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Fractional order Volterra equations with applications to elasticity
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
139, 448-464
APPLICATTONS
(1989)
Fractional Order Volterra Equations with Applications to Elasticity* U JIN CHOI Department
of Mathematics, Korea Institute Taejon, Choong Nam, Korea
of Technology,
AND
R. C. MACCAMY Department
of Mathematics, Carnegie Pittsburgh, Pennsylvania Submitted
Mellon lS.?I3
University,
by V. Lakshmikantham
Received October 13, 1987
A class of Volterra integro-differential equations with singular kernels is studied. It is chosen so that as the order of singularity approaches zero or one the equation approaches an analog of the equation for viscoelasticity or a Kelvin-Voigt solid. Qualitative results are obtained and estimates given for the effect of the order of singularity on errors in Galerkin’s method. The role of the quasi-static approximation is studied. 0 1989 Academic Press, Inc.
1. IN~R00ucT10N In the paper [2] we described some continuum mechanics situations in which one interpolates between models of different types with fractional order Volterra operators. The situation studied in detail in [2] was an interpolation between the usual heat flow model (a parabolic problem) and the Gurtin-Pipkin heat flow model (a problem of hyperbolic type) [6]. We abstracted this situation to obtain a class of singular Volterra problems in Hilbert space. We then investigated the effect of the order of singularity on the time smoothness of error estimates for Galerkin approximations. In the present paper we give a parallel analysis for a model interpolating between a viscoelastic solid and a Kelvin-Voigt solid. The work of [2] and the present paper was stimulated by two sets of * This work DMS 8601288.
was supported
by the National
448 0022-247X/89 Copyright All rights
$3.00
0 1989 by Academic Press, Inc. of reproduction in any form reserved.
Science Foundation
under
Grant
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ideas. The first is the recent work of Hrusa and Renardy [7, 111. They studied a class of viscoelastic models containing a singular kernel and discovered that increasing the order of singularity had a smoothing effect on solutions. The second idea comes from the work of Douglas and DuPont on parabolic problems [S]. They found that one could obtain symmetric error estimates for Galerkin approximation provided one uses fractional order Sobolev norms with respect to time. Let us describe the elastic situation. We are concerned with the longitudinal motion of an elastic bar of uniform cross section, length L, and fixed ends. If p, U, C, and b are density, displacement, stress, and body force, the dynamic equation is P,,(x,
t) = CJ,(X, t) + W, t),
(1.1)
u(O,t)=u(L,t)=0,O0.
It is assumed that the bar is at rest for t < 0. Two common linear theories relating a(x, t) to the strain u,(x, t) (for an initial rest state) are a(x, t) = &[u:(x,
.)] = Eu,(x,
t) - 1; a(t - z) u,(x, 5) dz
(viscoelasticity ) fJ(x, t) = F, [24:(x, .)] = F&(X, t) + h,,(x,
t)
(Kelvin-Voigt
(1.2)
).
Here E, F, ;1 are positive constants and a is a positive decreasing function often taken as a positive sum of exponentials. Substitution of one of Eqs. (1.2) into (1.1) yields an evolution equation problem. For & one has finite propagation speed and for 9i infinite propagation speed. For both one has asymptotic stability. We define our interpolation model by the formula
where A > 0 and r is the gamma function. Formally, when a -+ 0, $$ tends to Y0 with E=A+b(O) and a(t)= -b(t) and when a-l,Sfm tends to s1 with F= A and A = b(0). Substitution of (1.3) into (1.1) yields a problem with infinite propagation speed and asymptotic stability, provided suitable hyptheses are placed on 6. We now abstract this situation. We start with the familiar differential equation setting of three Hilbert spaces H, c H, t HP i with H_ I the dual of H, with respect to H, ((A, u) = (h, u)* for h E H,,). Let & denote the set
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of bounded, symmetric linear maps from H, to H- , and let d + denote the subspace of coercive maps. We assume that we are given A e d + and a collection of mappings B, : [0, cc ) + d, 0 < a < 1. We define operators LB,, by
and consider the problems D2u+Au+DLBd[u]=f;
t>O, u(O)=Du(O)=O,
(Pa)
where D denotes the time derivative. The idea is that the operators B, are to be “of order tMa” near t = 0 in such a way that as a + 0 or 1 we obtain, formally, D2u+Mu+Lso[u]=f, D’u+Nu+Dl[u]=f,
u(0) = Du(0) = 0, B. “regular” at t = 0
(PO)
u(O)=Du(O)=O,
(PI)
with M, N, and I in d+. Our conditions on B, are described in the next section. There are two conditions, both stated in terms of the Laplace transform fi, of B,. The first is familiar in the study of Volterra equations [lo]. It yields dissipativity of L,Bmand produces asymptotic stability. The second involves the behavior of B,(s) for large s and abstracts the order t-” singularity. It is this second condition which controls the smoothness, both in our work and in [7, 11). In order to put the problems (1.1) and (1.3) into our general setting one makes the following identifications:
H, = H:(O, L),
Ho =
L2(0,
L),
H-, =H-1(&L)
(4~ 0) = j; cp’(x) u’(x) dx
In a similar way one can include general three dimensional problems (see [S] for the viscoelastic and Kelvin-Voigt cases). A common engineering procedure for Eq. (1.1) is called the quasi-static approximation in which one assumes that u varies slowly with time and
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sets D*u = 0 (see [S]). We consider this approximation study the problem Au + DL,Ju]
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for (P,), that is, we
=f
(A,)
with no initial condition. Under our hypothesis on B,, (P,) and (Q,) will have unique generalized solutions and one can obtain stability results as follows. For a map 4: [0, co) -+ X, where X is any Hilbert space we define, for any y, (I.61
where I(s) is the Laplace transform of 4 and 8 is the complexification of X. For our solutions of (P,) we will be concerned with norms of the form
IIu II;,a = IIu IIL(O,cc:Ho)+ II2.d IIffqo,a :H,)3
(1.7)
where y and S will depend on a. A main goal is the analysis of Galerkin approximations I!?‘, in a sequence of finite dimensional spaces Sh, to both (P,) and (QJ. We obtain symmetric error estimates for u - Uh in the fractional order norms (1.7). These depend on CI but are analogous to the ones in [3] for parabolic problems. It is well known for elliptic problems that a straightforward error analysis does not give the best convergence rates in weak norms like L,. This difficulty is overcome in elliptic theory by what is known as Nit&e’s trick. It is shown by Mary Wheeler [ 123 that a similar improvement can be achieved in parabolic theory by making use of the steady state theory to define what is sometimes called the elliptic projection. In Section 4 we present an extension to (P,) by using our approximate equation (Q,) to define what we call a quasi-static projection. We want to make some remarks on the Renardy-Hrusa model. For the elasticity case they use, instead of (1.3), the formula
4% t) = q&,
.)I
= Au,(x, t) + J6r 7
(u,(x, t -
7)
- u,(x, t))
d7,
(1.8)
where 0 < y < 2. (Notice that the normalizing factor r( 1 - y) does not appear. Recall we are assuming u(x, t) E 0 for t < 0.) When y --+0, (1.8) tends to the viscoelastic model. For 0 < y < 1, (1.8) produces finite wave speed but the speed tends to infinity as y --* 1. For 1 d y < 2, (1.8) produces infinite propagation speed and is similar to our
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model (1.3) except that it does not have a limit as y --f 2. We have not as yet completed an error analysis for Galerkin’s method as it applies to (1.8). We observe that passage to the limit tl=O in our model (1.3) is quite complicated. Although the equation formally tends to that for viscoelasticity the convergence of solutions for tx > 0 to those for CI= 0 has some type of non-uniformity in time which we do not yet completely understand. This is reflected in our error estimates in Section 2 which degenerate as CI+ 0. It is also reflected in the fact that at ct - 0 the wave speed suddenly drops from infinity to a finite value. Our model (1.3) is, on the other hand, very well behaved with respect to the passage to the limit CI= 1, that is, Kelvin-Voigt. In contrast (1.8) gives problems which are well behaved on passage to the viscoelastic limit y = 0 but do not yield KelvinVoigt materials as y + 2.
2. STATEMENT OF RESULTS We begin with our conditions on B,. We assume B, E L,(O, co : &). Then B, has a Laplace transform b,(s). B, exists and is continuous in Re s>O and is analytic for Re s > 0 as a bounded linear map from w, to w-i. Here and in what follows we denote the complexilication of any space X by x Our conditions are: (B.l)
For any N>O there exists a a(N)>0
(Re~,(irt)u,u)~6(N)I(uJI: (B.2) Re ~20.
forany
B,(s) = soL- ’ B,+s*-‘B,(s),
such that
u~Hi B,,E~+,
and B,(s)=o(l)
VE[-N,N]. as IsI + 0~) in
In Section 5 we indicate how to generate B, satisfying our conditions and the connection to behavior of B,(t) at t = 0. We define L, by the formula LBz[u](t)
= I,’ B,(t - T) u(t) dt.
(2.1)
L, takes values in A-, . The transform of DL,[u] is sj,(s) a(s). The regularity of DL,[u] in the fractional order spaces is then controlled by (B.2) for s = iv; that is, one has an estimate of the form II s&(s) fi(s)ll~-~ < 42 + Irl 12Y’2IIC(itl)llp, This yields the following result.
for some c.
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2.1. DLBX maps H*(O, 00 : H,) into L,(O, 00: H-_,).
On the basis of Proposition 2.1 we can define generalized solutions (I’,) and (Q,). We introduce the following hypothesis on f: fEL,(O,
co:H-,)nL,(O,
co:K,).
DEFINITION 2.1. u is a solution of (P,), u E H”(0, cc : H,) n H’(O, cc : HP 1) and
D*U+Au+DL&]=f
on
(J")
for f satisfying
t > 0, u(0) = Du(0) = 0.
2.2. u is a solution of (Q,), for f satisfying co: H,) and on t>O. Au + D&&4] =f
DEFINITION
u~H”(0,
Remark 2.1. Equations (2.2) and (2.3) make sense L,(O, 00 : ZL,). For (2.2) we have UEL,(O, cc : H,) while in &(O, 00 : H-,). It follows, by interpolation, that continuous as maps from [O, co) to [H,, HP 1] 1,2= H,, conditions in (2.2) make sense. Remark 2.2.
of
if
(F’),
(2.2)
(F’),
if
(2.3)
as equations on DU and D2u are u and Du are hence the initial
For (I’,) we will strengthen (F’) to
fe
L1(O,
00: H,)
n L,(O, 00 : H,)
(4
and require of the solution of (2.2) that u E H2(0, co : H,). In this case (2.2) has the variational formulation
(D*u, u)o+ (Au, u> + @L&4
0) = (f, do
for any u E L,(O, co : H,), u(O) = 0, Du(0) = 0. Similarly
(2.4)
(2.3) has the formulation
0) = (f, u>
for any
u E L,(O, cc : H,).
(2.5)
We are concerned with Galerkin approximation. We suppose that we have a family of finite dimensional suspaces Sh = (&, .... dh,,) of H, , 0 < h, with the following approximation property: Given any E there exists h,,(s) such that for any u E H, and any h
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We can now state our results. THEOREM
2.1.
(i) Forfsatisfying
(I;), (P,) has a unique solution u with
UEH2(0, co:H*).
(ii)
There is a constant c, such that )(u (I2,Ni c, )If I(L2C,,, co:H,,j.
THEOREM
(ii)
2.2. (i) For
f satisfying (F), (Q,) has u unique solution u.
There is a constunt c, such that
THEOREM
2.3. There exist unique Gulerkin approximations uh to (P,)
and HA). THEOREM 2.4. There exists a constant c,, independent of h such that if u is the solution of (P,) and uh is its Gulerkin approximation,
0u - UhII1,(.x/2) d cmIIu - WhIII,(a/Z) for any
wh: CO,~0) --t Sh, IIwhII,,(a,*)
<
~0.
(2.6)
THEOREM 2.5. There exists a constant c, independentof h and a such that if u is the solution of (Qb) and uh is its Gulerkin approximation and y is any number.
II24- UhIIHY(O, m: If,) Gc IIu for
whIIffqo,co: HI) any wh: lW4+Sh,
Il~hll~2~o,s,:n,~<~. (2.7)
Remark 2.3. We will see that as a + 0 the constants c, + 00 so that our estimates degenerate. Remark 2.4. The estimate (2.6) is called symmetric since the same norms appear on both sides. This notation was introduced in [3] for parabolic problems. We emphasize that our error estimates involve only smoothness with respect to time: they will all be in the norms Hk. Let us consider again our example ( 1.1 ), (1.4). Let us choose for Sk the standard piecewise linear finite element space of mesh length h. It is known then that the following approximation results hold: there exists a constant c independent of h such that if u E H,(O, 1) n @(O, 1) then there exists a wh E Sh such that [4] IIu - WhIIH,(o,1)G ch IIu IIjiz(o,1,;
II~-‘+‘~ll~~~o.~~~~h* ll~llh~~o,~~~(2.8)
The estimate (2.6) can then be used to show that if u is a solution of (l.l),
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(1.4) and U” is its Galerkin approximate then, provided the solution u is smooth enough, )(u - uh (1H~t0,m: H0) and (1ZJ- uh 11P~~C0,m: H,J will be order h. There is a defect in the previous estimate. From (2.8) one would conjecture that, in II . II L2(0,m: Ho), u - uh should be order h*. In elliptic differential equation theory this defect is overcome with what is known as N&he’s trick [4]. For parabolic equations it was overcome in [2] by using the idea of elliptic projection. In Section 4 we provide an analog for our problem using (QE) to develop what we call a quasi-static projection.
3.
PROOFS
In this section we indicate the proofs of Theorems 2.1-2.5. Some of these will be quite sketchy and we refer the reader to [t] for further details. The strategy is to use Laplace transforms. We formally transform (2.2) and (2.3) to obstain s*ii + Ati + s&s)
ti =f,
(3.1)
Aii + s&(s) t =“f
(3.2)
The idea is to solve (3.1) and (3.2) and then define u by the inversion formula u(t)= (277-l /I,”
e’%(iq)
dq
(3.3)
and show that this indeed gives solutions of (P,) and (Q,). To be a little more precise what we do is to transform (2.4) and (2.5) to obtain the variational problems: (s%, q. + (A& 5) + (d,(s) (A,
5) + (s&(s),
i&6) = (j: 8),
for all
C E t?,
(3.1)’
6, 6) = (i: IT)
for all
fi E i?, .
(3.2)’
(Recall we are taking f(t)
and D’u(t)
in Ho for (P,).)
LEMMA 3.1. Equations any s with Re s 2 0.
(3.1)’ and (3.2)’ have a unique solution i(s) for
ProojI From (F), p(s) is defined as an element of e. for any s in Re s > 0. We observe first that the fact A is coercive and B, is continuous in Re s > 0 means that we can solve (3.1)’ and (3.2)’ uniquely by successive 409/139:2-II
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approximations for s small. Hence we consider these equations for 1s ( > E for some E> 0. We divide (3.1)’ by s and consider the bilinear form
P(s: ti, t?)o= (SC,u)(J+ f ii, B + (B,(s) 22,6). ( > We assert that for any s in Re s > 0, JsJ 2 E, for E sufficiently small, there is a 8(s) such that Re P(s, 6, r?) 2 b(s)(J ti )I$, .
(3.5)
If (3.5) is established then the existence of a solution of (3.1)’ follows from the Lax-Milgran lemma. To prove (3.5) we observe that for any q, (B.l) yields, for some 6,’
Moreover given any B,,, 0 < 8, < n/2, we see from (3.3) and the coercivity of A that for E sufficiently small (3.4) holds on s = seie, - 8, < 6 < t$. Then (3.4) follows easily from the maximum principle. The argument for (3.2)’ is essentially the same. The following result is quite easy and is proved in [i 1. LEMMA 3.2. For both (3.1)’ and (3.2)‘, C(s) is continuous in Re s>,O, analytic in Re s > 0, and real for s real. LEMMA
(l+v*)*
3.3.
There exists a constant c, such that
II~(irl)ll&o+(l +v*Y Il(i~Nl~,~c, IIPCi~~lI~o (1 + v’)* IIfi(irl)ll B, < c, IIf(iq>ll Be,
for
(3.1)’
for
(3.2)‘.
ProoJ: Once again a successive approximation argument, with the coercivity of A, shows that for some E sufficiently small one has, for (3.1), Ilfi(iq)llR,
&WI ~(irl)ll$, < =Re
(
iv llirll&--~
(Ati(
ri(iq))+
= - Re f @(iv), fi(iq)), = Re(](iq),
’ This statement
uses the symmetry
of B,(f)
(and
(fi,(iq)J(iq), a(@)),
hence of h,(iq)).
r(c)>)
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457
Hence the conclusion for (3.1)’ holds for (n ( < N. The crucial calculation involves the behavior at infinity. We define y, and pt, by y, = sin : c( y,=cos;a, and note that for q > 0, (iv)“= q >o,
(3.6)
~“(y, + ip,). We have then from (3.2)‘, for
We take the imaginary part of this equation. Using the coercivity of B, and the fact that B,(s) is small when s is large we deduce that for q sufficiently large we have
for some constant c’. Next we take the real part and obtain, from (3.7) an estimate of the form,
Hence we have established the conclusion for (3.1)‘. For (3.2)’ we note that for large q we can calculate as for (3.7) and obtain
This yields the result for q large and for q on bounded sets the result is immediate. With Lemmas 3.1-3.3 in hand the proofs of Theorems 2.1 and 2.2 are fairly straightforward. One defines ZJ by the formula (3.3). One can then show that this function will satisfy (2.4) or (2.5) together with the appropriate initial conditions for (2.4). The details appear in [ 11. The estimates in Theorems 2.1 and 2.2 are an immediate consequence of (1.6), (1.7), and Lemma 3.3. The proof of Theorem 2.3 can be obtained once again by Laplace transforms. One transforms the variational formulations and obtains finite dimensional versions of (3.1)’ and (3.2)‘. The same argument as before (on the finite dimensional spaces) will show that these have solutions and if uh
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is defined by the inversion integral one obtains the Galerkin approximations. We turn now to the optimality results (2.6) and (2.7). Once again we work in the transform domain. Let u be the solution of (p,) and uh its Galerkin approximation with transforms 6, lib. For an arbitrary element 6h in Sh set E^h= ih - ah, ;h=c-$h. We put 6 = 17~~3;~ in (3.1)’ and subtract the result from the corresponding equation for tih. The result is (s*&h, Bh), + (ah,
eh> + s@,(s) ih, fib)
= (Ah, tih)() + (‘4Ch, fTh) + s(8,(s) ih, oh).
(3.8)
We put Ch= Zh and s = iv in (3.8) and estimate, as before for large (q I. The imaginary part of the left side is bounded below by C,U~)q 1a 1)lh(iq)ll 5,. On the other hand the imaginary part of the right side is bounded above by a constant time IJ* (Ith ((R,, (I.? [I~~ + I rl I’ IICh[lA, lI‘ZhIIR,. We get a similar estimate by considering the real part and arrive at the two inequalities
lrlla lle”llB,~~ {q’ Il@hII&IIE^hlI&+l?la l18hlIR,IIWIR,) Id2 Ilb”ll~occ’:{~* Ilthll& IIE^hIIHo+I?(” lPhIIR, lIE^hIIRI+IrllO’II~hII!-?J. (3.9) Put
~‘lrlP2 IIE^hllR,9 A= Id=‘* llehllR,, b= I?1lI~hll~~~ B= Irll lI~hllfi~. Then (3.9) becomes
b*
of the form
.* + b2 < $ (A2 + P). a that is,
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One can obtain estimates on finite 4 intervals as before. Then if we note that, in any norm, Jjf-iihl) 6 (I$- ti’j) + (1Gh - till we arrive at the estimate
Translation back to the time domain yields the estimate (2.6). For (Q,) Eq. (3.8) is replaced by (A.?‘, 13~) + s{&(s)
Zh, @) = (A@, Ch) -t s@,(s) gh, v^“>
for any Bh. We estimate for s = iv, q large. Divide the equation by s’, put s = iv, and I?” = @(iq). This yields an estimate of the form )I@iq)ll~, 6 c I( g(iq) (InI with a constant c which is independent of tl. A similar estimate holds for bounded q. Again the estimate 11u - uh 11< 11u - wh I( + II wh - uh (I yields (2.7). 4. THE QUASI-STATIC
PROJECTION
Let u be the solution of (I’,). We make the following definition. DEFINITION 4.1. The quasi-static Uh : [0, co) -+ Sh such that
(AL@ i- DL,[
projection
Uh], uh) = (Au + DLB,[u],
u”)
of u is
the function
for any uh E L,(O, co : S”).
The existence and uniqueness of Uh can be established in the same way as the Galerkin approximation for (pa). Further one obtains an estimate of the form
II UhIIW(0.cc: H,)Gc II2.4 IIW(0,32:HI)
(4.1)
for any y. THEOREM
4.1.
There exists a constant independent of h such that
IIUh-
Proof. An easy computation the formula (D2(Uh-uh),
uh),,+
llu- ~hIlH2(o,oC:HO).
Uhl12,01eC,
using the definitions
(A(Uh-u’),
= (P( Uh - u), Vh)Ho
of uh and Uh yields
uh) + (DLBz[Uh-u’],
for any
vh E sh.
(4.2)
vh) (4.3)
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Then one can go to the transform plane and argue as in the proof of Theorem 2.2 to estimate 6” - tih in terms of oh - 0. Since only the zero norm appears on the right one obtains the formula (4.2) in the time domain. Formula (4.2) is the basis of our version of Nitche’s trick as we describe now. We begin by defining a new function 8 as the solution of the problem (AC + d,(s)
z?,(8) = (ti - Oh, O),
for any
v^ERi.
(4.4)
Because of the symmetry of A and B,, (4.4) is the same as the problem with d and 6 interchanged on the left and one can argue as in Theorem 2.1 that it has a unique solution. Now put B = ti - oh in (4.4) and obtain Illi - 0” I( j& = ((A + s&(s))@ - Oh), 4). By construction
(4.5)
we have
((A +sS,(s))(ti-
Oh), Gh)=O
for any
fi” E gh
and if we subtract this from (4.5) we obtain I)a- z7hII~o= ((A +sB,(s))(tiThen condition
IIa-~hll$@
Oh), &Bh).
(B.2) yields
IvIa IIa-fihII
dR, Il&mIA,
for any
@E 3”
(4.6)
for some constant c. Now observe that, for any Oh,Gh E Sh, ((A + sB,(s))[
Oh - Gh], fib> = ((A + sS,(s))[ti
- Gh], uh>.
(4.7)
Choose fib = 0” - Gh and obtain, for any tih E $h, ((A+sB,(s))[oh-~h],
irh--h)=((A+S~‘,(S))[U--h],
Oh-tih).
The arguments used before yield
II Oh(Q) - ~“(~vlh, G c IIG(h) - ~h(mla, and thus,
IIw?)--hG4QllA,~C
IIw+~h~~rl)llAl.
(4.8)
We substitute (4.8) into (4.6) and obtain
Illi(i~)-~h(i~)ll~~~cl~la
II~(irl)-~h(~rlNl~, lI$(~r)-~hlI~
(4.9)
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for any tih and 6’ in 5”. Finally translating yields IIU-
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back to the time domain
Uhl12L2(0.,:Ho)~C llU-whllH’l(O,df,)
ll(b-~hI/mx3:H,),
(4.9) (4.10)
for any vh and wh in &(O, cc : sh). Formula (4.10) is the form of Nitsche’s trick we want. It estimates u - iJh in the H, space in terms of u - wh and 4 - vh in the H, norm. In concrete examples this can be used to remedy the defect indicated in Section 2. We illustrate with our example (1.1 ), (1.4). For that example we have (Au, v ) = A loL u’v’ dx,
<&(t)
b,(r)=b(t)/r(l
= b,(t) loL u’u’ dx,
Thus the problem
w v>
-lx) P.
(4.4) becomes loL (A +6,(s)
6’8 dx) = joL p(‘(a-)
dx.
This says 6 is a weak solution of the problem
-@(A+~,(s))=z+ Elliptic
Oh,
q?(O) = J(L) = 0.
theory states that 6 satisfies an estimate of the form (4.11)
II&Q)IIA2(0.L)~C Illi-- UhlIE*(O,L). From (2.8) one can then infer that there is a Sh such that (I&fih(I1
l)ti-OhI(jJo
or, in the time domain,
II4 - ohIIL2(0,co: H,)d 4 IIl4- UhIILI(O,cc: Ho). We insert this estimate into (4.10) and obtain
IIu - UhIIho, a,: Ho)f c IIu - WhIIW(0,m: H,)h II24- UhIIL*(O,
co : Ho)
or II u - uh II L*(O, co: Ho)G 4 II24- whIIffyo,00:H,)’
(4.12)
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CHOIANDMACCAMY
We can use (2.8) again to estimate the term on the right side of (4.12). By the first of inequalities (2.8) we have )I u - wh ))HaC,,m:H,j d C/J II u II Hb(O,oo:Hzj and hence (4.12) yields IIu - Uh IIL2(0,co: Ho)6 A2 IIu - whIInyo, a3:“2)’
The same argument can be used to show that
But then (4.2) yields II u - Uh II Lz(O, a!: Ho)G IIu - cJhIIL2(0, cc: Ho)+ II uh - UhIIL2(0, m: Ho) < (1 + c,) 11 u - uh )I~(0, m: t,,,) G ch* IIu IIH2+ a(o,m: m
which is an order h* estimate in the Ho norm. The term 1)u )I H~+ a(o,oo:Hzj can be estimated in terms of the data.
5. SINGULAR
KERNELS
In this section we indicate how to generate classes of kernels B, satisfying our conditions (B.l) and (B.2). We begin with what we term product kernels which are illustrated by our elasticity example. Suppose M is a fixed operator from H, to H-, which is symmetric and coercive. For example, take HI = H,(O,L) and H-, = H-,(0, L) as in Section 1 and define M by (Mu, v) = joLu’v’ dx.
Now define B,[t] by B,[t] = b,(t) M, where b, is a scalar function which is in L,(O, co)nL,(O, co). Then (B.l) translates into conditions on 6,: for all q
Re b,(iq) > 0 &)=s
‘-=bo+s’-oL61(s), b,>O,
and
(5.1)
6,(s)=u(l)as
(s( *co.
(5.2)
We recall some facts from [9, 10 J. For a E &(O, co) one has (-l)ku(k)(t)>O,
k=O, 1 * -~Irn&i~)>O
(- I)h a(“)(t) > 0, h = 0, 1, 2 => Re 6(iq) > 0
for for
n#O all q.
(5.3)
FRACTIONAL
463
ORDER VOLTERRA EQUATIONS
We generate a class of kernels b, by taking 1 r(1 -a)
a(z)dz+me-‘,
b,(t) = ____
where a E L,(O, oo), (- 1)“ 8’) (t)>O,k=O, that
(5.4)
1,2, and m>O. We observe
6,(s)= (s+s),-A~)+~.
(5.5)
For s = iq we have ((iv) + 1 )‘- ’ = (1 + q2)(a’2)- ’ (cos 6, - i sin %,),
0 < 6, < ?c/2.
Thus (sin 6, Re 6(iq) - cos 6, dn &ill)) + -!?1 +q2’
Re 6,(iq) =
Under our hypothesis. (5.3) shows that the right side is bounded below on any compact set 1q ( d N and (B.1) is satisfied. If we assume that UE Cc2’(0, co) and that &‘EL,(O, co), j=O, 1, 2, then one has 6(s) = (a(O)/s) + (a(0)/s2) + o(l/s2) as )s) --, 00 and one verities that (B.2) is also satisfied. One verifies easily that b,(t)-f$?
near t = 0,
confirming that (B.2) represents singular behavior at t = 0. The above idea can be extended to general kernels. Let A(t) be a family of mappings from HI to H- 1 with ( - l)k Ack’( t) coercive for k = 0, 1, 2 and let M be coercive. Then if one defines B, by B,(t)=--
___
A(z) do + Me-’
one will have a family satisfying (B.l) and (B.2).
REFERENCES 1. U JIN CHOI, “Fractional Order Volterra Equations in Hilbert Spaces,” Thesis, Dept. of Mathematics, Carnegie Mellon University, July, 1987.
464
CHOI AND MACCAMY
2. U JIN CHOI AND R. C. MACCAMY, Fractional order Volterra equations, in “Proceedings, Conf. on Volterra Equations, Trento, Italy, February, 1987,” to appear. 3. J. DOUGLAS AND T. DUPONT, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7, No. 4 (1970), 575-626. 4. G. FIX AND F. STRANG, “An Analysis of the Finite Element Method.” Prentice-Hall, Englewood Cliffs, NJ, 1973. 5. A. C. ERENGEN,“Mechanics of Continua,” Wiley, New York, 1967. 6. M. E. GURTIN AND A. C. PIPKIN, A general theory of heat condution with finite wave speed, Arch. Rational Mech. Anal. 31 (1968), 113-126. 7. W. J. HRUSA AND H. RENARDY, On wave propagation in linear viscoelasticity, Quart. Appl. Math. 48, No. 2 (1985), 237-253. 8. R. C. KOELLER, Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51 (1984), 299-307. 9. R. C. MACCAMY, Nonlinear Volterra equations on Hilbert space, J. D@firential Equations 10 (1974), 373-398. 10. R. C. MACCAMY AND J. S. W. WONG, Stability theorems for some functional equations, Trans. Amer. Math. Sot. 164 (1972), l-37. 11. M. ~NARDY, Some remarks on the propagation and non-propagation of discontinuties in linearly viscoelastic liquids, Rheology Acta 21 (1982), 251-254. 12. M. F. WHEELER, A priori L, estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10, No. 5 (1973), 723-759.
OF MATHEMATICAL
ANALYSIS
AND
139, 448-464
APPLICATTONS
(1989)
Fractional Order Volterra Equations with Applications to Elasticity* U JIN CHOI Department
of Mathematics, Korea Institute Taejon, Choong Nam, Korea
of Technology,
AND
R. C. MACCAMY Department
of Mathematics, Carnegie Pittsburgh, Pennsylvania Submitted
Mellon lS.?I3
University,
by V. Lakshmikantham
Received October 13, 1987
A class of Volterra integro-differential equations with singular kernels is studied. It is chosen so that as the order of singularity approaches zero or one the equation approaches an analog of the equation for viscoelasticity or a Kelvin-Voigt solid. Qualitative results are obtained and estimates given for the effect of the order of singularity on errors in Galerkin’s method. The role of the quasi-static approximation is studied. 0 1989 Academic Press, Inc.
1. IN~R00ucT10N In the paper [2] we described some continuum mechanics situations in which one interpolates between models of different types with fractional order Volterra operators. The situation studied in detail in [2] was an interpolation between the usual heat flow model (a parabolic problem) and the Gurtin-Pipkin heat flow model (a problem of hyperbolic type) [6]. We abstracted this situation to obtain a class of singular Volterra problems in Hilbert space. We then investigated the effect of the order of singularity on the time smoothness of error estimates for Galerkin approximations. In the present paper we give a parallel analysis for a model interpolating between a viscoelastic solid and a Kelvin-Voigt solid. The work of [2] and the present paper was stimulated by two sets of * This work DMS 8601288.
was supported
by the National
448 0022-247X/89 Copyright All rights
$3.00
0 1989 by Academic Press, Inc. of reproduction in any form reserved.
Science Foundation
under
Grant
FRACTIONAL
ORDER
VOLTERRA
449
EQUATIONS
ideas. The first is the recent work of Hrusa and Renardy [7, 111. They studied a class of viscoelastic models containing a singular kernel and discovered that increasing the order of singularity had a smoothing effect on solutions. The second idea comes from the work of Douglas and DuPont on parabolic problems [S]. They found that one could obtain symmetric error estimates for Galerkin approximation provided one uses fractional order Sobolev norms with respect to time. Let us describe the elastic situation. We are concerned with the longitudinal motion of an elastic bar of uniform cross section, length L, and fixed ends. If p, U, C, and b are density, displacement, stress, and body force, the dynamic equation is P,,(x,
t) = CJ,(X, t) + W, t),
(1.1)
u(O,t)=u(L,t)=0,O
It is assumed that the bar is at rest for t < 0. Two common linear theories relating a(x, t) to the strain u,(x, t) (for an initial rest state) are a(x, t) = &[u:(x,
.)] = Eu,(x,
t) - 1; a(t - z) u,(x, 5) dz
(viscoelasticity ) fJ(x, t) = F, [24:(x, .)] = F&(X, t) + h,,(x,
t)
(Kelvin-Voigt
(1.2)
).
Here E, F, ;1 are positive constants and a is a positive decreasing function often taken as a positive sum of exponentials. Substitution of one of Eqs. (1.2) into (1.1) yields an evolution equation problem. For & one has finite propagation speed and for 9i infinite propagation speed. For both one has asymptotic stability. We define our interpolation model by the formula
where A > 0 and r is the gamma function. Formally, when a -+ 0, $$ tends to Y0 with E=A+b(O) and a(t)= -b(t) and when a-l,Sfm tends to s1 with F= A and A = b(0). Substitution of (1.3) into (1.1) yields a problem with infinite propagation speed and asymptotic stability, provided suitable hyptheses are placed on 6. We now abstract this situation. We start with the familiar differential equation setting of three Hilbert spaces H, c H, t HP i with H_ I the dual of H, with respect to H, ((A, u) = (h, u)* for h E H,,). Let & denote the set
450
CHOI
AND
MACCAMY
of bounded, symmetric linear maps from H, to H- , and let d + denote the subspace of coercive maps. We assume that we are given A e d + and a collection of mappings B, : [0, cc ) + d, 0 < a < 1. We define operators LB,, by
and consider the problems D2u+Au+DLBd[u]=f;
t>O, u(O)=Du(O)=O,
(Pa)
where D denotes the time derivative. The idea is that the operators B, are to be “of order tMa” near t = 0 in such a way that as a + 0 or 1 we obtain, formally, D2u+Mu+Lso[u]=f, D’u+Nu+Dl[u]=f,
u(0) = Du(0) = 0, B. “regular” at t = 0
(PO)
u(O)=Du(O)=O,
(PI)
with M, N, and I in d+. Our conditions on B, are described in the next section. There are two conditions, both stated in terms of the Laplace transform fi, of B,. The first is familiar in the study of Volterra equations [lo]. It yields dissipativity of L,Bmand produces asymptotic stability. The second involves the behavior of B,(s) for large s and abstracts the order t-” singularity. It is this second condition which controls the smoothness, both in our work and in [7, 11). In order to put the problems (1.1) and (1.3) into our general setting one makes the following identifications:
H, = H:(O, L),
Ho =
L2(0,
L),
H-, =H-1(&L)
(4~ 0) = j; cp’(x) u’(x) dx
In a similar way one can include general three dimensional problems (see [S] for the viscoelastic and Kelvin-Voigt cases). A common engineering procedure for Eq. (1.1) is called the quasi-static approximation in which one assumes that u varies slowly with time and
FRACTIONAL
ORDER
VOLTERRA
sets D*u = 0 (see [S]). We consider this approximation study the problem Au + DL,Ju]
451
EQUATIONS
for (P,), that is, we
=f
(A,)
with no initial condition. Under our hypothesis on B,, (P,) and (Q,) will have unique generalized solutions and one can obtain stability results as follows. For a map 4: [0, co) -+ X, where X is any Hilbert space we define, for any y, (I.61
where I(s) is the Laplace transform of 4 and 8 is the complexification of X. For our solutions of (P,) we will be concerned with norms of the form
IIu II;,a = IIu IIL(O,cc:Ho)+ II2.d IIffqo,a :H,)3
(1.7)
where y and S will depend on a. A main goal is the analysis of Galerkin approximations I!?‘, in a sequence of finite dimensional spaces Sh, to both (P,) and (QJ. We obtain symmetric error estimates for u - Uh in the fractional order norms (1.7). These depend on CI but are analogous to the ones in [3] for parabolic problems. It is well known for elliptic problems that a straightforward error analysis does not give the best convergence rates in weak norms like L,. This difficulty is overcome in elliptic theory by what is known as Nit&e’s trick. It is shown by Mary Wheeler [ 123 that a similar improvement can be achieved in parabolic theory by making use of the steady state theory to define what is sometimes called the elliptic projection. In Section 4 we present an extension to (P,) by using our approximate equation (Q,) to define what we call a quasi-static projection. We want to make some remarks on the Renardy-Hrusa model. For the elasticity case they use, instead of (1.3), the formula
4% t) = q&,
.)I
= Au,(x, t) + J6r 7
(u,(x, t -
7)
- u,(x, t))
d7,
(1.8)
where 0 < y < 2. (Notice that the normalizing factor r( 1 - y) does not appear. Recall we are assuming u(x, t) E 0 for t < 0.) When y --+0, (1.8) tends to the viscoelastic model. For 0 < y < 1, (1.8) produces finite wave speed but the speed tends to infinity as y --* 1. For 1 d y < 2, (1.8) produces infinite propagation speed and is similar to our
452
CHOIANDMACCAMY
model (1.3) except that it does not have a limit as y --f 2. We have not as yet completed an error analysis for Galerkin’s method as it applies to (1.8). We observe that passage to the limit tl=O in our model (1.3) is quite complicated. Although the equation formally tends to that for viscoelasticity the convergence of solutions for tx > 0 to those for CI= 0 has some type of non-uniformity in time which we do not yet completely understand. This is reflected in our error estimates in Section 2 which degenerate as CI+ 0. It is also reflected in the fact that at ct - 0 the wave speed suddenly drops from infinity to a finite value. Our model (1.3) is, on the other hand, very well behaved with respect to the passage to the limit CI= 1, that is, Kelvin-Voigt. In contrast (1.8) gives problems which are well behaved on passage to the viscoelastic limit y = 0 but do not yield KelvinVoigt materials as y + 2.
2. STATEMENT OF RESULTS We begin with our conditions on B,. We assume B, E L,(O, co : &). Then B, has a Laplace transform b,(s). B, exists and is continuous in Re s>O and is analytic for Re s > 0 as a bounded linear map from w, to w-i. Here and in what follows we denote the complexilication of any space X by x Our conditions are: (B.l)
For any N>O there exists a a(N)>0
(Re~,(irt)u,u)~6(N)I(uJI: (B.2) Re ~20.
forany
B,(s) = soL- ’ B,+s*-‘B,(s),
such that
u~Hi B,,E~+,
and B,(s)=o(l)
VE[-N,N]. as IsI + 0~) in
In Section 5 we indicate how to generate B, satisfying our conditions and the connection to behavior of B,(t) at t = 0. We define L, by the formula LBz[u](t)
= I,’ B,(t - T) u(t) dt.
(2.1)
L, takes values in A-, . The transform of DL,[u] is sj,(s) a(s). The regularity of DL,[u] in the fractional order spaces is then controlled by (B.2) for s = iv; that is, one has an estimate of the form II s&(s) fi(s)ll~-~ < 42 + Irl 12Y’2IIC(itl)llp, This yields the following result.
for some c.
453
FRACTIONALORDERVOLTERRAEQUATIONS PROPOSITION
2.1. DLBX maps H*(O, 00 : H,) into L,(O, 00: H-_,).
On the basis of Proposition 2.1 we can define generalized solutions (I’,) and (Q,). We introduce the following hypothesis on f: fEL,(O,
co:H-,)nL,(O,
co:K,).
DEFINITION 2.1. u is a solution of (P,), u E H”(0, cc : H,) n H’(O, cc : HP 1) and
D*U+Au+DL&]=f
on
(J")
for f satisfying
t > 0, u(0) = Du(0) = 0.
2.2. u is a solution of (Q,), for f satisfying co: H,) and on t>O. Au + D&&4] =f
DEFINITION
u~H”(0,
Remark 2.1. Equations (2.2) and (2.3) make sense L,(O, 00 : ZL,). For (2.2) we have UEL,(O, cc : H,) while in &(O, 00 : H-,). It follows, by interpolation, that continuous as maps from [O, co) to [H,, HP 1] 1,2= H,, conditions in (2.2) make sense. Remark 2.2.
of
if
(F’),
(2.2)
(F’),
if
(2.3)
as equations on DU and D2u are u and Du are hence the initial
For (I’,) we will strengthen (F’) to
fe
L1(O,
00: H,)
n L,(O, 00 : H,)
(4
and require of the solution of (2.2) that u E H2(0, co : H,). In this case (2.2) has the variational formulation
(D*u, u)o+ (Au, u> + @L&4
0) = (f, do
for any u E L,(O, co : H,), u(O) = 0, Du(0) = 0. Similarly
(2.4)
(2.3) has the formulation
0) = (f, u>
for any
u E L,(O, cc : H,).
(2.5)
We are concerned with Galerkin approximation. We suppose that we have a family of finite dimensional suspaces Sh = (&, .... dh,,) of H, , 0 < h, with the following approximation property: Given any E there exists h,,(s) such that for any u E H, and any h
454
CHOIANDMACCAMY
We can now state our results. THEOREM
2.1.
(i) Forfsatisfying
(I;), (P,) has a unique solution u with
UEH2(0, co:H*).
(ii)
There is a constant c, such that )(u (I2,Ni c, )If I(L2C,,, co:H,,j.
THEOREM
(ii)
2.2. (i) For
f satisfying (F), (Q,) has u unique solution u.
There is a constunt c, such that
THEOREM
2.3. There exist unique Gulerkin approximations uh to (P,)
and HA). THEOREM 2.4. There exists a constant c,, independent of h such that if u is the solution of (P,) and uh is its Gulerkin approximation,
0u - UhII1,(.x/2) d cmIIu - WhIII,(a/Z) for any
wh: CO,~0) --t Sh, IIwhII,,(a,*)
<
~0.
(2.6)
THEOREM 2.5. There exists a constant c, independentof h and a such that if u is the solution of (Qb) and uh is its Gulerkin approximation and y is any number.
II24- UhIIHY(O, m: If,) Gc IIu for
whIIffqo,co: HI) any wh: lW4+Sh,
Il~hll~2~o,s,:n,~<~. (2.7)
Remark 2.3. We will see that as a + 0 the constants c, + 00 so that our estimates degenerate. Remark 2.4. The estimate (2.6) is called symmetric since the same norms appear on both sides. This notation was introduced in [3] for parabolic problems. We emphasize that our error estimates involve only smoothness with respect to time: they will all be in the norms Hk. Let us consider again our example ( 1.1 ), (1.4). Let us choose for Sk the standard piecewise linear finite element space of mesh length h. It is known then that the following approximation results hold: there exists a constant c independent of h such that if u E H,(O, 1) n @(O, 1) then there exists a wh E Sh such that [4] IIu - WhIIH,(o,1)G ch IIu IIjiz(o,1,;
II~-‘+‘~ll~~~o.~~~~h* ll~llh~~o,~~~(2.8)
The estimate (2.6) can then be used to show that if u is a solution of (l.l),
FRACTIONAL
ORDER
VOLTERRA
455
EQUATIONS
(1.4) and U” is its Galerkin approximate then, provided the solution u is smooth enough, )(u - uh (1H~t0,m: H0) and (1ZJ- uh 11P~~C0,m: H,J will be order h. There is a defect in the previous estimate. From (2.8) one would conjecture that, in II . II L2(0,m: Ho), u - uh should be order h*. In elliptic differential equation theory this defect is overcome with what is known as N&he’s trick [4]. For parabolic equations it was overcome in [2] by using the idea of elliptic projection. In Section 4 we provide an analog for our problem using (QE) to develop what we call a quasi-static projection.
3.
PROOFS
In this section we indicate the proofs of Theorems 2.1-2.5. Some of these will be quite sketchy and we refer the reader to [t] for further details. The strategy is to use Laplace transforms. We formally transform (2.2) and (2.3) to obstain s*ii + Ati + s&s)
ti =f,
(3.1)
Aii + s&(s) t =“f
(3.2)
The idea is to solve (3.1) and (3.2) and then define u by the inversion formula u(t)= (277-l /I,”
e’%(iq)
dq
(3.3)
and show that this indeed gives solutions of (P,) and (Q,). To be a little more precise what we do is to transform (2.4) and (2.5) to obtain the variational problems: (s%, q. + (A& 5) + (d,(s) (A,
5) + (s&(s),
i&6) = (j: 8),
for all
C E t?,
(3.1)’
6, 6) = (i: IT)
for all
fi E i?, .
(3.2)’
(Recall we are taking f(t)
and D’u(t)
in Ho for (P,).)
LEMMA 3.1. Equations any s with Re s 2 0.
(3.1)’ and (3.2)’ have a unique solution i(s) for
ProojI From (F), p(s) is defined as an element of e. for any s in Re s > 0. We observe first that the fact A is coercive and B, is continuous in Re s > 0 means that we can solve (3.1)’ and (3.2)’ uniquely by successive 409/139:2-II
456
CHOIANDMACCAMY
approximations for s small. Hence we consider these equations for 1s ( > E for some E> 0. We divide (3.1)’ by s and consider the bilinear form
P(s: ti, t?)o= (SC,u)(J+ f ii, B + (B,(s) 22,6). ( > We assert that for any s in Re s > 0, JsJ 2 E, for E sufficiently small, there is a 8(s) such that Re P(s, 6, r?) 2 b(s)(J ti )I$, .
(3.5)
If (3.5) is established then the existence of a solution of (3.1)’ follows from the Lax-Milgran lemma. To prove (3.5) we observe that for any q, (B.l) yields, for some 6,’
Moreover given any B,,, 0 < 8, < n/2, we see from (3.3) and the coercivity of A that for E sufficiently small (3.4) holds on s = seie, - 8, < 6 < t$. Then (3.4) follows easily from the maximum principle. The argument for (3.2)’ is essentially the same. The following result is quite easy and is proved in [i 1. LEMMA 3.2. For both (3.1)’ and (3.2)‘, C(s) is continuous in Re s>,O, analytic in Re s > 0, and real for s real. LEMMA
(l+v*)*
3.3.
There exists a constant c, such that
II~(irl)ll&o+(l +v*Y Il(i~Nl~,~c, IIPCi~~lI~o (1 + v’)* IIfi(irl)ll B, < c, IIf(iq>ll Be,
for
(3.1)’
for
(3.2)‘.
ProoJ: Once again a successive approximation argument, with the coercivity of A, shows that for some E sufficiently small one has, for (3.1), Ilfi(iq)llR,
&WI ~(irl)ll$, <
(
iv llirll&--~
(Ati(
ri(iq))+
= - Re f @(iv), fi(iq)), = Re(](iq),
’ This statement
uses the symmetry
of B,(f)
(and
(fi,(iq)J(iq), a(@)),
hence of h,(iq)).
r(c)>)
FRACTIONAL
ORDER
VOLTERRA
EQUATIONS
457
Hence the conclusion for (3.1)’ holds for (n ( < N. The crucial calculation involves the behavior at infinity. We define y, and pt, by y, = sin : c( y,=cos;a, and note that for q > 0, (iv)“= q >o,
(3.6)
~“(y, + ip,). We have then from (3.2)‘, for
We take the imaginary part of this equation. Using the coercivity of B, and the fact that B,(s) is small when s is large we deduce that for q sufficiently large we have
for some constant c’. Next we take the real part and obtain, from (3.7) an estimate of the form,
Hence we have established the conclusion for (3.1)‘. For (3.2)’ we note that for large q we can calculate as for (3.7) and obtain
This yields the result for q large and for q on bounded sets the result is immediate. With Lemmas 3.1-3.3 in hand the proofs of Theorems 2.1 and 2.2 are fairly straightforward. One defines ZJ by the formula (3.3). One can then show that this function will satisfy (2.4) or (2.5) together with the appropriate initial conditions for (2.4). The details appear in [ 11. The estimates in Theorems 2.1 and 2.2 are an immediate consequence of (1.6), (1.7), and Lemma 3.3. The proof of Theorem 2.3 can be obtained once again by Laplace transforms. One transforms the variational formulations and obtains finite dimensional versions of (3.1)’ and (3.2)‘. The same argument as before (on the finite dimensional spaces) will show that these have solutions and if uh
458
CHOI
AND
MACCAMY
is defined by the inversion integral one obtains the Galerkin approximations. We turn now to the optimality results (2.6) and (2.7). Once again we work in the transform domain. Let u be the solution of (p,) and uh its Galerkin approximation with transforms 6, lib. For an arbitrary element 6h in Sh set E^h= ih - ah, ;h=c-$h. We put 6 = 17~~3;~ in (3.1)’ and subtract the result from the corresponding equation for tih. The result is (s*&h, Bh), + (ah,
eh> + s@,(s) ih, fib)
= (Ah, tih)() + (‘4Ch, fTh) + s(8,(s) ih, oh).
(3.8)
We put Ch= Zh and s = iv in (3.8) and estimate, as before for large (q I. The imaginary part of the left side is bounded below by C,U~)q 1a 1)lh(iq)ll 5,. On the other hand the imaginary part of the right side is bounded above by a constant time IJ* (Ith ((R,, (I.? [I~~ + I rl I’ IICh[lA, lI‘ZhIIR,. We get a similar estimate by considering the real part and arrive at the two inequalities
lrlla lle”llB,~~ {q’ Il@hII&IIE^hlI&+l?la l18hlIR,IIWIR,) Id2 Ilb”ll~occ’:{~* Ilthll& IIE^hIIHo+I?(” lPhIIR, lIE^hIIRI+IrllO’II~hII!-?J. (3.9) Put
~‘lrlP2 IIE^hllR,9 A= Id=‘* llehllR,, b= I?1lI~hll~~~ B= Irll lI~hllfi~. Then (3.9) becomes
b*
of the form
.* + b2 < $ (A2 + P). a that is,
FRACTIONAL
ORDER
VOLTERRA
459
EQUATIONS
One can obtain estimates on finite 4 intervals as before. Then if we note that, in any norm, Jjf-iihl) 6 (I$- ti’j) + (1Gh - till we arrive at the estimate
Translation back to the time domain yields the estimate (2.6). For (Q,) Eq. (3.8) is replaced by (A.?‘, 13~) + s{&(s)
Zh, @) = (A@, Ch) -t s@,(s) gh, v^“>
for any Bh. We estimate for s = iv, q large. Divide the equation by s’, put s = iv, and I?” = @(iq). This yields an estimate of the form )I@iq)ll~, 6 c I( g(iq) (InI with a constant c which is independent of tl. A similar estimate holds for bounded q. Again the estimate 11u - uh 11< 11u - wh I( + II wh - uh (I yields (2.7). 4. THE QUASI-STATIC
PROJECTION
Let u be the solution of (I’,). We make the following definition. DEFINITION 4.1. The quasi-static Uh : [0, co) -+ Sh such that
(AL@ i- DL,[
projection
Uh], uh) = (Au + DLB,[u],
u”)
of u is
the function
for any uh E L,(O, co : S”).
The existence and uniqueness of Uh can be established in the same way as the Galerkin approximation for (pa). Further one obtains an estimate of the form
II UhIIW(0.cc: H,)Gc II2.4 IIW(0,32:HI)
(4.1)
for any y. THEOREM
4.1.
There exists a constant independent of h such that
IIUh-
Proof. An easy computation the formula (D2(Uh-uh),
uh),,+
llu- ~hIlH2(o,oC:HO).
Uhl12,01eC,
using the definitions
(A(Uh-u’),
= (P( Uh - u), Vh)Ho
of uh and Uh yields
uh) + (DLBz[Uh-u’],
for any
vh E sh.
(4.2)
vh) (4.3)
460
CHOI
AND
MACCAMY
Then one can go to the transform plane and argue as in the proof of Theorem 2.2 to estimate 6” - tih in terms of oh - 0. Since only the zero norm appears on the right one obtains the formula (4.2) in the time domain. Formula (4.2) is the basis of our version of Nitche’s trick as we describe now. We begin by defining a new function 8 as the solution of the problem (AC + d,(s)
z?,(8) = (ti - Oh, O),
for any
v^ERi.
(4.4)
Because of the symmetry of A and B,, (4.4) is the same as the problem with d and 6 interchanged on the left and one can argue as in Theorem 2.1 that it has a unique solution. Now put B = ti - oh in (4.4) and obtain Illi - 0” I( j& = ((A + s&(s))@ - Oh), 4). By construction
(4.5)
we have
((A +sS,(s))(ti-
Oh), Gh)=O
for any
fi” E gh
and if we subtract this from (4.5) we obtain I)a- z7hII~o= ((A +sB,(s))(tiThen condition
IIa-~hll$@
Oh), &Bh).
(B.2) yields
IvIa IIa-fihII
dR, Il&mIA,
for any
@E 3”
(4.6)
for some constant c. Now observe that, for any Oh,Gh E Sh, ((A + sB,(s))[
Oh - Gh], fib> = ((A + sS,(s))[ti
- Gh], uh>.
(4.7)
Choose fib = 0” - Gh and obtain, for any tih E $h, ((A+sB,(s))[oh-~h],
irh--h)=((A+S~‘,(S))[U--h],
Oh-tih).
The arguments used before yield
II Oh(Q) - ~“(~vlh, G c IIG(h) - ~h(mla, and thus,
IIw?)--hG4QllA,~C
IIw+~h~~rl)llAl.
(4.8)
We substitute (4.8) into (4.6) and obtain
Illi(i~)-~h(i~)ll~~~cl~la
II~(irl)-~h(~rlNl~, lI$(~r)-~hlI~
(4.9)
FRACTIONAL
ORDER
VOLTJZRRA
for any tih and 6’ in 5”. Finally translating yields IIU-
461
EQUATIONS
back to the time domain
Uhl12L2(0.,:Ho)~C llU-whllH’l(O,df,)
ll(b-~hI/mx3:H,),
(4.9) (4.10)
for any vh and wh in &(O, cc : sh). Formula (4.10) is the form of Nitsche’s trick we want. It estimates u - iJh in the H, space in terms of u - wh and 4 - vh in the H, norm. In concrete examples this can be used to remedy the defect indicated in Section 2. We illustrate with our example (1.1 ), (1.4). For that example we have (Au, v ) = A loL u’v’ dx,
<&(t)
b,(r)=b(t)/r(l
= b,(t) loL u’u’ dx,
Thus the problem
w v>
-lx) P.
(4.4) becomes loL (A +6,(s)
6’8 dx) = joL p(‘(a-)
dx.
This says 6 is a weak solution of the problem
-@(A+~,(s))=z+ Elliptic
Oh,
q?(O) = J(L) = 0.
theory states that 6 satisfies an estimate of the form (4.11)
II&Q)IIA2(0.L)~C Illi-- UhlIE*(O,L). From (2.8) one can then infer that there is a Sh such that (I&fih(I1
l)ti-OhI(jJo
or, in the time domain,
II4 - ohIIL2(0,co: H,)d 4 IIl4- UhIILI(O,cc: Ho). We insert this estimate into (4.10) and obtain
IIu - UhIIho, a,: Ho)f c IIu - WhIIW(0,m: H,)h II24- UhIIL*(O,
co : Ho)
or II u - uh II L*(O, co: Ho)G 4 II24- whIIffyo,00:H,)’
(4.12)
462
CHOIANDMACCAMY
We can use (2.8) again to estimate the term on the right side of (4.12). By the first of inequalities (2.8) we have )I u - wh ))HaC,,m:H,j d C/J II u II Hb(O,oo:Hzj and hence (4.12) yields IIu - Uh IIL2(0,co: Ho)6 A2 IIu - whIInyo, a3:“2)’
The same argument can be used to show that
But then (4.2) yields II u - Uh II Lz(O, a!: Ho)G IIu - cJhIIL2(0, cc: Ho)+ II uh - UhIIL2(0, m: Ho) < (1 + c,) 11 u - uh )I~(0, m: t,,,) G ch* IIu IIH2+ a(o,m: m
which is an order h* estimate in the Ho norm. The term 1)u )I H~+ a(o,oo:Hzj can be estimated in terms of the data.
5. SINGULAR
KERNELS
In this section we indicate how to generate classes of kernels B, satisfying our conditions (B.l) and (B.2). We begin with what we term product kernels which are illustrated by our elasticity example. Suppose M is a fixed operator from H, to H-, which is symmetric and coercive. For example, take HI = H,(O,L) and H-, = H-,(0, L) as in Section 1 and define M by (Mu, v) = joLu’v’ dx.
Now define B,[t] by B,[t] = b,(t) M, where b, is a scalar function which is in L,(O, co)nL,(O, co). Then (B.l) translates into conditions on 6,: for all q
Re b,(iq) > 0 &)=s
‘-=bo+s’-oL61(s), b,>O,
and
(5.1)
6,(s)=u(l)as
(s( *co.
(5.2)
We recall some facts from [9, 10 J. For a E &(O, co) one has (-l)ku(k)(t)>O,
k=O, 1 * -~Irn&i~)>O
(- I)h a(“)(t) > 0, h = 0, 1, 2 => Re 6(iq) > 0
for for
n#O all q.
(5.3)
FRACTIONAL
463
ORDER VOLTERRA EQUATIONS
We generate a class of kernels b, by taking 1 r(1 -a)
a(z)dz+me-‘,
b,(t) = ____
where a E L,(O, oo), (- 1)“ 8’) (t)>O,k=O, that
(5.4)
1,2, and m>O. We observe
6,(s)= (s+s),-A~)+~.
(5.5)
For s = iq we have ((iv) + 1 )‘- ’ = (1 + q2)(a’2)- ’ (cos 6, - i sin %,),
0 < 6, < ?c/2.
Thus (sin 6, Re 6(iq) - cos 6, dn &ill)) + -!?1 +q2’
Re 6,(iq) =
Under our hypothesis. (5.3) shows that the right side is bounded below on any compact set 1q ( d N and (B.1) is satisfied. If we assume that UE Cc2’(0, co) and that &‘EL,(O, co), j=O, 1, 2, then one has 6(s) = (a(O)/s) + (a(0)/s2) + o(l/s2) as )s) --, 00 and one verities that (B.2) is also satisfied. One verifies easily that b,(t)-f$?
near t = 0,
confirming that (B.2) represents singular behavior at t = 0. The above idea can be extended to general kernels. Let A(t) be a family of mappings from HI to H- 1 with ( - l)k Ack’( t) coercive for k = 0, 1, 2 and let M be coercive. Then if one defines B, by B,(t)=--
___
A(z) do + Me-’
one will have a family satisfying (B.l) and (B.2).
REFERENCES 1. U JIN CHOI, “Fractional Order Volterra Equations in Hilbert Spaces,” Thesis, Dept. of Mathematics, Carnegie Mellon University, July, 1987.
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CHOI AND MACCAMY
2. U JIN CHOI AND R. C. MACCAMY, Fractional order Volterra equations, in “Proceedings, Conf. on Volterra Equations, Trento, Italy, February, 1987,” to appear. 3. J. DOUGLAS AND T. DUPONT, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7, No. 4 (1970), 575-626. 4. G. FIX AND F. STRANG, “An Analysis of the Finite Element Method.” Prentice-Hall, Englewood Cliffs, NJ, 1973. 5. A. C. ERENGEN,“Mechanics of Continua,” Wiley, New York, 1967. 6. M. E. GURTIN AND A. C. PIPKIN, A general theory of heat condution with finite wave speed, Arch. Rational Mech. Anal. 31 (1968), 113-126. 7. W. J. HRUSA AND H. RENARDY, On wave propagation in linear viscoelasticity, Quart. Appl. Math. 48, No. 2 (1985), 237-253. 8. R. C. KOELLER, Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51 (1984), 299-307. 9. R. C. MACCAMY, Nonlinear Volterra equations on Hilbert space, J. D@firential Equations 10 (1974), 373-398. 10. R. C. MACCAMY AND J. S. W. WONG, Stability theorems for some functional equations, Trans. Amer. Math. Sot. 164 (1972), l-37. 11. M. ~NARDY, Some remarks on the propagation and non-propagation of discontinuties in linearly viscoelastic liquids, Rheology Acta 21 (1982), 251-254. 12. M. F. WHEELER, A priori L, estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10, No. 5 (1973), 723-759.