Zt]leads to differenceequationsof unbounded order

A PARABOLIC DIFFERENTIAL

^EQUATION

WITH UNBOUNDED PIECEWISE CONSTANT DELAY

JOSEPHWIENER

Department

ofMathematics TheUniversityof

Texas-Pan

American

Edinburg,

Texas

78539 LOKENATHDEBNATH

Department

of Mathematics University of CentralFlorida

Orlando,

Florida 32816

(Received February

26, 1991and in revisedform

May 1, 1991)

ABSTRACT. A partial

differentialequationwiththe

argument [Z.t]

isstudied,where

[.]

^denotesthe

greatest integerfunction. The infinitedelay

-[ Zt]

leads to differenceequationsof unbounded order.

KEY WORDS AND PHRASES.

Partial differentialequation, piecewise^constantdelay, boundary^value

problem,

initialvalue

problem.

1991

AMS SUBJECT SIFICATION CODE. 35A05, 35B25, 35L10,

34K25.

1.

INTRODUCTION.

Functional differentialequations

(FDE)

^with

delay

providea mathematical modelforaphysicalor biological systemin which therateofchangeof thesystem

depends upon

itspast history. The theory of

FDE

with continuousargumentiswell

developed,

^andhasnumerousapplicationsin natural andengineering sciences. Thispapercontinuesour earlier work

[1-5]

in anattempttoextend thistheorytodifferential equationswith discontinuousargumentdeviations.

In

these

papers,

ordinarydifferentialequations having intervalsofconstancyhave been studied. Such equations representa

hybrid

^ofcontinuousand discrete dynamical systems andcombinepropertiesof both differentialanddifference

equations. They

include as particularcases loaded andimpulse

equations,

hence their

importance

incontrol

theory

andin certain biomedical

problems.

Indeed,we considertheequation

x’(t ax(t

+

bx([t]), (1.1)

where

It]

denotes thegreatest integer function,and write it as

x’(t)-ax(t)+ Y. bx(i)(H(t-i)-H(t-i- I)), (1.2)

where

H(t) I

for >0and

H(t)

0for <

O.

Ifwe admit distributionalderivatives, thendifferentiating thelatter relationgives

x"(t)-ax’(t)+ Y. bx(i)(6(t-i)-6(t-i- I)), (1.3)

340 J. WIENER AND L. DEBNATH

where 6 isthe delta functional. Thisimpulse equationcontainsthevalues of the unknown solution for the integralvalues oft. Within intervalsof certainlengths,differential equations^withpiecewise constant argument

(EPCA)

describe continuousdynamical systems.

Continuity

ofa solution at apointjoining

any

twoconsecutive intervalsimpliesrecursion relationsfor the values of the solutionatsuchpoints. Therefore,

EPCA

areintrinsicallyclosertodifferenceequations ratherthan differentialequations. Themainfeature ofequationswithpiecewiseconstantargumentisthat it isnaturaltoformulateinitial andboundary value problemsfor them not on intervals but at anumber ofindividualpoints.

In [6]

boundaryvalueproblems^{for linear}

EPCA

inpartialderivatives were considered and the behavior of their solutions studied. Theresults werealso extended toequationswithpositivedefiniteoperatorsin Hilbertspaces.

In [7]

^initialvalueproblemswere studiedfor

EPCA

inpartialderivatives.

A

class of loaded equationsthat arise insolvingcertain inverseproblemswasexploredwithin thegeneral framework of differentialequations^withpiecewiseconstant

delay. Integral

transformswere

successfully

used to find the solutionsof initial valueproblems^{for linear}partialdifferentialequationswithpiecewiseconstantdelay.

It

has beenshown in

[6]

and

[7]

^thatpartial differentialequations

(PDE)

^withpiecewiseconstanttime naturallyarise in theprocessofapproximating

PDE

by simpler

EPCA.

_Thus,if in theequation

u,-a-u=-bu, (1.4)

wichdescribesheatflowin a rod withbothdiffusion

a2u,=

alongthe rod and heat loss

(or gain)

acrossthe lateral sidesoftherod,thelateralheat

change

ismeasured atdiscretemomentsof time,then we

get

an equationwithpiecewiseconstantargument

u,(x,t) a2u,=(x,t) bu(x,

^nh

), (1.5)

_[nh,(n +l)h],

ⁿ -0,1 whereh >0issome constant. Thisequationcan be written inthe form

u,(x,t) a2u=(x,t) bu(x,[t/h ]h (1.6)

The

purpose

of thepresent

paper

is toinvestigate boundaryvalueproblemsand initial value

problems

for linear

PDE

withthepieeewiseconstantargument kt/h

]h,

^where

.

^andh>0are constants and0

<.

<1.

Such equationsareofboth theoretical andappliedinterest.

For

instance,theequation

y’(t ay(t

+

by(

kt

(1.7)

arises asa mathematical idealization of an industrial

problem

involvingwave motion in the overhead

supply

linetoan electrifiedrailway system. The

profound

study

[8]

^of

Eq. (1.7) has

led to numerous works in thisdirection,someofwhichwere reviewed in

[9]. In

particular,in

[10]

^and

[11]

distributional andentire solutionswere

explored

for generalclassesof equations of type

(1.7)

^{withpolynomial}coefficients. While ofconsiderableimportancein their ownright,solutionsof

EPCA

with theargument

[

^.t/h

]h

canbeused to

approximate

solutionsof

equations

^{of form}

(1.7)

^{ash 0.}

^Obviously,

^thelags

.t

and

.t/h ]h

become infinite as 2.

MAIN RFULTS.

We

consider the

boundary

value

problem (BVP)

consistingof theequation

Ou(x,t)+p(O)

^Ot

x ^u(x,t)-Q (O) -- u(x,[kt/h]h), (2.1)

where

P

and

Q

arepolynomialsof thehighestdegreem with coefficients that

may depend only

onx,the

boundary

conditions

Liu t. ^{(Mitu- )(0)}

⁺

^Ntu - ^l’(1))

^0,

^(2.2)

where

Mj,

^and

Nj,

are constants,j 1 m;and the initial condition

u(x,0) Uo(X (2.3)

where

(x, t)

tE

[0,1] [0, oo),

andh >0, 0<

,

<1are constants. Equations

(2.2)

can be written as

Lu

-0.

Following

[6],

^weintroduce thefollowingdefinition.

DEFINITION

2.1.

A

function

u(x, t)

iscalled a solutionofthe above

BVP

if it satisfies the conditions:

(i)

^u

(x, t)

is continuous in

G [0,1 [0, =); (ii)

^{8u and}

Otu/Oxt(k O,

1 m exist and are continuous in

G,

withthepossible exception^{of the}points

(x, nh/.),

where one-sided derivatives exist

(n

0,1, 2,

...);

(iii) u(x,t)

^{satisfiesequation}

(2.1)

in

G,

withthepossible exception^{of the}points

(x,nh/.),

^{andconditions}

(2.2)-(2.3).

Let u,(x,t)

be the solution of thegiven

problem

on the intervalnh/k <

(n

+

1)h/.,

then

8u,(x,t)/#t

+

Pu,(x,t) Qc,(x)

where

We

nextwrite

whichgivestheequation

andrequirethat

c.(x) u(x,

^nh

u.(x,t) w.(x,t)

+

v.(x) ow./ot

+

Pw.

⁺

^{Pv.(x) Qc.(x)}

(2.2’)

witha solution

T,(t e-’

^-’/)

and the

BVP

P(d/dx)X- oX ^{O, LX O (2.8)}

where

L

isdefined in

(2.2)

^and

(2.2’).

^If

^BVP (2.8)

has an infinitecountablesetof

eigenvalues Ix

^and

corresponding

eigenfunctions X(x) ^{tE C=[0,1],}

^{thenthe series}

--t-IX.)....,

w,(x,t)-

_i-

_t.,ce

,itx),

C.i-

^const

^(2.9)

representsaformalsolution of

problem (2.5)-(2.2’)

^and

-. ^{t.,.e P-Qc.(x) (2.10)}

Ow./Ot

+

Pw. ^{O, (2.5)}

Pv,,(x) Qc,,(x) (2.6)

Assumingboth

w,,

^and

v,

satisfy

(2.2’)

leads to anordinary

BVP (2.6)-(2.2’),

whose solution is denotedby

v.(x)-P-Qc.(x),

and to

BVP (2.5)-(2.2’),

whose solution is

sought

in theform

w,(x,t) X(x)T,(t) (2.7)

Separation

^ofvariables

produces

the

ODE

T,’

⁺

txr, o

(2.4)

342 J. WIENER AND L. DEBNATH

isaformalsolution of

(2.1)-(2.2). ^At

^nh/,wehave

s.(x) c.(x)

⁺

e-’Qc.(x)

where

(2.11)

Since

so(x) Co(X) u0(x),

substitutingthe initial function

u0(x) C’[0,1

in

(2.12)

^{as n} 0 producesthe coefficients

Coj,

^andputting themtogetherwith

Uo(X)

in

(2.10)

asn-0 givesthe solution

Uo(X,t)

^of

^BVP (2.1)-(2.3)

^onthe interval 0 <h/..Since

Uo(X,h cx(x)

and

Uo(X,h/)) sx(x),

^{wecan findfrom}

(2.12)

^thenumbers

C1i

^{and then}

substitute themalong^with

(x)

in

(2.10)

asn

I,

toobtain the solution

ut(x,t)

^onh/. 2h/.. This methodof stepsallowstoextend the solution toanyintervalnh/k ^ffi

(n

+

1)h/..

Furthermore, continuity of the solution

u(x,t)

implies

u.(x,(n

+

)h/.)=u,/(x,(n

+

)h/.)=s,/(x),

hence,at

(n

+

1)/h .

^wegetthe recursion relations

s. ^:(x -i.t ^C,e?/xXi(x

⁺

^{P-Qc.(x <2.13)}

Finally,from

(2.11)

and

(2.13)

^weobtain

s. ^{(x) s.fx)} -i. ^C.i(l e/’/fX/fx).

Thisconcludes the

proof

of thefollowingtheorem:

THEOREM

2.1. Formula

(2.10),

with coefficients

C.i

^defined

^by

recursion relations

(2.12), represents

aformalsolutionof

BVP (2.1)-(2.3)

ⁱⁿ

[0, I]

^x

[nh/k,(n

+

l)h/.],

forn

0, I,...,

ifBVP

(2.8)

has acountable numberof eigenvalues_{ix and a}

complete

orthonormal setof

eigenfunctionsX(x) ^IE C’[0, I]

andtheinitialfunction

u0(x) IE C’[0,1]

satisfies

(2.2).

Thesolutionof

Eq. (2.1)

onnh/., <

(n

+

1)h/.

can bealso

sought

inthe form

u.(x, t) i. ^{X(x )T.i(t (2.14)}

where

X(x)

^{are the}eigenfunctions ofthe

operator P. Upon multiplying (2.14) ^{by X,(x),}

then integrating between0and

I

andchanging k^toj,we obtain

T,o’(t)

⁺

IxiT.i(t) ^q,

#

’ I ^{X(x )Q (a /ax )c.(x)ax}

c.(x) u(x,nh ),

whence

s,(x) u,(x, nh/,)

Therefore, assumingthesequence

{X(x) ^}

^iscompleteand orthonormal in

C’[0,1

yieldsfor the coefficients

C,i

^theformula

C,i- f (s,(x)-P-IQc,(x))Xi(x)d.x, (n 0,1,2,...). (2.12)

T.(nh/ s.(x(x s,(x)-u(x, nh/)

e

principalrole of theoperator

P emerges om e

methods ofconstcting solution.

t

-0

where

i

are real-valued nctions of claes

C" -

^{on 0 x 1and}

⁽⁾

^{0 on}

^{[0,1 ].}

^uming

^C’[O,

isembedded in

L 0,1]

with

e

inner

,z)- y(x(x,

BVP (2.8)

iscalledself-adjointif

(ey,z (y,ez

forall

y,z

tE

C"[0,1

thatsatisfytheboundaryconditions

Ly -Lz

-0.

If

BVP (2.8)

isself-adjoint,then all itseigenvaluesarereal and form at most a countable set without finite limitpoints. Theeigenfunctions correspondingtodifferenteigenvaluesare

orthogonal.

The

proof

^{of the} followingtheorem is omitted since it

parallels

theproofof Theorem2.3in

[6].

THEOREM

2.2.

BVP (2.1)-(2.3)

has a solution in

[0,1] [nh/:k,(n

+

1)h/:k],

^{for each n} -0,1 given

by

formula

(2.10)

^{if the}following

hypotheses

holdtrue.

(i) BVP (2.8)

^isself-adjoint,all itseigenvalues

ti

^arepositive.

(ii) ^For

^each

t,

^{the rootsof theequation}

P(s) _Ixi

0 have non-positiverealparts.

(iii)

^Theinitial function

uo(x) E C"[0,1]

satisfies

(2.2).

EXAMPLE

2.1. Thesolution

u.(x,t)

of theequation

ut(x,t) a-u=(x,t)

+

bu(x,[

.t/h

]h (2.15)

in

[0,1][nh/k,(n

+

1)h/.],

with the

boundary

conditions

u,(O,t)-u,(1,t)-O

and initial condition

u,(x, nh/.) s,(x),

is

sought

inform

(2.14). Separation

^ofvariables

produces X(x) ^/sin(njx)

^and

T,/(t)---a2j2T, i(t)+bT(nh), ^{(nh/.t <(n}

⁺

^{1)h/.) (2.16)}

whence

T.i(t C.ie

^{-’it-’’x)+}

a_j=

b

^{Ti(nh (2.17)}

Thefollowingremark is in order. The subindex n is omitted from theterm

T(nh)

ⁱⁿ

(2.16)

and

(2.17)

because the

point

nh doesnot

belong

tothe interval

[nh/,, (n

+

1)h/, ].

^{Since 0}

<.

<

I,

the

delay

nh in

Eq. (2.16)

becomes infinite as +oo.

As

mentionedabove,

u.(x,t)

^isthe restrictionofthe solution

u(x,t)

of

problem (2.1)-(2.3)

^{totheinterval}

^[nh/k,(n

⁺

^{1)h/k]. Therefore,}

^ff

^u(x,t)

^is

^sought

^inform

(2.14), T.i(t)

^istherestrictionof

T(t)

^totheindicated interval.

Furthermore, putting -nh/k

in

(2.17)

gives

344 J. WIENER AND L. DEBNATH

whence

and

T,,(nh/,)-C,,.

⁺ b

aj T(nh),

% r././x)-

b

a

;0,_ ^r/,a,)

T|jCt)

T.]Cnh/k)e

^{-*’xb’-’/*}

At (n "" ^1)h/X

^wegetfrom

^(2.18)

We

denote

b

e,% _,o,/X))Ti(n

^h

+

aj(1 ^).

T,,i(h(n

⁺

1)1.)

^e

’"T"i(nh/k)+a’(l-e"A"’i’/X)Ti(nh)’-.l

(2.18)

(2.19)

IS._l,jlsM. ), It._,,jl’:M. ^’), Aj+IBiI<q,

andfrom

(2.20)

^weget

t. ^qg. ,

^whilethe condition b <

,e:

implies

q

<1.

By

induction,we con- cludefrom

(2.20)

^that

It,, +,i[ ^qM. ),

1.

Furthermore,

itfollows from

(2.16)

^thaton

^every

^interval

[nh/X,(n

+

1)/]

the function

T,,i(t)l

attains its maximum at anendpoint ofthis interval.

Hence,

the inequality

tt.,l q."

ad to

g3 q.’>. Thfo. t2 ^qgtk

^{and the}

^proof

^is

^completed

by

lowering thesubindex

[1/X ]

timessuccessively.

We

also note that the functions

T(t) ^decay

^slowerfor

equation

(2.15)

^thanfor the equationwithout

delay

then

Ai

^{Ie-’22hi2/x}

Bi

r.i(nh/,) t. T.i(nh

^s

Sin

contuity

of

e

lutionimplies

r.i(h(n

⁺

1)/k)-r.+,,i(h(n

⁺

^1)IX),

equation

(2.19)

^becomes

t.

,i

"Ait

⁺

B ^(2.20)

e

difference

uation (2.20)

^with

reset ^m t

^isof bounded order bee it eontai

s ^(),

where

[lk,(N

+

l/k), at

,Nis

e _teal

ofnk.

For smee,

k-

1 en s ^(), t4 T4()-

s,i, d is

. ^(2.20)

^becomes

s

zi

A:,

⁺

B:.i

Fuaheore,

at

(

+

1,

foula

(2.18) ves

s

,i e

"i

^b

s,

⁺

_aj (1-

e

O

2.

lb I< a 2, en

allnetio

(t) e

expa0sion

(2.14)

forthe lutionof equation

(2.15)

^with

homogeneo unda

^nditioexnentially nd

m

roas +.

PROOF. note

)-

^max

_(t) I, ^{[(n 1, ),}

q

-e

+:a-’

u,(x,t) aU=(x,t)

+

bu(x,t) (2.21) THEOREM

2.4. If 0<b<

aa 2,

thenthe functions

T(t)

^{tend to}zero monotonicallyas +oo,and none of them has a zero in

(0, ,).

PROOF. Assuming,

for instance,

T0j(0)

^>0we resort toequation

(2.16)

^andthecondition b<

aa

to show that the function

Toj(t)

^ismonotonically decreasing on

[O,h/.). ^Moreover,

^{sinceb>0weconclude}

from

(2.18)

^that

Toi(t)

^{>0 on}

[0,h/.]. Hence, tl

^{>0, Sl]>}

^0,

^{and from}

(2.20)

we see that

t

^{>0.Therefore,}

Ti(t)

^{isdecreasingandpositiveon}

^[h/.,2h/.

^{anditremains touse}

(2.20)

successivelytoobtainthe sameresult oneachinterval

[nh/.,(n

+

1)h/.].

THEORF 2.5. For

b<

0,

eachfunction

T(t)

^is

oscillatory,

that is,ithas

infinitely large

zeros.

PROOF. Assume

that a certain function

T(t)

^isnonoscillatory,

say,

positive for

large

^{t. Then}

t

^and

s,jarepositive forlarge n,andtherefore itfollowsfrom

(2.20)

^thatt, _1.

<A/,

^with0

<Ai

^<1.

^Hence,

t,i

^{tends tozerofasterthan}

A

^{as n}

--

^{% whereas}

^{s decays}

^{ata slowerrateof}

^A:

^{as n}

-

^{oo. This}

contradicts

(2.20)

^and

^proves

^that

T(t)

^isoscillatory. Thistheoremrevealsastrikingdifference between the behaviorofthe functions

T(t)

^forequations

^(2.15)

^and

^(2.21)

^{when b<0: forequation}

^(2.21)

^without

delay,the

T(t)

^are

^always

nonoscillatory.

THEOREM 2.6.

Ifb>

aa’-m",

thenthe functions

T(t) T,(t)

areunbounded.

EXAMPLE 2.2. For

the equation

u(x,t) au=(x,t)

⁺

bu=(x,[ .t/h ]h ), (2.22)

the functions

T/(t) ^satisfy

the relation

T,/(t -ajT(t bjs

from which thefollowingconclusion can be derived.

THEOREM 2.7.

If b

I< ^a’,

^thenall functions

T(t)

for equation

(2.22) exponentially

^tendtozero as +.If-a²<b<

0,

thenall

T(t)

tend to zeromonotonicallyas +oo,andnone of themhas a zero in

(0, oo). For

^b>

0,

each function

T(t)

^isbounded andoscillatory.

ACKNOWLEDGMENT.

This research was

partially supported by U.S. Army Grant DAAL03-89-G-0107,

and

by

theUniversity of CentralFlorida.

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Lakshmikantham,

V. (editor),

Marcel Dekker,

New York, 1983,

547-552.

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and

WIENER, J.

Retardeddifferentialequationswith

piecewise

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delays, J.

Math.Anal.

Appl. 99(1), (1984),

^265-297.

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SHAH, S. M.

and

WIENER, J.

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with

piecewise

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Internat. J.

Math

&

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671-703.

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and

WIENER, ^J. An

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Proc.

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⁹⁹

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301-321.

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DEBNATH, L.

Partialdifferential equationswith

piecewise

constant

delay,

Intcmat. ^,l,

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^&

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¹⁴

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y’(x) ay (gx)

+

by (x),

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346 J. WIENER AND L. DEBNATH

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and

WIENER, J.

Distributional and entire solutions ofordinarydifferential and functional differentialequations,

Internat..I.

Math.

&

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(1983),

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,.

^Math.

&

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Distributional andanalytic solutions of functional differential equations,

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