately approximating u(P) in the metric c(G,). proof: Suppose it be necessary to obtain a solution with accuracy E and for this let it be necessary for the distance between the boundaries to be of order a. If the distance between S1 and s2 is greater than a, then the integral (2) will be the solution required [4] . Otherwise we place the

Short communications

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boundary 5’Sat a distance 2a from the boundary $2. At points of the domain G, situated at a distance not greater than a from S2 the integral (4) gives an approximation of the function u of order E. At points of the domain G2 situated at a greater distance from S2 the required approximation is obtained by means of the integral (2). It is shown in [4] that when continuous boundary conditions are specified on the cont~uo~ boundary S, there exists for the Laplace equation and the quasi-elasticity equation a function satisfying the condition of the lemma and an approximat~g solution of the boundary value problem with the specified accuracy. Together with the rest& of the lemma this enables us to dismiss the problem of the existence of an approximate solution of Eq. (3) for these problems. An equation of the type (3), an integral equation of the first kind, is efficiently solved by the method of regularization [6,4]. The following calculations were performed to finish off the technique. 1. The exterior IX&let problem in a plane. The domain Gu is a square with side 1. Tne boundary con~tion is f= (&-y2) (L+y2f -2. The sources are arranged within the square on a circle of radius RncT= 0.4. The resulting mean-square error of approximation of the solution on the boundary was &.p=1.1.10-5. Thereby the accuracy of the solution at the control points dispersed over the plane was better by a factor of two than the results obtained at the same points in [7], where the exterior domain was conformally mapped onto the interior domain. After performing a similar transformation of the domain the procedure for interior problems [3] was applied, RBCT=2.5. The result obtained was 6,,,=8, 1O-8 and the accuracy of the solution at the control points was better by a factor of one and a half than the results of [7] . 2. The tree-tensions

exterior I&i&let problem. The domain GO is a sphere of radius Thereby &,=O.~O/owas obtained and the similar error at the control points was 6,=78% on a sphere of radius ‘RH==2.

Rcg= 1 f=sin x cos yezf2. The sources are arranged on a sphere of radius &~=0.6.

In addition calculations were performed for RxcT>RK. For RsC,=2.4 we obtained 6rp=0.080/0, &,=23%. For RWCT=4.2 we obtained similarly ~~~=0.03%, S,=9%. For R,Tc,=6 we obtained 6,P=0.02~/o,6,=6%. In conclusion the author takes this opportunity to thank A. Kh. Rakhmatulin for a number of useful discussions. ~anslate~

by

J. Berry

REFERENCES 1, SAMARSKII,A. A.,Zntroduction to the theory of difference &hem), Nauka,Moscow,1971.

schemes Wvedenie v

teoriyu raznostnykh

2, IMSHENNIK,V. S., Some non-linear problems of the dynamics of a dense high-temperuture plasma (Nekotorye nelineinye zadachi dinamiki plotnoi vysokotemperaturnoiplazmy),Diss. dokt. fii.-matem. n., IPM Akad. Nauk SSSR, Moscow, 1967. 3.

FRYAZINOV, I. V., Economic schemes for the equation of heat conductionwith a boundary condition of the third kind. Zh. vjkhisl. mat. mat. F&L, 12,3,612-626,1972.

4.

KOCHETOV, I. I. and R~~MA~LINA, A. Kh., On a method of solving boundary value problems for elliptic equations. Preprint, IMP Akad Nauk SSSR, No. 5% 1972.

Yu. N. Drozhzhinov and N. I. Kozlov

230 5.

RAKHMATULIN, Kh. A. and TKACHEVA, G. D., Solution by the method of sources and sinks of the problem of the impact of an elastic body against a rigid obstacle. Zh. vjbhisl. Mat. mat. Fiz., 12,3, 814-819,1972.

6.

TIKHONOV, A. N., On the solution of incorrectly posed problems and a method of regularization. Dokl. Akad. Nauk SSSR, 151,3,501-504,1963.

7.

GREENSPAN, D., Lectures on the numerical solutionof linear, singular,and non-linear differential equations. Prentice-Hall Inc., London, 1968.

ASYMPTOTIC BEHAVIOUR OF THE SOLUTION OF THE WAVE EQUATION WITH A PIECEWISE-DISCONTINUOUS COEFFICIENT* Yu. N. DROZHZHINOV and N. I. KOZLOV

Moscow (Received 28 September 1973)

THE PROPAGATION of small perturbations in an infinite medium consisting of two different media is studied. The half-space z>O consists of a medium in which the speed of propagation of sound equals A, and in the half-space za. It is shown that in such a medium Huygens principle is violated, namely: at any point of space some time after an instantaneously acting point source a leading wave front arrives, but the trailing edge of the wave is absent. The perturbation at a spatial point decays with time as tm3.

1. Statement of the problem In this paper we study the propagation of small perturbations in an infmite medium (x, y, z), consisting of two different substances. The half-space z>O consists of a medium in which the speed of propagation of sound is A, and in the half-space zt0 the sound propagates at a speed a, where A>;. We show that in this medium Huygens principle is violated, namely: at any point of space some time after an instantaneously acting point source a leading edge of a perturbation arrives, but the trailing edge of the wave is absent. However, the perturbation at a spatial point decays as t-3. We consider that the instantaneously acting source is situated at the origin of coordinates and acts at the instant t = 0. In each half-space (z>O and zcO)the bicharacteristics are straight lines along which the perturbations from the origin of coordinates will be propagated. It is easy to understand that the trailing edge of the wave in the four-dimensional space (t, x, y, z) is a conoid whose cross section at the instant t = 1 by the plane zx is shown in Fig. 1. The whole perturbation is strictly contained within the cone ~ZS=U (x2+y2+z2). We note in particular that for any fured z our perturbation (in the three-dimensional space time t, x, y) lies strictly within the cone lY={t, I, y:t2>a2r2, wherer2=

(.f+y”)“~}.

Our perturbation satisfies the equation

a;“(z) equals A2 for z>O anda 2 for 1~0. We take the initial conditions as zero. We will seek a solution of Eq. (1) in the space of generalized functions S’(R 3, slowly increasing with respect to the variables t, x, y and continuous with respect to z as parameter. For any z our solution is

where

Vh. &h&Z. Mat. mat. Fiz.. 15,3,782-785,

1975.