x (t ax(t + bx([t]), (1.1)

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1 Internat. J. Math. & Math. Sci. VOL. 15 NO. 2 (1992) A PARABOLIC DIFFERENTIAL EQUATION WITH UNBOUNDED PIECEWISE CONSTANT DELAY JOSEPH WIENER Department of Mathematics The University of Texas-Pan American Edinburg, Texas LOKENATH DEBNATH Department of Mathematics University of Central Florida Orlando, Florida (Received February 26, 1991 and in revised form May 1, 1991) ABSTRACT. A partial differential equation with the argument [Z.t] is studied, where [.] denotes the greatest integer function. The infinite delay -[ Zt] leads to difference equations of unbounded order. KEY WORDS AND PHRASES. Partial differential equation, piecewise constant delay, boundary value problem, initial value problem AMS SUBJECT SIFICATION CODE. 35A05, 35B25, 35L10, 34K 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. This paper continues our earlier work [1-5] in an attempt to extend this theory to differential In these papers, ordinary differential equations having equations with discontinuous argument deviations. 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 problems. Indeed, we consider the equation x (t ax(t + bx([t]), (1.1) where It] denotes the greatest 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 > 0 and H(t) 0 for < O. If we admit distributional derivatives, then differentiating the latter relation gives x"(t)-ax (t)+ Y. bx(i)(6(t-i)-6(t-i- I)), (1.3)

2 340 J. WIENER AND L. DEBNATH where 6 is the delta functional. This impulse equation contains the values of the unknown solution for the integral values of t. Within intervals of certain lengths, differential equations with piecewise constant argument (EPCA) describe continuous dynamical systems. Continuity of a solution at a point joining any two consecutive intervals implies recursion relations for the values ofthe solution at such points. Therefore, EPCA are intrinsically closer to difference equations rather than differential equations. The main feature of equations with piecewise constant argument is that it is natural to formulate initial and boundary value problems for them not on intervals but at a number of individual points. In [6] boundary value problems for linearepca in partial derivatives were considered and the behavior of their solutions studied. The results were also extended to equations with positive definite operators in Hilbert spaces. In [7] initial value problems were studied for EPCA in partial derivatives. A class of loaded equations that arise in solving certain inverse problems was explored within the general framework of differential equations with piecewise constant delay. Integral transforms were successfully used to find the solutions of initial value problems for linear partial differential equations with piecewise constant delay. It has been shown in [6] and [7] that partial differential equations (PDE) with piecewise constant time naturally arise in the process of approximating PDE by simpler EPCA. Thus, if in the equation u,-a-u=-bu, (1.4) wich describes heat flow in a rod with both diffusion a2u,= along the rod and heat loss (or gain) across the lateral sides of the rod, the lateral heat change is measured at discrete moments of time, then we get an equation with piecewise constant argument u,(x,t) a2u,=(x,t) bu(x, nh ), (1.5) _[nh,(n +l)h], n -0,1 where h > 0 is some constant. This equation can be written in the form u,(x,t) a2u=(x,t) bu(x,[t/h ]h. (1.6) The purpose of the present paper is to investigate boundary value problems and initial value problems for linear PDE with the pieeewise constant argument kt/h ]h, where and h > 0 are constants and 0 <. < 1. Such equations are of both theoretical and applied interest. For instance, the equation y (t ay(t + by( kt (1.7) arises as a mathematical idealization ofan industrial problem involving wave motion in the overhead supply line to an electrified railway system. The profound study [8] of Eq. (1.7) has led to numerous works in this direction, some of which were reviewed in [9]. In particular, in [10] and [11] distributional and entire solutions were explored for general classes of equations of type (1.7) with polynomial coefficients. While of considerable importance in their own right, solutions of EPCA with the argument [.t/h ]h can be used to approximate solutions of equations of form (1.7) as h 0. Obviously, the lags.t and.t/h ]h become infinite as 2. MAIN RFULTS. We consider the boundary value problem (BVP) consisting of the equation Ou(x,t)+p(O) (O) u(x,[kt/h]h), (2.1) where P and Q are polynomials of the highest degree m with coefficients that may depend only on x, the boundary conditions Liu t.(mitu- )(0) + Ntu-l (1)) 0, (2.2) Ot x u(x,t)-q --

3 PARABOLIC EQUATION WITH PIECEW[SE CONSTANT DELAY 341 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), and h > 0, 0 <, < 1 are constants. Equations (2.2) can be written as Lu -0. Following [6], we introduce the following definition. DEFINITION 2.1. A function u(x, t) is called a solution ofthe 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, with the possible exception of the points (x, nh/.), where one-sided derivatives exist (n 0,1, 2,...); (iii) u(x,t) satisfies equation (2.1) in G, with the possible exception of the points (x,nh/.), and conditions (2.2)-(2.3). Let u,(x,t) be the solution of the given problem on the interval nh/k < (n + 1)h/., then where We next write which gives the equation and require that with a solution and the BVP where L is defined in (2.2) and (2.2 ). 8u,(x,t)/#t + Pu,(x,t) c.(x) u.(x,t) u(x, nh Qc,(x) w.(x,t) + v.(x) ow./ot + Pw. + Pv.(x) T,(t e- - /) corresponding eigenfunctions X(x) te C=[0,1], then the series Qc.(x) (2.2 ) P(d/dx)X- ox O, LX O (2.8) If BVP (2.8) has an infinite countable set of eigenvalues Ix and --t-ix.)..., w,(x,t)- t.,ce,itx), C.i- i- Ow./Ot + Pw. O, (2.5) Pv,,(x) Qc,,(x) (2.6) Assuming both w,, and v, satisfy (2.2 ) leads to an ordinary BVP (2.6)-(2.2 ), whose solution is denoted by v.(x)-p-qc.(x), and to BVP (2.5)-(2.2 ), whose solution is sought in the form w,(x,t) X(x)T,(t) (2.7) Separation of variables produces the ODE T, + txr, const (2.9) represents a formal solution of problem (2.5)-(2.2 ) and -. t.,.e P-Qc.(x) (2.10) o (2.4)

4 342 J. WIENER AND L. DEBNATH is a formal solution of (2.1)-(2.2). At nh/, we have where Since s.(x) c.(x) + e- Qc.(x) so(x) Co(X) u0(x), (2.11) substituting the initial function u0(x) C [0,1 in (2.12) as n 0 produces the coefficients Coj, and putting them together with Uo(X) in (2.10) as n -0 gives the solution Uo(X,t) of BVP (2.1)-(2.3) on the interval 0 < h/.. Since Uo(X,h cx(x) and Uo(X,h/)) sx(x), we can find from (2.12) the numbers C1i and then substitute them along with (x) in (2.10) as n I, to obtain the solution ut(x,t) on h/. 2h/.. This method ofsteps allows to extend the solution to any interval nh/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 we get the recursion relations Finally, from (2.11) and (2.13) we obtain s. :(x -i.t C,e?/xXi(x + P-Qc.(x <2.13) s. (x) s.fx) -i. C.i(l e/ /fx/fx). This concludes the proof of the following theorem: THEOREM 2.1. Formula (2.10), with coefficients C.i defined by recursion relations (2.12), represents a formal solution of BVP (2.1)-(2.3) in [0, I] x [nh/k,(n + l)h/.], for n 0, I,..., ifbvp (2.8) has a countable number of eigenvalues ix and a complete orthonormal set of eigenfunctionsx(x) IE C [0, I] and the initial function u0(x) IE C [0,1] satisfies (2.2). The solution of Eq. (2.1) on nh/., < (n + 1)h/. can be also sought in the form where X(x) are the eigenfunctions of the operator P. between 0 and I and changing k to j, we obtain T,o (t) + IxiT.i(t) q, # I X(x )Q (a/ax )c.(x )ax whence s,(x) u,(x, nh/,) Therefore, assuming the sequence {X(x)} is complete and orthonormal in C [0,1 yields for the coefficients C,i the formula C,i- f (s,(x)-p-iqc,(x))xi(x)d.x, (n 0,1,2,...). (2.12) u.(x, t) i. X(x )T.i(t (2.14) c.(x) u(x,nh ), Upon multiplying (2.14) by X,(x), then integrating

5 PARABOLIC EQUATION WITH PIECEWISE CONSTANT DELAY 343 T.(nh/ s.(x(x s,(x)-u(x, nh/) e principal role of the operator P emerges - om e methods of constcting solution. t -0 where i are real-valued nctions of claes C" on 0 x 1 and() 0 on [0,1 ]. uming C [O, is embedded in L 0,1] with e inner,z)- y(x(x, BVP (2.8) is called self-adjoint if for all y,z te C"[0,1 (ey,z (y,ez that satisfy the boundary conditions Ly -Lz -0. If BVP (2.8) is self-adjoint, then all its eigenvalues are real and form at most a countable set without finite limit points. The eigenfunctions corresponding to different eigenvalues are orthogonal. The proof of the following theorem is omitted since it parallels the proof of Theorem 2.3 in [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 hold true. (i) BVP (2.8) is self-adjoint, all its eigenvalues ti are positive. (ii) For each t, the roots of the equation P(s) Ixi 0 have non-positive real parts. (iii) The initial function uo(x) E C"[0,1] satisfies (2.2). EXAMPLE 2.1. The solution u.(x,t) of the equation 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 in form (2.14). Separation of variables produces X(x) /sin(njx) and T,/(t)---a2j2T,i(t)+bT(nh), (nh/.t <(n + 1)h/.) (2.16) whence b T.i(t C.ie - it- x) + a_j= Ti(nh (2.17) The following remark is in order. The subindex n is omitted from the term T(nh) in (2.16) and (2.17) because the point nh does not belong to the interval [nh/,, (n + 1)h/, ]. Since 0 <.< I, the delay nh in Eq. (2.16) becomes infinite as +oo. As mentioned above, u.(x,t) is the restriction of the solution u(x,t) of problem (2.1)-(2.3) to the interval [nh/k,(n + 1)h/k]. Therefore, ff u(x,t) is sought in form (2.14), T.i(t) is the restriction of T(t) to the indicated interval. Furthermore, putting -nh/k in (2.17) gives

6 344 J. WIENER AND L. DEBNATH b T,,(nh/,)-C,,. + aj T(nh), whence and T jct) b % r././x)- a;0,_ r/,a,) T.]Cnh/k)e -* xb - /* b e,% _,o,/x))ti(nh + aj(1 ). (2.18) At (n "" 1)h/X we get from (2.18) We denote T,,i(h(n + 1)1.) e "T"i(nh/k)+a (l-e"a" i /X)Ti(nh) -.l Ai Ie - 22hi2/x Bi (2.19) then r.i(nh/,) t. T.i(nh s Sin contuity of e lution implies 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 is of bounded order bee it eontai s (), where [lk,(n + l/k), at,nis e teal ofn k. For smee, k- 1 en s (), t4 T4()- s,i, d is s zi A:, + B:.i Fuaheore, at ( + 1, foula (2.18) ves s,i e "i b s, + (1- e aj O 2. lb I< a2, en all netio (t) IS._l,jlsM.), It._,,jl :M. ), Aj+IBiI<q,. (2.20) becomes e expa0sion (2.14) for the lution of equation (2.15) with homogeneo unda nditio exnentially nd m ro as +. PROOF. note )- max (t) I, [(n 1, ), q -e +:a- and from (2.20) we get t. qg., while the condition b <,e: implies q < 1. By induction, we conclude from (2.20) that It,, +,i[ qm.), 1. Furthermore, it follows from (2.16) that on every interval [nh/x,(n + 1)/] the function T,,i(t)l attains its maximum at an endpoint of this interval. Hence, the inequality tt.,l q." ad to g3 q. >. Thfo. t2 qgtk and the proof is completed by lowering the subindex [1/X ] times successively. We also note that the functions T(t) decay slower for equation (2.15) than for the equation without delay

7 PARABOLIC EQUATION WITH PIECEWISE CONSTANT DELAY 345 u,(x,t) au=(x,t) + bu(x,t) (2.21) THEOREM 2.4. If 0 < b < aa2, then the functions T(t) tend to zero monotonically as +oo, and none of them has a zero in (0,,). PROOF. Assuming, for instance, T0j(0) > 0 we resort to equation (2.16) and the condition b < aa to show that the function Toj(t) is monotonically decreasing on [O,h/.). Moreover, since b > 0 we conclude 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) is decreasing and positive on [h/.,2h/. and it remains to use (2.20) successively to obtain the same result on each interval [nh/.,(n + 1)h/.]. THEORF 2.5. For b < 0, each function T(t) is oscillatory, that is, it has infinitely large zeros. PROOF. Assume that a certain function T(t) is nonoscillatory, say, positive for large t. Then - t and s,j are positive for large n, and therefore it follows from (2.20) that t, 1. <A/, with 0 <Ai < 1. Hence, % whereas s decays at a slower rate of A: as n oo. This contradicts (2.20) and proves that T(t) is oscillatory. This theorem reveals a striking difference between the behavior of the functions T(t) for equations (2.15) and (2.21) when b < 0: for equation (2.21) without t,i tends to zero faster than A as n -- delay, the T(t) are always nonoscillatory. THEOREM 2.6. If b > aa -m ", then the functions T(t) T,(t) are unbounded. 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 the following conclusion can be derived. THEOREM 2.7. If b I< a, then all functions T(t) for equation (2.22) exponentially tend to zero as +. If-a 2 < b < 0, then all T(t) tend to zero monotonically as +oo, and none of them has a zero in (0, oo). For b > 0, each function T(t) is bounded and oscillatory. ACKNOWLEDGMENT. This research was partially supported by U.S. Army Grant DAAL03-89-G-0107, and by the University of Central Florida. 1. WIENER, I. Differential equations with piecewise constant delays, in Trends in the Theory_ and Practice of Nonlinear Differential Eo_uations. Lakshmikantham, V. (editor), Marcel Dekker, New York, 1983, COOKE, K. L. and WIENER, J. Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99(1), (1984), SHAH, S. M. and WIENER, J. Advanced differential equations with piecewise constant argument deviations, Internat. J. Math & Math Sci. 6(4), (1983), COOKE, K. L. and WIENER, J. Neutral differential equations with piecewise constant argument, Bolletino Unione Matemati Italiana 7 (1987), COOKE, K. L. and WIENER, J. An equation alternately of retarded and advanced type, Proc. Amcr. Math. Soc. 99 (1987), WIENER, J. Boundary-value problems for partial differential equations with piecewise constant delay, lnternat. J. Math. & Math. Sci. 14 (1991), WIENER, J. and DEBNATH, L. Partial differential equations with piecewise constant delay, Intcmat.,l, Math. & Math. ScL 14 (1991), KATO, T. and McLEOD, J. B. The functional differential equation y (x) Amer. Math. Soc. 77 (1971), ay(gx) + by (x), Bull.

8 346 J. WIENER AND L. DEBNATH 10. II. SHAH, S. M. and WIENER, J. Distributional and entire solutions of ordinary differential and functional differential equations, Internat..I. Math. & Math. Sci. 6 (1983), WIENER,.I. Distributional and entire solutions of linear functional differential equations,,. Math. & Math. Sci. 5 (1982), COOKE, K. and WIENER, J. Distributional and analytic solutions of functional differential equations, I. Math. Anal. Ap_pl. 98 (1984), II

9 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from Qualitative Theory of Differential Equations, allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Center for Applied Dynamics Research, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Hindawi Publishing Corporation