CC BY
14
UDC 517.972.5
Variational Principles for the Differential Difference Operator
of the Second Order
I. A. Kolesnikova
Department Mathematical Analisys And Theory Of Functions Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, Russia, 117198
The purpose of the present paper is to investigate the potentiality of the operator of differential difference equations and to construct of the functional, if the given operator is a potential on a given set relatively to the some bilinear form.
Key words and phrases: differential difference equations, functional differentional equations, inverse problem of the calculus of variations, variational principles, equations with deviating arguments.
1. Introduction
The construction of variational principles in the investigation of the differential equations N(u) = f is connected with the inverse problem of the calculus of variations. The investigation of the solutions of this inverse problem as the construction of the functional F[w] for which the set of critical (extremal or stationary) points coincides with that of a solution of the given equation N(u) — f = 0.
Differential difference equations or functional difference equations have already appeared in mathematical papers of the XVIIIth century, for example, in the Euler solution of the problem connected with a search of the general form of a line similar to its evolute.
The search of a functional F that admits some given equations as its Euler-Lagrange equations is known as the classical inverse problem of the calculus of variations. Since the end of the XlXth century there has been a great deal of activity in this field (see Helmholtz [1], Volterra [2], Santilli [3], Tonti [4], Filippov, Savchin and Shorokhov [5] and refs. therein).
There is a practical need to develop different approaches to the construction of integral variational principles for equations with deviating arguments.
It is possible to investigate the problem of the construction of the variational multiplier if the operator of the given equation is not potential on the given set with respect to some bilinear form.
The main aim of this paper is to investigate the potentiality of the operator N(u) of the differential difference equation and to construct the functional F[w], if the operator N(u) is potential on the given set relatively to some bilinear form.
2. Some auxiliary notations and definitions
Let U, V be normed linear spaces over the field of real numbers R, and Ou ,Ov be their zero elements.
Take any operator N : D(N) ^ R(N), where D(N) C U, R(N) C V .A limit
lim ^[N(u + eh) — N(u)] = 5N(u,h), u e D(N), (u + eh) e D(N),
e
if it exists, is called the Gateaux differential of N at the point u. If it is linear relative to h, then the operator 5N(u, ■) : U ^ V is called the Gateaux derivative of N at u and will be denoted by N'u. Its domain of definition D(N'U) consists of elements h e U such that (u + eh) e D(N) for all e sufficiently small.
Received 2nd July, 2012.
Let us consider the equation N(u) = Ov, u G D(N) with the Gâteaux differentiable operator N, and a convex set D(N).
In order to consider the existence of its variational formulation we need a non-degenerate bilinear form
11
( u, „) =jju(x, t), v(x, №. (1)
n to
Definition. The operator N : D(N) ^ V is said to be potential on the set D(N) relative to a given bilinear form {■, ■) : V x U ^ R, if there exists a functional F^ : D(FN) = D(N) ^ R such that 5FN[u, h] = {N(u),h) Vu e D(N), Vh e D(N'U).
The functional FV is called the potential of the operator N, and in turn the operator N is called the gradient of the functional Fm. As it is known (see Volterra [2]) the condition for potentiality of the operator N takes the form
{ N'ah,g) = {N'ag,h) Vu e D(N), Vh,g e D(N'U). (2)
Under this condition the potential Fn is given by
1
Fn[u] = J {N(u0 + A(u — u0)),u — u0) dA + const,
0
where u0 is a fixed element of D(N).
Let us consider the differential equation with deviating arguments
N(u) = ^ a\(x, t)(x,t + At) — b^(x, t) ® ^ (x,t + Ar)+ x=—1 Xl xj
+ f(x, t + At, u(x, t + At), uXj (x, t + At),ut(x, t + At)) = 0, (3)
where u is an unknown function; (x, t) e Q = Q x (t 1 ,¿2); h — t1 > 2r; a\(x, t) e
(Q), b^(x, t) e clj(Q), Vi,j =
The domain of definition D(N) is given by the equality
D(N) = ju e U = C2/t (Q x [io — t, t1 + r]) : dku(x, t)
dtk dku(x, t)
= <pik(x, t), (x, t) G Ei = fi x [t0 - t, i0], k = 0,1, = V2k(x, t), (x, t) G E2 = ÏÏ x [t 1, ti + r], k = 0,1,
= $v, v = 0, 1 S , (4)
dtk
dvu dxv
,
where Q C 1", rT = 3Q x (t0 — t, t1 +t), mp10, pt0, — are given sufficiently smooth 3km ■
functions, mjk = 3tjj0, (7 = 1,2; k = 0,1). The formulation of the problem:
1) to investigate the potentiality of the operator N of the equation (3) on the set D(N) (4) relatively to the some bilinear form (1);
r
2) if the operator N is potantial, then to construct the variational principle for the operator N of the equation (3).
The investigation of the potentiality of the operator N of the equation (3).
Theorem. For the potentiality of the operator N (3) on the given .set D(N) (4) with respect to the bilinear form (1), it is necessary and sufficient that the following conditions hold:
a\(x, t) = a-\(x,t + At), (x, t) = b%lx(x,t + At) VA = —1, 0,1, I\ f(x,t,U,Ux- ,Ut) = f(x,t,U,Ux- ,ut),
I\f (x,t,u,ux- ,Ut) = dax(x ^ Ut (x,t + At ) — - Q x ^ Uxi (x,t + At ) + Ix^(x,t,u),
X3 (5)
where ^ are sufficiently smooth functions, I\g(x, t) = g(x,t + At). Proof. We denote
1 d2h •• d2h N'uh = a\(x, t)(x,t + At) — (x, t) -x-x . (x,t + At)+
A=—1 1 3
M d^h+h-+^ +At )
Vu e D(N), Vh,g e D(N'a), Vi,j = 1,n. (6) Taking into account formulas (1) and (6), we get
{NLh, g) = J\a^(x, l) -Q-fi(x,t + Ar) — &a3 (x, i) (x,t + Ar)+
A=-1n to ^ 13
+ { d^h + Qhhh^ + Q^ht} (x, t + At) |9(x, i)didx
Vu e D(N), Vh,g e D(N'U), Vi,j = 1,n. (7) We denote in the items of the formula (7) thus
1 f f d2h I I ax(x,t]
x=-i:
1 ti
1 f f d2h h = ^2 J J a\(x, t)(x,t + At)g(x, t)dtdx,
Q t o 1 ti
1 { { d2h h b\ (x, t) Qx-x . (x,t + At)g(x, t)dtdx,
X=-1Q to 1 3
A=-1Q to
h = I I (x,t + At)g(x, t)dtdx,
1 ti
1 Q
I4 ——ht (x,t + Ar)g(x, i)dida
, J J -Ut
A=-1Q to
d u
A=-in to
We know that
I5 = ^^ I I d-h(x,t + Xt)g(x, t)dtdx.
to tl+T
Ih(xi)di= /h(x' i)di=°'
o- T 1
il io+T
jh(x,t - r)dt= j h(x.t + = 0 Vx G n,
1- T o
from the formula Ii — I5 we get
i ii i ii Ii = ^ J J h(x,t + Xt )D2(a\(x, t)g(x, t))dtdx = ^ J J h(x,t + Xr)x A i n to A i n to
(8)
(
d2a\(x, f) dax(x±t) dg(x, f) d2g(x f) A
-g(x, t) + 2-—--—--1--—2—ax(x, t) didx.
dt2 ^ ' dt dt dt2 By using the change t' = t + At , we obtain
1 ti + xt
h = £/ / h(x, f)(d^ — Xr)S(x, ^ — Xr) +
aA (x, t' — Xtdidx.
A=-in to+\T
dax(x, t' — At) dg(x, t' — At) d2g(x, t' — At) + dt dt + dt2
Denoting t' by t and taking into account formula (8), we get /1 = £ / / h(x, t) (dtax(dt— Ar)g(x,t — At) +
A=-in t,
or
+ 2 ^^ — Xr ) ^Kxd^AT) + d (M ~ Xr) aA(x, i — XT)) dtdx
h = j h(x, t)(d a-x(dx^2 + XT),(x,t + Xr) +
A=-in to
+ 2da-A(xdjtt + Xr) dg(xd+ Xt) + d2g(x^ + Xr) a-x(x,t + Xr)) didx. (9)
/2 = £ y y h(x,t + Xt)DXiXj (bA(x, t)g(x, t))dtdx = ^y j h(x,t + Xt) A i n ¿o A i n ¿o
(d2 b3 (x, t) . . ndbX (x, t)dg (x, t) d2g (x, t)li3/ , ,
x J g(x, t) + 2 y ' ; ' ; + /V ; 633 (x, iHdidx.
\ dxidxj dxi dxj dxidxj X I
Reasoning by analogy as for I1, we obtain
ti
* f r s id2bi3x(x,t + At) /
/2 = zjjh(x, +At)+
1 Q to
db-x(x,t + At)dg(x,t + At) , d2g(x,t + At) i3
+2 -AV-- k -L + \\- b-x(x,f + A^ didx
dxi dx3 dxidx3 '
>)
Vi, j = 1,n. (10)
i *i h = - ¿y+ Ar)x
A=-1n to
i<9/(x, t + At, u(x, t + At), u«j (x, t + Ar), u« (x, t + Ar)) \ _
x^-^ 9ul3 ^Jdidx =
1 «i
[ [hi i | , \ fdf(x f + Aт, u(x f + At ), U«j (x f + At ), "tQ^ f + At )) ,
- ^ «(x, i+AT ) I g«(x, i)+
to V
^ <9/(x t + u(x t + Ar), (x t + Ar ^ "tpx t + Ar)) \\ + 9(x, t^«,^ 9ul3 JJdidx.
By using the change t' = t + At
1 ti + XT
t [ [ h t +'\(df(x, t',u(x, (x, t' ),Ut(x, t'»
h = h(x, t)( ---^ (x,t —At )+
\__1 j j \ xj
+Xt
+ g(x,t —At )DXi{--))
didx.
Denoting t' by t and taking into account formula (8), we get 1 ti
T ^ f [hi 4-\(df(x,t,u(x, t),U'Xj (x, t),Ut(x, t»
h = — h(x, t) I ---(x,1 — Ar)+
X=-1Q to v Uxj
id f(x,t,u(x, t),uX-(x, t),ut (x, t))\\ + g(x,t — At)DXj ^ M ' ' V ' du J J didx
or
1 ti
t ^ m hi A f w&^ufa t),uxj (x, t),Ut(x, t»
h = ^^ h(x, tU -d^-9xj (x,t + At)+
\ — — 1 i Y ^ Xi
- Q t o
id f(x,t,u(x, t),ux-(x, t),ut(x, t))\\ ,
+ g(x,t + At)Dxj ^ M ' ' V ' du jjdtdx. (11)
h = - £ J J h(x,t + Xt)x A=-1n to
D ( 9f (x>t + ^^ u(x>t + xt )> Uxi (x>t + xt )> Ut(x't + xt )) (x t)\ dfdx =
v dut ' J
1 fl
f [,, ,fdf(x,t + XT,U(x,t + XT),Ux. (x,t + XT),Ut(x,t + XT))
= - J J h(x, t + Xr)^-duft-gt(x, t)+
x=-1n to
, ^ + *T,u(x,t + xt),ux-(x,t + Xr),ut(x,t + xt))\\ + g(x, t)Dfi -^-j I didx.
By using the change = + A
1 tj + xt
h = —^J J h(x, t -dut-gt(x,f —At )+
a=-i n îo+at
/df (x, t',u(x, t'),uXj (x, t'),ut(x, t')) \\
V dût ))
+ g(x, t' — Xt)Dt ' „ x ' didx
Denoting t' by t and taking into account formula (8), we get 1 ti
f f , ^ id f(x,t,u(x, t),ux .(x, t),ut(x, t)) h = — ^ J J h(x, t)^ M ' ' V ' ^ 9t(x,t — At )+
A=-in to
or
i ^
id f(x,t,u(x, t),uX-(x, t),ut (x, i))W + g(x, t — Xt)Dt f M V du ) ) d^
T f Î v.f ( dî(x,t,u(x, t),ux,(x, t),ut(x, t)) h = — ^ h(x, i) I-d-1-gt(x,t + Xt)+
A=-i o / ^ ut
- n o
+ g(x,t + Xt )Dt ( df ^^ ^(x, ^^ )) didx. (12) h = £ f / h(x, i)^fixA^x^Axi^Ox^^x^ + xr)didx. (13)
A=-in i
Thus from (7) and (9)-(13) we obtain
m'^ \ / L Jidta-x(x,i + Ar) d26-x(x,i + AT)
{Nuh, 9) =
È f / h(x,
A=-i n /„ I V
dt2 dxidxj
A=-i n io ^ v
_D fdf (x,t,u(x, t),uxJ (x, t),ut(x, t))\ d (df (x,t,u(x, t),ux(x, t),ut(x, t))\ + X V dux, /A dut J
+ df Q^ t, u(x, t), Ux(x, t),ut(x, t»j ^(x t + AT)+
d2 (x, + A ) d2 (x, + A )
+ a-X<x, i + AT) 9[ dt2 ' - 0-X<x. « + ) ^x.dx,
( db-x(x,t + at) , df (x,t,u(x, t),Uxj (x, t),ut(x, t))\ — 2-dxi-+-d^i- ^(x, i + Ar)+
/2 da-x(x^t +A*) — df (x,t,u(x, t)duux(x, t),ut(x, t)y) gt (x^t + AT) >dtdx
gt (x,t + at )|(
Vu e D(N), Vh,g eD(N'U), Vi,j = 1,n. (14)
Using the Gateaux derivative of the operator (3) , we get
d2 (x, + A ) d2 (x, + A )
1 ti
№,h) = E //{ax(x,0d 9(x^AT) — $(x,0"d^^) +
X=-1Q to ^ 13
t| (x, t + At) | h(x, i
d d d + \dki9 + d^- gxj + (x,t + Ar) ^h(x, i)didx
Vu e D(N), Vh,g e D(N'U), Vi,j = 1,n. (15)
We mean IX f(x, t, u, ux,ut) = f(x, t + At, u(x, t + At), ux(x, t + At), ut(x, t + At
The equations (14) and (15) are equal if and only if
Ixf (x, t, U, Ux, Ut) = f(x, t, U, Ux, Ut), (16)
ax(x, t) = a-x(x,t + At), b%X(x, t) = b%lx(x,t + Ar) VA = —1, 0,1,
vi ? Mr* !( d 2ax(x, t) d2 b*x3 (x, t)
hj J H""X2 dTdx- -
X=-1Q to 1 v °3
_D ( df (x, t, u(x, t), Uxj (x, t),Ut(x, t)) \ Xj V duxj J
fdf(x,t,U(x, t),Uxj ^^ t),Ut(x, t))W
— DA-^- 9(x,f + Ar)—
— 2( ^ + -j (x^ + - ) + < ^ — ^(M + Ar )}didx = 0
Vu e D(N), Vh,g e D(N'U), Vi,j = 1,n.
The condition (16) is fulfiled for some periodic functions, for example. For the potentiality of the operator N of the equation (3) on the set D(N) (4) relatively to some bilinear form (1), it is necessary and sufficient that the following conditions hold
for all h and
db%Aj (x, t)
dxj + iH- = 0,
daA(x, t) — dj_ = 0 (17)
dt dut
D / da a(x, t) — df_ \ —Dx ( db A (x, t) + _df_\ =Q \ dt dut) XM dxi dux, '
The third condition is a consequence of the first and the second conditions.
dax(x t)
Ixf (x,t,u,ux ,ut) =-—ut(x,t + At) + hp(x,t,u,ux), (18)
where p is sufficiently smooth function
Substituting (18) for f in the second condition of the sistem (17), we get
dbZj (x ~t)
Ixp(x, t, u, ux) =--\ ' ux(x, t + Ar) + Ix^(x, t, u), (19)
d x
where ^ is sufficiently smooth function.
Thus from equalities (18) and (19), we obtain
Ixf (x,t,u,ux,ut) = ^X^ ^ ut(x, t+Ar) — d Xd(X, ^ ux (x, t+Ar) + Ix^(x,t,u), (20)
The proof of the theorem is completed. □
The construction of the functional Fn [u] for the operator N of the equation (3)
We can write (3), making use of (20)
1 d 2 u d 2 u
N(u) =22 ax(x, t)(x,t + At) — bl<j(x, t) dx.dx . (x^ + At)+
A=— i
+ dax(^> ut(x, t + At) — dbxd(X, ^ ux(x, t + At) + Ix^(x, t, u) = 0. (21)
We can find the functional Fn [u]
1
Fn[u] = J {N(u0 + ^(u — u0)),u — u0) d^ =
0
! t 1
j I I (aA(x, t)[uo11 + l(utt — uoti)] — bA(x, t)[uoxixj + M(uxixi — uoxixj)] +
A=-in to 0
+ a At (x, t)[uot + l(ut — uot )] — blAjxi (x, t)[uoxj + l(uxj —uoxj )])(u — uo )d|dtdx+ i il i
+ ^^ J J J IA^(x,t,u)(u — u0)d|dtdx, u = u0 + |(u — u0).
A=-in to 0
We integrate the first integral (21) in the variable ^ and get
Fn [u] =
i ii 1 £//■/ (
A— — 1 o fl
n i 0 0
- bA (x, t)
a\(x, t)
1
U0ii + 2 (Mii - U0ii)
^Oxixi + -¿(UxiXj - UOxiXj )
+ a Ai (x, t)
UOi + 2(Ui - U0i)
- C (X, t)
1
U0xj + ö(ux,- - U0xj )
2
(u - U0)d^dtdx+
i 1
+ E
A 1
IA^(x, t, U)(u - u0)d^dtdx
n io 0
or
fnM = 1 2 (ax(x, t)[uott +utt] - bA (x, t)[uçÎXiXj + uXiX.]+
1 n t0
+ axt(x, t)[uot +ut] - bA>x. (x, t)[uoxj + Ux])(u - uo)didx+
t1 i
+ J J J h^(x,t,ù)(u - uo)d/j,dtdx = 1 22 J J Dt(a\(x, t)(uot +u)(u - uo)) -
- n 10 o
- n t0
- Dxi(blA (x, t)(uoxj +Uxj )(u - uo)) - ax (x, t)(uot +Ut)(ut - Uot )+
1 tt 1
+ bX (x, t)(uoxä +Uxj )(uxi-Uoxi )didx + 2 J J J Ix^(x,t,U)(u - Uo)dndtdx. (22)
A=-1
n 10 o
Thus, the unknown functional for the operator (21) (from the equality (22)) is obtained in the following form
Fn[u] = -1 ¿7 J(aA(x, t)u2t - b%(x, t)uxiUx0 )dtdx+
A=-1n io
i 1
+ E
A 1
IA^(x, t, U)(u - u0)d^didx.
n io 0
References
1. Helmholtz H. Ueber die physikalische Bedeutung des Prinzips der kleinsten Wirkung // J. Reine und Angew. Math. — 1887. — Vol. 100. — Pp. 137-166.
2. Volterra V. Leçons sur les Fonctions de Lignes. — Paris: Gautier-Villars, 1913.
3. Santilli R. M. Foundations of Theoretical Mechanics l. The Inverse Problem in Newtonian Mechanics. — Springer-Verlag, 1978.
4. Tonti E. A General Solution of the Inverse Problem of the Calculus of Variations // Hadronic J. — 1982. — Vol. 5, No 4. — Pp. 1404-1450.
5. Filippov V. M., Savchin V. M., Shorokhov S. G. Variational Principles for Nonpotential Operators // J. Math. Sci. — 1994. — Vol. 68, No 3. — Pp. 275-398.
УДК 517.972.5
Структура дифференциально-разностного оператора второго порядка, допускающего вариационный принцип
И. А. Колесникова
Кафедра математического анализа и теориии функций Российский университет дружбы народов ул.Миклухо-Маклая, д.6, Москва, Россия, 117198
В статье исследуется на потенциальность оператор на заданной области определения и относительно некоторой билинейной формы. В случае потенциальности строится соответствующий функционал.
Ключевые слова: дифференциально-разностные уравнения, функционально-дифференциальные уравнения, обратная задача вариационного исчисления, вариационный принцип, уравнения с отклоняющимися аргументами.
