Boundary value problems for the diffusion equation with piecewise continuous time delay

Internat. J. Math. & Math. Sci. VOL. 20 NO. (1997) 187-195

187

BOUNDARY

VALUE PROBLEMS

FOR

THE DIFFUSION

EQUATIONWITH

PIECEWISE CONTINUOUS TIME DELAY

JOSEPH WIENER

Department

ofMathematics Uraversity ofTexas PanAmerican Edinburg,Texas78539, U.S.A

LOKENATH DEBNATH

Department

of Mathematics Umversity of Central Florida Orlando,Florida32816,USA

(ReceivedNovember 30, 1995 and inrevisedformJanuary20, 1996)

ABSTRACT. A study is made of partial differential equations with piecewise constant arguments.

’Boundary

valueproblemsfor three types of equationsarediscussed delayed; alternately ofadvanced and retarded type; and most importantly, an equation of neutral type (that is, including the derivative at differentvaluesoftimet).

KEY WORDS AND PHRASES: Partial differential equations, piecewise constant arguments, oscillations,andstability.

1991AMSSUBJECTCLASSIFICATION CODES: 35A05, 35B25, 35L10,34K25.

1. INTRODUCTION

Functional differential equations (FDE)with delay provide a mathematical model fora physical or biological system in which therateofchange ofthe systemdepends uponits pasthistory. Thetheory of FDE with continuous arguments is well developed and has numerous applications in natural and engineering sciences. This paper continues our earlier work in an attempt to extend this theory to differential equations with discontinuous argument deviations. In

articles

[1-5], ordinary differential equations with argumentshavingintervals ofconstancy have been studied. Such equations representa hybrid ofcontinuous and discrete dynamical systems and combine properties of both differential and difference equations. They include as particular cases loaded and impulsive equations, hence their importanceincontroltheoryand in certain biomedicalmodels Continuity ofasolutionat apoint joining any two consecutive intervals implies recursion relations for the values of the solution at such points. Therefore, differential equationswith piecewisecontinuousargument

(EPCA)

areintrinsically closer to difference rather thantodifferentialequations. In [6]boundaryvalueproblemsforsome linearEPCAin

2. ACOMPARISONOF TWO EPCA Theequation

ut(z,t)

a2u=(z,t)

bu(z,t)

(2.1)

describes heatflow ina rod with both diffusion

a2uzz

alongthe rod andheat loss (orgain) across the lateral sides of the rod. Measuring the lateral heat change at discrete moments oftime leads to the equationwithpiecewisecontinuousdelay

ut(z,t)

a2u(x,t)

bu(z,

[t]),

(2 2)

which wasinvestigatedin[6]. Here

[-

designatesthegreatest integerfunctionand

(x, t)

E

[0, 1]

x

[0,

oo) Theproblem posedin[6]forEq. (2.2)consistsoftheboundaryconditions

(0. t)

0.

,(. t)

0 (23)

and the initial condition

and the solution issoughtintheform

(, o)

uo(z),

(2 4)

u(z, t)

X(z)T(t).

(2,5)

Thenseparation ofvariablesleadstothe BVP

X’

+A2X

O, withtheorthonormalsetof solutions

X(O)

X(1)

O, (2.6)

X(z)

V

sin(zrjz) (2.7)

on

[0, 1],

andtotheequation

r(t)

a27r2j2T:(t)

bT:([t]).

(2.8)

Let

T(t)

denote a solutionof

(2.8)

onthe intealn

<_

t

<

n

+

1, where n is anonnegative integer Then

T:(t)

a2r2j2T.j(t)

bT.j(n),

(2.9)

and the general solution ofthisequationis

T(t)

C.e

--Weputheret nand get

thatis,

where

At n

+

1 we have

b

a27r2.7.

T,j(n). (2.10)

T(t)

G(t- ,)T,(,).

(2.11)

T,,(n

+

1)

G(1)T,3(n)

E(t)

_oe,

b

(

_o,,).

BOUNDARYVALUEPROBLEMS 189

and since

T,a(n

+

1)

T,+,,a(n +

1), then

T+,a(n

+

1)

Ea(1)Ta(n)

and

Ts(n

E(1)Toa(O).

(2 13)

Therefore,

and

To(t)

Ea(t

n)E;(1)To,(O)

u,,(x,t)

Z

E’(1)Ta(O)Ea

(t

n)sin(rrjx),

3=1

(214)

(2 15) where

un(x,t)

designatesthe solution ofBVP (2.2), (2.3), (2.4) in

[0,1]

x

[n,n

+

1].

Putting 0, n 0gives

uo(x)

T0a(0)V

sin(Trjx)

=1

To,(O)

V/

uo(x)sinQrjx)dx. and

AlongwithEq. (2.2),westudytheequation

([1

u,

(z, t)

a

u=

(z, t)

bu z, t

+

-

(2 16)

underconditions(2.3), (2 4)

DEFINITION. A function

u(x,t)

is said to be a solution ofthe above BVP if it satisfies the conditions (i)

u(x, t)

is continuousinG

[0,1]

[0,

oo);

(ii)utand

u

exist and arecontinuous inG, withthe possible exception ofthe points

(x,

n

+

),

whereone-sided derivatives exist

(n

0,1, 2,

...);

(ii)

u(x,t)

satisfiesEq. (2.16)in

G,

withthe possible exception of the points

(x,

n

+

1/2),

andconditions (2.3),(2.4).

Again, the solution is soughtinform (2.5),and separation ofvariables generates the eigenfunctions (2.7)andleadstotheEPCA

T’(t)

a27r2j2T(t)

bT

(

[t+l

)

(217)

\L .j/

Eq.

(2.17)

isofconsiderableimerest, sincethe argument deviation 1

(t)

t-

t+

changesthesignineach interval

(n

,

n+),

withintegern. Indeed,

r(t) <

0 forn

<

<

nand

r(t)

>

0 for n

<

t

<

n

+

,

which meansthat Eq. (2.17)is alternately ofadvanced and retarded type. Assume that

T(t)

is a solutionof(2.17) on theinterval

[n- 1/2,n

+

].

ThenEq. (2.17) changesto

and since

then

Tn+l,,(:)

Ea(;-

rz-

1)Tn+l,2(r/,

+

1),

(1)

()

T,.,

n+-

E

Tr+.a(n

+

1).

Furthermore,continuityofthe solution

T(t)

implies

andtherefore

whence

From here,

and

Ea(l/2)

Tn/l’a(n+l)

Ea(-

1/2)T,a(n).

1

(218)

where

E(t)

is definedby

(2.12).

TItEOREM2.1. For

(2.21) the solution(2.15)ofEq. (2.2)tendstozero as oo,uniformlywith respect to z

PROOF. From (2.15)itfollows thatthe assertion is true if

IE(1)I

<

1. Solvingtheinequalities 1

<

e_,,,

b

(

a,,)

a27r3,

2 1

e-

<

1

forbprovesthe proposition. Furthermore,

u(x, t)

approacheszero withanexponentialrate

TIIEOREM:L2. For

b

>

-a27r

2

(2.22) the solution(2.20)of

Eq. (2.16)

tendstozero ast oo,uniformlywithrespectto x.

PROOF. From (2.20)we seethat theassertion is valid

iflEj(/2)E(

/2)

<

,

where

E(t)is

given by(2.12). Consideringthe case

Ea(

1/2)

>

0gives

b

>

2#ae

’

/

(e

’

1),

(2.23)

1

T,.,a(t)

Ea(

n)E’

(

)

E’

(

-

)

Toa(O).

(2.19)

The solution

un(z,t)

of BVP (2.17), (2.3), (2.4) in theregion

[0,

1]

x

[n-

,n+1/2]

is givenby the formula

u(x,t)

Z

v/E;

E;

To,(O)&(t-

n)sin(rjx), (2.20)

BOUNDARYVALUE PROBLEMS 191 withthe notation

i/A2

a2

71_2

j2/2.

(2 24)

From

E2(1/2

<

E2(

1/2)

weobtain theinequality

b

>

2#2

(225)

which is stronger than(223), and so(2 23)should beomitted Furthermore,

E2(1/2

>

E2(

1/2)

implies that

b

>

-21.z2(e

m

+

e-m)/(e

m

+

e-m-2), and since this inequality is weaker than (2.25) it may be disregarded E

1/2) <

0 isequivalentto

b

<

2#je

u’/(e

u’

1),

On the other hand, the case

(2.26) andthe inequality

Ea

(1/2) >

E

1/2)

leadsto

b

<

2#.

The latterinequality maybeignoredbecause it isweakerthan(2.26) From

E(1/2) <

E(

1/2)

it

f611owsthat

b

<

2#,(e

m

+

e-u,)/(e

u,

+

_e-u,

2).

The latterinequality shouldbe omitted since (2.26)is morestringent. Onthe otherhand, (2.26)cannot holdtruefor all values ofj, and wemust retainonly(2.25). Thisinequalityis validforall j if(2.22)takes place. Comparing inequalities(2.21) and(2.22) shows that the stabilityintervalforEq. (2.17) islarger than thestabilityimervalforEq. (2.8).

TEIEOREM2:.3. Each solutionofEq. (2.8)hasazerointheinterval

In,

n

+

1]

if

The same istrueforEq (2.17)if

and

PROOF. From

Eq.

(2.13)wehave

(,

+

)

7/()T0(0)

T,,,(n)T,,,(n +

1)=

g2n+l

(1)T.(O).

Hence,

T,v(n)T,,,(n

+

1) <

0 if

E(1)

<

0.

The latterinequalityisequivalentto

b

>a27r2j2/(ear’-1),

whichholdstrueforall3under condition(2 27).

inequality

Onthe otherhand, (2.28)isderivedby examiningthe

Inequalities(2.27)and(2 28)represent sufficient conditionsofoscillationfor

Eqs

(2 8)and(2 17),forall values ofjsimultaneously

THEOREM2.4. Each solutionofEq. (2.8)isnonoscillatory, forallj, if b

<

0 Foranyb

>

0 and sufficientlylargej, thefunctions

T(t)

areoscillatory.

PROOF. Theinequality

E

(1) >

0 isequivalentto

b

<

aTrgjg-/(e’_

l)

and holds true for all j only if b

<

0 Since its fight-hand side tends to zero as j

,

the above inequality breaks down forb

>

0and allsufficiently largej. Therefore,in this case the solutions

T(t)

of (2 8) oscillate, in sharp comrast tothe functions

T(t)

inthe Fourier expansion for the solution of the equation

u

ag-u

bu without timedelay.

TItEOREM2.5. Each solution ofEq. (2.17)isnonoscillatory, forall j, if

-aTr/(e’’/-l)

<b<0. (2.29) Foranyb

>

0and sufficiently largej, thefunctions

T(t)

areoscillatory Furthermore,if

then the functions

T

(t),

Tnm

(t)

oscillate.

PROOF. Fromtheinequalities

E(

1/2)

>

0and

E(1/2)

>

0itfollowsthat

2#e/(e

’

1)

<

b

<

2#/(e

,

1),

where_{# is}given by(2.24). Thelettofthese inequalities holdstreeforall j if b

>

-a27r/(e’’/- 1),

and thefight one is satisfied forall j if b

<

0. The inequalities

E(-

1/2) <

0 and

E(1/2)

<

0 are contradictory. Onthe otherhand,therightside oftheinequality

b

< 2#/(e

’

1)

tendstozero as j

--

o% andthereforetheinequality breaks down for any givenb

>

0 and allsufficiently largej. Moreover,theleftsideoftheinequality

-2#je

/(e

’

1) <

b

monotonicallydecreasesto ooas j oo. Hence,if thisinequalityfailsforj m, thesame istreealso forj

<

m.

TEIEOREM2.6. If,for some integerm

_>

1,

b=

a2Tr2m2(e’’m’+ 1)/(e

’’m’

1),

_(2.30) then the solution

Tin(t)

of

Eq.

(2.8)is aperiodic functionwithperiod2. The same istrueforthe solution ofEq (2.17)if

b 2/,

(e

+

e -u

/ (2

e

"

e-’"

),

(2.3 l) where#,isgivenin(2.24).

PROOF. Sincetheinitialvalue problem for

Eq.

(2.8)has auniquesolutiononeach unit interval with imegral endpoints,wehave onlytoshow that

T(n

+

2)

T,(,),

,

0, ,2,

BOUNDARY VALUE PROBLEMS 19 3

and

T(t)

E.(t- n)E(1)To.(O),

Try(t)

Em(t-

-

2)E,/

(1)To(0),

n<t<n+l

n+2<t<n+3.

Therefore, whence

At t n

+,

Eq (3.3)gives intervaln

<

t

<

n

+

1is Then the series

and conditions (2.3) and (2.4). This equation is of neutral type since it includes the derivative ut at

differentvalues oft.

Let

un(x,t)

be the solution of the given problem on the interval n

<

t

<

n

+

1. Then

un(x,t)

satisfies theequation

Ou(z,t)

a

+b

(z,n

+

at Oz (32)

withboundaryconditions(2.3)and the initial condition

,(:,,+)

,(:),

where

u(z)

is yetunknown. Thesolution of

(3.2)

issought inform(2.5), and separation ofvariables generatesBVP(2.6)witheigenfunctions(2.7)and theODE

T’,

,=aT,(t)

+

T’,(,+),

un(x,t)

E

X3(x)Tn(t)

(3 4)

3=1

represents a formal solution of problem(3.2),

(2.3).

Turning toEq. (3.3), itsgeneral solution onthe

A 7rj. (3.3)

b ₊

m,(t)

c.,-’,

(’-")

+

T,;,(,

).

2

+ a

,

T,(n),

T,.,,(n

)=b-1

b

Tna(n).

T(t)

CeV,

(’-)

+

bZ

₁

Since hypothesis (230) implies

E,(1)

1, then T,

(n

+

2)

T,(n),

which proves the theorem for Eq (2.8) The proof for Eq (2 17) is analogous Condition (2.31) is equivalent to

Em(1/2)

E,(

1/2)

Itremainsto notethatthe value orb in(2.30)ispositive, and the valueorb in(2.31)isnegative.

3. APARABOLICEPCAOF NEUTRAL TYPE

Consider theboundaryvalueproblem(BVP)consisting oftheequation

ut(x, t)

a2uzz(x,

t)

+

but(x,

[t])

(3 1)

At nand n

+

2theseformulasrespectively yield

At n,wehave

that is,

and

Denote

At n

+

1,wehave

1

C,,

1 b

T,(n)

l

(b e-d’(t-’))T,.u(n)

T’v(t)

b- 1

F3(t)

_{b- 1} (3.5)

T,.u(n

/

1)=

F3(1)T,(n

and since, by hypothesis(i)oftheabove definition,

T,,3(n

+

1)

Tn+lj(n

+

1),

then

and

T,.,+,a(n

+

1)

Fa(1)T,2(n)

Ta(n)

Ff’

(1)Toa(0).

(3 6)

Consequently,

T,.,(t)

Fj(t-

n)Fj(1)To.(O)

and

(3 7)

u,(x,t)

E

VF;(1)Ta(O)Fa

(t

-n)sin(rjx). (3.8) Thefollowing theoremsillustratethefarmorecomplicatedsolution

tructure

of

Eq.

(3.1)comparing withthe diffusionequationwithout timedelay.

TREOREM3.1. Forb

>

1, eachfunction

T,(t)

ismonotoneunboundedas

PROOF. If b

>

1, then (3.5) implies that

F:(t)>

1, for all t

>

0. Hence,

F"(1)

grows exponentiallyast

+

oo,and theprooffollows from(3.7).

TI]EOREM3.2. For

(1

+

e

-Lx)

<

b

<

1, (3.9)

2

eachfunction

T,,

(t)

isunboundedandoscillating.

PROOF. Frominequalities(3.9)itfollowsthat

F3(1)

<

1, andtherefore

FJ

t]

(1)

isunboundedand hasazeroineachunit interval withinteger endpoints.

"rl]EOREM3.3. Forb

<

0, each function

T,(t)

monotonically tendstozero as PROOF. Theinequalities

0

< F(1) <

1 (3 10)

BOUNDARYVALUEPROBLEMS 195

Clearlythe right-hand sideof(3 11) vanishes asj oo, and therefore theconditionb

<

0 isnecessary and sufficientforall

Tn3(t)

toapproachzero monotonically.

TItEOREM3.4. If

1

a2A

>In2 and e-’2x <b<

,

(3 12) then every solution(3 7)tendstozeroand oscillatesast

+

oo

PROOF. Fromtheinequalities

-1<F3(1

<0, (3 3)

whichimply asymptotic stabilityandoscillatorybehaviorofsolution(3 7),itfollowsthat

e-’’-a,

<b<

l+e-These inequalities holdtrueforall jsimultaneouslywith(3.12)

TtlEOREM 3.5. Foranycoefficientbthat satisfies theinequalities 1

-<b<l, 2

there existinfinitely manysolutions

T,3

(t)

whichareunbounded andoscillating.

PROOF. For b

<

1, the inequality

F(1) <

1 whichimplies unboundedness and oscillation of

T3(t

),

isequivalentto

Clearlythefight-handsideapproaches

1/2

as j oo.

THEOREM3.6. Ifb

<

e

-ax,

thenthesolutions

T1

(t),

T,(t)

of

Eq.

(3.3)monotonically tendto zeroas -+

+

oo.

ACKNOWLEDGEMENT. Thisresearchwaspartially supportedby NASAGrantNAG9-553

[]] [2]

[3] [4]

REFERENCES

AFTABIZADEH,

A.R. and

WIENER, J.,

Oscillatory properties of first order linear functional differential equations,ApphcableAnal.20(1985), 165-187.

AFTABIZADEH, A.R.,

WIENEK, J. and

XU, J.-M.,

Oscillatory and periodic solutions of delay differential equations withpiecewise constantargument, Proc. Amer.Math. Soc. 99 (1987), 673-679.

COOKE, K.L. andWIENER, J., Retarded differentialequationswithpiecewiseconstantdelays,J. Math. Anal.

App.

99(1984),265-297.

COOKE, K.L. and

WIENER,

J., Anequationalternatelyof retarded and advanced type,Proc.Amer. Math.Soc. 99(1987),726-732.

[5]

WIENER,

J. and

COOKE, K.L.,

Oscillations in systems ofdifferential equations with piecewise constantargument,J.Math.Anal.Appl.137(1989),221-239.

[6]

WIENER,

J., Boundary-value problems for partial differential equations with piecewise constant delay,lnternat.J.Math.andMath.Sci. 14(1991),301-321

[7] WIENER, J. and DEBNATH, L., Partial differential equations with piecewise constant delay, Internat. J.Math. and Math.Sct. 14(1991),485-496.