A Tree Combinatorial Structure on the Solution of a ...

A Tree Combinatorial Structure on the Solution of a Delay Differential Equation: A Generating Function Approach M. Fatima Fabiao*, Paulo B. Brito^ and Antonio StAubyn** *CEMAPRE ISEG Technical University of Lisbon (T. U. Lisbon), 1200-781 Lisboa, Portugal "^ UECE ISEG Technical University of Lisbon (T U Lisbon) 1249-078 Lisboa, Portugal "Matemdtica Aplicada/USA ISA, Technical University of Lisbon (T U Lisbon) 1349-017 Lisboa, Portugal Abstract. This paper introduces a new approach for obtaining expUcit solutions for a first order Unear delay differential equation with constant coefficients. We conjecture that there is a generating function defined over of a specific class of polynomials in the delay that solves the equation, and prove in the main theorem that the conjecture is valid. We also show the advantage of our method as regards the traditional Method of Step Algorithm (MSA). Keywords: Delay Differential Equations, Method Step Algorithm, Generating Functions PACS: 02.30.Ks,02.30.Sa,02.30.Hq

INTRODUCTION Delay differential equations (DDEs) are a special class of functional differential equations. The most important results on existence, uniqueness, and the properties of solutions for linear and nonlinear DDEs can be found in [1, 2, 3]. In this paper we present a new approach to obtain the exact solution for a particular linear DDE with constant coefficients, based upon the generating function concept. The results presented in the Main Theorem concern the solutions of the Basic Initial Problem (BIP), •r),

t>0

te[-r,0],

"

where B and r are constants, r > 0 is the delay, and ^ (f) is a given continuous function on [-r, 0]. Assuming ^(f) is constant, on [-r,0], and applying the MSA to the BIP, the solutions x„{t) defined on A„ = ((« - l)r, nr],n> 1, reveal a kind of a tree structure for the solution, x(f), of the problem. This allows to formulate a conjecture concerning the solution for the BIP: x{t) is the generating function for a sequence of polynomials in the delay PJ{rB). In order to prove this, a new specific formulation of the solution of the problem is required. As far as we know, the approach via generating function is new to the relevant literature. When compared with the MSA, the advantage of the Main Theorem is to provide an exphcit formula for x„(f) on the An interval without the use of the information on all solutions Xn-i{t) defined on the previous intervals /l„_i. If we used the MSA, each solution x„ (f) would depend upon the solution x„_i(f). Our main theorem proves that this is possible if we introduce polynomials Pj{rB).

PRELIMINARIES By the Method of Step Algorithm, we can define the solution of

fx'(0=/(x(f-r)), \x{t) = (j>{t),

f>0 tG[-r,0]

on each interval An = [ ( « - l)r,«r], « > l,by

x„{t)=x„-i{{n-l)r)+

/

f{x„-i{s-r))ds,

(n-l)r

CP1048, Numerical Analysis and Applied Mathematics, International Conference 2008 edited by T. E. Simos, G. Psihoyios, and Ch. Tsitouras © 2008 American Institute of Pliysics 978-0-7354-0576-9/08/$23.00

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where xo(.) = ^ (.). For j = 1,2,3,4, letXj{t) be the solution of (1) defined on the interval Ay obtained with the MSA. If we assume ^(f) = ^(0) is a constant, then XI (0

=

(p{0)[l+^

X2{t) =

(^(0)

X3(0 =

m [ ^

X4(f)

HO)

2!

^

-Bt{\-Br)

{Brf 2!

+ ^{l-fBr)+Bt(l-Br+'-i{Br)')-^^2^ + ^{l-'-iBr)

+^

+

+1

^-^[l-fBr+ 0, where the derivative at f = 0 represents the righthand derivative. Two different types of conditions must hold. On one hand, we are concerned with the differenciabihty at each point t = nr, which will guarantee the continuity of the solution. On the other hand, we want x'{t) = Bx {t - r) to be satisfied on any interior point of An. In order to do that, we determine which conditions the iterates w^ (r) in equation (2) should satisfy in terms of 9 (f). Meaning (p'(0)=5(p(-r), x'n (t) = Bxn-i {t — r) for t G /!„,« > 1, where XQ = ^•

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(3)

We state two fundamental propositions which establish sufficient conditions on w": (r), in order for (3) to hold. From now on we consider ^{t) = ^ (0) = C for f G [-r, 0], where C is a real constant. Proposition 1 If wl (r) = C, w\ (r) = BC and w] (r) = 0/or j >1 (4) then xi{t)=Y,w]{r)f j>o

=

C{l+Bt)

is the solution ofproblem (1) defined on Ai = (0, r]. Proposition 2 For n>2 the solution x„ (f) = Lj>o^/ ('') t^' defined on each interval An, is obtained through the application of the following formulas, in the following order

and w"o+i(r) = wl(r) - £ [w^+i {r)-Mf'j (r)! {nry . Theorem The solution ofproblem (1) with {t) =C ift G [—r,0], can be written as Xir,t)=l^vjir)tJ, j>o for t > 0. The sequence Vj (r) is defined by Vj{r)=CjP]{rB), where the polynomials PJ (rB) are defined by

\l+Ltt'''-^ii

+ jr\

Pj('-B)=\l,

ifjn+l.

Hence, if we fix « G N, we can calculate x„ (t) with t GA„ = {{n-l)r,

nr] using

RJ

x„{t)=Y^Wj{r)t}=CY^-Pj{rB)t}, jl i>o

(6)

j>o '

where PJ (rB) are given in equation (5).

AN APPLICATION Suppose we want to study a population P(f) = {x{t),y{t)), where x(f) denotes the average height and ^(f) the average weight. It was observed that x{t) depends on the height of the previous generation through x'{t) = Bx{t - r), where r is the size (in units of of time) of a generation. We can determine explicitly the behavior of this variable regarding the fourth generation. This means that we want to compute X4(f), given x{t) = C for t G [-r, 0], where C is the average height. Using equation (6) we can determine it directly, without having to compute the the height for the previous generations. DJ

X4{t)=l^wj{r)t^ = Cl^-Pf{rB)tK j>0

i>oJ-

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where Pj (rB) are computed applying theorem 4. From equation (6), as P^irB) = l 0. i-rBy+'

Pf(r5) = l + t ^ ^ J ^ ( / + i y + i = l + ( - r 5 ) + | ( - r 5 f + | ( - r 5 )

Po^r5) = 1 + E ^ f ^ ' ' + ' = 1 + ^ ^ ^ + - ( - ' • 5 ) ' + - (-'•5)'• °^ ^ AQ ('•+!)' 2! 3! ^ ^ 4! ^ ' for J = 0,1,2,3,4, then we have

= c