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An approximate method of solving the mixed boundary value problems of the theory of momentless elastic spherical envelopes
Theory of momentless elastic spherical envelopes
143
{cp,,(z, t) , F,* (2, p, t) } be the solutions of the system (12) constructed in section 6. On multiplying the left sides of the system (12) by finite functions, integrating by parts and passing to the limit as n-+ 00, it is easy to see that we obtain that the functions (8 (z, t) ,F’ (2, p, t) } constructed in section 7 are the required generalized solution of the system (12). The author thanks all those who participated in A. A. Samarskii’s and A. G. Sveshnikov’s seminar for discussing the results. Translated by J. Berry REFERENCES 1.
VLASOV, A. A., Many particle theory (Teoriya mnogikh chastits), Gostekhizdat,
2.
IORDANSKII, S. V., The Cauchy problem for the kinetic equation of a plasma. Tr. matem. in-taAkad NaukSSSR, 60,181-194,196l.
3.
CHALJUB-SIMON, A., Existence globale d’une solution du problkme de Cauchy pour le syst’eme d’&uations de&es partielles de Liouville-Newton. C. R. Acad. Sd, A276.20, 1343-1346,1973.
Mowcow, 1950
AN APPROXIMATE METHOD OF SOLVING THE MIXED BOUNDARY VALUE PROBLEMS OF THE THEORY OF MOMENTLESS ELASTIC SPHERICAL ENVELOPES* L. S. KLABUKOVA
Moscow (Received 19 February 1973) (Revised version 31 January 1974)
THE SOLUTION by the mesh method of the boundary value problems of the theory of momentless elastic spherical envelopes with boundary conditions containing variables and forces is discussed. Sufficient conditions are determined for the unique solvability and stability of the problem, and the convergence of the method as the mesh step tends to zero is proved. 1. Formulation of the problem and its reduction to a boundary value problem for a first-order system of equations of elliptic type
For the well-known [ 1 ] system of equations of elastic spherical envelopes dT ----
a,x
as ap
ZthaT-
g+$--22th&+ au
--ila
&Y_IE_O,
atI r(l+4 ap + th au +
Eh
g+thilc-
r(l +a) E/l
2
ab
1 XT=07
(1.1)
1 -&-$s=o
we consider a boundary value problem with conditions connecting the variables and forces at the edge of the envelope, of the form *Zh. v?chisl Mat. mat. Fiz., 15, 1, 14&162, 1975.
L. S. Klabukova
144
where a=ln tg (O/2), 0 and fl are spherical coordinates on the envelope which represents a domain G on a sphere, in the general case multiply connected (obviously, O<%n and -x<~+c);o, Eh, r are constants (r is the radius of the envelope); T and S are expressed in terms of the forces TI , Tz, S,, Sz by the formulas T1=-T-$Z,
S1 = - s, = S;
T%=T-+Z,
0.3)
u, v are variables; X=X (a, 8) , Y = Y (a, p) , Z=Z (a, 8) are specified external loads; un and ur are the normal and tangential variables at the edge of the envelope; T, and 7’r are the normal and tangential forces at the edge of the envelope; Xi=%<(s) , (si=cli (8), i= 1, 2, are specified smooth functions of the arc-length s on the boundary of the envelope. We write the system of equations (1 .l) and the boundary conditions (1.2) in abbreviated form, introducing the independent variable 7 =a+ip, the complex functions T+iS,u+iv and generalized derivatives. We write down expressions for the boundary conditions. Let the equation of the boundary of the domain I’be cr=cr(s), where s is the arc-length on I’. Then u,+iu,=y’(
u-iu)
B=p (s) or y=r (s), where
,
The tensor of forces is given by the following matrix (see Fig, 1):
FIG. 1 Accordingly, on the boundary of the envelope we have T,=T,
cos (n, a) -S, cos (n, p) =T~j3’+&a’,
TB=S, cos (n, a) +T, cos (n, p) =S$-Tza’
(141)
145
Theory of momentless elastic spherical envelopes
(n is the direction of the outward normal on I’). Consequently, we obtain T,fiT,,=
(T&5’-tT,p’)
fi(T,fi’-Tpa’)
= (Ta-LT,)
7’.
From this, taking into account Eqs. (1.3), we easily obtain T tiiT,=i(r’)2(T+iS)--i$Z.
(1.5)
By Eqs. (1.4) and (1 S), the boundary conditions (1.2) can be written in the form Re [r,,~‘(u+iv)+io,(y’)‘(T+iS)]=‘l,o,rRe
(Z), (1.6)
Re [ ix2y’( ufiu)
-I-02( y’)’ (T+iS)
1=‘lzw-
Re 2.
Accordingly, our problem reduces to the solution of the system of equations (1 .l) with the boundary conditions (1.6). Just as in the case where the variables on the boundary of the envelope are specified (considered in [2]), it is here useful to make the same change of variables z=ev
(1.7)
and change of the unknown functions by the formulas
v (2)
-3
(-g”’ (u +
y-
Y(z)
iu),
= 2
4r
(1 + zq2
(T + is).
(1 .f9
After this substitution the domain in which the solution of the problem is sought will be bounded in the case where 8=0 (and accordingly the corresponding a=--oo) belongs to the envelope. The domain of definition of the solution on the z-plane will be denoted by G, and its boundary by I’. In the general case G is an (m + 1)connected domain with boundary r=I’,-l-I’,+ . . . +rrn, where I’u is the external contour and the curves ri do not intersect one another. For the unknown functions V(z) and W (z), we obtain, by (1.7), (1.8), from (1.1) and (1.6) the following system of equations in the domain G: (1.9) where
r(lf4
“=
(F)“, f’ ) "'(X FziJs (
2Eh
fi =
(1
z=x+iy, iU) -
2r i3Z -7 (1 + 22)2 dz
fi = 0,
and the derivatives with respect to Y and z are generalized derivatives with respect to the corresponding variables and are defined by the formulas
af -=-
ar
1 2
af
(
atfiX
af i
)
aj 1 af -=---ii' a5 2 c.a~
af ay)
with the boundary conditions on r Re[Bxi~‘T/‘+io,(t’)‘Y]=hl(s),
Re [i8x2T’V+oz(t’)“Y]=h,(s),
(1.W)
146
L. S. KIalxikova
where
Here and below we denote by t points of the boundary I’, and t’ is the derivative oft with respect to the arc length s. We obtain sufficient conditions for the unique solvability and stability of problem (1.9), (1.10). We consider three particular forms of the boundary conditions (1 .lO). We suppose that everywhere on I’we have xi =-xz=l (case l), or 01=02=1 (case 2), or xi=02=l (case 3). We consider each of the three cases separately. From what follows it will be obvious that all the results obtained are also valid in the general case when boundary conditions of a different form are imposed on individual parts of the boundary.
2. U~queness of the solution of problem (19), (1.1(l) h, =h,=O, that is, there are We consider the homogeneous problem (1.9), (1 .lO) (fi=f2=0, no external loads). We find sufficient conditions for the uniqueness of the solution for each of the three cases. From the system of equations (1.9) we have
From this, separating the real part, we obtain 0 =\\~IYI~dxdy-~$
[Re((t’)2Y)Re(iZ’V) (2.1)
+ Re’(i (t’)l- Up)Re (Hf)] d: Case 1. x~=x~=I.
Then Eq. (2.1) for the homogeneous conditions (1.10) assumes the form
0 = S~“jYjgdsdy.++~
8I {cr [Re(i (t’)” Y)]* + ce [Re ((t’)” Y?)12}ds. r
G
obvious from this that Y (z) =0 for oj (s) , csz(s) 20 in the domain G. From the boundary conditions (1 .lO) for W (z) 4l we obtain V(t) -0 on G, consequently,V (z) ~0, since by (19) V(z) is an analytic function for Y (z) =O Accordingly, problem (1.9) (1 .lO) has only a trivial solution. Conditions oI, oz>O are sufficient for the solution of the problem to be unique.
It is
Case 2. B,=EF~.
For the homogeneous conditions (1.10) Eq. (2.1) obviously assumes the
form 0 ==~~o~Y~“&dy+ G
fs
0 {xl (Re (5’ V)]” + xz [Re(iZ’V)]2) ds. r
Theory of momentless elastic SpherfcaIenvelopes
147
It follows from this that V (z) =O in the domain G for 3t,, X,&O Consequently, by (l.lO), for the analytic function v(z) on I’ we have x1 Re (?I’) =0 and x2 Re (i?‘V) -0. If xi+xz>O everywhere on I’ and x*, ‘x2+0, this implies that I’( 1) =0 on some part (interval) of I’, and accordingly V(z) =0 in the domain G. But if the domain G is more than doubly connected (m>l), the condition x,+x?O, is sufficient, since on I’we haveRe [ (~~i-ix~) t’V] =0 and ind [ (xl--i’~l~)t’]=l--ml accordingly, V(z) =O in G. (we assume that the point z = 0 is situated within the domain G.) Therefore, for the uniqueness of the solution of the problem the conditions xi, x&O, xl+x2>0 are sufficient if m>l; if m
For the homo~neous con~tions (1.10) we obtain from (2.1) s1
(Re [i (t’)” Y1)2 -+ Qx, (Re fiZ’V])2} da.
Fromthisweobtain oi(.s), x,(s)>0 in thedomainGfor Y(z)=0 Accordingly,by(l.lO), for the analytic function V(z) on I’we have Re (?‘V) -0 and xa Re (iS’V) =O. If scP0 on some part of the boundary r, this implies that V(z) =0 in the domain G. But if the domain G is more than doubly connected (nz>‘l) , the additional requirement x.?O is superfluous, since the boundary condition Re (t/V) =O has index n=l-ml, and this implies that V(z) ==O inthe domain G. Consequently, the conditions CT%, xz&O for m>l; are sufficient for the solution of the problem to be unique; for m--Cl sufficient conditions are oi,x230 on I’ and x2)0 on part of r.
3. Existence of a solution of problem (1.9), (1.10) It has been shown (see [3,4]) that problem (1.9), (1 JO) is always solvable if the corresponding conjugate homogeneous problem has only a trivial solution. We write down the correspon~ng homogeneous conjugate problem and find sufficient conditions for its solution to be unique. We write problem (1.9), (1.10) in matrix form: -
8% - Bm as
= F
inthedomain
G,
Re (g(s) w) ==B(s)on the boundaryI’, where
We consider the homogeneous problem conjugate to problem (3.1):
ixv* fjr
+ B’W
= 0 in the domain
G,
Re [ 2’(g’) -‘W*.] -0 on the boundary I’. Here B * andg* are the transposed matrices of B and g respectively. It is easy to see that
(3-l)
148
L. S. Klabukova
B* =
0
I
0
--w 0’
-
I
(g*)-’ =
ix, (5’)2
&- a,t’
1 Xl%
$X2% I
Xl(T)2
-
L,t'
0
Consequently, for A=xIo2+x2cr1#0 the conjugate problem is defined everywhere on r and for ‘4 *, I’* we have the following boundary value problem: in the domain G dY */az-moV*=o,
Re {a Re{+(
(-
dVlZ=O;
(3.2)
ix,t’Y” + & 5, (t’)” V*)] = 0,
xrtQP* - +
61 (t’)” v*,}
I
(3.3) = 0.
The uniqueness conditions for the solution of problem (3.2), (3.3) are investigated by the same method as for problem (1.9), (l.lO), in all the three cases considered above. As a result of these investigations we obtain that these two conjugate problems have common uniqueness conditions. We notice that in the definition of the conjugate problem (3.2), (3.3) it is s assumed that A+0 everywhere on r, that is, in the three cases considered the expressions oi+o~, x~-I-x~, l+alxn.respectively, must be non-zero. Therefore, we finally obtain that the following conditions, satisfied everywhere on I’, are sufficient for the unique solvability of problem (1.9), (1 JO): in the first case.&, (J&O, G+G-O 3ci+3c2>0 for m>l (the case ol, 02= 0 was considered in [2]); in the second case xi, x&, and also xi, I.&O for ml and also a>0 on part of r for m
4. Stability of problem (1.9), (1.10) We will call problem (1.9) (1 JO) stable if its solution satisfies the inequality (4.1) where
149
Theory of momentless elastic spherical envelopes
Here and below C denotes a constant quantity. We will show that the conditions of unique solvability of the problem are simultaneously the conditions of stability of the problem. 1. Stability of problem (1.9), (1 .lO) with respect to the boundary conditions. We consider the homogeneous system of equations (1.9) with the inhomogeneous boundary conditions (1 .lO). To investigate the problem of the stability of the solution of problem (1.9), (1.10) we reduce it to a system of singular integral equations. It is known (see [3,5]), that the analytic function Y (z) can be represented in the form (4.2) where p(u(t)is a real function of t~I’,and c is a real constant. By [3], the representation (4.3) holds for v(z), where I;=c+iq~G,
@ (z) is a function analytic in G.
Representing 0 (z) in the form cqz)
=
_!_~2i!kE + r
t--z
id,
(4.4)
where v(t) is a real function of t=I’, d is a real constant, we obtain by (4.2)-(4.4) the following representation for v(z):
where k,( z, t) =kO* (z, t) / 1z--t IL, O
and k,,*(z, t) are continuous.
Now substituting the expressions (4.2) and (4.5) in the boundary conditions (1 .lO), we obtain the following system of singular integral equations for p(r) and v(t):
13~= A (to)P (to)+
s
B PO) ’ -PWdf +
ni
r t-
to
sk’ (4 to)
P (q dt
= g P”),
(4.6)
,’
where to is any point belonging to the boundary l-‘; p (t) is the required vector, p(t) = {p(t) , ~(t)};g(~)isthevectoroftherightside,g(t)={g,(t), g?(t)} gl(t)
-
= hI + C[Q Re [(t’)‘] + xlOKe
l&is”_
c: 5--t
d&q)]
dx, Re (3’) 0,
g, (t)=
hz--c
+ c&O Re (V);
(4.7) [anRe[i(t’)2]+~20Reit-sS~
3X% 5 - t dEdq
L. S. KIabukova
150 A(t), B(t),
K(t, to) arematrices: q Re [i (t’)“]
A(t) =
mdK(t,
to)=llkap(t,
x20 He (if’) il’
I 32 Re ]@‘)“I icr, Re [(t’)“]
B (t) =
ke(t,
x10 Re (3’)
I-
is,
-
ix,9
Re [i (t’)“]
(4.8)
ix,@ Re (iZ’)l
Re (3’) (I
to)II, a, p= 1, 2, where the kaQ(t, to) are known roots of the form O
to) =kz,* (4 kl) / 1t--t, 1I,
We notice that because of the solvability of problem (1.9), (1.10) and the possibility of representing its solution in the form (4.2) and (4.5), real constants c and d always exist which when substituted into the right side g(t) of Rq. (4.6), the latter has a real solution p (t) = {p(t) , v ( t) } . We show that in all three cases considered above the equation Ap=g [6]) and has a solution satisfying the condition
We calculate the determinants of the matrices S=A+B s =
&(V azttr>2
i;;;:+
is of normal type (see
andT=A-B.By
+;;;;:
(4.8) we obtain,
_Lz2;,iv
H
from which we have det S=-8
(uix2+x102)
t’, det T--8
(oIx2i-x~o~) t’.
Since in all the three cases considered above A=oix2+x102+0 everywhere on F, det S#O and det T#O on I’, accordingly the operator A and Eq. (4.6) are of normal type. We calculate the total index of the equation Ap=O (see [6]), which is defmed by the formula x =&argdetT-argdetS]. It is easy to calculate that x=2 (m-l),
where m + 1 is the connectivity of the domain G.
It is known (see [6]), that if I and Z’denote, respectively, the number of linearly-independent solutions of the associated equations Ap=O and A’o=O, we have (4.10)
l-r=%, where x is the total index of the equation Ap=O. In our case (see [3,5]) Z=2m, consequently, by (4.10), I’ = 2. It is also known [6], that the necessary and sufficient conditions for equation Ap=g solvable are that
sg
(t) ba (t) dt = 0,
a=l,2
,...,
1’.
to be
(4.11)
r
where oa, a=l, 2, . . . , I’, is the complete system of linearly-independent solutions of the associated homogeneous equationA’o=O. Here g(t) CF(t) is the inner product of vectors, that is,
Theory of momentless elastic spherical envelopes
151
where n is the dimension of the vector. In the problem considered the equation R’o=O has two solutions o’ (t) and o’(t) . Therefore, by (4.11) and (4.7), the solvability conditions of problem (4.6) are a system of two linear algebraic equations for c and d, whose right side is a vector with the components s (hl (t) al1 (t) + h, (t) g,l (t)) & r
s (h, (t) 31’ (t) + h, (t) gz’(t)) &. r
By the above this system is solvable, and from the form of its right side we conclude that
We consider equation (4.6) in whose right side g(t) we have substituted c and d, found from the condition of solvability of the equation, and in what follows we regard g(t) as known. From (4.7) and (4.12) we obtain for&) the following estimate:
We show that the equation Rp=g has a solution satisfying the condition (4.9). We mention that this equation obviously has several solutions, and we will discuss one of them. We regularize Eq. (4.6), that is, we reduce it to a Fredholm equation equivalent to it in the generalized sense (see [3,5,6]). We will adhere to the method of regularization proposed in [6]. It is easy to verify that for an operator
of normal type we can use as regularizer the operator
P$ + ;
GZ
$
[S-l (to) + T-l (to)] I# (to) (4.14)
[S-l (to) - T-l (to)] -&
’ ; ‘“’ p” 0 ’ a
where S=A-l-B, T=A-B. We denote the partial indices of the operator P by k,‘, . . . , k,’ and take an integral non-negative number k such that kj’+k>O for all j. We consider the operator
A1Wq+--,\EP(to)
’ r
-c~(t)dt t-to ’
(4.15)
where E is the unit matrix, j3(to) = (I--C&-~) / ( I+cx~,-~),where a#0 is a constant such that i+atO-k+O everywhere on r. Then l+b(to) and I-P(t 0) are non-zero everywhere on r. It can be shown that the partial indices of A, are equal to -k, therefore, it is known (see [6]) that I$. (4.6) is equivalent to the equation
152
L. S. Klabukova
A,Ap=A,g.
(4.16)
For equation (4.16) we construct a regularizer P* in the same way as the regularizer P was constructed for Eq. (4.6); we obtain
As shown in [6], the partial indices of the equation P?#=O are k,f+k, j= I, 2, . . . , n, that is, they are non-negative by the definition of k, consequently, the equation P*$=f is solvable for an arbitrary right side f. Taking into consideration the fact that Eqs. (4.6) and (4.16) are equivalent, it is easy to verify the validity of the follo~ng theorem.
Equation (4.6) and the Fredholm equation .wF’~=Arg
(4.17)
are equivalent in the sense that they are simultaneously solvable or nonsolvable, and in the case when they are solvable their solutions are connected by the relation p*$=p.
(4.18)
Following I. N. Vekua, we say that Eqs. (4.6) and (4.17) are equivalent in the generalized sense. Instead of E$. (4.6) we will consider the Fredholm equation (system) (4.17) equivalent to it in the generalized sense, which is solvable, since Eq. (4.6) is solvable. We show that the estimate Lzcr,~‘CllgllL,cr,. 11~11 holds for at least one of the solutions of Eq. (4.17). We consider the general inhomogeneous system of n Fredholm equations
For the case of one equation, if it is solvable, we have proved the possibility of representing its solution (more precisely, one of its solutions) in terms of a generalized Fredholm solution (see [3]), which enables us to estimate the norm of this solution in terms of the norm of the right side by means of the inequality
II*11~2(F)~wllL*tFf. To obtain the generalized Fredholm solution a new Fredholm equation was constructed for the given equation, which was uniquely solvable, and its solution was a solution of the equation considered, if the latter is solvable. The solution of the equation constructed was the generalized solution of the original equation.
153
Theory of momentless elastic spherical envelopes
Following this plan we construct for the system of equations (4.19) another system of Fredholm equations which is uniquely solvable and whose solution is a solution of (4.19), if the latter is solvable. All the constructions and proofs are similar to those used in the one-dimensional case (see [5]). We denote by Q’, . . . , $” the complete system of linearly independent solutions of the homogeneous system of equations (4.19) and by X’Y. . . 9 Xkthe complete system of linearly independent solutions of the homogeneous system of equations N’X=O, associated with the system (4.19). We construct the two systems of vectors: E’ ( t) , . . . , g”(t) and vi ( t) , . . . , qk (t) , a=l,2,.. ., k, j=l, 2,. . . , TZ,suchthat E”(t)={gja(t)}, r”(t)={~ja(t)},
= dap,
~xyt)qyt)dt-dap,
(4.20)
1‘ where
$3 a#B,
I9
61=
1
&p=
0,
and the integrands are inner products, that is, for example, *a(t)P((t)
=
t$ja(t)b8(t)* j=l
We construct the system of Fredholm equations
NlJ(to)+
Sfiy’(” to)*((t)dt = f(to),
(4.2 1)
r a-1
where ~“(t, to), a=l, 2,. . . , k, are matrices of order n, y”(t, to) =I]y$(t, elements are defmed by the formula yija (t, to) Yij"(t,
to) I], whose
tO)=~ia(t)~ja(tO),
We prove two theorems. Theorem 1 Any solution of the system of equations (4.21) is a solution of the system of equations (4.19) if the latter is solvable. Indeed, let II,(t) be a solution of (4.21), then k
c caTla(to) = f (to)7
Iv+ (to) +
a=1
where the c, are constant coefficients, n 1
c, =
SC (t) Ej"
r
j=l
This implies that for any function XB(t) we have
$j
(t) OS!*
L. S. Klabukova
154
(Here and below in similar expressions the integrands are inner products).
s$ (to) N’x” a, = 0
$
N2C,(t,) x0 (t,) d&J=
r
and from the conditions of solvability of the system (4.19) we have
therefore
k
ccv ca
a=1
Kl) x0(4J& = 0,
fl = 1, 2, . .
.)
k.
ti
From this, by (4.20), we have k z
Gz6czp =
0,
p=1,2
)...,
k,
a=1
and c,=O for all a=1, (4.19).
2,. . . , k. Consequently, 9 (t) is a solution of the system of equations
Theorem 2 The system of equations (4.21) has only a single solution. We consider the homogeneous system of equations (4.2 1) and show that it has only a trivial solution Let 11,(t) be a solution of the homogeneous system (4.21), then 4(t) = &&a(t), a=1
the b, are constants. In the proof of Theorem 1 we obtained that all the a = 1, 2, . . ., k. Ea((t)$(‘)dt = 0, s! Accordingly, substituting 9 (t) here we obtain by (4.20) for all cr c, =
o=s
E”(t)tb,&fi(t)dt=&
r
fi=l
SE”(t)~(t)dt=fiqa,=b.. fJ=l
r
fJ=l
Accordingly, b,=O, a;=i, 2, . . . , k, and $(t)=O. By Theorem 2, the system of Fredholm equations (4.21) is uniquely solvable, consequently its solution satisfies the estimate
L*(r)4lfllLt(nf 111111
(4.22)
which can be obtained in the usual way by expressing the solution of the system in terms of a Fredhohn solution or by regarding the system (4.2 1) as an equation with a completely continuous operator, which is uniquely solvable. By Theorem 1, the solution of the system (4.21) is a solution of the system (4.19). Accordingly, the system of equations (4.19) has a solution $ (t)for which the estimate (4.22) holds.
Theory of momentless elasticspherical envelopes
1.55
We return to our system of equations (4.17). This is a system of Fredholm equations with right side f(t) =&g(t) . Accordingly, by what was said above, it has a solution $ (t)whose norm, by (4.22) and the boundedness of the operator A,, is estimated by the formula
llgllL,cr,~cllgllL,cr,.
(4.23)
As mentioned above, Eqs. (4.17) and (4.6) are equivalent in the generalized sense, accordingly, p (t) =P*$( t) is a solution of equation (4.6) for which, because of the boundedness of the operator P* and the estimates (4.23) and (4.13), the estimate (4.9) holds. Since the components of the vector p(t) are p(t) and u(t), in terms of which the solution of problem (1.9) (1 .lO) is expressed by formulas (4.2) and (4.5), we obtain for Y (z) and V(z), by the estimates (4.9) and (4.12),
Il~ll+ll~II~CII~II~,~r,,
(4.24)
which proves the stability of problem (1.9), (1.10) to the boundary conditions. 2. Stability of the solution of problem (1.9), (1.10) with respect to the right side. We consider the inhomogeneous system of equations (1.9) with the homogeneous boundary conditions (1 .lO). We prove stability with respect to the right side on the assumption that the problem is stable with respect to the boundary conditions. We transfer the inhomogeneity from the right side of the system (1.9) to the boundary conditions. For this we introduce the operator
It is known [3] that dP- (f)
/az=f, consequently, introducing the functions Y o(z) =F (f,)
,weobtainforthedifferencesY((z)=Y(z)-Y,(z),~(z)=l/(z) andV,(z) =T(fz-cd-(fi)) - Vg( z) in the domain G the homogeneous system of equations
(4.25)
with the inhomogeneous boundary conditions on r Re [Ox,t’~+io,(t’)“~]=-Re[O~it’Vo+ioi(t’)2Yy,l, (4.26) Re [iox,tl~+o,(t’)‘\Y]=-Re
[i0~~t’~~+oe(t’)‘Y~l.
Since, by hypothesis, problem (4.25) (4.26) is stable to the boundary conditions, we obtain
and accordingly, (4.27) Since the operator ,9- is bounded as an operator acting from L2(G) into L2(I’) and from L2(G), into LZ(G), we have IIYYoll+II~oll~CII~II~,~o,.
156
L. S. Kkabukova
Consequently, from (4.27) we obtain (4.28) that is, stability with respect to the right side. It follows from (4.24) and (4.28) that (4.1) holds for the general problem (1.9), (l.lO), that is, it is stable. Since in the proof of stability no additional constraints on xj and oj, j= 1, 2, were used, apart from those used in investigating the solvability of the problem, the sufficient conditions for the unique solvability of the problem given at the end of section 3 are also sufficient for stability.
5. Solution of problem (1.9), (1 .lO) by a difference method We consider finding an approximate solution of problem (1.9), (1 .lO) by the mesh method. We construct the quadratic functional
+
+
a
’ (Re [OxlZ’V
’ (Re [i@xJ’V
$ iq (f)”
Y] - l&1}*ds
(5.1)
+ s2 (t’)“Y?] - h2}2ds.
a We cover the domain G by a net of right-angled isosceles triangles with equal sides h. We denote the domain composed of triangles by G,. We take a covering such that EcG,, and the intersection of any triangle of G, with the domain G is not empty. We consider the class of functions v (z), $ (z) , which is defined and continuous in the domain Gh such that in every triangle of GZ,they are linear, accordingly, they are determined by their values at the vertices of the triangle. To construct the corresponding difference problem we write down the conditions for a minimum of the functional Z (V, Y) in the class of functions v( z),Y (2). We obtain a system of linear algebraic equations in which the unknowns are the values of V (z) and Y (2) at the nodes of the net. This system of equationsis uniquely solvable, since problem (1.9), (1.10) is uniquely solvable. (It is assumed that the sufficient conditions for unique solvability are satisfied.) Indeed, to investigate the solvability of a difference problem it is sufficient to consider the corresponding homogeneous system of equations which is obtained on minimizing the functional Z( G, Y) , written down for f i (z) ~0, fz (z) ~0, h, (s) ‘0, hz (s) ~0. It is easy to see that A the functions realizing this minimum give Z (v, Y ) =0, from which we obtain for v^(z) and Y? (z) the homogeneous problem (1.9), (l.lO), which has only a trivial solution, Consequently, f(z) =O and Y (z) 50. Consequently, the homogeneous difference system of equations also has only a trivial solution. The unique solvability of the difference problem is proved. We estimate the error of the difference solution and prove its convergence to the solution of problem (1.9), (1 .lO) as h-+0. We denote by Vh( z$ and ?Yh(z) piece&se-flat functions realizing the minimum of the functional Z (e, Y ) . We call the pair V,(z) , Y h( Z) the difference solution of the problem.
Theory of momentless elastic spherical envelopes
157
We denote by v’,(z) and ‘I!‘,(z) piecewise-flat functions, which at the nodes of the net zp assume the values V( z,) and W (2,) respectively. It is obvious that for smooth v(z) and Y (z)
=
max -$(fo--V)
0 (h),
G
max 1PO r
v
I = 0(h2),
max &T,-y)/=O(h), G mix 1 ‘@‘,,-
Y 1 = 0 (h2).
Accordingly, we have
+
) {Re [0x11’ (v,, -
V) + is1 (t’)” (‘p, - ‘1”)1}2ds
r +
’ (Re [iO?c,%’(v’, -
! k
V) + c2 (t’)? (‘k^“,- Y)],” ds=O(h?).
Since v,(z) and rI?h(z) give a minimum of the functional 1 (V, Y) in the class of piecewise-flat functions in Gh, we have
I(Vh, Yh)
(5.2)
From (5.1) and (5.2) we obtain
+
s{Re [0x1’?’(V, -
V) + is1 (t’)” (Y,
-
Y))2 ds
I? +
1 {Re [iOx2f’(VI, r
-
V) + c&v (t’)” (YPh -
Y)]}2 ds=O(h2).
Returning now to the condition of stability of problem (4.1), we obtain the stability estimate we require IIV,-VllSIIYrY
II=O(h).
(5.3)
Equation (5.3) proves the convergence in the mean of the difference solution to the solution of problem (1.9), (1.10) as h-+0. Tmnslated by J. Berry REFERENCES 1.
GOL’DENVEIZER, A. L., Theory of elnstic thin envelopes Ueoriya uprugikh tonkikh obolochek), Gostekhizdat, Moscow, 1953.
2.
KLABUKOVA, L. S., Approximate method of solving a boundary value problem of the theory of momentless elastic spherical envelopes. Zh. vjkhisl. Mat. mat. Fiz., 13, 3, 698-711, 1973.
158
V.M. Krivtsov, et al.
3.
VEKUA, I. N., Generalizedanalytfcfunctions (Obobshchennye analiticheskie funktsii), Fizmatgiz, Moscow, 1959.
4.
DANILYUK, I. I., Some properties of the solutions of first-order elliptic systems and boundary value probZems(Nekotorye svoistva reshenii ellipticheskikh sistem l-go poryadka i kraevye zadachi), Diss. Kand. fiz.-matem. nauk, Matem. in4 Akad. Nauk SSSR, Moscow, 1958.
5.
MUSKHELISHVILI, N. I., Singular integral equations (Singulyamye integml’nye uravneniya), Fizmatgiz, Moscow, 1962.
6.
VEKUA, N. P., Systems of singularintegral equations (Sistemy singulyamykh integral’nykh uravnenii), Fizmatgiz, Moscow, 1962.
CHECK OF TWO METHODS OF CALCULATING RADIATIVE TRANSFER* V. M. KRIVTSOV, I. N. NAUMOVA, Yu. D. SHMYGLEVSKII and N. P. SHIJLISHNINA Moscow (Received 16 March 1974) TWO METHODS, due to the authors, are used to check the penetration of radiation into a half-space occupied by matter at a constant temperature and with an absorption coefficient dependent on frequency only. The errors of the calculation are obtained for three versions of the relation between the frequency and the intensity of the external radiation. [2] is checked in the case of the penetration of radiation into an isothermal medium. The absorption coefficient of this medium is so designed that the selectivity in part of the interval affects the results more strongly than when the absorbing properties of air are used [3 J. Three examples of the frequency dependence of the external radiation are studied. Detailed data on the errors of the methods of calculating the values determining the heating of the medium are given. The accuracy of the method proposed in [I] and of Gal&kin’s improved method
1. The problem on which the accuracy of the methods is tested is as follows. The half-space ~20 is filled with a gas at the temperature T = 10 000 K.
The dependence of the absorption coefficient x on frequency is shown in Fig. 1. The quantity x=hv/kT, where ZJis the frequency, h is Plan&s constant, and k is Boltzmann’s constant. In Fig. 1 *Zh. vjkhisL Mt. mat. Fiz., 15, 1, 163-171, 1975.
143
{cp,,(z, t) , F,* (2, p, t) } be the solutions of the system (12) constructed in section 6. On multiplying the left sides of the system (12) by finite functions, integrating by parts and passing to the limit as n-+ 00, it is easy to see that we obtain that the functions (8 (z, t) ,F’ (2, p, t) } constructed in section 7 are the required generalized solution of the system (12). The author thanks all those who participated in A. A. Samarskii’s and A. G. Sveshnikov’s seminar for discussing the results. Translated by J. Berry REFERENCES 1.
VLASOV, A. A., Many particle theory (Teoriya mnogikh chastits), Gostekhizdat,
2.
IORDANSKII, S. V., The Cauchy problem for the kinetic equation of a plasma. Tr. matem. in-taAkad NaukSSSR, 60,181-194,196l.
3.
CHALJUB-SIMON, A., Existence globale d’une solution du problkme de Cauchy pour le syst’eme d’&uations de&es partielles de Liouville-Newton. C. R. Acad. Sd, A276.20, 1343-1346,1973.
Mowcow, 1950
AN APPROXIMATE METHOD OF SOLVING THE MIXED BOUNDARY VALUE PROBLEMS OF THE THEORY OF MOMENTLESS ELASTIC SPHERICAL ENVELOPES* L. S. KLABUKOVA
Moscow (Received 19 February 1973) (Revised version 31 January 1974)
THE SOLUTION by the mesh method of the boundary value problems of the theory of momentless elastic spherical envelopes with boundary conditions containing variables and forces is discussed. Sufficient conditions are determined for the unique solvability and stability of the problem, and the convergence of the method as the mesh step tends to zero is proved. 1. Formulation of the problem and its reduction to a boundary value problem for a first-order system of equations of elliptic type
For the well-known [ 1 ] system of equations of elastic spherical envelopes dT ----
a,x
as ap
ZthaT-
g+$--22th&+ au
--ila
&Y_IE_O,
atI r(l+4 ap + th au +
Eh
g+thilc-
r(l +a) E/l
2
ab
1 XT=07
(1.1)
1 -&-$s=o
we consider a boundary value problem with conditions connecting the variables and forces at the edge of the envelope, of the form *Zh. v?chisl Mat. mat. Fiz., 15, 1, 14&162, 1975.
L. S. Klabukova
144
where a=ln tg (O/2), 0 and fl are spherical coordinates on the envelope which represents a domain G on a sphere, in the general case multiply connected (obviously, O<%n and -x<~+c);o, Eh, r are constants (r is the radius of the envelope); T and S are expressed in terms of the forces TI , Tz, S,, Sz by the formulas T1=-T-$Z,
S1 = - s, = S;
T%=T-+Z,
0.3)
u, v are variables; X=X (a, 8) , Y = Y (a, p) , Z=Z (a, 8) are specified external loads; un and ur are the normal and tangential variables at the edge of the envelope; T, and 7’r are the normal and tangential forces at the edge of the envelope; Xi=%<(s) , (si=cli (8), i= 1, 2, are specified smooth functions of the arc-length s on the boundary of the envelope. We write the system of equations (1 .l) and the boundary conditions (1.2) in abbreviated form, introducing the independent variable 7 =a+ip, the complex functions T+iS,u+iv and generalized derivatives. We write down expressions for the boundary conditions. Let the equation of the boundary of the domain I’be cr=cr(s), where s is the arc-length on I’. Then u,+iu,=y’(
u-iu)
B=p (s) or y=r (s), where
,
The tensor of forces is given by the following matrix (see Fig, 1):
FIG. 1 Accordingly, on the boundary of the envelope we have T,=T,
cos (n, a) -S, cos (n, p) =T~j3’+&a’,
TB=S, cos (n, a) +T, cos (n, p) =S$-Tza’
(141)
145
Theory of momentless elastic spherical envelopes
(n is the direction of the outward normal on I’). Consequently, we obtain T,fiT,,=
(T&5’-tT,p’)
fi(T,fi’-Tpa’)
= (Ta-LT,)
7’.
From this, taking into account Eqs. (1.3), we easily obtain T tiiT,=i(r’)2(T+iS)--i$Z.
(1.5)
By Eqs. (1.4) and (1 S), the boundary conditions (1.2) can be written in the form Re [r,,~‘(u+iv)+io,(y’)‘(T+iS)]=‘l,o,rRe
(Z), (1.6)
Re [ ix2y’( ufiu)
-I-02( y’)’ (T+iS)
1=‘lzw-
Re 2.
Accordingly, our problem reduces to the solution of the system of equations (1 .l) with the boundary conditions (1.6). Just as in the case where the variables on the boundary of the envelope are specified (considered in [2]), it is here useful to make the same change of variables z=ev
(1.7)
and change of the unknown functions by the formulas
v (2)
-3
(-g”’ (u +
y-
Y(z)
iu),
= 2
4r
(1 + zq2
(T + is).
(1 .f9
After this substitution the domain in which the solution of the problem is sought will be bounded in the case where 8=0 (and accordingly the corresponding a=--oo) belongs to the envelope. The domain of definition of the solution on the z-plane will be denoted by G, and its boundary by I’. In the general case G is an (m + 1)connected domain with boundary r=I’,-l-I’,+ . . . +rrn, where I’u is the external contour and the curves ri do not intersect one another. For the unknown functions V(z) and W (z), we obtain, by (1.7), (1.8), from (1.1) and (1.6) the following system of equations in the domain G: (1.9) where
r(lf4
“=
(F)“, f’ ) "'(X FziJs (
2Eh
fi =
(1
z=x+iy, iU) -
2r i3Z -7 (1 + 22)2 dz
fi = 0,
and the derivatives with respect to Y and z are generalized derivatives with respect to the corresponding variables and are defined by the formulas
af -=-
ar
1 2
af
(
atfiX
af i
)
aj 1 af -=---ii' a5 2 c.a~
af ay)
with the boundary conditions on r Re[Bxi~‘T/‘+io,(t’)‘Y]=hl(s),
Re [i8x2T’V+oz(t’)“Y]=h,(s),
(1.W)
146
L. S. KIalxikova
where
Here and below we denote by t points of the boundary I’, and t’ is the derivative oft with respect to the arc length s. We obtain sufficient conditions for the unique solvability and stability of problem (1.9), (1.10). We consider three particular forms of the boundary conditions (1 .lO). We suppose that everywhere on I’we have xi =-xz=l (case l), or 01=02=1 (case 2), or xi=02=l (case 3). We consider each of the three cases separately. From what follows it will be obvious that all the results obtained are also valid in the general case when boundary conditions of a different form are imposed on individual parts of the boundary.
2. U~queness of the solution of problem (19), (1.1(l) h, =h,=O, that is, there are We consider the homogeneous problem (1.9), (1 .lO) (fi=f2=0, no external loads). We find sufficient conditions for the uniqueness of the solution for each of the three cases. From the system of equations (1.9) we have
From this, separating the real part, we obtain 0 =\\~IYI~dxdy-~$
[Re((t’)2Y)Re(iZ’V) (2.1)
+ Re’(i (t’)l- Up)Re (Hf)] d: Case 1. x~=x~=I.
Then Eq. (2.1) for the homogeneous conditions (1.10) assumes the form
0 = S~“jYjgdsdy.++~
8I {cr [Re(i (t’)” Y)]* + ce [Re ((t’)” Y?)12}ds. r
G
obvious from this that Y (z) =0 for oj (s) , csz(s) 20 in the domain G. From the boundary conditions (1 .lO) for W (z) 4l we obtain V(t) -0 on G, consequently,V (z) ~0, since by (19) V(z) is an analytic function for Y (z) =O Accordingly, problem (1.9) (1 .lO) has only a trivial solution. Conditions oI, oz>O are sufficient for the solution of the problem to be unique.
It is
Case 2. B,=EF~.
For the homogeneous conditions (1.10) Eq. (2.1) obviously assumes the
form 0 ==~~o~Y~“&dy+ G
fs
0 {xl (Re (5’ V)]” + xz [Re(iZ’V)]2) ds. r
Theory of momentless elastic SpherfcaIenvelopes
147
It follows from this that V (z) =O in the domain G for 3t,, X,&O Consequently, by (l.lO), for the analytic function v(z) on I’ we have x1 Re (?I’) =0 and x2 Re (i?‘V) -0. If xi+xz>O everywhere on I’ and x*, ‘x2+0, this implies that I’( 1) =0 on some part (interval) of I’, and accordingly V(z) =0 in the domain G. But if the domain G is more than doubly connected (m>l), the condition x,+x?O, is sufficient, since on I’we haveRe [ (~~i-ix~) t’V] =0 and ind [ (xl--i’~l~)t’]=l--m
For the homo~neous con~tions (1.10) we obtain from (2.1) s1
(Re [i (t’)” Y1)2 -+ Qx, (Re fiZ’V])2} da.
Fromthisweobtain oi(.s), x,(s)>0 in thedomainGfor Y(z)=0 Accordingly,by(l.lO), for the analytic function V(z) on I’we have Re (?‘V) -0 and xa Re (iS’V) =O. If scP0 on some part of the boundary r, this implies that V(z) =0 in the domain G. But if the domain G is more than doubly connected (nz>‘l) , the additional requirement x.?O is superfluous, since the boundary condition Re (t/V) =O has index n=l-m
3. Existence of a solution of problem (1.9), (1.10) It has been shown (see [3,4]) that problem (1.9), (1 JO) is always solvable if the corresponding conjugate homogeneous problem has only a trivial solution. We write down the correspon~ng homogeneous conjugate problem and find sufficient conditions for its solution to be unique. We write problem (1.9), (1.10) in matrix form: -
8% - Bm as
= F
inthedomain
G,
Re (g(s) w) ==B(s)on the boundaryI’, where
We consider the homogeneous problem conjugate to problem (3.1):
ixv* fjr
+ B’W
= 0 in the domain
G,
Re [ 2’(g’) -‘W*.] -0 on the boundary I’. Here B * andg* are the transposed matrices of B and g respectively. It is easy to see that
(3-l)
148
L. S. Klabukova
B* =
0
I
0
--w 0’
-
I
(g*)-’ =
ix, (5’)2
&- a,t’
1 Xl%
$X2% I
Xl(T)2
-
L,t'
0
Consequently, for A=xIo2+x2cr1#0 the conjugate problem is defined everywhere on r and for ‘4 *, I’* we have the following boundary value problem: in the domain G dY */az-moV*=o,
Re {a Re{+(
(-
dVlZ=O;
(3.2)
ix,t’Y” + & 5, (t’)” V*)] = 0,
xrtQP* - +
61 (t’)” v*,}
I
(3.3) = 0.
The uniqueness conditions for the solution of problem (3.2), (3.3) are investigated by the same method as for problem (1.9), (l.lO), in all the three cases considered above. As a result of these investigations we obtain that these two conjugate problems have common uniqueness conditions. We notice that in the definition of the conjugate problem (3.2), (3.3) it is s assumed that A+0 everywhere on r, that is, in the three cases considered the expressions oi+o~, x~-I-x~, l+alxn.respectively, must be non-zero. Therefore, we finally obtain that the following conditions, satisfied everywhere on I’, are sufficient for the unique solvability of problem (1.9), (1 JO): in the first case.&, (J&O, G+G-O 3ci+3c2>0 for m>l (the case ol, 02= 0 was considered in [2]); in the second case xi, x&, and also xi, I.&O for m
4. Stability of problem (1.9), (1.10) We will call problem (1.9) (1 JO) stable if its solution satisfies the inequality (4.1) where
149
Theory of momentless elastic spherical envelopes
Here and below C denotes a constant quantity. We will show that the conditions of unique solvability of the problem are simultaneously the conditions of stability of the problem. 1. Stability of problem (1.9), (1 .lO) with respect to the boundary conditions. We consider the homogeneous system of equations (1.9) with the inhomogeneous boundary conditions (1 .lO). To investigate the problem of the stability of the solution of problem (1.9), (1.10) we reduce it to a system of singular integral equations. It is known (see [3,5]), that the analytic function Y (z) can be represented in the form (4.2) where p(u(t)is a real function of t~I’,and c is a real constant. By [3], the representation (4.3) holds for v(z), where I;=c+iq~G,
@ (z) is a function analytic in G.
Representing 0 (z) in the form cqz)
=
_!_~2i!kE + r
t--z
id,
(4.4)
where v(t) is a real function of t=I’, d is a real constant, we obtain by (4.2)-(4.4) the following representation for v(z):
where k,( z, t) =kO* (z, t) / 1z--t IL, O
and k,,*(z, t) are continuous.
Now substituting the expressions (4.2) and (4.5) in the boundary conditions (1 .lO), we obtain the following system of singular integral equations for p(r) and v(t):
13~= A (to)P (to)+
s
B PO) ’ -PWdf +
ni
r t-
to
sk’ (4 to)
P (q dt
= g P”),
(4.6)
,’
where to is any point belonging to the boundary l-‘; p (t) is the required vector, p(t) = {p(t) , ~(t)};g(~)isthevectoroftherightside,g(t)={g,(t), g?(t)} gl(t)
-
= hI + C[Q Re [(t’)‘] + xlOKe
l&is”_
c: 5--t
d&q)]
dx, Re (3’) 0,
g, (t)=
hz--c
+ c&O Re (V);
(4.7) [anRe[i(t’)2]+~20Reit-sS~
3X% 5 - t dEdq
L. S. KIabukova
150 A(t), B(t),
K(t, to) arematrices: q Re [i (t’)“]
A(t) =
mdK(t,
to)=llkap(t,
x20 He (if’) il’
I 32 Re ]@‘)“I icr, Re [(t’)“]
B (t) =
ke(t,
x10 Re (3’)
I-
is,
-
ix,9
Re [i (t’)“]
(4.8)
ix,@ Re (iZ’)l
Re (3’) (I
to)II, a, p= 1, 2, where the kaQ(t, to) are known roots of the form O
to) =kz,* (4 kl) / 1t--t, 1I,
We notice that because of the solvability of problem (1.9), (1.10) and the possibility of representing its solution in the form (4.2) and (4.5), real constants c and d always exist which when substituted into the right side g(t) of Rq. (4.6), the latter has a real solution p (t) = {p(t) , v ( t) } . We show that in all three cases considered above the equation Ap=g [6]) and has a solution satisfying the condition
We calculate the determinants of the matrices S=A+B s =
&(V azttr>2
i;;;:+
is of normal type (see
andT=A-B.By
+;;;;:
(4.8) we obtain,
_Lz2;,iv
H
from which we have det S=-8
(uix2+x102)
t’, det T--8
(oIx2i-x~o~) t’.
Since in all the three cases considered above A=oix2+x102+0 everywhere on F, det S#O and det T#O on I’, accordingly the operator A and Eq. (4.6) are of normal type. We calculate the total index of the equation Ap=O (see [6]), which is defmed by the formula x =&argdetT-argdetS]. It is easy to calculate that x=2 (m-l),
where m + 1 is the connectivity of the domain G.
It is known (see [6]), that if I and Z’denote, respectively, the number of linearly-independent solutions of the associated equations Ap=O and A’o=O, we have (4.10)
l-r=%, where x is the total index of the equation Ap=O. In our case (see [3,5]) Z=2m, consequently, by (4.10), I’ = 2. It is also known [6], that the necessary and sufficient conditions for equation Ap=g solvable are that
sg
(t) ba (t) dt = 0,
a=l,2
,...,
1’.
to be
(4.11)
r
where oa, a=l, 2, . . . , I’, is the complete system of linearly-independent solutions of the associated homogeneous equationA’o=O. Here g(t) CF(t) is the inner product of vectors, that is,
Theory of momentless elastic spherical envelopes
151
where n is the dimension of the vector. In the problem considered the equation R’o=O has two solutions o’ (t) and o’(t) . Therefore, by (4.11) and (4.7), the solvability conditions of problem (4.6) are a system of two linear algebraic equations for c and d, whose right side is a vector with the components s (hl (t) al1 (t) + h, (t) g,l (t)) & r
s (h, (t) 31’ (t) + h, (t) gz’(t)) &. r
By the above this system is solvable, and from the form of its right side we conclude that
We consider equation (4.6) in whose right side g(t) we have substituted c and d, found from the condition of solvability of the equation, and in what follows we regard g(t) as known. From (4.7) and (4.12) we obtain for&) the following estimate:
We show that the equation Rp=g has a solution satisfying the condition (4.9). We mention that this equation obviously has several solutions, and we will discuss one of them. We regularize Eq. (4.6), that is, we reduce it to a Fredholm equation equivalent to it in the generalized sense (see [3,5,6]). We will adhere to the method of regularization proposed in [6]. It is easy to verify that for an operator
of normal type we can use as regularizer the operator
P$ + ;
GZ
$
[S-l (to) + T-l (to)] I# (to) (4.14)
[S-l (to) - T-l (to)] -&
’ ; ‘“’ p” 0 ’ a
where S=A-l-B, T=A-B. We denote the partial indices of the operator P by k,‘, . . . , k,’ and take an integral non-negative number k such that kj’+k>O for all j. We consider the operator
A1Wq+--,\EP(to)
’ r
-c~(t)dt t-to ’
(4.15)
where E is the unit matrix, j3(to) = (I--C&-~) / ( I+cx~,-~),where a#0 is a constant such that i+atO-k+O everywhere on r. Then l+b(to) and I-P(t 0) are non-zero everywhere on r. It can be shown that the partial indices of A, are equal to -k, therefore, it is known (see [6]) that I$. (4.6) is equivalent to the equation
152
L. S. Klabukova
A,Ap=A,g.
(4.16)
For equation (4.16) we construct a regularizer P* in the same way as the regularizer P was constructed for Eq. (4.6); we obtain
As shown in [6], the partial indices of the equation P?#=O are k,f+k, j= I, 2, . . . , n, that is, they are non-negative by the definition of k, consequently, the equation P*$=f is solvable for an arbitrary right side f. Taking into consideration the fact that Eqs. (4.6) and (4.16) are equivalent, it is easy to verify the validity of the follo~ng theorem.
Equation (4.6) and the Fredholm equation .wF’~=Arg
(4.17)
are equivalent in the sense that they are simultaneously solvable or nonsolvable, and in the case when they are solvable their solutions are connected by the relation p*$=p.
(4.18)
Following I. N. Vekua, we say that Eqs. (4.6) and (4.17) are equivalent in the generalized sense. Instead of E$. (4.6) we will consider the Fredholm equation (system) (4.17) equivalent to it in the generalized sense, which is solvable, since Eq. (4.6) is solvable. We show that the estimate Lzcr,~‘CllgllL,cr,. 11~11 holds for at least one of the solutions of Eq. (4.17). We consider the general inhomogeneous system of n Fredholm equations
For the case of one equation, if it is solvable, we have proved the possibility of representing its solution (more precisely, one of its solutions) in terms of a generalized Fredholm solution (see [3]), which enables us to estimate the norm of this solution in terms of the norm of the right side by means of the inequality
II*11~2(F)~wllL*tFf. To obtain the generalized Fredholm solution a new Fredholm equation was constructed for the given equation, which was uniquely solvable, and its solution was a solution of the equation considered, if the latter is solvable. The solution of the equation constructed was the generalized solution of the original equation.
153
Theory of momentless elastic spherical envelopes
Following this plan we construct for the system of equations (4.19) another system of Fredholm equations which is uniquely solvable and whose solution is a solution of (4.19), if the latter is solvable. All the constructions and proofs are similar to those used in the one-dimensional case (see [5]). We denote by Q’, . . . , $” the complete system of linearly independent solutions of the homogeneous system of equations (4.19) and by X’Y. . . 9 Xkthe complete system of linearly independent solutions of the homogeneous system of equations N’X=O, associated with the system (4.19). We construct the two systems of vectors: E’ ( t) , . . . , g”(t) and vi ( t) , . . . , qk (t) , a=l,2,.. ., k, j=l, 2,. . . , TZ,suchthat E”(t)={gja(t)}, r”(t)={~ja(t)},
= dap,
~xyt)qyt)dt-dap,
(4.20)
1‘ where
$3 a#B,
I9
61=
1
&p=
0,
and the integrands are inner products, that is, for example, *a(t)P((t)
=
t$ja(t)b8(t)* j=l
We construct the system of Fredholm equations
NlJ(to)+
Sfiy’(” to)*((t)dt = f(to),
(4.2 1)
r a-1
where ~“(t, to), a=l, 2,. . . , k, are matrices of order n, y”(t, to) =I]y$(t, elements are defmed by the formula yija (t, to) Yij"(t,
to) I], whose
tO)=~ia(t)~ja(tO),
We prove two theorems. Theorem 1 Any solution of the system of equations (4.21) is a solution of the system of equations (4.19) if the latter is solvable. Indeed, let II,(t) be a solution of (4.21), then k
c caTla(to) = f (to)7
Iv+ (to) +
a=1
where the c, are constant coefficients, n 1
c, =
SC (t) Ej"
r
j=l
This implies that for any function XB(t) we have
$j
(t) OS!*
L. S. Klabukova
154
(Here and below in similar expressions the integrands are inner products).
s$ (to) N’x” a, = 0
$
N2C,(t,) x0 (t,) d&J=
r
and from the conditions of solvability of the system (4.19) we have
therefore
k
ccv ca
a=1
Kl) x0(4J& = 0,
fl = 1, 2, . .
.)
k.
ti
From this, by (4.20), we have k z
Gz6czp =
0,
p=1,2
)...,
k,
a=1
and c,=O for all a=1, (4.19).
2,. . . , k. Consequently, 9 (t) is a solution of the system of equations
Theorem 2 The system of equations (4.21) has only a single solution. We consider the homogeneous system of equations (4.2 1) and show that it has only a trivial solution Let 11,(t) be a solution of the homogeneous system (4.21), then 4(t) = &&a(t), a=1
the b, are constants. In the proof of Theorem 1 we obtained that all the a = 1, 2, . . ., k. Ea((t)$(‘)dt = 0, s! Accordingly, substituting 9 (t) here we obtain by (4.20) for all cr c, =
o=s
E”(t)tb,&fi(t)dt=&
r
fi=l
SE”(t)~(t)dt=fiqa,=b.. fJ=l
r
fJ=l
Accordingly, b,=O, a;=i, 2, . . . , k, and $(t)=O. By Theorem 2, the system of Fredholm equations (4.21) is uniquely solvable, consequently its solution satisfies the estimate
L*(r)4lfllLt(nf 111111
(4.22)
which can be obtained in the usual way by expressing the solution of the system in terms of a Fredhohn solution or by regarding the system (4.2 1) as an equation with a completely continuous operator, which is uniquely solvable. By Theorem 1, the solution of the system (4.21) is a solution of the system (4.19). Accordingly, the system of equations (4.19) has a solution $ (t)for which the estimate (4.22) holds.
Theory of momentless elasticspherical envelopes
1.55
We return to our system of equations (4.17). This is a system of Fredholm equations with right side f(t) =&g(t) . Accordingly, by what was said above, it has a solution $ (t)whose norm, by (4.22) and the boundedness of the operator A,, is estimated by the formula
llgllL,cr,~cllgllL,cr,.
(4.23)
As mentioned above, Eqs. (4.17) and (4.6) are equivalent in the generalized sense, accordingly, p (t) =P*$( t) is a solution of equation (4.6) for which, because of the boundedness of the operator P* and the estimates (4.23) and (4.13), the estimate (4.9) holds. Since the components of the vector p(t) are p(t) and u(t), in terms of which the solution of problem (1.9) (1 .lO) is expressed by formulas (4.2) and (4.5), we obtain for Y (z) and V(z), by the estimates (4.9) and (4.12),
Il~ll+ll~II~CII~II~,~r,,
(4.24)
which proves the stability of problem (1.9), (1.10) to the boundary conditions. 2. Stability of the solution of problem (1.9), (1.10) with respect to the right side. We consider the inhomogeneous system of equations (1.9) with the homogeneous boundary conditions (1 .lO). We prove stability with respect to the right side on the assumption that the problem is stable with respect to the boundary conditions. We transfer the inhomogeneity from the right side of the system (1.9) to the boundary conditions. For this we introduce the operator
It is known [3] that dP- (f)
/az=f, consequently, introducing the functions Y o(z) =F (f,)
,weobtainforthedifferencesY((z)=Y(z)-Y,(z),~(z)=l/(z) andV,(z) =T(fz-cd-(fi)) - Vg( z) in the domain G the homogeneous system of equations
(4.25)
with the inhomogeneous boundary conditions on r Re [Ox,t’~+io,(t’)“~]=-Re[O~it’Vo+ioi(t’)2Yy,l, (4.26) Re [iox,tl~+o,(t’)‘\Y]=-Re
[i0~~t’~~+oe(t’)‘Y~l.
Since, by hypothesis, problem (4.25) (4.26) is stable to the boundary conditions, we obtain
and accordingly, (4.27) Since the operator ,9- is bounded as an operator acting from L2(G) into L2(I’) and from L2(G), into LZ(G), we have IIYYoll+II~oll~CII~II~,~o,.
156
L. S. Kkabukova
Consequently, from (4.27) we obtain (4.28) that is, stability with respect to the right side. It follows from (4.24) and (4.28) that (4.1) holds for the general problem (1.9), (l.lO), that is, it is stable. Since in the proof of stability no additional constraints on xj and oj, j= 1, 2, were used, apart from those used in investigating the solvability of the problem, the sufficient conditions for the unique solvability of the problem given at the end of section 3 are also sufficient for stability.
5. Solution of problem (1.9), (1 .lO) by a difference method We consider finding an approximate solution of problem (1.9), (1 .lO) by the mesh method. We construct the quadratic functional
+
+
a
’ (Re [OxlZ’V
’ (Re [i@xJ’V
$ iq (f)”
Y] - l&1}*ds
(5.1)
+ s2 (t’)“Y?] - h2}2ds.
a We cover the domain G by a net of right-angled isosceles triangles with equal sides h. We denote the domain composed of triangles by G,. We take a covering such that EcG,, and the intersection of any triangle of G, with the domain G is not empty. We consider the class of functions v (z), $ (z) , which is defined and continuous in the domain Gh such that in every triangle of GZ,they are linear, accordingly, they are determined by their values at the vertices of the triangle. To construct the corresponding difference problem we write down the conditions for a minimum of the functional Z (V, Y) in the class of functions v( z),Y (2). We obtain a system of linear algebraic equations in which the unknowns are the values of V (z) and Y (2) at the nodes of the net. This system of equationsis uniquely solvable, since problem (1.9), (1.10) is uniquely solvable. (It is assumed that the sufficient conditions for unique solvability are satisfied.) Indeed, to investigate the solvability of a difference problem it is sufficient to consider the corresponding homogeneous system of equations which is obtained on minimizing the functional Z( G, Y) , written down for f i (z) ~0, fz (z) ~0, h, (s) ‘0, hz (s) ~0. It is easy to see that A the functions realizing this minimum give Z (v, Y ) =0, from which we obtain for v^(z) and Y? (z) the homogeneous problem (1.9), (l.lO), which has only a trivial solution, Consequently, f(z) =O and Y (z) 50. Consequently, the homogeneous difference system of equations also has only a trivial solution. The unique solvability of the difference problem is proved. We estimate the error of the difference solution and prove its convergence to the solution of problem (1.9), (1 .lO) as h-+0. We denote by Vh( z$ and ?Yh(z) piece&se-flat functions realizing the minimum of the functional Z (e, Y ) . We call the pair V,(z) , Y h( Z) the difference solution of the problem.
Theory of momentless elastic spherical envelopes
157
We denote by v’,(z) and ‘I!‘,(z) piecewise-flat functions, which at the nodes of the net zp assume the values V( z,) and W (2,) respectively. It is obvious that for smooth v(z) and Y (z)
=
max -$(fo--V)
0 (h),
G
max 1PO r
v
I = 0(h2),
max &T,-y)/=O(h), G mix 1 ‘@‘,,-
Y 1 = 0 (h2).
Accordingly, we have
+
) {Re [0x11’ (v,, -
V) + is1 (t’)” (‘p, - ‘1”)1}2ds
r +
’ (Re [iO?c,%’(v’, -
! k
V) + c2 (t’)? (‘k^“,- Y)],” ds=O(h?).
Since v,(z) and rI?h(z) give a minimum of the functional 1 (V, Y) in the class of piecewise-flat functions in Gh, we have
I(Vh, Yh)
(5.2)
From (5.1) and (5.2) we obtain
+
s{Re [0x1’?’(V, -
V) + is1 (t’)” (Y,
-
Y))2 ds
I? +
1 {Re [iOx2f’(VI, r
-
V) + c&v (t’)” (YPh -
Y)]}2 ds=O(h2).
Returning now to the condition of stability of problem (4.1), we obtain the stability estimate we require IIV,-VllSIIYrY
II=O(h).
(5.3)
Equation (5.3) proves the convergence in the mean of the difference solution to the solution of problem (1.9), (1.10) as h-+0. Tmnslated by J. Berry REFERENCES 1.
GOL’DENVEIZER, A. L., Theory of elnstic thin envelopes Ueoriya uprugikh tonkikh obolochek), Gostekhizdat, Moscow, 1953.
2.
KLABUKOVA, L. S., Approximate method of solving a boundary value problem of the theory of momentless elastic spherical envelopes. Zh. vjkhisl. Mat. mat. Fiz., 13, 3, 698-711, 1973.
158
V.M. Krivtsov, et al.
3.
VEKUA, I. N., Generalizedanalytfcfunctions (Obobshchennye analiticheskie funktsii), Fizmatgiz, Moscow, 1959.
4.
DANILYUK, I. I., Some properties of the solutions of first-order elliptic systems and boundary value probZems(Nekotorye svoistva reshenii ellipticheskikh sistem l-go poryadka i kraevye zadachi), Diss. Kand. fiz.-matem. nauk, Matem. in4 Akad. Nauk SSSR, Moscow, 1958.
5.
MUSKHELISHVILI, N. I., Singular integral equations (Singulyamye integml’nye uravneniya), Fizmatgiz, Moscow, 1962.
6.
VEKUA, N. P., Systems of singularintegral equations (Sistemy singulyamykh integral’nykh uravnenii), Fizmatgiz, Moscow, 1962.
CHECK OF TWO METHODS OF CALCULATING RADIATIVE TRANSFER* V. M. KRIVTSOV, I. N. NAUMOVA, Yu. D. SHMYGLEVSKII and N. P. SHIJLISHNINA Moscow (Received 16 March 1974) TWO METHODS, due to the authors, are used to check the penetration of radiation into a half-space occupied by matter at a constant temperature and with an absorption coefficient dependent on frequency only. The errors of the calculation are obtained for three versions of the relation between the frequency and the intensity of the external radiation. [2] is checked in the case of the penetration of radiation into an isothermal medium. The absorption coefficient of this medium is so designed that the selectivity in part of the interval affects the results more strongly than when the absorbing properties of air are used [3 J. Three examples of the frequency dependence of the external radiation are studied. Detailed data on the errors of the methods of calculating the values determining the heating of the medium are given. The accuracy of the method proposed in [I] and of Gal&kin’s improved method
1. The problem on which the accuracy of the methods is tested is as follows. The half-space ~20 is filled with a gas at the temperature T = 10 000 K.
The dependence of the absorption coefficient x on frequency is shown in Fig. 1. The quantity x=hv/kT, where ZJis the frequency, h is Plan&s constant, and k is Boltzmann’s constant. In Fig. 1 *Zh. vjkhisL Mt. mat. Fiz., 15, 1, 163-171, 1975.