This article was downloaded by: [University of California Santa Cruz]On: 18 November 2014, At: 18:10Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Applicable Analysis: An International JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gapa20

A first order asymptotic expansion of the solutionof a singularly perturbed problem for the telegraphequationsLuminiţ Barbu a & Gheorghe Moroşanu b

a ∗)Department of Mathematics , “Ovidius” University , Constanţa, România, RO–8700b ∗∗)Department of Mathematics , “A. I. Cuza” University , Iaşi, România, RO-6600Published online: 02 May 2007.

To cite this article: Luminiţ Barbu & Gheorghe Moroşanu (1999) A first order asymptotic expansion of the solution of asingularly perturbed problem for the telegraph equations, Applicable Analysis: An International Journal, 72:1-2, 111-125, DOI:10.1080/00036819908840732

To link to this article: http://dx.doi.org/10.1080/00036819908840732

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in thepublications on our platform. However, Taylor & Francis, our agents, and our licensors make no representationsor warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of the authors, and are not theviews of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor and Francis shall not be liable for any losses,actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoevercaused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Applicable Analysis. Vol. 72(1-Z), pp. 11 1-125 Repr~nts ava~lable directly from the Pubhsher Photocopying permitted by license only

Q 1999 OPA (Overseas Publ~shers Associat~on) N.V. Published by license under

the Gordon and Breach Science Publishers imprint.

Printed in Malaysia.

A First Order Asymptotic Expansion of the Solution of a Singularly Perturbed Problem for the Telegraph Equations Communicated by P.D. Panagiotopoulos

Luminita Barbu *) and Gheorghe Moroganu '*), *) "Ovidius" University, Department of Mathematics,

Bd. Mamaia 124, RO-8700 Constantja, Romhia **) "A. I. Cuza" University, Department of Mathematics,

Bd. Copou 11, RO-6600 Iagi, Rominia Abstract. The boundary value problem (S), (IC), (BC) below is considered, where c

is a positive small parameter. This problem is nonlinear because fo in (BC) is assumed to be a nonlinear function. We indicate a first order expansion of the solution (see

(1.1) below), including some correctora (boundary layer functions). In Section 2 of this paper we formally compute the correctors Vo, Vl and indicate the problems verified by UI and R (the remainder of order 1). Section 3 is devoted to the existence and higher order regularity of the solutions of the problems involved in our treatement. Finally, in Section 4, estimates of the remainder components are performed in order to validate our asymptotic expansion.

AMS: 35L50,35K20,47H05,34E05,34E15

KEY WORDS: Asymptotic expansion, boundary layer function, monotone operator, contraction semigroup.

(Received for Publication January 1999)

1. INTRODUCTION

Let us consider in the rectangle DT := ((2, t ) ; 0 < x < 1, 0 < t < T ) the system of the telegraph equations

eut t v , t ru = fl ,

Vt + U, + 9v = f2,

with which the following initial and boundary conditions are associated: Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

112 L. BARBU AND G. MORO2ANU

Here fi, fi : DT + R, uo, vo : [O, 11 + R, fo : D(f0) = R + IR, e : [0, T] + IR are given functions while r, g, ro, c are given positive constants. The number E is a positive small parameter. The above problem (S), (IC), (BC) will shortly be called (P,). Since fo is in general nonlinear, (P,) is a nonlinear problem.

Let us also remark that (BC) are boundary conditions of non-local type because of the presence of vt(l,t). For e > 0, (P,) is a hyperbolic problem, while (Po) is a parabolic problem.

In G. Moroganu & A. Perjan [9], L. Barbu [2], L. Barbu & G. Moroganu [3] the parabolic model (Po) has been validated by showing that the solution of (P,) approximates the solution of (Po), as e + 0, with respect to certain topologies. In those papers, a zero order expansion of the solution of (P,) is found, involving a corrector (or boundary layer function), due to the singular behaviour of the solution of (P,) in a neighbourhood of the segment {(x,O); 0 5 x 5 1).

In this paper, we indicate a first order expansion of the solution of (PC) :

where r = t/E;

UO = col(X(x,t), Y(x,t)) is the solution of (Po);

Ul = col(Z(x, t), W(x, t)) verifies a similar problem;

are boundary layer functions;

R = col(Rl(x, t, E), Ra(x, t, E)) is the remainder of order 1.

More precisely, in Section 2 of this paper, we formally compute the correctors V,, K and indicate the problems verified by Ul and R. Then, Section 3 is devoted to the existence and higher order regularity of the so- lutions of the problems under discussion. Of course, in order to get higher order regularity, some higher order compatibility conditions are required. Finally, in Section 4, estimations of the remainder components are per- formed to validate the asymptotic expansion (1.1). Clearly, our treatement could be adapted to higher order asymptotic expansions of U, but paying the price of some additional technical difficulties.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

THE TELEGRAPH EQUATIONS

2. THE FORMAL FIRST ORDER ASYMPTOTIC EXPANSION OF U,

The well known method of Vishik and Lyusternik (see Vasilieva, Bu- tuzov & Kalashev [lo]) leads us to consider for the solution Uc of the problem (P,) a first order expansion of the form (1.1), where the meaning of its terms are explained in Section 1. In this section we use heuristic arguments (assuming as much regularity of the involved functions as we need) to find out the terms of (1.1). So, we impose to Uc given by (1.1) to satisfy formally (PC) and then identify the coefficients of the same power of E . Of course, we have to distinguish between the coefficients depending on t and those depending on T = t/e.

First of all, we can write the equation Yl, = 0, with which we associate the condition Yl 4 0 for T + m , which means that Yl should exhibit a null effect far from the boundary layer (which is a neighbourhood of the segment {(x, 0); 0 5 x 5 1)). Therefore,

We also have the equation XI, + rXl = 0 and hence XI has the form

where the function cu will be determined later. As the components of Vl verify the system

21, + WI, + rZ1 = 0, w1, + XI, = 0,

we can easily see that

21 = P( x) exp(-rr) - ( l /r)dl(x)r exp(-rr), (2.4

where /? remains to be determined. To write (2.3) we have used the natural condition Wl (x, m ) = 0. On the other hand,

and continuing the heuristic identification procedure, we can write

and

( S 4

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

114 L. BARBU AND G. MOROqANU

Clearly, (2.5) and (2.6) can be equivalently rewritten as

and, respectively, = -(l/r)(Xt + Wz), (2.8)

Now, from (IC) we can write that Xl(s,O) + X(z,O) = uo(x) and this means that

a(,) = uo(.) + (l/r)M.) - f1(~,0)1. (2.9)

We can also write

(IC. W) W(x, 0) = -(l/r)af(x),

and Z(z, 0) + Zl(x, 0) = 0, which means that

In addition, we have

(IC. R) R ~ ( x , 0, e) = Rz(x, 0, E ) = 0.

Now, let us look at (BC). We first write the equation roX(0, t)+Y(O, t) = 0, which yields

Also, by identification, we obtain Xl(O, T ) = 0, i.e., a(0) = 0, which means (see (2.9))

vA(0) + ruo(0) = fl(0,O). (2.11)

Clearly, (2.11) is a compatibility condition. We can also deduce from (BC) that roZ(O, t ) + W(0, t ) = 0 and, consequently,

Another equation is roZl(0, T ) + WI(O, r ) = 0

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

THE TELEGRAPH EQUATIONS

which is satisfied if and only if

On the other hand, we have

which implies

Another equation given by the identification procedure is

that is a(1) + &(l) = 0.

We shall see that (2.13) and (2.15) are compatibility conditions. Now, a natural boundary condition for Ul is

which yields

Finally, we should have

3. EXISTENCE AND REGULARITY OF THE SOLUTIONS FOR PROBLEMS (PC), (S.Y) - (1C.Y) - (BC.Y),

(S.W) - (1C.W) - (BC.W)

In this section we formulate problem (P,) as an abstract Cauchy p r e blem in the space H := (L2(0, 1)12 x IR, which is a Hilbert space with the inner product (., a ) :

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

116 L. BARBU AND G. MOROQANU

for every x; = col(p;, q;, a;) E H, i = 1,2. Denote by 1) - 1 ) the corresponding Hilbertian norm. We define an operator B : D ( B ) c H + H, by

B(col(p, q, a ) ) = col(e-'q' + re-'p, p' + gq, -c - '~(1) t c - ' f o ( ~ ) ) ,

D ( B ) = { c o l ( ~ , q , a ) E H ; p,q E H1(O, I ) , rop(0) + d o ) = 0, a = q(1)).

Denote by Bo the B operator in the case fo r 0. Then problem (PC) can be expressed as the following Cauchy problem in H :

y:(t) + By,(t) = F,(t), 0 < t < T ye(0) = Yo,

We have the following proposition which we state without proof (for the proof we refer to G . Moroganu & A. Perjan [9], Prop. 3.2 and Prop. 3.4)

Proposition 3.1. Assume that

c, g, r , ro are positive constants; (3.2)

hold. Then, there exists a unique strong solution ye of problem (3.1) such that

Ye E W2'm(0 , T ; H ) and ( ( t , 6) = v(1, t , E ) for every t E [0, TI.

Moreover, if f o E w , " d d , " ( ~ ) , f ; > _ O ; (3.3)

2

f l ( . , t ) , f 2 ( . , t ) E W31m(0, T ; La(O, 1 ) ) n 0 w ~ - ~ - ~ - ~ ( o , 2'; ~ ~ ( 0 , I)), j=1

e ( t ) E W 3 f m ( ~ , T ) ; (3.4)

Yo, Y l o E D ( B ) , Y2o := F:(O) - Boylo - col(0,0, c-' f ~ ( v o ( l ) ) v t ( l , 0, E ) )

E D ( B ) , (3.5)

then 3

u, E n W ~ - ~ , ~ ( O , T ; H ~ ( o , I ) ) . k=O

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

THE TELEGRAPH EQUATIONS 117

Clearly, the last condition in (3.5) can be expressed in terms of initial data.

Now, we shall show that problem ( S . Y ) - ( B C . Y ) - ( 1 C . Y ) has a unique smooth solution. For this, we make the following change of function:

Then problem ( S . Y ) - ( B C . Y ) - ( I C . Y ) becomes

where

We shall regard problem (3.6)-(3.8) as a Cauchy problem in the Hilbert space X = L2(0, 1) x IR, with the scalar product

and the associated norm I l . l l x . We now define a nonlinear operator A : D ( A ) C X + X, by

A(c01 ( p , a ) ) = col (-r-lp1I + gp, r-'c-'~'(1) + c-'fo(a)),

D ( A ) = {col(p, a ) E X ; p E H2(0, I ) , a = p ( l ) , ropf(o) = rp(0)) .

Then problem (3.6) - (3.8) reduces to the following Cauchy problem in X :

where ~ ( t ) = c o l ( y ( . , t ) , t ( t ) ) , h ( t ) = c o l ( h l ( . , t ) , ~ - l h 2 ( t ) ) , D

ownl

oade

d by

[U

nive

rsity

of

Cal

ifor

nia

Sant

a C

ruz]

at 1

8:10

18

Nov

embe

r 20

14

118 L. BARBU AND G. MOROqANU

If c,g, r , ro are positive constants and fo : R + R is maximal monotone then A is maximal monotone (see G. Moroganu [lo]). Moreover, A is the subdifferential of the function iP : X + (-00, +m], defined by

where 10 : R + R is convex and continuous, 310 = fo, and

Denote by A. the A operator in the case fo 0.

We have the following proposition:

Proposition 3.2. Suppose that (3.2) holds,

h E W a f ( O , T ; X ) ;

zo € D(A) , wo := h(0) - AZO E D(A).

Then problem (3.9) has a unique strong solution,

z E C2([0, TI; X ) , and t ( t ) = y(1, t ) , for every t E [O, TI.

Moreover, i f fo E W Z ~ ) , f; 2 0; (3.10)

h E W4*2(0, T ; x); (3.11)

20, wo f D ( A ) , woi := x(0) - Aowo E D(A) , w02 := z ( 0 ) - Aowol E D(A),

where E(t) := hl ( t ) - col(0, S t ( t ) ) , S ( t ) := ~ - l f ~ ( ~ ( l , t ) ) , - h( t ) := h"(t) - col(0, P ( t ) ) , (3.12)

then

(Clearly, (3.12) can easily be expressed in terns of initial data.)

Sketch of the proof. Arguments estabishing the existence and reg- ularity of strong solution are readily adapted from results appearing in G. Moroganu & A. Perjan (9, Prop. 4.31. It is known that -Ao is the infinitesimal generator of a contractions semi- group T ( t ) , t 2 0. Then problem (3.9) has a unique strong solution z f W1pm(O, T ; X ) , z ( t ) f D(A0) for every 0 < t 5 T and therefore

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

THE TELEGRAPH EQUATIONS

&) = y ( l , t ) , 0 I t I T . The strong solution of (3.9) may be represented as

t z ( t ) = T( t ) zo + 4 T ( t - s )X(s)ds . (3.13)

We also have

where H ( s ) = col(hl(.,t), t (h2(t) - d ( t ) ) . Now, we consider the Cauchy problem

where w ( t ) = col(g(., t ) , f ( t ) ) , E(t) = ht( t) - col(0, ( l / c ) f ~ ( ~ ( 1 , t ) )c ( t ) ) . Reasoning as in G. Moropanu & A. Perjan [9, Prop. 4.31 we can deduce that the problem (3.15) has a unique strong solution w E W1*"(O, T ; X ) , w ( t ) E D ( A ) , O s t I T and

From (3.14) and (3.16) using Gronwall's lemma we obtain that z t ( t ) = w ( t ) , t E [0, TI. Hence z E W2@(0, T ; X ) , z t ( t ) E D ( A ) for 0 5 t 5 T and

Let us remark that (3.17) can be derived from (3.9) by a formal diffe- rentiation. Using similar arguments we can show that d3)(t) is the strong solution of the problem

and therefore z E W4t2(0, T ; X ) and yttt, E L2(DT). We can differentiate (3.18) and we obtain the Cauchy problem:

where m(t) = col(ml(t), g2 ( t ) ) . As the right hand side of (3.19)1 is a function of L2(0, T ; X ) problem (3.19) has a unique mild solution ii7 E C([O, TI; X ) (see N . U. Ahmed [ I ] , p. 168)

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

120 L. BARBU AND G. MOROqANU

and ~ ( t ) = dS)( t ) for every 0 5 t 5 T . Moreover, wl E L2(0, T ; H1(O, 1)). In conclusion yttttx E L2(DT).

Corollary 3.1. Assume that (3.2) and (3.10) hold. If

f 2 E W412(0, T ; L2(0, I ) ) , fi f W412(0, T ; H1(O, 1)) f i (0, .) E W512(0, T ) , e E W412(0, T ) (3.20)

and conditions (3.12) are satisfied, then the conclusions of Proposition 3.2 hold.

By the substitution

problem ( S : W ) , ( IC. W ) , (BC. W ) becomes:

where

We define the operator C ( t ) : D(A) C X + X ,

C(t)(col(p, a ) ) = ~ o l ( - r - ' ~ " + gp, r-'c-'pl(l) + c-'f;(Y(l, t ))a) .

It is obvious that problem (3.21) - (3.23) may be written as a Cauchy problem in X :

Remark 3.1. From Proposition 3.2 and (2.5) we get

Xt, E W2p2(0, T ; L2(0, 1)) and X(0 , t ) E W4t2(0, T ) .

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

THE TELEGRAPH EQUATIONS 121

Remark 3.2. From the compatibility conditions yo, ylo, y2o E D ( B ) and (3.12) we obtain the relations (2.11), (2.13), (2.15) and moreover, a0 E D ( A ) , l (0) - C(0)ao E D(A).

Reasoning as in Proposition 3.2., using Remarks 3.1. and 3.2. we have the following proposition:

Proposition 3.3. Assume that (3.2), (3.10), (3.11), (3.12) hold. Then problem (3.24) has a unique strong solution a E C2([0 , TI; X ) . Moreover, btt, E L 2 ( D ~ ) .

4. ESTIMATES OF THE REMAINDER COMPONENTS

In this section we shall formulate and prove the main result of this paper.

Theorem 4.1. Assume that (3.2), (3.4), (3.5), (3.10), (3.11)) (3.12) hold. Then, for each e > 0, problem (P,) has a unique strong solution Us of the form (1.1), where Uo is determined from (2.5) and (S.Y), ( IC .Y ) , (BC.Y) , Ul = col(a(x)exp(-rr),O) with a ( x ) given by (2.9)) Vo is deter- mined from (2.6) and (S .W) , (ZC.W), (BC.W) ,

with /3 given by (2.10) and R(x , t , c ) is determined from (S .R) , ( IC .R) , (BC.R) . Moreover, we have the following estimates:

)IRit(., ., e)I(c([o,q;~q0,1)) I IIRzt(., -, ~)llc([o,q;~2(o,i)) I M E ,

IIRls(., ., e)IIc([o,q;~~(o,l)) I Me, 1lR2z(., ., ~ ) ~ I C ( [ O , ~ ; L ~ ( O , ~ ) ) I where M is a positive depending on r , ro,c,g, T , f l , fi, e and fo.

Proof. For each e > 0, problem (S.R), ( IC .R) , ( 3 C . R ) has a unique smooth solution R. The function

satisfies the Cauchy problem

where L$) =

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

122 L. BARBU AND G. MOROqANU

e(t,e) := Y(1,t) + ~W(1 , t ) + eWl(1, r ) , D(Ec(t)) = D(3),

Ee(t)(col(p, q, a) ) := Bo(col(p, q, a)) + col(O,O, c-'(fo(a + e(t, s))-

-fo(W, 4 ) ) .

Let us multiply scalarly Eq, (4. l), by Re(t) and then integrate on 10, t] :

(1/2)IIRe(t)l12 + / t ( ~ e ~ ~ ) ~ e ( ~ ) , 0 ~e(s ) )ds = / t (~e ( s ) , 0 Q(s))~s. (4.2)

Because (E,(s) Re (s), R,(s)) 2 0 we obtain that

This implies that

(4.3)

On the other hand,

where K t , is a point between Y(l, t) and O(t,e). Therefore

5 ~~a~ + M7t2 exp(-2rs/e)(l + s ~ / E ~ ) .

Because ~:(1+ s/e)e-"l'ds = O(e) we obtain that

From (4.3) and (4.4) we get

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

THE TELEGRAPH EQUATIONS 123

where 1) . ) I o denotes the norm of L2(0, 1). Qc(t) := R:(t) is the strong solution of the Cauchy problem

where

uc(t) = c-l(fi(e(t, 6 ) +R2(l, t , ~))(R2t( l , t , 6 ) + Bt(t1 4) - fW, e))et(t, 4). Clearly,

On the other hand, l l ~ ~ ( o ) l l ~ 5 h a 2

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

124 L. BARBU AND G. MOROSANU

Clearly,

-eztt(., t)

(P1(.)r + r-la"'(.) - aN(.)7 + go!'(-)) exp(- r~)

c-'(-rp(1) + d'( l) /r - ratl(l))exp(-rr) + S.(t)e

where Sc : [O,T] + IR such that IS,(t)l < oo, for every 0 < s << 1 and O I t I T . Since

From (4.9) we have ((Qc(t)((2 5 Mlas2, that is

Because

from (4.2), (4.4) and (4.5) we obtain that

Analogously, from (4.8), (4.10) and (4.11) we get

Also, we have ll&(., t , a)& 5 Ma4~. (4.16)

Now, (4.5), (4.14), (4.15), (4.16) imply

%(x, t , s ) I %(I, t , e) + 211R2(., t, ~)llollR2=(.1t, 4 l l o I hfEnf4. Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014

THE TELEGRAPH EQUATIONS

REFERENCES

[I] N.U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman, Research Notes in Math. 246, Longman, Harlow, 1991, pp.143-175.

[2] L. Barbu, On a singularly perturbed problem for the telegraph equa- tions, Comm. Appl. Analysis, in print.

[3] L. Barbu and G. Moroganu, Asymptotic analysis of the telegraph equa- tions with non-local boundary value conditions, Pan American Math. J . 8(1998), no 4, pp. 13-22.

[4] H. Brkzis, Analyse Fonctionelle. Thkorie et Applications, Masson, Paris, 1983.

[5] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19(4) (1967), pp. 508-520.

[6] R. Luca and G. Moroganu, On a class of nonlinear hyperbolic systems, Memoriile Sechiilor Stiintifice ale Academiei Rom&ne, Seria IV, Tomul XIV, nr. 2 (1991), pp. 219-237.

[7] G. Moroganu, Nonlinear Evolution Equations and Applications, Reidel, Dordrecht, 1988.

[8] G. Moroganu, Parabolic problems associated to integrated circuits with time dependent sources, in vol. Differential Equations and Control Theory, Longman Scientific and Technical, Boston-London-Melbourne, 1991.

[9] G . Moroganu and A. Perjan, The singular limit of telegraph eequations, Comm. Appl. Nonlinear Anal. 5(1998), no. 1, pp. 91-106.

[lo] A.B. Vasilieva, V.I. Butuzov and L.V. Kalashev, The Boundary Func- tion Method for Singular Perturbation Problems, SIAM, Philadelphia, 1995.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 18:

10 1

8 N

ovem

ber

2014