Práger, M.

Taufer, J.

Vitásek, E.

Czech digital mathematics library

Transfer of boundary conditions for two-dimensional problems

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Milan Práger; Jiří Taufer; Emil VitásekTransfer of boundary conditions for two-dimensional problemsIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 293--296.Persistent URL: http://dml.cz/dmlcz/702393Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czTRANSFER OF BOUNDARY CONDITIONS FOR TWO-DIMENSIONAL PROBLEMS PRAGER M./TAUFER J./VTTASEK E., PRAGUE, Czechoslovakia 1. Formulation of the problem and main results Let us consider a boundary value problem for a differential equation in a domain H , Very often such a situation occurs that the solution of this problem is physically interesting only on a part of -Q. and the original domain is in fact introduced only for the reason of simple formulation of boundary conditions. It is, there-fore, interesting and maybe also effective to reformulate the origi-nal problem only on the domain which we are interested in, in other words, to find the equivalent boundary conditions on the reduced domain* Since the attempt to generalize the idea of the one-dimensio-nal transfer of boundary conditions (see, e.g. [l]) leads to the same problems we will speak also about transferring boundary condition. Thus, let the domain -ft and its subdomain ft 2 be given. Put -ft 1 = JTL\jn2 and suppose that -ft j is also a domain. Suppose, further, that the boundaries of ft f ftjf and -ft2 are all Lipschitzian. Let the original problem be the problem of solving the differential equation Lu = f in n , where L is an elliptic differential operator of the second order, with the boundary condition u = g on 9ft . Let us recall now the weak formulation of this problem. Let there be given a function w € H (ft) such that w = g on 3 fl and a bilinear form a(u,v) defined on Hj(ft) X HJ(.Q) which is conti-nuous and V-elliptic. We have the well-known result: There exists one function v £ H Q ( A ) such that a(vf f ) = -a(w, f) + {f$f) for any <f € Hj(ft); the weak solution of the original problem is then defined as the function u = v + w. Let us introduce now a problem on .Q2: Let V be the space of 293 such functions u of H ^ H ^ for which u = 0 on 3 -Q. H 3JTL2. Let us define first the operator of continuation VI : V -^ H«(.0.) in such a way that for y3 £ V the function Hcf is the solution of the homogeneous problem on A . ; thus, it holds (1) a ( n ^ , y ) = 0 for any y G Hj(n). The mapping l~| is infective and continuous and it holds II <f (I v = ^ ii n f II f H1(a) by the equation -= II n »p H # Further, let us define a bilinear form A on V H1(0.) A(u,v) = a(f1 u, flv). It is obvious that A is continuous and V-elliptic. Again, we can assert that there exists one and only one function v such that A(v, if ) = -a(w, n <f ) + If, lly) for any cf £ V. The function u = v + w|^ is then considered to be the weak solu-tion of the boundary value problem defined on H 2 ' Now we can formulate the following theorem. Theorem 1. The restriction of v on fl 2 equals to v, i.e., v l n = ^ and> consequently, also u|^ = u. 2. Example It is very important in actual situation to be able to divide the restricted bilinear form on a differential operator and boundary conditions. This is, generally, a rather difficult problem. In order to indicate the form of equivalent (or transferred) conditions, we will investigate a simple example. Let there be given the Poisson equation 2 2 (2) — 2 + - — + -j — j = f(r,y ) 3r r 3 if on a circle K = {(r, LP ); 0 * r < R, 0 = <f < 2^:} with the boundary condition (3) u(R,y) = 0. Let us try to restrict this problem on the annulus K „ = = ((r, if)^ r- = r = Rf and to find on its inner boundary an equi-valent boundary condition. To avoid some difficulties accompanying the straightforward approach indicated above we use here the results about the one-dimensional transfer of boundary conditions. For this reason, let us write the solution of the problem (2), 294 (3) in the form of a Fourier series (4) u(rf <f ) = -j aQ(r) + Z_ [ a-^rjcos k if + bk(r)sin k if ] . If we denote the Fourier coefficients of the right-hand term of (2) by ck and dk we can write for the coefficients a- and bk of u the differential equations 2 (5) -(ra£(r))» + ̂ - ak(r) = -r ck(r)f k = 0,1,... k2 -(rb£(r))» • & . bk(r) = -r dk(r)f k = 1,2,... with appropriate boundary conditions at r = 0 and r = R. Since the classical theory of transfer of boundary conditions is elaborated in detail for a differential equation of the form (6) -(py')' + qy = f with 1/p, q, and f being Lebesgue integrable or having a singula-rity in one of the coefficients 1/p and q (see [ij, [2]), we must first introduce a theorem covering the case of equations (5), i.e., the case admitting the singularities in both 1/p and q. Theorem 2. Let p, q: [a,b] -> ft are such that p > 0, q = 0 almost everywhere in [a,b] , 1/pfq € ,£(a+£fb) for any 0 < i < b-a and 1/p,q j£ X (afb). Further, let % be a nonpoai-tive (absolutely continuous) solution of the differential equation (?) v * h 2 - « -Then any solution of (6) with the Lebesgue integrable right-hand term satisfies the differential equation (8) />>(t)y(t) + p(t)y'(t) = | (t) for any t G [a,b] and f i s defined by the differential equa-tion (9) | » = 1 ^ - f with the i n i t i a l condition (10) £ (a) = 0. Let us note that under the hypothesis of Theorem 2 any solu-tion of (6) satisfies p(a)y'(a) = 0 (and, also, y(a) = 0). Thus, the equation (8) may be looked at as the result of transferring this 295 condition. Let us turn back to the problem ( 2 ) , (3). In our special situa­tion the equation (7) has the form and a function satisfying the assumptions of Theorem 2 is, obviously, the function % ~ ~k. Computing the corresponding function f by solving (9) with the initial condition (10) we obtain from (5) ^(r)-|-i(r) ...4 / ( | > V k ( p đ ? , 0 r Vг ) " f V1"' " " 4 / ( r ) kf đ k ( ľ } ăf ' k = *»-»••• 0 Substituting these results into (4) we have (after some manipula­tion) the relation 2« (11) u(r, y ) = 7 — J u(r,y ) d y + 0 • -J^T / In 2[l-cos k(y-y> )] ur(r,y )d y = f (r, y> ), 0 0 « r « Rf where r 25T ° ' ""Sir/ i {?f(j> ,y)ln[l-2| cos(c^-y) ^ ( 1 )2] d|> dy. 0 0 Thus, in equation (11) we have found a form of the equivalent boundary condition on the inner circular boundary of the annulus. Bibliography [l] Taufer, J.: Losung der Randwertprobi erne fur Sjysteme von linearen Different!algleichungen. Rozpravy fiSAV 83 (1973), No 5. [z] Pfikryl, P., Taufer, J., Vitasek, E.: Transfer of Conditions for Singular Boundary Value Problems. Aplikace matematiky 32 (1989), 246-258. 296 

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Transfer of boundary conditions for two-dimensional problems

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Institute of Mathematics AS CR, v. v. i.