Determination of source functions in composite type system of equations Текст научной статьи по специальности «Математика»

УДК 517.9

Determination of Source Functions in Composite Type System of Equations

Yury Ya. Belov* Vera G. Kopylova^

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 20.04.2014, received in revised form 05.05.2014, accepted 28.05.2014 The problem of identification of the source function for semievolutionary system of two partial differential equations is considered in the paper. The investigated system of equations is obtained from the original system by adding the time derivative containing a small parameter e > 0 to the elliptic equation. The Cauchy problem and the second boundary-value problem are considered.

Keywords: identification, inverse problem, parabolic equation, the method of weak approximation, small parameter, convergence.

We obtain a priori (uniform in e > 0) estimates of solutions of approximate problems. We prove convergence of solutions approximating the inverse problems to solutions of original problems when e ^ 0 on the basis of the obtained a priori estimates. We obtain that the rate of convergence of solutions of approximate problems is O(ei/2) in class of continuous functions. The case of the first boundary-value problem has been studied by Yu.Ya.Belov.

An identification problem of source functions in the composite type system is considered in [1-3].

1. Formulation of the problem and reduction it to the direct problem

Consider in the strip G[0,T] = {(t, x) | 0 < t < T, x e E1} the problem of determining real-valued functions (u(t,x),v(t,x),g(t)j, satisfying the system of equations

ut(t, x) + aii(t)u(t,x) + ai2(t)v(t,x) = giuxx(t,x) + g(t)f (t, x), evt(t,x) + a2i(t)u(t,x) + a22(t)v(t,x) = g Vxx(t,x) + F (t,x),

(1)

where constant e e (0,1]. Initial conditions are

u(0, x) = uo(x), V(0, x)= vo(x), (2)

and the overdetermination condition are

u (t,x°) = <p(t), у e c2 [0, t], (3)

where y(t) is a given function, 0 ^ t ^ T and x0 is some fixed point.

*ybelov@sfu-kras.ru 1 kopylova.vera@mail.ru

© Siberian Federal University. All rights reserved

Let the functions = 1,2, be defined on the interval [0, T] and let the functions

f (t,x), F(t,x) be defined on the strip G[0,T]. Let gi; g2 > 0 be given constants.

Let the relationship

f (t, x0) | ^ S > 0, t G [0, T], S > 0 — const. (4)

is fulfilled.

Assuming sufficient smoothness of the input data

- we prove the solvability of the problem (1)-(3) for each fixed e G (0,1];

- provided periodic in x and smooth input data f,F,u0,v0 we prove the existence of a sufficiently smooth solution u, v, g in QT = [0, T] x [0, l] for the boundary conditions

^(t, 0) = Vx(t, 0) = Ux(t, l) = Vx(t, l) = 0, t G [0, T];

- we prove the existence of solution u, v,g to the second boundary problem (10) (20), (30), where

u = lim u, v = lim v, g = lim g,

and (10), (20), (30) denote (1), (2), (3) with e = 0 (as u, = u, v = v, g = g);

- we obtain an estimate of the rate of convergence of u, v, g to u, v,g, respectively , when e ^ 0.

Let us assume that the following consistency condition is fulfilled

u0 (x0) = v(0),

functions aj(t), i, j = 1, 2, are of class C2[0, T]. Let us also assume that matrix

(5)

A(t) = ( a11(t) a12(t) A

A(t) = ^ a2i(t) a22(t) )

generates a symmetric and coercive bilinear form a(t, £, x) = (A(t)£, x) and

a(t,c,x) = ^tx^t V^X G £2

a(t,£,£) > к |£|2 V£ = (£i,&) G E2, t G [0,T], к> 0 — const.

(6)

Let us reduce inverse problem (1)-(2) to an subsidiary direct problem. In system (1) we set

0

f y>t(t) + au(t)^(t) + ai2(t)v(t, x0) = j)i^xJt, x( + g(t)f (t, x0), |evt(t,x0) + a2i(t)^(t) + a22(t)v(t,x^ = ^xx^x0) + F(t,x0).

From (7) we obtain

(7)

£( ) ^(t) + ai2(t)^t, x0) — xx{t,x )

g(t) =

where

^(t) = Vt (t) + an(t)g(t)

is known.

Substituting expression for g(t) in (1) we obtain the following direct problem:

ut(t, x) + an(t)u(t, x) + a\2(t)v(t, x) —

= w kx.(t,x) + m + ^lUxx(t-X°> f (t,x), (9)

j [t,x )

evt(t,x) + a2i(t)u(t, x) + a22(t)v(t, x) — ^2vxx(t, x) + F(t,x),

и(0, x) — иo(x), V(0, x) — vo(x).

(10)

(11)

2. Proof the solvability of problem (1)-(3) for £ £ (0,1]

We use the method of weak approximation [4,5] to prove the existence of a solution of direct problem (9)-(11). We reduce problem (9)-(11) to the problem

T

xx

T

xx

Ut (t,x) = 3VlUxx(t,x)

£ T £ T / 1 \

evt (t,x) = 3F2vxx(t,x), jT <t £ (j + 3)

(12)

ut (t, x) + 3an(t)u (t, x) — 0,

£Vt (t,x) + 3a22 (t)V (t,x) — 0, (j + 3) т <t £ (j + §) т,

(13)

£T £T T ut (t,x) = -3ai2(t)v (t — §, x) + 3

Ф(t) + a12(t)v (t — §, x0) — F3uxx(t — §,x0) f (t x)

f (t, x),

f (t,x0)

£-vt (t, x) — —3a2i(t)u (t — §,x) + F(t, x),

j + 3 )т <t £ {j + 1) т, (14)

и (t,x)\t^o — u0(x')i

£T

v (t,x)\t^o — vo(x).

(15)

(16)

Here j — 0,1,... ,N — 1 and тМ — T.

The input data are sufficiently smooth functions. They have all continuous derivatives occurring in given below relations (17)-(19) and satisfy these relations:

\aij(t)\ £ C, i — 1, 2, j — 1, 2,

d k

ax;f Xx) +

d k

a?F (t'x)

+

dhMx)

+

dbvoix)

£ C, к — 0,. .. ,p + 6,

\p(t)\ + p (t) + p (t) £ C\ (t,x) £ G[o,T].

(17)

(18)

(19)

Next we consider some proofs assuming, for convenience, that the constant C is greater than unity and the constant p > 5 is an odd number.

We obtain uniform estimates with respect т

dk £T, , dk £T n

dxsи Xx) + dxkv (t’x)

£ C(£),

к

0,...,p + 6, (t,x) £ G[o,t],

(20)

dj £t , eT, n

dx». (M) + || 5GJ ”■ ((’x)

^ C(e) j — 0, ...,p + 4, (t,x) G g[q,t]-

(21)

Taking into account (20), (21), the Arzela’s theorem [6] and the convergence theorem of the weak approximation method [4], we can prove the following theorem.

Theorem 1. Let conditions (4)-(6), (17) —(19) are fulfilled. Then there exists a unique solution u(t,x),v(t,x),g(t) of problem (1)-(3) in the class

Z(T) — {U(t,x),V(t,x),g(t)|U(t,x) G Ct1;X’+4(G[q,t]), V(t,x) G c1’X,+4(G[qjT]), g(t) G C([0,r\)} , and the following relations hold

p+6

E

k=Q

where

d k

a?u«,x)

+

d k

ae{«.x>

+

C![Q,T ]

dtu(t,x)

+

+

d e(t )

< C(e) (t,x) G g[q,t], (22)

c1;X’+4(G[q,T]) — { f (t,x,z)|ft G C(G[q,ti), Цf G C(G[q,ti), k — 0, ...,p + 4 j . In the general case, the constant C(e) in (22) depends on e and the input data.

3. Periodicity

Assumption 1. Let us assume that the constant xQ G (0, 1), the input data wQ(x),vQ(x), f (t, x),F(t, x) are periodic in x functions with period 21 > 0 and the series

kn

Uq (x

(x) ai cos—x,

k=Q

k=Q

Vq(x) — ^2 At cos ^x,

kn

(23)

f (t, x) —£ fk (t) cos —x

k=Q

kn

F (t, x) — ^ Fk (t)cos —

k=Q

converge uniformly on [0,1\ and QT, together with their derivatives with respect to x of order p + 4.

Solution w (t,x), v (t, x) converges uniformly in G|MT] for any fixed M > 0 together with its derivatives with respect to x up to order p + 4 to w(t, x), V(t, x). Considering Assumptions 1, the components of solution w(t,x),V(t,x) are periodic functions with respect to variable x with period 21. Then we have the following theorem.

Theorem 2. Let Assumption 1 and the conditions of Theorem 1 hold. Then for any fixed e > 0 the components w, v of the solution ^w,v,gj to problem (1)-(3) are periodic functions with respect to variable x with period 21 and they satisfy the following relations

d2m+1U (t, 0) d2m+1U (t,1) d 2m+1V(t, 0) d 2m+iV (t,i)

dx2m+1

dx2m+1

dx2m+1

dx2m+1

0, m — 0,1,...,P+3. (24)

x

Remark 1. It follows from relation (22) and system (1) that U(t,x), dL^.V(t, x),

2m + k < p + 6 exist, they are continuous in G[0,T] and

Qm. Qk

dtm dxk

Let us assume that

u(t, x)

+

dm d k

dtm dxk

V(t, x)

^ C(e) (t,x) G G[0,T] •

Mi,j = max

QT

M2,j = max

Qt

dj

7—— f (t, x) dxj v 7

d j

dj F (t’x)

r1

M-J j = max

Qt

M2 j = max

Qt

dt dxj f ( ,x)

d _ .

— ——F (t, x) dt BxJ v ’ '

1

M1 = max {M1,j, Mf,-} , M2 = max {M2j, Mf,} •

0<J<P+2

0<J<P+2

In what follows we assume that the conditions of Theorems 1 and 2 are fulfilled.

(25)

(26)

4. The solution existence of the second boundary problem

Let us consider problem (1)-(3) in G[0,T] with boundary condition

j(t,o) = Vx(t,o) = Ux(t, i) = Vx(t, i) = 0, t g [0,T]•

(27)

Let us prove that the uniform with respect to e family of estimates jwj = jи,vj solutions of (1)-(3), (27) exists under conditions (5)-(13).

Let us introduce

dj e dj eA (e

и, v

w ^ ,vj = 1 dF U>dFv

dxj

Let us differentiate j times (j < p) problem (1), (2) with respect to x. Then we multiply the

d

( d

d

result of differentiating by e —w,+2 = e ivdtMj+2’ д^'+2у)’ where constant в > 0, and

integrate over Qt = (0, t) x (0, l), t G (0, T). This can be done by virtue of Remark 1.

We have the following relations

f —Qv d e d e f —Qv d e d e

e —U d_Mj'+2dx®v + e e —v, —Vj+2dxdv+

J Qt J Qt

f —Qv ( e d e \ , , f —QvE d e

+ e a v, w,, —Wo+2 dxdv — uw e Wo+2^—Wd^dxav—

Gt v dv w ./Qt dv

l e d e l e d e l d e

— U2 e—Qv v,+^— v,+2 dxdv = e—Qv gf,— Mj^dxdv + e—Qv F,— vj^dxdv, (28)

•/Qt dv -/Qt dv ./Qt dv

■ [ e—Qv — и,— Uj+2dxdv = f e~Qv (—dxdv, (29)

3Qt dv dv jQt V^v ;

: J e—Qv—v,dj—v,+2dxdv = e^ e—Qv ^—v,+^ dxdv, (30)

11 = —

/2 = —e

/3 = —J e Qv a ^v, w,, — dxdv = J e Qv ^ ^v, w,+ 1, — dxdv =

- r d

2JQt dv

— 6v ,

e a v, Wj+i,

dxdv + - f e 6va (v, Wj+i, Wj+i^ dxdv— J Qt

j e-6va ^v,Wj+i,Wj+^ dxdv =— e-6t j a (t, Wj+i(t, x), Wj+i(t, x)^ dx---f a (v, Wj+i(x), Wj+i(x^ dx +— ( e-6v a(v,Wj+1 jWj+Л dxdv—

2 J0 ^ ' 2 JQt ^ '

e 6va lv, Wj+i,Wj+0 dxdv,

I4 = Ti J e 6Vdv (Uj+0 dxdv = ^-Te 6t J (Uj+2(t,x^ dx+

+ _-T /" e 6v(Uj+2)2dxdv — Ti f ^Uj+2(x^ dx, J Qt J 0

5 = w2 f e 6v-d(vj+^ dxdv = ^e 6t l (Vj+2(t,x)

/5 = f Jqt ^dvlVj + 2j

2Q

Substitution of (29)-(33) into (28) gives

j (Vj+2(t,x)^ dx+

+ _y J e-6^Vj+^ dxdv — (0+2(x)) dx-

X, V jdxdv+cL

el e 6v (——Vj+i ] dxdv+ Qt Vdv j

- e 6t / a

j a (v, Wj+i(t, x), Wj+i(t, x)^ dx + +— j e 6va (v, Wj+i, Wj+^ dxdv—

— f e 6va fv,Wj+i,Wj+i^j dxdv + — e 6t ( Uj+2(t,x)dx + Tl— f e 6vUj+2dxdv+ 2 Qt 2 0 2 Qt

Mi_ f - 6ve 2

2 Jq t

a

T2_ f -6ve2

+ TTe 6t I Vj+2(t,x)dx + I e 6vVj+2dxdv =

2 0 2 Qt

= f e 6vgfj—Uj+2dxdv + ( e 6vFj—Vj+2dxdv+

JQt dv J Qt dv

- fl fn 0 , , 0 , Л Ti fl 02 , . T2 fl 02 , .

- a i(),Wj+i(x),Wj+i(xU dx + + — Uj+2(x)dx + — Vj+2(x)dx.

2 0 2 0 2 0

In system (1) we set x = x0 e (0, l) and evaluate function g(t). We obtain e p (t) + anp(t) + ai2V(t,x0) — Tiuxx(t,x0)

g(t) =--------------------fxx>----------------•

-

2

-

2

Q

(31)

(32)

(33)

(34)

By virtue of (4) and conditions imposed on a12, Mi function g satisfies inequality

g(t) < Ci ^1+ V (t,x0) + uxx(t,x0)

(36)

where

к Mi |ai21 1

Ci = ma^ j-t-—}

к = max

[0,T ]

f (t) + \aiif{t)\

There is the following Embedding theorem [7].

Theorem 3. Let 0 be bounded sidereal relative to some sphere domain and 0 C Rn. If f G Hm(0) and n < 2m then f is continuous function everywhere in the region 0 including the boundary of 0. Herewith we have

where ||f||C(Q) = max \f\ and K is a constant independent of the choice of function f.

\\f\\c(o) ^ K||f ||Hm(^>

a constant i

By virtue of Theorem 3 we obtain from (36)

(t,x° )|) < Ci t + ? |vJ w||C(0J) + Y, !“>(t)

t Г' 3 2 \ 1/2 l ,‘ 3 2 \

1 + K Y1 Uj (t,x)dx I + K Y1 Vj (t,x)dx I

) < c^ 1+k^|U(t)

(37)

g(t) < Ci (l+ V(t,x0) l

< C i

j=о

i/2

C(0,l)

<

V

= Cd 1 + K

u(t)

H3(0,l)

v(t)

H3(0,l)

Hm+1 (0,l)

v(t)

Hm+1(0,l)/

m ^ 2. (38)

where K is a constant that depends only on l.

Upon substituting (38) into (34) and summing over j from 0 to m, we obtain

+Y

e 9va I v,

=0 Qt

m p / d \ 2 1 lit pl

Y1 e~9v { — Uj+ij dxdv +— e-9t Y1 a ^v, Wj+i(t, x), Wj+i(t, x)^ dx+

j=0-' ь 2 j=Y0

^v,Wj+i,Wj+^ dxdv-----Y1 e-9va ^v,Wj+i,Wj+^ dxdv+

2 j=VQt

M m r l 2 . — m c 2

+ mt e-6t^^ Uj+2(t,x)dx +—Y1 e-9v Uj+2dxdv+

2 j=0j0 2 j=0JQt

, m r l 2 , a m r 2

+ e~6t^ Vj+2(t- x)dx + e~9VVj+2dxdv <

2 j=0j0 2 j=0JQt

< Y f e-evcA 1 + K U(v) + .

jbjQt V H m+40>l)

(v)

, fj— Uj+2dxdv+ Hm + 1(0,l) J dv

m t- я

+ Y e-9v Fj —Vj+2dxdv + C2. (39)

j=Y Qt v

j=Y Qt

where C2 is a constant that depends only on the input data and constants T and l.

We have the following relations

2 t -Svf 2

"v г 2 Г1 m pi 2 f

j> / e-evUj(v,x)dxdv = / e-evdv / Uj(v,x)dx = / ~0J Qt JO j~QJ° J°

j=0 " 0

m „ i

im p 2 /*t lm pi 2 /*

Y' / e-evVj (v, x)dxdv = e-evdvY} / Vj (v, x)dx = /

Qt j 0 0 j 0

-0V

-0V

j=0 Qt 0 j=0

By virtue of (6) we obtain from (39)

U(v)

V(v)

H m(0,i)

Hm(0,i)

dv, (40)

dv. (41)

1

-0V

d u( ) dvu(v >

Hm+1(0,i)

dv + ( — mCiMiK

К at

+ 2 e-et

i(v)

+ I 7T — a ) e

Hm+1(0,i) V2

-0v

)(v)

u(v )

2

2

Hm+1(0,i)

+

dv+

+ (—(^ — a) — mM\C\K — a^ J e ev v(v)

H m+1(0,i)

dv+

Hm+1(0,i)

+ ^ e-et + 2

u(v)

+ ^ e-et

Hm+2(0,i) 2

V(v)

2 + —Ul/ e-ev

Hm+2(0,i) 2 J0

u(v)

dv+

+ e-V

(v)

Hm+2(0,i)

H m+2(0,i)

dv ^ C3(l,T, a, Mi,M2,Ci), (42)

where C3 is a constant depending on l, T, J, a, иь y>, M1}j, M2,j and constant a > 0.

Let us assume that a =4. Then we choose — = — such that all the coefficients before integrals in the left-hand side of previous inequality are the coefficients more than

. r 1 к и 1 /U2\ 7 = mm\2, 4 , T, T Г

Then we obtain the following inequality

Ye

-ST

d u( ) TvU[v >

dv +

Hm+1(0,i)

U(v)

H m+2(0,i)

+

V(v)

<

Hm+2(0,i)

^ C3(1, TJ aj M1,j, M2,j, Cl)j t G [0, T ].

Hence the following estimates is implied

5 U )

dvu(v >

dv + /

Hm + 1(0,i) 00

(v)

dv + / Hm+2 (0,i) ,/0

(v)

U(t)

Hm+2(0,i)

+

V(t)

Hm+2(0,i)

Hm+2(0,i)

^ M, Vt G [0, T], m = 0,p

dv ^ M, Vt G [0, T], (43) (44)

where M = C3(l, T, a, Mij, M2,j, Ci)eeT/7.

From the last estimate we obtain for t G [0, T] the following uniform with respect to e inequality

Uj (t)

+

C([0,i])

By virtue of (44) we obtain

Vj (t)

C([0,i])

g(t)

^ C, j = 0, ...,p + 1, t G [0,T].

C[0,T ]

< C.

(45)

(46)

t

2

2

e

2

t

t

2

0

2

к

2

2

2

2

2

0

2

t

t

2

2

0

2

2

This inequality is uniform with respect to e.

We assume the following compatibility conditions for the input data

М2Vxx(x) + F(0, x) - a2i(0)U(x) - a.22(0)v(x) = 0. Upon differentiating system (1) with respect to t, we obtain the system

(47)

д £' £ £ f £ f £ £

Жи + aii u + ai2v + aiiU + a^v = = Miuxx

д£ £ £ f £ f £ £

e c%V + a2iu + a22v + a2iU + a22v : = M2vxc

(48)

with initial data

u'(0,x) = -aii(0)u(x) - ai2(0)v(x) + MiUxx + g(0)f (0,x), (49)

£

v'(0,x)=0. (50)

Let us differentiate problem (48)-(50) j times (j < p) with respect to x , multiply the result of differentiating by — e~9t(jk (U’j+2 ), i(vj+2)) and integrate Qt,t G (0, T). This can be done by virtue of Remark 1. Then we have the following relations

I e 9v—uj—uj+2dxdv - e I e 9u —vj—vj+2dxdv—

Qt dv OV JQ, dv dV

д

f — 9V( £' d £ ' \ , , f — 9V ' ( £ d e' \

e a v,Wa,^—Wa+2 dxdv - e a v, wa,^—Wa+2 dxdv+

jQt V j dv jQt V dv j+2J

д e'

д

dv' д e'

+ МП e 9vUj+2— Uj+2dxdv + М2 e 9vVj+2 — Vj+2dxdv

JQt dv JQt dv

f e~9u g fj d!-Uj+2 dxdv -( e~9v gfj d-U.^dxdv - f e—9v Fj d-vj+2 dxdv, (51)

Qt Qt Qt

16 = -

I7 = -e

[ e 9v —uj—uj+2dxdv = f e 9v (—U.-.Л dxdv, iQt dv j dv JQt \dv )

-9v

д £' д £'

П e~'dTvv> avj-dxdv = \Jq t

/ д ! \ 2

e-9\ ~dvVj+i) dxdv

(52)

(53)

- I e 9va ( v,w, -^Wa.o I dxdv

JQt

1 Г д * JQt dt

/ д £'

j ,дvwj+2

— 9v . e a v, Wj+i, w

д £'

e va { v,Wj+i, — Wj+i I dxdv

, wj+i, wj+i

dxdv + ^ J e 9v a I v, Wj+i, Wj+i j dxdv -

— I e 9va ^v,Wj+i,Wj+ij dxdv = ■*e 9t j a |t, Wj+i(t, x), Wj+i(t, x) J dx-

f' ( 0 , . 0 , —9v( £ £' \ i i

j ayv,Wj+i(x),Wj+i(x)j dx + ^ J e a yv,Wj+i,Wj+ij dxdv-

e va I v,Wj+i,Wj+i I dxdv, (54)

8

Q

—

2

Q

ь = -

f ' ( £ d £' \ d d f r ( £ d £' A d d

e a v,Wj, — Wj+2 dxdv = e a v, Wj+i, — w,-+1 dxdv JQt V dv ) JQt \ dv J

'Qt

1 Г d

2 JQt dt

— dv ' ,

e a v,Wj+i,Wj.+ 1

в

dxdv +2 j e eva | v, Wj+i, Wj+i j dxdv-1

Qt

l

— I e eva I v, Wj+i, Wj+i I dxdv = — e 2 d Qt ' ' 2

-et

t,Wj+i(t,x),Wj+i(t,x)\ dx-

l

в

2Q

a I v, Wj+i(x), Wj+i(x) 1 dx +— I e eva ( v, Wj+i, Wj+i 1 dxdv—

2J Qt

e va I v, Wj+i, Wj+i I dxdv, (55)

т = 2i / e—ev

Ii0 Le

T 22 f —ev d ,

In = T 4e avl'j+2

— fuj+G dxdv = 2!e et ( Uj+2(t,x)j dx+

2

+ ^ ^ e-(U

r \ 2 p l / ^ r \ 2

j+G dxdv - "2-J (uj+2(x)J dx (56)

dxdv = 22e et I I Vj+2 (t, x)j dx+

+ в22 f e—ev I V

2 JQt

(\+2^ dxdv - (\'+2(x)^ dx■ (57)

On the basis of (17)-(19), (49), (50), (51)-(57) and repeating arguments used to obtain inequality (44), we have

£ . . u (t)

+

Hm+2(0,l)

£ . . v (t)

^ C, Vt G [0, T], m = 0,p.

H m+2(0,l)

From (58) we obtain uniform with respect to e inequality

^ C, j = 0,+ 1, t G [0,T]■

C([0,l])

uj(t) + vj(t)

C([0,l])

(58)

(59)

By virtue of (45), (59) and taking into account Arzela’s theorem [6], we can choose the

(60)

subsequence (u, vj such that it converges to the vector function (u, v) as 2 ^ 0: Uj ^ uj, Vj ^ Vj uniformly in C(QT), j = 0, ■■■,p - —

Taking the limit 2 ^ 0 in system (1) (with e = 2) and taking into account estimate (59) (with j = 0), by virtue of (60) we find that the vector (u, v) satisfies in QT the following system of equations

U(t) + an(t)u(t,x) + ai2(t)v(t, x) = 2iu®x(t,x) + g(t)f (t,x),

(61)

\ a2i(t)u(t,x) + a22(t)v(t,x) = 22vKx(t,x)+ F(t,x), (t,x) G QT

with initial conditions

u(0,x) = uo(x), v(0, x)= vo(x), x G [0, l], (62)

a

—

2

0

—

2

2

2

boundary conditions

ux (t, 0) = vx(ti 0) ux(ti l) vx(ti l) 0 (63)

and the overdetermination condition u(t, x0) = p(t). (64)

Based on overdetermination condition (64), we have

p + anp + ai2v(t, x0) — ^iuxx(t, x0)

g(t) =---------------Ш0)--------------------

in system (61).

Hence, by virtue of (60) we have

g(t) ^ g(t), uniformly in C[0, T]. (65)

Let us prove the uniqueness of the solution to problem (61)-(64).

111222 __ i 2

Let us assume that (u,v,g), (u,v,g) are two solutions of problem (61)-(64) and u = u — u,

12 ____ 12

u = v — v, u = g —g.

The vector (u, u, u) satisfies the system of equations

1 ut(t) + a11u + a12u = M1uxx + uf, \ a21 u + a.22v = g2'Uxx- (66)

with initial conditions u(0, x) = u(0, x) = 0, (67)

boundary conditions ux(t 0) 0) ux(ti 1) vx(ti 1) 0 (68)

and the overdetermination condition u(t, x0) = 0. (69)

By virtue of conditions (67)-(69) we obtain

~ = a12u(t, x0) — gruxx(t, x0) 5 f (t,x0) ' (70)

Let us differentiate problem (66)-(69) j times (j < p) with respect to x, multiply the result of differentiating by e-etWj = e-et (uj,vj) and integrate over Qt = (0,t) x (0,1),t G (0,T). This can be done by virtue of Remark 1. Then we have the following relations:

e

Qt

uj— uj dxdv + dt

e

a (v, Wj, Wj) dxdv—

Qt

— P1

e

6v

Uj+2'uj dxdv — g2

e Uj+2Ujdxdv

e 9v ufj uj dxdv, (71)

Qt

Qt

Qt

I12 = / e-0vu

d ^ 1 f d _ 1 f„

—Uj dxdv = — e~ev—u2dxdv = — e-et u2(t,x)dx+

dt 2 JQt dt 0 2 J0 0

Of 1 fl 0 2

4— e-6vujdxdv--u, (x)dx, (72)

2 Qt j 2 0 j

Q

-ev~

j e g2+idxdv, Qt (73)

1 e-evvj2+idxdv. Qt (74)

Taking into account (70), conditions (4) and the condition imposed on pi, function g satisfies the following inequality

|g(t)| < C4(|g (t,x0) | + \uxx (t,x0) |),

, n i Mi |ai21

where C4 = max < —

S’ 6

Hence, by virtue of Theorem 3 we obtain

|g(t)| ^ C4K (||g(t) 1 \ит(0,1) + 1 |g(t) 1 |Hm(0,l^ • Considering relationships (40), (41), we obtain

в mMiC4Ka 1 \ ft

—+ к — 2 +

-6v ц~i|2

2

+ к —

2a J

mMiC4Ka

llHm(0,1) 1

2e-0t УдУИт(0,0

t

+

2

^ 6 -V ||g||Hm(0,;)dV + Mi^ e -V ||д||Ит + 1(0,г)^^+

+ М2 j e ||g|lHm+1(0,I)dV ^ (75)

We choose the constant a > 0 to be sufficiently small such that

mMiC4Ka

2

> 0.

Then we choose the constant в to be sufficiently large such that

в mMiC4Ka 1

- + к-----—--------> 0.

2 2 2a

From relation (75) we obtain

||u||L2(Qt) + ||g||L2(QT) ^ 0.

(76)

(77)

(78)

It follows that u = g = 0 in QT.

Thus we have proved the following result.

Theorem 4. Let us assume that conditions (3)-(6), (24), (25), (47) and Assumption 1 are satisfied. Then there exists a unique solution (u,v,g) of problem (61)-(64) in the class

X(T) = {g(t,x),v(t,x),g(t)|u(t,x) G C^Qt) , v(t,x) G C^^Qt) , g(t) G C ([0,T])} •

The solution (u,v,g) to problem (61)-(64) is unique and belongs to the class of X(T). The sequence (u,v, ^ converges to (u,v,g) as well as the subsequence (u,v,g^ given above. There is the following theorem.

Theorem 5. Let us assume that the conditions of Theorem 4 are satisfied. When e ^ 0

uj^ uj, vj^ vj in C(Qt) , j = 0, ...,p — 1,

g^ g in C([0,T]).

(79)

(80)

Q

j

Q

к—

5. Degree of convergence as £ ^ 0

(£ £ \ ( £( 1) e(2)\

Let us subtract system (61) from system (1) and denote (u — u, v — vj = r ,r I

Then we obtain the following system of equations

£(1)

d £ (1) £ (1) £ (2) d2r * _ ,

—r + anr + ai2r = ^1 dx2 + G(t)f (t,x),

£ £(2)

5V £ (1) £(2) д2 Г

+ a21r + a22r = М2 gx2 ,

for the vector r = V \r^ ) that satisfies conditions

(81)

r(0, x) = 0, x £ [0, l],

d2m+1 d2m+1

Ir(t 0)= о 2m+1 r(t,1)=0, m = 0, 1> 2-

dx2m+1

dx2m+1

In (81) the function G is of the following form:

G(t) = g(t) — g(t)

£(2У 0\ £(1C 0\

a12r (t,x0) — M1fxx (t,x0) f (t,x0) ‘

(82)

(83)

(84)

Let us differentiate problem (81)-(83) three times with respect to x, multiply the differentiated system by e-etr3 and integrate over Qt,t £ (0,T). Repeating arguments used in obtaining relation (44), we have the inequality

£(1)(t) r (t)

+ e

H3(0,l) J0

-9v

£(1)(t) r (t)

dv + / e--v H4(0,l) Jo

e(2)(t) r (t)

dv < eC.

H4(0,l)

(85)

By virtue of (85) it follows that

um

< e1/2C, j = 0, 2, (86)

C(GT) < e1/2C, m = 0, 4. (87)

L2 (Qt

we have from (84)

— g(t) < e1/2C. L2([0,T ]) (88)

Thus we have proved the following result.

Theorem 6. Let us assume that the conditions of Theorem 4 are satisfied. Then relations (86), (87) hold.

Let us consider problem (1)-(3) and assume that the conditions of Theorem 2 are satisfied. By virtue of the periodicity of the input data there is the following theorem.

Theorem 7. Let us assume that the conditions of Theorem 4 are satisfied. Then the solution (u,v,g) of the problem

(ut(t) + au(t)u(t, x) + a12(t)v(t, x) = M1Uxx(t, x) + g(t)f (t,x),

1 a21(t)u(t, x) + a22(t)v(t, x) = M2vxx(t,x) + F(t, x),

2

2

2

t

u — u

j

u(0, x) = uo(x), v(0, x) = vo(x), x G [0, l], u(t, x0) = <^(t)

exists and it is unique in the class

X (T )

u(t,x),v(t,x),g(t)\u(t,x) G С1’ 1(G[ojT]),v(t,x) G C°£

t,x 1 (G[0 ,T]),

g(t) g c([0, t])} .

Relations (86), (87) are satisfied and when e ^ 0 Uj^ Uj, £ £

0, ...,p — 1, g^ g, uniformly in C[0,T], u(t,x) — u(t,x)

Vj^ Vj uniformly in G[0,T],

^ e1/2c, (t, x) G G[0,T].

References

[1] A.I.Prilepko, D.G.Orlovsky, I.A.Vasin, Methods for Solving Inverse Problems in Mathematical Physics, New York, Marcel Dekkar, Inc., 1999.

[2] V.G.Romanov, Inverse problems for differential equations, Novosibirsk, Novosobirsky Gos. Universitet, 1978 (in Russian).

[3] Yu.Ya.Belov, V.G.Kopylova, On some identification problem for source function to one semievolutionary system, Journal of Inverse and Ill-posed Problems, 20(2012), no. 5-6, 723-743.

[4] Yu.Ya.Belov, S.A.Cantor, The method of weak approximation, Krasnoyarsky Gos. Univer-sitet, 1990 (in Russian).

[5] N.N.Yanenko, Fractional steps for solving multidimensional problems of mathematical physics, Novosibirsk, Nauka, 1967 (in Russian).

[6] L.V.Kantorovich, G.P.Akilov, Functional analysis, Moscow, Nauka, 1977 (in Russian).

[7] S.L.Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Moscow, Nauka, 1988 (in Russian).

Определение функций источника систем уравнений составного типа

Юрий Я. Белов Вера Г. Копылова

Рассмотрена задача идентификации функций источника одномерной полуэволюционной системы уравнений для двух уравнений в частных производных. Исследована система уравнений, полученная из исходной системы, в которой в эллиптическое уравнение добавлена производная по времени, содержащая малый параметр е > 0. Рассмотрены задача Коши и вторая краевая задача.

Ключевые слова: идентификация, обратная задача, уравнение параболического типа, метод слабой аппроксимации, малый параметр, сходимость.