Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 363-380
363
BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS
WITH PIECEWlSE CONSTANT DELAY
JOSEPH WIENER
Department of Mathematics
The University of Texas
Pan American
Edinburg, Texas
78539
(Received February 21, 1990)
ABSTRACT.
The influence of certain discontinuous delays on the behav-
ior of solutions to some typical equations of mathematical physics is
studied.
KEY WORDS AND PHRASES.
Partial Differential Equation, Piecewise Con-
stant Delay, Boundary Value Problem, Fourier Method, Positive Opera-
tor, Weak Solution.
1980 AMS Subject Classification Codes.
i.
35A05, 35B25, 35LI0, 34K25.
INTRODUCTION.
Functional differential equations(FDE) with delay provide a mathematical model for a physical or biological system in which the rate
of change of the system depends upon its past history.
The theory of
FDE with continuous argument is well developed and has numerous applications in natural and engineering sciences.
earlier work
[1-5]
This’note continues our
in an attempt to extend this theory to differential
equations with discontinuous argument deviations.
In these papers,
ordinary differential equations with arguments having intervals of
constancy have been studied.
Such equations represent a hybrid of
continuous and discrete dynamical systems and combine properties of
both differential and difference equations.
They include as particular cases loaded and impulse equations, hence their importance in
control theory and in certain biomedical models.
Continuity of a
solution at a point joining any two consecutive intervals implies recursion relations for the values of the solution at such points.
Therefore, differential equations with piecewise constant argrument
(EPCA) are intrinsically closer to difference rather than differential
364
J. WIENER
Here boundary value problems for some linear EPCA in
equations.
partial derivatives are considered and the behavior of their solutions
studied.
The results are also extended to equations with positive
definite operators in Hilbert spaces.
2.
BOUNDARY VALUE PROBLEMS.
The equation
u
a
t
2
u
xx
u
b(u
0
describes heat flow in a rod with both diffusion a
2
u
xx
along the rod
and heat loss (or gain) across the lateral sides of the rod.
loss (b > 0) or gain (b < 0)
Heat
is proportional to the difference between
the temperature u(x,t) of the rod and u
0
of the surrounding medium.
In chemistry where u may stand for concentration, the above equation
says that the rate of change u
2
diffusion a u
xx
t
of the substance is due both to the
(in the x-direction) and to the fact that the sub-
tance is being created (b < 0) or destroyed (b > 0) by a chemical reaction proportional to the difference between two concentrations
u to u and consider the equation
We may change u
u and
u016._
0
u
a
t
2
bu.
Uxx
Measuring the lateral heat change (or substance change due to a chemical reaction) at discrete moments of time leads to an equation with
piecewise constant argument
a
ut(x,t
t
[nh,
2
Uxx(X,t)
(n +
l)h],
bu(x,nh),
n
0,i
This equation can be written in the
where h > 0 is some constant.
fol-m
a
ut(x,t)
where
[-]
2
Uxx(X,t)
[t/h]h),
(2.1)
designates the greatest integer function. Ordinary differen-
tial equations with arguments
gated in
bu(x,
[1-4],
with
[t
+
1/2]
[t], [t
in
[5],
n], [t
+
and with
Furthermore, EPCA have been used recently in
[8]
n]
have been investi-
[t/h]h
in
[7,8].
to approximate solu-
PARTIAL DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT DELAY
365
tions of equations with continuous delay.
The diffusion-convection equation
u
a
t
2
u
ru
xx
x
describes, for instance, the concentration u(x,t) of a pollutant carried along in a stream moving with velocity r.
diffusion contribution and -ru
The term a
2
u
xx
is the convection component.
x
is the
If the
convection part is measured at discrete times nh, the process results
in the equation
ut(x
a
t)
2
u
xx
ru
(x t)
x (x,[t/h]h).
(2.2)
We consider the boundary value problem (BVP) consisting of the
equation
[t/h]h),
Q
(2.3)
u(x,
u(x,t)
@t
where P and Q are polynomials of the highest degree m with coeffi-
.m[
cients that may depend only on x, the boundary conditions
L.u
3
k=l
(Mjk
and
Mjku(k-1) (0) + Njku(k-1) (i)
Njk
are constants, j
(2.4)
O,
l,...,m)
and the initial condition
U(X,O)
[.]
Here
and h
U
designates the greatest integer function,
const > 0.
DEFINITION 2.1.
[0,1]x[0,m),
(2.4’)
0.
A function u(x,t) is called a solution of the
above BVP if it satisfies the conditions:
[0,1]x[0,);
(x,t)E
Equations (2.4) will be writte briefly as
Lu
G
(2.5)
O(x)
(ii)u/t and
oku/xk
(k
(i)u(x,t) is continuous in
0,1...,m) exist and are
continuous in G, with the possible exception of the points (x,nh),
where one-sided derivatives exist (n
0,1,2..); (iii) u(x,t) satis-
fies equation (2.3) in G, with the possible exception of the points
(x,nh), and conditions (2.4)-(2.5).
Let u (x,t) be the solution of the given problem on the interval
n
nh < t < (n
+ l)h, then
OUn(X,t)/Ot
+
PUn(X,t)
Qu
n(x),
(2.6)
366
J. WIENER
where
u
u (x)
n
Write
n(x,nh)-
w (x,t) + v
n
u (x,t)
n
n(x),
which gives the equation
PWn + PVn(X)
+
OWn/Ot
Qu
n(x),
and require that
PWn
+
OWn/@t_
Qu
PVn(X)
Assuming both w
n
and v
n
(2.7)
0,
(2.8)
n(x)
satisfy (2.4’) leads to an ordinary BVP (2.8)-
whose solution is denoted by
(2.4
v
n
p-i Qu
(x)
n
(x)
and to BVP (2.7)-(2.4’), whose solution is sought in the form
w (x t)
n
(2.9)
X(x)T n(t).
Separation of variables produces the ODE
T
n
+ IT
0
n
with a solution
e
T (t)
n
-I (t-nh)
and the BVP
P(d/dx)X- IX
LX
0,
(2.10)
0
If BVP (2.10) has an infinite
where L is defined in (2.4) and (2.4’).
countable set of eigenvalues I. and corresponding eigenfunctions
3
X. (x) C
3
m
[0, i]
.
then the series
Wn (x, t)
9=1
.
Cnje
I j (t-nh)
Xj (x)
Cn3
const
represents a formal solution of problem (2.7)-(2.4’) and
Un(X,t)
j=l
Cnje-lj(t-nh) Xj (X)
.
is a formal solution of (2.3)-(2.4).
U
n
(x)
CnjXj(x
Therefore, assuming the sequence
{Xj}
At t
+
+
P-IQun(x)
(2.11)
nh we have
P-iQUn(X ).
(2.12)
is complete and orthonormal in
PARTIAL I)IFFERENTIAL EQUATIONS WITH
m
L0, lj yie ids for the coefficients
C
i
Cn-’3 -[0 X j (x) (I
Substituting the initial function
n=0
-i
gives the solution
val 0 < t < h.
u0(x,t
(2.13)
Q)u (x)dx
n
TM
u0(x)6 C [0,i]
in (2.13) produces the
and putting them together with
C0j
367
0,1,2...)
(n
coefficients
CONSTANT DELAY
the formula
n3
P
P|ECEWISE
in (2.11) as
u0(x
of BVP (2.3), (2.4), (2.5)
on the inter-
Since u (x,h)
(2.13) the numbers
CI_.3
u (x,h)
u (x), we can find from
0
1
1
and then substitute them along with
in
Ul(X
(2.11), to obtain the solution u (x,t) on h < t < 2h.
This method of
steps allows to extend the solution to any interval nh < t < (n+l)h.
Furthermore, continuity of the solution u(x,t)
u (x, (n + l)h)
n
u
implies
+ l)h)
n+ 1 (x, (n
u
n+ 1 (x),
(n + l)h we get from (2.11) the recursion relations
hence, at t
Z Cnje- Xjhx.3 (x)
Un+l(X)
+
P-iQu n (x)
(2.14)
Therefore,
u
n+l
u (x)
n
(x)
j=l
-Xjh) Xj (x)
Cnj (I
e
c
e -Xjh )(l-p-
and
iQ)u n+l (x)
(I-p
Multiplying by
(I-P
Xk(X
iQ)u n (x)
Z
j=l
nj
(I
IQ )xj(x).
and integrating between 0 and 1 yields the recur-
sion formulas
C
n+l,k
Z Cnj (i
Cnk- j=l
e -Xjh
)Xjk,
where
Xjk
THEOREM 2.1.
;01 Xk(X) (I p-IQ)xj (x)dx.
Formula (2.11), with coefficients C
n3
and functions
u (x) defined by recursion relations (2 13) and (2 14), represents a
n
formal solution of BVP (2.3), (2.4), (2.5)
[O,l]x[nh,
l)h]
in
(n +
for
0,i,..., if BVP (2.10) has a countable number of eigenvalues X.
3
and a complete orthonormal set of eigenfunctions X. (x) C TM
and the
i]
3
n
[0
368
J. WIENER
initial function
u0(x)Ecm[0,1]
satisfies (2.4).
A different method can be used if we look for a solution with
continuous derivatives
tE (nh, (n + l)h).
o2u/ot2
ok+lu/toxk
and
(k
0,1..,m) for
In this case we differentiate (2.6) with respect to
t and obtain the equation
Oyn/Ot
+ P(O/Ox)y
.
0
n
whose solution is sought in form (2.9).
ables produces T (t) and BVP (2.10).
n
Yn(X’t)
Yn
Ou /%t
n
Again, separation of vari-
Integrating the solution
Bnje- j (t-nh) X. (x)
(2.15)
between nh and t gives
u (x t)
n
u (x) +
n
j=l
u (x) +
n
From (2.6) and (2.15) at t
(t-nh)
)Xj (x)/j.
(2.16)
(n + l)h implies
Continuity of the solution at t
Un+ 1 (x)
e -j
Bnj(l
e -jh
Z Bnj (I
j=l
(2.17)
Xj (x)/ 3
nh we have
Yn(X’nh)
(Q
Yn (x,nh)
j=l
.
p)u
n(x),
BnjX j (x),
and consequently,
Bnj
THEOREM 2.2.
;01 Xj (x) (Q
P)u n(x)dx.
Series (2.16), with coefficients B
u (x) defined by (2.17) and (2.18)
n
continuous in
[0,1]x[nh, (n
[O,l]x(nh,(n
+
n3
and functions
formally represents a solution of
BVP (2.3), (2.4), (2.5) whose derivatives Ou /Ot
n
are continuous in
(2.18)
l)h]
and 0
okun/O x k
(k
0,1,..m)
2un/ot 2, ok+lun/OtO xk are
+ l)h) if, in addition to the other condi-
tions of Theorem 2.1, the initial function
u0(x
and (Q-
isfy (2.4).
The solution u (x,t) of the nonhomogeneous equation
n
P)u0(x
sat-
PARTIAL DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT DELAY
0u(x,t) + p
u(x,t)
t
u(x, [t/h]h) + f(x,t)
Q
.
369
(2.19)
on nh < t < (n + l)h is also sought in the form
Un(X,t)
where
(2.20)
Xj (X)Tnj (t),
j=l
Upon multiply-
are the eigenfunctions of the operator P.
Xj (x)
ind (2.19) by
then integrating between 0 and 1 and changing k
Xk(X),
to j, we obtain
T
nj
(t) +
jTnj (t)
1
X
0
j
qnj
+
qnj
fj (t)
(x)Q(d/dx)u (x)dx
n
I Xj (x) f(x t)dx
f. (t)
3
0
whence
Tnj (t)
-i
(Tnj (nh)
+ X
-
j qnj)e
t
-I
+
j qnj
;nh e
-j (t-nh)
J (t-s) f. (s) ds,
3
(nh <_ t < (n + l)h)
T
nj
1
0Un (x) Xj (x) dx,
(nh)
that is,
;
Tnj (t)
+
i
0
Xj(x)(I-
iQ)Un(X)dx] e -j(t-nh) + k3 O1
-i
3
Xj (x)Qu n(x)dx
]tnh e-j (t-s)f.3 (s)ds.
(2.21)
The principal role of the operator P emerges from the above three
methods of constructing the solution.
m
PY
j=0
where
P0(X)
pj
Let
pjy(m-j)
are real-valued functions of classes Cm-3 on 0 < x < 1 and
0 on
[0 I]
Assuming
cm[0 i]
is embedded in L
inner product
(y,z)
[01
y(x) z(x)dx,
BVP (2.10) is called self-adjoint if
(Py,z)
(y,Pz),
2
[0, i]
with the
370
J. WIENER
for all y zE C
m
[0 ,I]
that satisfy the boundary conditions
Lz
Ly
O.
is self-adjoint, then all its eigenvalues are real and
If BVP (2.10)
The eigenform at most a countable set without finite limit points.
functions corresponding to different eigenvalues are orthogonal.
THEOREM 2.3.
[0,1]x[nh, (n
+
BVP (2.3),(2.4),(2.5) has a solution in
l)h],
for each n
0,i,..., given by formula (2.11) if
the following hypotheses hold true.
(i) BVP (2.10) is self-adjoint, all its eigenvalues
tive.
_.,
(ii) For each
the roots of the equation P(z)
3
.
.
are posi-
have
0
3
non-positive real parts.
(iii) The initial function
PROOF.
Qu
0
.
v0(x)=P -iQu 0 (x)
satisfying the boundary conditions
0(x)
Then the difference u
0.
0
satisfies (2 4)
According to (2.8), we find the solution
of the equation Pv (x)
Lv
m
u0(x)E C [0, i]
0(x)
P-IQu 0(x)Cm[0,1]
satisfies
(2.4’), and therefore we conclude from (2.12) that the Fourier series
C0jXj(x
Xj(x)}
.
on
converges to it absolutely and uniformly on
[0,I],
where
is the set of the orthonormal eigenfunctions of (2.10).
Since
> 0, the series in (2.11) also converges absolutely and uniformly
[0,1]x[0,h].
Furthermore, the same is true on
in (2.14) at n=0, and
Ul(X
satisfies (2.4).
used now to find the solution
Qu
l(x)
satisfying Lv
I
(2.13) and the solution
cording to (2.11).
P-IQul(x
Hence,
for the series
Ul(X
should be
of the equation
PVl(X
0, then to calculate the coefficients C
Ul(X,t
of the given BVP on
[0,1]x[h,2h],
Ij
by
ac-
This procedure can be continued successively to
construct the solution u n (x,t)
that all u (x) satisfy (2.4’)
n
Vl(X
[0,i]
for
any n > o.
From (2.12) we conclude
Differentiating (2.11) term by term
PARTIAL DIFFERENTIAL EQUATIONS WITH PIECEWIES CONSTANT DELAY
with respect to t produces a series which converges to
ly on
[0,1]x[nh
> 0.
,
+
uniform-
> 0, since
for sufficienty small
it follows from (2.13) that
Furthermore,
P- IQ )Un (x)dx
1
-i
0 (PXj)(I
and P-iQUn(X
Cnj
and since
l)h],
(n +
OUn/Ot
371
xj
Xj(x), Un(X),
-i
j
Cn j
I Xj (x)
0
satisfy (2.4’), then
(P
Q)u
n(x)dx.
Hence,
<
Cnj
Let
P0(X)
;0
j
1 and
p
m
-i
< c I.
n 3
QUn
(Pun
0
(2.22)
then in any domain T of the complex p-plane
the equation
P(d/dx)y
has m linearly independent solutions
Ym
Yl
pl,
respect to pT, for sufficiently large
(r-l)
Yk
r-i
(x)
ek
l’’’’’m
x
which are regular with
and satisfy the relations
[k
r-i
+ O(
-i]
l,...,m)
(k,r
where
0
ly
are the different m-order roots of unity
[9].
There-
fore, by virtue of condition (ii) and estimates (2.22), differentiat-
I,...,) with respect to x
ing series (2.11) term by term r times (r
..
produces series that converge iniformly on
for sufficiently small
and large
[0,1]x[nh
Letting t
+ ,(n +
(n+l)h in each of
these series and taking into account (2.14) shows that u
if u (x) C
n
m
REMARK
[0, I].
i.
By virtue of (iii)
P0(X)
is the leading coefficient of the operator P(d/dx).
whose leading coefficient is I.
y
If
n+l
(x)cm[0 i]
the proof is complete
We assumed in this theorem that
then dividing the equation Py
l)h],
0 by
P0(X)
P0
I, where
If
p0
const
P0(X)
I,
produces an equation
const on
[0,i]
and retains
372
J. WIENER
its sign, then we may assume
;X0 pi/m
Xl
[9]
> 0 and use the substitution
P0(X)
(s)ds /
0 p 1/m
1
(s)ds
to reduce the above equation to a new one in the interval 0 < x
1
< I,
with a constant leading coefficient.
REMARK 2.
The Fourier coefficients used in the above three
methods of solving BVP (2.3)
(2.5) are closely interrelated.
(2.4)
Indeed, differentiating (2.11) with respect to t and comparing
Furthermore, comparing (2.11) with
(2.15) shows the B. =-.C
n3
with
3 n3
(2.20) and (2.13) with (2.21), we have to prove that
;01 Xj (x)p-IQu n (x)dx.
Xj (x)Qu n(x)dx
since
P-IQu n (x)
satisfies (2.4), then
P
iQu n (x)
.
1
(x)dx
Z
Xk(X)
0 Xk(X)p-IQu n
k=l
and applying the operator P to this equation yields
QUn(X)
IkXk(X);01 Xk(X)p-iQu n(x)dx.
k=l
It remains to multiply this expansion by
tween 0 and
Xj (x)
and to integrate be-
i.
EXAMPLE 2.1.
[0,1]x[nh, (n
+
The solution u (x,t) of equation (2.1)
l)h],
n
with the boundary conditions
and initial condition u (x,nh)
in
Un(0,t)=Un(l,t)=0
u (x) is sought in form (2.20).
n
n
paration of variables produces
X. (x)
3
-sin(,jx)
T’nj(t)
a2, 2 j 2 Tnj (t)
+
-bTnj (nh)
whence
-a 2,2 j (t-nh)
2
Tnj(t)
We put t
Cnje
b
a
22 j ’2 Tnj (nh).
nh in this equation and get
1 +
(nh),
a2,2j2 Tnj
that is,
Tnj (t)
Ej (t
nh)Tnj (nh),
Se-
t,QUA’rIONS WIItl
PARtiAL DIFFERENTIAl,
CONTANT DELAY
PIEC1,WISE
373
where
e
E. (t)
-a
22 j 2 t
e
1
-a
22 j 2 t
a
At t
.
(2.23)
22.2
3
(n + l)h we have
+ l)h)
Tnj ((n
Ej (h)Tnj (nh)
and since
Tnj ((n
+ l)h)
+ l)h)
Tn+l, j ((n
then
Tn+l,j ((n
+ l)h)
E_.j (h)Tn_.j (nh)
and
n
Tnj (nh)
Ej (h)T0j (0)
Therefore,
-
Tnj (t)
and
j=l’
Un(X,t)
Putting t
O, n
n
Ej (t
nh)Ej
(h)T
0j
(0)
En’(h)3 T0j (0)Ej(t- nh)sin(,jx).
(2.24)
, T0j(0)q--sin(,jx)dx
0 gives
u0(x
and
1
0 u0(x) sin(,jx)dx
T0j (0)
If
< i, then solution (2.24) decays exponentially as t
Ej (h)
uniformly with respect to x.
,
From (2.23) it follows that this is true
if
-a
,
2 2
< b <
a2, 2
e
a
+ 1
e
/
a2, 2h
1
Furthermore, from the equations
Tnj (nh)
we see that
n
mj (h)T0j (0) Tnj ((n
Tn-’3 (nh)Tn-’3 ((n
+ l)h) <
0 if
+ l)h)
E
E. (h) < 0.
n+l
j
(h) T0j (0)
The latter
inequality holds true if
b > a
22
/
[a2"2h
e
1
]
(2.25)
374
J. WIENER
Hence, under condition (2.25), each function Tnj(t (j
[nh,
zero in the interval
tions
ut
Tj
(n +
l)h],
1,2,..) has a
in sharp contrast to the func-
(t) in the Fourier expansion for the solution of the equation
a2Uxx
bu without time delay.
Moreover, the inequality Ej (h)<0
takes place for sufficiently large j and any b > 0.
Therefore, for
b > 0 and sufficiently large j, the functions
EXAMPLE 2.2.
are oscillatory.
Tnj(t
Equation (2.2) on nh < t < (n + l)h becomes
rU’n(X ),
a22Un(X,t)/x 2
Un(X,t)/t
and we differentiate the latter with respect to t to obtain the equation
a22yn/X2, Yn
Yn/t
whose solution is sought in form (2.9).
@Un/t,
Separation of variables leads
to the equations
X’’ (x)
+ X(x)
T’n(t)
0
and posing the boundary conditions
+ a 2 T n(t)
0,
Un(l,t)=0 gives
Un(0,t
.=
j2,2
and
Yn(X,t)
Z
j=l
Yn(X,nh)
a
Tnj (nh)e
-a2, 2j 2 (t-nh)
sin(. jx)
Since
2
u
,,
n
.
then
a
u
n
r u (x)
n
(x)
j=l
and
;01
Tnj(nh) =-a2"2j2-
+
Finally,
u (x t)
n
u (x) +
n
.
r,j-
Tnj(nh)
j=l
Given the initial function u(x,0)
cients
-
(x) -r u (x), u (x)
n
n
1
0
1
)__
Un(X,nh
Tnj (nh)’sin(,jx)
Un(X)sin(,jx)dx
Un(X)Cs("jx)dx"
e
_a2, 2j 2 (t-nh)
u0(x ), we
u0(x,t on 0 _<
sin(,jx)/a2,2j 2
can find the coeffi-
t <
T0j (0) and the solution
u l(x), we can calculate the coefficients
TIj (h)
h.
Since
u0(x,h
and the solution
PARTIAL DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT DELAY
be exBy the method of steps the solution can
u (x,t) on h < t < 2h.
I
[nh,
tended to any interval
EXAMPLE 2.3.
(n +
l)h].
Separation of variables for the equation
a
u(x,t)/t
2 2
u(x,t)/ x
with the boundary conditions u(0,t)
functions X. (x)
2
2
b u (x, [t])/x
u(l,t)
2
0 produces the
eigen-
sin(,jx) and the equation
T’ (t)
+
a2,2j2T(t)
b,
2j2T([t]),
which on the interval n < t < n + 1 becomes
+ a 2.2j2 T nj(t)
Tnj(t)
b,
2 2
j T nj(n)"
From here,
Tnj(t)
Fj(t
n)Tnj(n),
where
e
Fj (t)
At t
-a2"2j2t
+
I
i
e
-a2"
2j2t]
b/a
2
n + 1 we have
and since
Tn_.j (n
Tnj (n + i) Fj (1)Tnj (n)
Tn+l,j (n + i), then
Tn+l,j (n + i) F_.3 (1)Tn-’3 (n)
+ i)
and
-
F
Tnj (n)
Hence,
u (x t)
n
Z
where
T0j(0)
n
j
(1)T0j (0).
Fnj (1)T0j (0)Fj(t-
n)sin(,jx),
;i0 u0(x)sin("jx)dx"
The inequalities
-a
are equivalent to
375
e
Fj (i)
+
1
/
1
< b < a
< 1 and ensure the exponential decay of
376
J. WIENER
u (x,t) as t
m, uniformly with respect to x.
n
_
b < -a
each function
Tn (t)
/ (e
a2, 2
i),
has a zero in the interval
impossible for the equation u
EXAMPLE 2.4.
2
For
(a
t
2
[n,
n +
i],
which is
b)Uxx.
The equation
2
u(x,t)
iq@t
2m
.,0
2
Ox
0
+ V(x)u(x, [t/h]h)
2
is a piecewise constant analogue of the one-dimensional Schr6dinger
equation
-q
iq#t(x,t
2
@xx(X,t)/2m0
+ V(x)#(x,t).
2, then sepa-
If u(x,t) satisfies conditions (2.4) and (2.5), with m
ration of variables produces a formal solution
for nh < t < (n
ator
+ l)h.
q2(d2/dx2)/2m0,
(t-nh)/q
Z
j=l Cn3e
u (x,t)
n
Here
Xj (x)
p-IQun(X
and
tion
q
2
that satisfies (2.4) and C
V
(X)
n
x j (x) + p-iQu n(x)
are the eigenfunctions of the operis the solution v (x) of the equan
2m0V(x)u n (x)
are given by (2.13)
n3
The Fourier method can be also used to find weak solutions of BVP
(2.3), (2.4), (2.5) and it is easily generalized to similar problems in
Hilbert space.
First, we remind a few well known definitions.
Let H
be a Hilbert space and let P be a linear operator in H (additive and
homogeneous but, possibly, unbounded) whose domain D(P) is dense in
H, that is D(P)
The operator P is called symmetric if (Pu,v)
H.
(u,Pv), for any u, v
D(P).
If P is symmetric, then (Pu,v) is a
symmetric bilinear functional and (Pu,u) is a quadratic form.
metric operator P is called positive if (Pu,u)
and only if u
0.
0 and (Pu,u)
A sym0 if
A symmetric operator P is called positive definite
2
if there exists a constant v > 0 such that (Pu,u)
every positive operator P a certain Hilbert space
ed, which is called the energy space of P.
D(P), with the inner product
(u,V)p
v
Hp
2
llull 2
With
can be associat-
It is the completion of
(Pu,v); u, v
D(P).
This pro-
PARTIAL DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT DELAY
duct induces a new norm
tive definite, then
llull
llUllp
_< 7
(Pu,u)
-i
....llUllp
I/2
377
D(P), and if P is posi-
U
is dense in H, it fol-
Since D(P)
lows by using the latter inequality that the energy space
Hp
of a
positive definite operator P is dense in the original space H.
Assuming P is positive definite, we may consider the solution
o
u(x,t)
BVP (2.3), (2.4), (2.5) for a fixed t as an element of
Hp.
If
D(Q) c H, then Qu(x,[t/h]h) may be treated as an abstract function
Qu([t/h]h) with the values in H.
to the abstract Cauchy problem
du
dt
+ Pu
Therefore, the given BVP is reduced
ut= 0
Qu([t/h]h), t > 0,
u
H.
0
(2.26)
If (2.26) has a solution, we multiply each term by an arbitrary func-
Hp
tion g(t)
in the sense of inner product in H and get on
nh <_ t < (n+l) h the equation
g
+
(u,g)p
(Qu
(2.27)
g),
n
1
where u
C ((nh, (n + l)h);D(P)) for all
Conversely if u
u(nh)
n
integers n > 0 and satisfies (2.27), then it also satisfies equation
Indeed, if u
(2.26).
D(P), then (u,g) p
(Pu,g), and (2.27) can be
written as
[d
Since
Hp
+ Pu
QUn
g]
o,
nh < t < (n + l)h.
is dense in H, then u(t) is a solution of equation (2.26).
DEFINITION 2.2.
An abstract function u(t):
[O,m3
H is called
a weak solution of problem (2.26) if it satisfies the conditions:
(i)u(t) is continuous for t > 0 and strongly continuously differentiable for t > 0, with the possible exception of the points t
one-sided derivatives exist;
nh where
(ii)u(t) is continuous for t > 0 as an
abstract function with the values in
Hp
and satisfies equation (2.27)
on each interval nh < t < (n + l)h, for any function g(t):
H
[O,m3
(iii)u(t) satisfies initial condition (2.26), that is,
Clearly, a weak solution u(t) is also an ordinary solution if
u(t)
D(P), for any t > 0, and u(x,t)
u0(x
as t
0 not only
in
p
378
J. WIENER
It is said
the norm of H but uniformly as well.
[I0]
that a symmetric
operator P has a discrete spectrum if it has an infinite sequence
j}
of eigenvalues with a single limit point at infinity and a sequence
Xj
of eigenfunctions which is complete in H.
Suppose the operator P
in (2.27) is positive definite and has a discrete spectrum and assume
existence of a solution u(t)
u
dition u(0)
u(x,t) to equation (2.27) with the con-
On the interval nh < t < (n + l)h this solution can
0.
be expanded into series (2.20), where
coefficents
Tj(t),
we put g(t)
X
k
(u(t),
Tj(t)
To find the
Xj).
in (2.27) and since X
k
does not de-
pend on t, then
Xk
’dt
(u,
Xk) p
(Pu,
--dt
Xk)
(u,
T k(t),
(u(t),
Xk)
PXk)
k(U,Xk)
kTk(t),
which again leads to the equation
Tnj(t)
+
jTnj(t)
and to a generalization of (2.21).
(Qu
n
Xj)
By selecting a proper space H, a
weak solution corresponding to conditions (2.4) can be constructed.
The proof of the following theorem is omitted.
If P and Q are linear operators in a Hilbert space
THEOREM 2.4.
and P is positive definite with a discrete spectrum, then there exists
a unique weak solution of problem (2.26).
ACKNOWLEDGMENT:
Research partially supported by U.S. Army Grant
DAAL03-89-G-0107.
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i.
COOKE, K. and WIENER, J. Retarded differential equations with
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2.
COOKE, K. and WIENER, J. Neutral differential equations with
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3.
WIENER, J. and COOKE, K. Oscillations in systems of differential
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379
4.
SHAH, S.M. and WIENER, J. Advanced differential equations with
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COOKE, K. and WIENER, J. Stability regions for linear equations
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i2A (1986), 695-701.
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9.
NAIMARK, M.A.
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MIKHLIN, S.G. Linear Equations in Partial Derivatives (Russian),
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