A boundary value problem for a hyperbolic equation with discontinuous boundary conditions

v.

CWEKHLOV

I.

h.I0 s c 0 w (Received

13

necember

1965)

Intraduetion IN this pager we consider a boundary value problem for a second-order linear hyperbolic equation with boundary conditions of the following form: on part of the lateral surface of a cylinder we are given the required function, and on the other part of it the conormal derivative. The existence and uniqueness of the general solution of such a problem is proved. The proof of the uniqueness is carried out by the usual method. The existence of the solution is proved by the method of finite difference with the extension of the coefficients of the equation into a wider region in the space R,“. They are continued through the boundary with Dirichlet conditions as l/e, and through the boundary with Neumann conThe solution of the original prohditions as E (E is a small par~eter). lem is obtained as a weak limit in II:, cl ). Similar problems in Cl - 41.

for elliptic

and parabolic

equations

are considered

Let 9 be a region in t3n (bounded or unbounded) and let its boundary f consist of two parts rl and Tz. In the cylinder ‘jr = Q x [O, 1'1we shall consider the boundary value problem for the hyperbolic equation

with init ial *

Zh.

conditions

vj;chisl.

Mat. mat. Fiz.

6, 6, 981 - 990, 1966. 49

50

V.I.

Chekhlou

UJt~=o, and boundary

-

al.2

I

at

= 0

+o

(2)

conditions UI

r,xto, Tl

=o,

-

dU

av

I rzxto, Tl

=

(3)

0.

Here

n is the outer

normal to I-. The condition

Uij =

Uji,

i

QT,

in

aijXihj>a[h(2

(4)

i, j=i

is assumed to be satisfied real vector. We shall

Definition.

where a > ‘3, h = (A,,

call u 1t=o

the function =

u(n,

u 1I-,x

0,

. . . . A,) is an arbitrary

t)

[O, Tl =

E

w2(1)(Ql)

such that

0

(5)

and T

W -$$+ 0

i

3

i, j=d

%a

(6)

Qij$z+b$q+ i

~+cucp}axdt= j \ fcpdxdt 0

for

any 9(x,

t) E

W,(l) U&) and such that (p1t=T

a general Theorem

solution

R

=

of problem

Q)(I‘,X[O,

0,

Tl =

0.

(1) - (3).

1

Let f(x,

t) E

L,&‘r)

bounded and continuously

and let

the functions

differentiable

aij(x,

with respect

t),

b(x,

to t in

t)

be

A hyperbolic

‘,).r = 2 x

IO,

equation

rl,

ai(n,

t)

with

discontinuous

boundary

be bounded and continuously

Theorem

of this

Theorem is carried

51

differentiable

with respect to xi in ‘1~ and c (x, t) bounded i’n ?r. solution of problem (1) - (3) is unique. The proof

conditions

Then the genera3

out in the usual

way

[?I.

2

Let the assumptions of Theorem 1 be satisfied for aij(x,_tf, f(n, t) and the functions ai(n, t 1, b(x, t 1, c(x, t) be bonded in 3~. Then the general solution of problem (If - (3) exists. Proof. 1. Let 9, be a region in En, “adjoining” 2 and having a common boundary rl with it. The region Sz’ 2 51IJ !A~,$&= 52’ \ (D U S&U 652 U d&). Let /I,, be a network in Rn with step h, and the segment [O, ~1 be divided into p. equal parts of length k; m = (ml, . , . , m,) is an integral vector mh = (Inlh, . . ., m,h) are the nodes of the network r?h. We shall denote the spacetime network by /%hk. We extend the coefficients of the equation in 2 ’ x [O, Tl in aocordante with cl3 (in 113 this is done for the elliptic and parabolic cases). Let (1

F(z) = l/h 1h &2,

t) =

in 52, in Qr, in Q2,;

I

t) inQ X [O, T], in’!& X [0, T], in& x [O, T]; J4j

U$j(X,

S+j(Z,.t)

f.zf(5, t) in Q x [0, T], l/h in Sib x [0, T], h in S& x [0, T]; T(X) t) =

=

(f/h)d,j

5(X,

t) =

b (5, t) in Q X [0, T], l/h irrS& x [O, 2’1, h in Qz x [0, T]; I

c (5, t) in !A x [O, T], l/h in !& x 10, T J, h in Q2 x [0, T].

We shall denote the function on the network Rhk by Uhk(pk;mh) = ZQ&(Pk; ratios by mlhr, .a., m,h) and its difference

ut (Pk) =

u((P+ W--u(W

ur ,(mh) =

k

u(?n*h,...,(mf+l)h

e,i(mh)=u,i(mth,.;.,(mi-~)h

Wk)

9

=

uc( (p -

1) k) I

,**., m,h)-u(mh) I

h ,...,

X’d).

V.I.

52

Chekhlov

Let Rh’ be the maximum region of cubes of the network /ih contained in R’. We introduce averaged coefficients as functions on the network i:hh for interior nodes of Qh’ x [!l, ;'I, viz.

pm=hn

s&)ch

1

at the node mh

m

(the integral with respect to x is taken over a cube of the network /1h with nodes mh nearest to the point O),

@+ak

1

aijmP

=

-

kh”

s s &j(s, pk m

t)da: dt st the node

(pk, mh),

and similarly for bmp, uPp, cmP. Instead of the right-hand f(x, t) on the network Rhk we shall consider (P+w

1

pP=_

2. We shall Qh’

side of

s s f(q pk MlQ

kh”

t) dz dt.

determine the fUnCtiO!I uhk(pk, the conditions

mh) on the

network C?hk in

x h, ‘1’1from

pmutF - ij=lIi {“4jmp&j)xi + bmPUt + mh is

an interior

$ U~wUx4 + cT”u i=l

=

FmP,

1
I, (7)

node of Rh’ z&k@,

mh) = 0,

(8)

WA@,mh) = 0,

uhk(Dk, A) = 0 at boundary nodes of $,’

for any p.

(Where this produces no ambiguities we shall omit the indices some non-essential arguments of the functions considered.) Formulae (‘7) - (9) give an ordinary explicit determining uhk(pk, mh). This system is closed in the obvious way. Later on we shall show that side R’ x [I), Tl , the functions some w. 3. We now obtain

-

(9) hk

and

difference scheme for and Uhk is found from it

if Uhk IS extended in a Similar way inso obtained converge weakly in Vz (1) to

the evaluations

necessary

for this.

We extend

Uhk

4

hyperbolic

with

equation

discontinuous

by zero on the whole Ijhk, o
(Ut + +”

-

boundary

/Jo. We multiply

conditions

(7) by

(u,(pk,

53

mh) +

{~ij”%,~};~ + bmPut + P’zL] = FmP (ut + zq).

i ij=l

Formula

(7) is true

for interior

nodes of (2,

x [o, Tl and the last - '1, by virtue of the extension of Uhk by zero. Multiplying by iizn we sum these ecgdalities over p and R (by the letter m we denote either the node A or the cube with a node I& nearest to the point II, according to the sense). Using the formula for the difference differentiation of a product, summing by parts and taking into consideration (7) - (9) we obtain eqUality iS tI7.E for all POintA Of qhh, O&p&p0

2; iS the sum over all Iir;

z

Z=

51

the nodes Of $,I,

right

up to the boundaries

.Let

m m:mnci#&f

PI-1 S

=

2

x

i

Uijq’U,j

(Ut +

p=1-,' h ij=l

It is immediately s=

where

verified

that

2 2 a;p'usj(p,)z~(~i)h~ L Ci,'ij=i

+Sz+S3,

Ut)si

kh”.

54

For the proof it is necessary to use the sane formulae a&pin and cOnsider (8) and (9). ~ubstitutin~ this result in the ec@ality being investigated we find that

PI--i Here

st=r,S,khfi(

)

are the terms containing difference ratios,

p==ip^

of

th% first order, of f&f&@, 8t). fn the left-hand side of (10) we replace the coefficients Of Uhk by the smaller n R

where

with a from condition (4). We evaluate the terms of the right-hand side

4 hyperbolic

+xnhg

equation

mes(mlQ2)

with

discontinuous

boundary

conditions

55

[5 ~+2(pl)+~q~(p$J+ k-1

Here 4 is defined by the condition

where pt =;

n2A in 52, n2/hin $21, n2h in&;

I a,’ = F s a’dx, m

The terms in 98 are evaluated by means of the Cauchy inequality and jxbf < '/2b2+ p). Here it is necessary to see to it that the sums over the regions 521, n, remain with the necessary coefficients (l/h, h). The terms with the difference ratios of the coefficients give the sum over R, For instance

where 4 1 is determined from the condition l&zij /atI < Ai 8%

X [O,T].

It remains to evaluate Uhk in terms of its difference consider

Let us

u (Ply m) =

9

p=1

Suppose that

UT (p,

m),

Pl<

PO.

ratios.

V.I.

56

I

p

zz

n

h

Chekhlou

s

c’ (5) dz, where r’(Z) =

m

C in!G?, l/h in Q1, h inC&,.

1

Here c is found from the condition

Ic(z, t) 1 G c in

multiplying

by vcy

the preceding

equality

CmU2(~~~

m)fPlk

f!

Substituting

all

m)k,

p$l g cmuT2khn, cmuF2khn.

cmu2(py m)kh’Kd~

2

(12)

h

h

these

(11)

h

‘h

p=1a

we find that

-j C%F2(p, 9=1

1~-3 cmu2 (~1,m)< T

c Xl LO,Tl. On

evaluations

xpm’) uf2 (pr) + I2 {(Pm-

in (10) we obtain

ux;(PI)} hn<

(a, - urn’~) i

i=l

xh

Here

Qrn

1 nh

in Q, a-nnxA =a0 (1- nx)/h = uo’/hin 511, h (1 - nx) = a,‘h in &,

&Iz, 77% dm Z&S

2&r, m

The constants M, Y, D, 1 depend only on the coefiicients and T, K = k/h is chosen from the condition: a) pm - xprnf = n?x = po’>o; l-

Pom > 0,

b) am - xam’ = aom > 0,

which gives

which gives

1 - n&4 = p. > 0,

1 -- nxA = a0 > 0, i - 11%=

o+’ > C. It is now easy to choose a constant

of the equation

l? such that

A

hyperbolic

equation

with

boundary

discontinuous

conditions

57

We shall use the notation

Then

f%m

C131 we have (y (pi) -

y (pi -

then y(pi) < cF(pi). y(t) nf%essary ~~~~al~ties = 0,

From (ll),

1)) / it <

E&i)

Together with (11),

-I- F(pil

I Bat since

(12) this gives the

(13) and (14) there follows the stronger evaluation

On d~v~~o~ing (14) we find that

V.I.

58

From (16)

-

ChekhEov

(18) we obtain

From (16’) - (17’) it follows that there existe such that &!$Q

a subsequence (hl,k 1)

W&, W&I converges weakly in jt;( (a tf S&) X,[O, T])

to K w, ~5, w” E&((Q IJ a,) x [O, .?‘])s h, k + o, while from (I?‘) it follows that these functions are zero in 52%X [O, TQ. Because of the

of the general SOfUtiOn all the Sa~SnGi!S %$a, k&kf,Wkho. will wholly converge to w, lai, wo weakly in &A(@ U a,) X [O, T-j as h, Ul’li~l.t6n@SS

k -* 0.

5. ‘if%Shall prove that

establish

w E W#‘( $2 u S&) X [0, T]), For instance we that ~w/&Q = u$. tit @(x, t ) have a compact medium In

(52u Szi) X [0, T] and be infinitely

differentiable.

j 5 w~(x~~)dxdt=bi;m_~~~*,w~~~~dxd~~ 0nlJ.Q,

Then

A hyperbolic

tpith discontinuous

equation

T

c5

. 0 QUO,

Similarly,

u” = 3, /at.

nr; con6eouently

boundary

conditions

59

T wWdxdt

z

-

ss

wgdxdt.

0 QUQ,

Thus W

Wjr,r[&,T]

=

E

wp(‘(a

u &)

i

X [o, T])

and w = 0 in

0.

If we take as 2Uhk(X,t) the continuous extension of f2hkt then the weak limit whk in ~~(i)(~(~ !J a,) X [O,T]) while UJhk(X,t) will const.

in L2(sZ U Qf)

converge

bl

will exist and be equal to w,

to w in view of (15) on every plane t =

and u(t)

will be a continuous function of t in

&z@ U 521) and w = 0 if t = 0. 6. It remains for us to prove that w(x, t) satisfies (6) in R x [O,rl. For the function on the network a~h(Pk, d) (7) is satisfied at interior nodes of !2h’ x [O, ?‘I. Let us take cD(x, t) as infinitely differentiable~ equal to zero near t = 7’ and in the neighbourhood of Q, x 10, 7’1, @(pk, &z) be a convolution of Q(x, t) on the network &A, We multiply (7) by hnk and ~(~~,~~ and sum with respect to p from 1 to pe - 1 and with respect to m, considering that Uhk(pk, I&) is extended outside f&’ x TO, rl by zero:

Using (a), (9) and the fact that Q,= 0 in 9, x CO, 71 we sum the lefthsnndside of the equality by parts:

I> = (0 (pk, mh) in the cube 1~.X tpk, (p + 1) kJ. In view of the choice of @(z, t) we have Let

a+*(z,

V.I.

me!&tov

A hyperbolic

Consequently Then as h +

0

equation

with

discontinuous

conditions

61

is O(lih) 40, h 4 0.

each term in the sum being considered the equality

n

‘II

+

boundary

2

Whkj@fhk

+

@+A’+

ij=1

2

Whki

+

CkC dt

L!Q,k

=

i=l T = ss

f(~,t)@~‘+,t)d~dt

k sa by virtue of the convergence of Whk, tihki, UJhk’, weakly in L,, the uniform convergence of ahk, @ihkt @ohk to the corresponding functions and by virtue of the convergence of the second term to zero, becomes the equality

This is true for any infinitely differentiable function which is equal to zero in the aeighbourhood of Q, x 10, 7’1 and near t = I’, and consequently also for any ~(2, t) E W&QfQ X [O, T]), tp lptxpA~] = 0, y&T = 0. Thus W(r, t> E %(*)(Q X [o, T]), satisfies (6). lem (1) - (3).

Consequently

w(x, t)

wlt=o = 0, wlr,x[rj,r, is the generalized

All the above argument can be directly carried non-uniform boundary and initial conditions.

0 and of prob-

over to the case of

In conclusion I wish to thank V.N. and help in my work.

Acknowledgements.

his direction

= solution

Translated

Maslennikov

by

for

H.F. Cleaves

V.I.

62

Chekhlou

REFERENCES 1.

LIONS, I.-L. aux limites.

Sur

l’spproximation

Rc.

Semin.

mat.

des Univ.

solutions Padova

32,

de certain problemes 3 - 54, 1962.

2.

FIKERA, G. On the oommon theory of boundary value problems second-order elliptico-parabolic equations. Matematika. i obz. in. period. lit., 7, 6, 99 - 122, 1963.

3.

PEETRE, I. Mixed problem for higher order elliptic ecWtiOnS on two variables. I. Ann. Scuola norm. super. Pisa. Sci. fis. e mat. 15, 4, 337 - 353, 1961.

4.

VISHIK, M.I. and ESKIN, G.I. Common boundary value problems discontinuous boundary conditions. Dokl. Akad. Nauk SSSR, 25 - 28, 1964.

5.

for Sb. pereb.

with 156,

1,

LADYZHENSKAY,O.A. A mixed problem for a hyperbolic equation. (Smeshannaya zadacha dlya giperbolicheskogo uravnenisa). Gostekhizdat, Moscow, 1953.