CC BY
10
Milogorodskii Aleksandr Andreevich, East China Normal University, PhD student, Department of Mathematics E-mail: amilogorodskii@mail.ru Berman David, Middle East University, Postdoctoral research assistant, Department of Physics E-mail: haimzilberberg@hotmail.com
Ming Kang Ni, East China Normal University, Professor of Math, Department of Mathematics
Shanghai, China
ON A CERTAIN PROBLEM OF A SINGULAR PERTURBATION OF THE WAVE EQUATION
Abstract: A complete asymptotic expansion with respect to a small parameter of the solution of a singularly perturbed Cauchy problem with initial jump for a multidimensional hyperbolic equation of the second order degenerating into a linear hyperbolic equation of the first order is constructed.
Keywords: optimal control, perturbation method, internal transition layers, singular perturbation, asymptotic approximations.
Introduction singularly perturbed problems for differential equa-
Many applied problems of physics, mechanics, tions with a small parameter for the highest deriva-engineering, chemistry and biology are described tives depend on the parameter both in a regular way
by means of differential equations containing small or large parameters called perturbed equations [1, 627-654; 2, 127-128; 3, 720-730; 4, 10-126]. Depending on the nature of the perturbation, such problems are subdivided into regularly perturbed and singularly perturbed problems. In comparison with regularly perturbed problems, the study of singularly perturbed problems presents a great difficulty. In such problems, the degenerate problem belongs to a different type than the original problem, and difficulties arise in qualitative investigation of the behavior of the solution of a non-degenerate problem. One of the difficult problems in the theory of singularly perturbed problems for differential equations is the construction of asymptotic expansions with respect to a small parameter of solutions for the problems under consideration. The solutions of
and in a singular way.
1. Statement of the problem
Consider the following singularly perturbed hyperbolic equation of the second order:
V
d2V
-AV
= A (t, x )— + v ' dt
\dV +pj(t 'x fe-
ci)
F (t, x ),
where £> 0 - small parameter, x = (x1,x2,...,xn) -n - dimensional vector,
d2V d2v AV = —- +
dx1
+ ... + -
d 2V
dx] dx2n
- Laplace operator.
Necessary to consider the behavior of the solution of equation (1) when s^ 0 with initial conditions
s
dV
V=o = v(x )> £ — = v(x ). (2)
dt t=o
The following conditions must be fulfilled:
1) Functions A(t,x),Bj (t,x), j = l,n,F(t,x) are sufficiently smooth in the domain
D = {(t,x): t > 0,x e Rn}.
2) Functions (p(x), y(x) smoothly enough Rn. 2. Calculation of the initial change
To determine the initial jump [5, 145-171] A(x ) multiply equation (1) by s and t = st. Then from (1), (2)
d2V u .6V —--£ AV = A (ex, x )-+
dT y ' dT
+e
n dV YB (eT,x)-+ F (t, x)
VT=o = v{x ) ^r =v(x ).
or t=o
(3)
From (3) when s = 0
d 2V
d 2 = A (0, x , d t d t
Vt=o = v{x ), ^T =v(x ).
dT t=o
(4)
From (4)
d2V
= A (o, x ^,
dt2 v ; dt
dV,
(5)
V (o, x,s) = ç(x ), — (o, x,s) = y(x ),
then
dV
V(x )
— (t, x ,£) =
dtx ' £
exp I A(o,x)—
Let t = te =-(elns)l£. Then from the expression dV /dt considering the properties 1), 2) when 0
dV —(t ^ x ,£)
dt
< M, M = const > o
(6)
for any bounded subset G e Rn.
Suppose that lim V(te, x ,s) = y(x ) + A(x) for any x e G. Integrating both sides of expression (5) from 0 to ts:
dV
£—(t e, x ,s)-y(x ) = A (0, x )(v (t £, x ,s)-y(x )) .(7)
dt
Passing to the limit in (7) when £ ^ 0 considering (6), for any x e G
v(x)
A(x ) = -
(8)
A (0, x)'
Thus, the initial jump for V defined. Then it is necessary to construct the asymptotics of the solution of problem (1), (2).
3. Construction of the asymptotics of the solution
The asymptotic expansion in the small parameter for the solution of the singularly perturbed problem (1), (2) is considered as the sum
V (t, x ,s) = Vs(t, x) + W£(t, x), (9)
where function V£ (t,x) - the regular part of the as-ymptotics is representable in the form of a series in powers of s
Vs(t, x ) = ^ekVk (t, x), (10)
k >0
function Wk (t,x) - the singular part of the asymptotics is representable in the form of a series in powers of s
W£(T,x) = ^kWk (t, x), (11)
k>0
The regular part of the asymptotics V (t, x) is determined by means of the initial singularly perturbed differential equation (1)
dt2
-AV.
= A (t,x )dVjL +
J
dV,L
+Ypt (t, x ) d
i=1 dx.
dt F (t,x),
(12)
and the singular part of the asymptotics We(r, x) is determined by means of the homogeneous equation corresponding to equation (1) (13):
ôr2 .dW£
-£2AW =
.ÔW
(13)
\dWF / .....
= A (£t, x I-- + £} Bi (£t, x I
V 1 dt t1'y ; dxt
Substituting the expansions (10), (11) respectively in (12), (13) and expanding the functions A (sr, x ),Bt (sr, x) in the series of degrees s and equating the coefficients for the same powers of £ in both sides of equations (12), (13), obtain a sequence
£
l
of differential equations for determining the coefficients of the expansions (10), (11).
For V0 (t,x) W0(t,x)
dV n dV
A (t,x)-V° + YBt(t,x+ F (t,x) = 0, (14)
dt i=i dxt
d2W , \ dW ° W = A (0,x) 0
Or1
dT
(15)
and for Vk (t, x ), Wk (t, x )
dV n dV
A (t,x+ YB1 (t,x= ®k (t,x), (16)
ox.
dt
i=i
d 2Wk .,n ,owk . . —t = A (0, x )—k + yk (r, x ), dr dT
(17)
where ®k already defined function V (t,x), i < k:
0k (t,x)=dVrL ~*Vk
and function expressed by
W (t, x ), 1 < k :
k-1
^k (T, x ) = -AWk-2 +X
1=0
-1+1
((+1)!
A(()(0, x )dWk -i-i
dT
+
_nk-1 1
+I I ^ }(0, x )
T B(1 )fn ^dWk-1 -1
;=1 1 =0
1!
dx.
To uniquely determine the coefficients of the expansions (10), (11), it is necessary to specify the initial conditions. To determine these conditions, we substitute the expansion (9) with (10), (11) into the initial conditions (2) and equate the coefficients for the same powers of £ in the expressions obtained. Then
W (0,x) = p(x)-V (0,x), (18)
dWn
-(0,x) = w(x),
or
Wk (0,x) = -Vk (0,x),
W (0, x ) = -dV-L (0, x ), k = 1,2,. Or y ' dt y '
(19)
Initial condition for V0(t, x)
V0 (0,x) = p(x) + A(x), (20)
where the initial jump is determined by means of formula (8).
Solution V0(t, x) of (14), (20) is constructed as follows [6, 235-236]. Integration of the equation
(14) is equivalent to integrating the characteristic system:
dV0 _ F(t,x) dx1 _ B1 (t,x) dt A (t, x)' dt A (t, x)'
i \ (21) dxn _ Bn (t, x )
' " dt A (t, x )'
Let
xi _ X1 ((, xi , • • •, xn ) , xn = Xn ( t, xi ,• * * , xn ) ,
v _ y v y - solution
Vo = Vo(t, x°,..., xn0, ç(xQ) + A(x o)) of the system of equations (21), satisfying the condition
Xi (0,xi,- • -,xn ) xi,- • -,Xn (0,xi,- • -,xn ) x„, Vo (xio,..0x0, ty(xo) + A(xo)) = ç(xo) + A(xo).
Then the unique solution Vo(t, x ) of the problem (14), (20) will be written in the form of an implicit formula Vo(t,x) = Vo(t,xo,^(xo) + a(xo)). The system of equations xi = Xi (t,xo),...,xn = Xn (t,x0 ) is uniquely solvable with respect to xo = (x°,..., x°) when t > o. For this it suffices to show that
det ||X (t, x o) = (t, x o)l =
6X,
d(x1 , ■ • - , xn )
sX,
ôx1
dX„
dx0
3X„
(t, x 0 ) 0.
where
dx° dxn
For this consider the system of identities
dX _ B(t, X) dt A(t, X)'
X = (Xi(t,x0),...,Xn (t,xo)), B(f,X) = = (Bi (t, X),..., Bn (t, X)). Differentiating this system by x0,
d dX _^BkA-A*BX i _ 12 n
dt dx0 k=1
A2 dx0 or in matrix form
d dX BA - AB dX
dt dxn
A dxn
(22)
The solution of the matrix equation (22), satisfying the initial condition
will
dX * ,
— (t, x o ) = exP
dx o
from where follows
■BXA - AXB A2
(t, X (t, x o ))dz
j dX , , n det-(t, x 0) ^ 0.
dx 0
Let x0 = x0(t, x) - solution of equationx = X(t, x0), then V,(t,x) = V(t,x0(t,x), @(x0(t,x)) + A(x0(i,x))) gives an explicit formula for solving the problem (14), (20).
Turn to the problem (15), (18). Integrating both sides of equation (15) from 0 to t and taking into account (18), (20) and (8),
dW , ,
n- = A(0,x)W0, W0 (0,x) =
dr
= ç(x ) - Vo (o, x ) = -a(x ) =
y(x )
A (0, x)'
The trivial solution of this equation is asymptotically stable when t ^ +00, i. e. lim W0(t,x) = 0 . Then equation implies that
lim W (t, x ) = 0.
dT
Thus, the zeroth approximation V0 (t, x), W0(t, x) constructed.
Turn to the definition Vk(t,x), Wk(t,x),k = 1,2,.... Equation (16) is solved by the method of characteristics. To determine the initial condition for Vk (t,x) consider (19). Suppose that Wk (t,x),dWk(t,x)/ dt limits to zero when t ^ +00. Then integrating both sides of (17) by t from 0 to +<» and considering (19),
dV m
(0,x) = A(0,x)vk (0,x) + JVk (t,x)dt.
Because A (0,x) ^ 0 , then the initial condition
for Vk (t,x):
dV
Vk (o, x ) = -&
^ (o, x )-{ (t, x )dt
A (0, x) ' (23)
Integrating now both sides of equation (17) from 0 to t with considering (19), (23),
dt
(t, x ) = A (o, x )Wk (t, x ) - jVk (t, x )dt, (24)
Wk (0, x ) = -Vk (0, x ). The trivial solution ofthe homogeneous equation corresponding to (24), asymptotically stable with t ^ +00. Thus, the formal expansion (9) for the solution of the original problem (1), (2) is constructed. 4. The validity of asymptotic expansions Defining the terms of the expansions (10), (11) n,n +1 inclusive and denote by Vn (t, x ,s) the corresponding partial sum of the expansion (9),
V (t,x,s) = Vn,£(t, x) + w^x) (25)
where
Vn,£ (t,x) = yskVk(t,x), Wn,s (t,x) = yfkWk (t,x), k=0 k=0 and the coefficients V0(t,x) and W0 (t,x) are uniquely determined from problems (14), (20) and (15), (18), coefficients Vk (t,x) and Wk (t,x) respectively, from problems (16), (23) and (17), (19), and the coefficient Wn+1 (t,x) is uniquely determined from equation (17) under the initial conditions
Wn+1 (t, x ) = o,
dWn+i(o, x) dVn
(o,x). (26)
dr dt
Suppose Rn (t,x,s) = V(t,x,s)- Vn (t,x,e), where
function V(t,x,e) - exact solution of the problem
(1), (2) and function Vn (t,x,e), expressed by the
formula (25),
n n+1
Vn (t,x,e) = YßkVk(t,x) + Y?kWk (t,x). (27)
k=o
k=o
Function Vn (t,x,e) in any characteristic cone Q equations (1) satisfies equation (1) with an accuracy of order sn+1,
dt2
- " V
(t, x ^ - F (t, x ) = Q (sn+1 ).
i=i dxl
- A (t, x)
V dt
(28)
s
The following is valid Theorem Suppose that the
following conditions are satisfied:
A (t, x), Bt (t, x), 1) functions , , . . .
i = 1,n;F(t,x) e Cn (D);
2) p(x),y/(x)^C L2j (Rn), and y/(x)* o; Then problem (1), (2) in Q - characteristic cone of equation (1) has a unique solution V (t, x ,s), which admits the following asymptotic expansion [7, 377-384]:
V (t, x ,s) = YjskVk (t, x )-
k=0
+
yjskWk (t,x) + Rn (t,x,j),
(29)
k=0
where the coefficients Vk (t,x),Wk (t,x),k = o,i,.. .,n are successively determined from problems (16), (23) and (17), (19), and the coefficient Wn+i (t, x) from the task (17), (26), and for the remainder term Rn and Qfor sufficiently small s the estimate
Rn ('•x•£)K,i(Q) =
J-JJ
2
I dt
+
S
2
rdRn
n
Kdxt J
dtdx
(30)
rdR
dt2
- " R
. dR ^ D dR = A—+ > B-+ g
dt v 1 öx
with zero initial conditions
dR,
R(0,x,e) = 0, -—(0,x,e) = 0,
(31)
(32)
where function g = g (t, x ,s) at sufficiently small s in Q has an estimation
g (t, x ,e) = O (n+1). (33)
= O (en+1).
Proof According to the conditions of the theorem, in any characteristic cone Q equation (1) when s* 0 exists [8, 178-179] the unique solution of problem (1), (2), having continuous derivatives appearing in equation (1).
To prove the estimate (30) we subtract from equation (1) the equation (28). Then, taking (27), (29) we obtain an equation for R = Rn (t, x ,s)
Figure 1.
Let Q the characteristic cone of equation (l) with vertex at point H0(t0, x0). Let put the cone Qin the cylinder with the upper and lower bases Гh and Г0, lying respectively on hyperplanes t = 10 and t = 0 and with a lateral surface S, lying on a cylindrical
n 2
hypersurface ^( - x0) = 32, where 3 > 10 - ar-
i=1
bitrary positive number, and Гt - arbitrary section of the cylinder Z hyperplane t e [0,t0 ] = const. Zt -part of the cylinder enclosed between the bases Г0 and Гt.
Multiplying both sides of(3l) by 2( ARt + ^BR )
/
after elementary transformations we obtain the following identity
d
s— dt
A
R +TK + 2TBARXi
-sY—\b1 (R2 + Ri ) + 2ARR
dx L ^ xi ' 1 x
V d
-£ y -
¿—I ^
1, j ('*j) dxj
2BtRxRx -BjRXi
(34)
= 2
2
ARt +YBR + 2g ■ ARt + YBR
V 1 / V 1 /
+e(Rt, Rv..., RXn )M ( ,RXi,..., RXn f where the matrix M
2
1=1
Q
£
M =
'A -TPtx,
B, - A.
Bnt - Ax
A -YpiXi
= (m tj )(t,x ).
We integrate the relation (34) by area Zt
№
d "dt
Rt+YrI + 2YbAK
t / J xt ^^ t t x
t
S (2BRR - BjRl )}dtdx =
t,j((*j) j
= 2{{{( ARt +YBR dxdt-
(35)
/
+ xjjj -(ARt + JBR )dxdt +
zt i
+£{{{(, Rxi,---, Rn )M (r , R^,..., Rn fdxdt.
Zt
Transforming the integral on the left-hand side of (35) by the Ostrogradsky-Gauss formula,
2 ii
rt
A
A
R +?& 1 + 2^BiRtRx
t J t S \
Rt + ÏK + 2^BtRtRx
dx -
i j i
+4f iE(B ( + K ) + 2ARtRXi )]
xi dx +
x - x t0
x — x 0
0 -}ds -
t ,j (t*j )
0
dxdt +
= 2iiil AR+lBiR
zt V t
+2{{{g -(AR, + ZBtRt )dxdt +
Z t
+siH(RtRx ,...,Rx )M(RtRx ,...,Rx )Tdxdt
7 1 n 1 n
(36)
where ds - surface element St, integration into St is produced on the outside of the surface St, determined by the direction of the normal to St:
n — J0, -
0
^, x x o —
0
=(!( - x")x
The integral over r0 on the left-hand side of (36) is equal to zero, since from (32) follows that Rt lr = 0, it remains to show
H ARx2 dx = 0
r0
jj AR2Xi dx = H AR2 dx1 ...dxn =
r0 f
= { + AR2xdxi
V r0
dxi... dx t—idx t+i... dxn.
Integrating by parts of the inner integral gives ¡AR2xdxt = ARRXi-¡R(ARXi) dxt, hence, by (32) obtain that
{{¿ARX. dx = 0.
r0 i=1
Let H = 10 + 28, where 5 any positive number, and y/ (x) e C 2(Rn) such function that 0 <y/ (x) < 1,
\ I 1, II x — x0 II— 10,
W (x) = < v ; [0, I Ix-x0II> t0 +5.
Instead of a function R(t, x ,s) consider the function R = R (t, x ,s) = R(t, x ,s)y/ (x). Obviously,
| R (t, x ,s)in Q ,
R (t, x,s) = •
0 0n Zstde.
Rewriting the equality (36) for function R = R(t,x,e) in the following way [9, 5-12]
'ii
- a(r + ~£r ;.) - 2-£b,r ,R x:
dx
+2iiif ar + tbr x
dxdt =
(37)
= -2ÎÎÎg 'I AR +TBR E
dxdt
i /
-fff(R, R v-, R xn )m (, R V-, R x„ fdxdt.
Zt
From the inequality Xcd <Sc2 + d2 / 5,5>0 and
i \ X i \x
±2g • ARt +YJBtRXi
V
< + 2
i /
V
AR +ZBR x
i
r
S. t
2
r
2
Z
Then from (37)
- a(r+ZR1 ) - I^bM x
dx< (38)
<
efff((, R v-, R n M (( , R V-, R n )Tdxdt-
+ÎÎÎ dxdt. Z 2
Quadratic form
w ((, rxi )=-a(r2+yrk ) - izbrâ
hence, taking into account the estimate (33)
/('Wife + TR2, )dx = O (2n+1). (42)
rt V i ) Then necessary to prove, that for R(t,x,e) with small s follows
JJJR2dxdt < CJJJR2dxdt, C = const > 0. (43)
Zt Zt
Considering the properties (32), for R in Qfollows
it A2 t
R2 = I jR2dt < t JR2dt.
V o y o Integrating it sequentially first on rt, then on t H = min j inf |a (t, x )|-2B (t, x ) ,inf|A (t, x )|-B j- , from t = const, ^ ^
is positive definite. We denote by
f i \
B = max sup B (t, x )|.
Then
/
W ((, RXi ) R2 + YrR2 . (39)
i /
Quadratic form (R, Rxi,..., R^) M (R, R^,..., R^ f is estimated from above in the following way [10, 181-191]
(R,RV-,Rxn )(R,Rv-,Rxn) ^
(40)
R2 +£r 2
<A
V i /
where
X = (n + 2)m, m = maxsuplm{j(t,x)l.
i ,j q ,
Introducing the notation
dx
J (t)=//(RR2+IR 2
T, V i
and taking into account (39), (40) from (38) t1
e/J (t )<elJ (t )dt + - JJJ"g2dxdt. (41)
0 2 zt
Applying the Bellman- Gronwall lemma to (41), we have
+
Hl^ 2dt <\s ffffifdtdx ds < — ffffifdtdx
Zt 0 | Zt J 2 Zt
From (42) using (43)
{{{R 2dtd,x = O (2n+1). (44)
Zt
From (42), (44)
{{{(Rt2+R2 + £Rx2 )dtdx = O (s2n+1). z, i
For rin q
II Rn (t,x^^(Q) = 1
f \2 = {{{(Rt2 + R2 + ^R2 )dtdx = O(s2n+1). V Q i )
Considering R _ Rn, for number n +1
II Rn+1 (t, x ,^)IU2(Q) = O (L5). (45)
Because
Rn = Rn+1 + (n+1 - V) = Rn+1 + ^ (Vn+1 + Wn+2), then from R2 <2(r2+1 + ((+1 -Vnf) using (45) followed the estimation
II Rn (t,x,e)l^(Q) = O(en+1)
J (t )<{{{g- dxdt ■ e »< 1 {№ dxdt ■ e » ,
2" ï e
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Q
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