Periodic solutions to systems of reaction-diffusion equations

Periodic Solutions to Systems of Reaction-@@sion by GERALD

Equations*

ROSEN

Defartment of Physics Drexel University, Philadelphia, Pennsylvania

: In

ABSTRACT

temporally reaction

this paper, necessary and suficient

periodic

“dissipative

rate terms dominant

structure”

in a generic system

Di V2 ci + Qi(c), where the enumerator tion or density

of reaction--diffusion

molecular or biological

and Qi(c), an algebraic

expresses the local rate of production

are derived for the existence

in cases of weak diffusion equations

of

with the

hi/at =

index i runs 1 to n, ci = c*(x, t) denotes the concentra-

of the ith participating

constant for the ith species

conditions

solutions

function

of the ith species

species,

Di is the diffusivity

of the n-tuple

due to chemical

c = (cl,..,, reactions

c,),

or bio-

logical interactions.

I.

Introduction

In a recent paper (1)the author has delineated the regime of applicability of the weak diffusion expansion (WDE) (2) for solutions to a generic system of reaction-diffusion equations &,/at = DiV2ci+Qi(c),

(1)

where the enumerator index i runs from 1 to n, ci = c&x, t) denotes the concentration of the ith participating molecular or biological species, Di is the diffusivity constant for the ith species and Q*(c) is an algebraic function of the n-tuple c = (cl, . . . , en) that expresses the local rate of production of the ith species due to chemical reactions or biological interactions. In cases for which the reaction rate terms dominate the diffusion terms in (l), the general solution to the initial value problem [with ci(x, 0) = ai assumed prescribed] can be given formally as an expansion in ascending powers of the Di’s, (2)

x

sIC$[s,

a(x)] D, V2ck”‘[s,a(x)] ds + O(D2).

(2)

In this so-called weak d@~sion expansion (WDE) given by (2), c:O) = clO)[t, a] denotes the solution to the system of ordinary differential equations ac;O’/at = Qi(c’o’) * This work was supported

by NASA

Grant NSG

307

(3) 3090.

Gerald Rosen subject to the initial conditions c:O)[O,at] = O(,~, and the associated quantities in (2) are defined by Eij[t, a] = @)[t, a]/&+ the quantities &![t, a] denoting the inverse matrix array. Rigorous upper bounds on the absolute value of the difference between the exact solution and the leading terms in the WDE for an approximate solution have been established, and it has been shown by example that the leading terms in the WDE provide an approximate solution which can be very accurate for an appreciable duration of time (1). The purpose of the present paper is to apply the WDE solutional method to the generic initial-value problem for temporally periodic but spatially nonuniform “dissipative structure” (3,4) solutions to systems of the form Eq. (1). Necessary and sufficient conditions for a periodic solution to the system of ordinary differential equations (3), c:O)[t+ T(O), a] = clO)[t,u], have been derived in the mathematical literature (5) for cases with n = 2 and rate expressions essentially restricted to a general quadratic form. Assuming that the system of ordinary differential equations (3) admits a periodic solution c:O)[t+ T(O), a(x)] = cjO)[t,a(x)] for all values of x in a spatial region R with a(x) prescribed suitably, necessary and sufficient conditions that must be satisfied by a(x) for the WDE solution (2) to yield a dissipative structure solution c&x, t + T) = ci(x, t) to first order in the diffusivity constants are given by (10) below, as shown by the analysis which follows. ZZ. Derivation of the Necessary Periodic Solution

and Suficient

Conditions

on

a(x) for a

For weak diffusion the period of the solution is analytic in the DiJs about D = 0, and thus we have T = T(O) + &T;l) where T!l) z E (8T/aD z.)D-O’

D, + O(D2),

(4)

Thus CJX, t + T) = ci(x, t) implies that

C&Gt + T(O)) + [ac,(x, t + T’“‘)/c%]k&Tp, D, + O(D2) = ci(x, t)

(5)

and by virtue of the induced periodicity fij[t +T(O), a(x)] = &[t,a(x)], follows from (2) that necessary and sufficient conditions are

4 =

kll[Qi(c(o)P9 WI) T,Y+ j$ij[t,a(x)1 ~,d~$b =0

to first order in the diffusivity

(6)

constants, where f,-,l[s, a(x)] V2 cfp)[s, a(x)] ds

(7)

are quantities independent of t as a consequence of the periodicity integrand. Now with the notation Qi j(~) = aQ,(C)/acj, we have

attiLt, al/at =

2 Qi,l(C'O') &&

I=1

308

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of the

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Periodic Solutions to Xystems of Reaction-Diffusion

Equations

owing to the definition &.(t, a) = &~:~)(t,a)/hi. Thus, the time derivative of the quantities defined in (6) can be expressed simply, and in fact we find aFi n at = lp,r(C(“))

4.

Hence, if Fi = 0 for all i at any instant of time, we must have (6) satisfied for all i for all values of time because of the linear form of (9). Requiring (6) to hold at the instant t = 0 produces the equivalent but drastically simplified necessary and sufficient conditions on a(x) for a dissipa,tive structure solution

5 (Baa)

k=l

T$) + P,k(x))Dk = 0,

(10)

where the constant Til)‘s are disposable. We illustrate (10) by way of example in the following section.

the application

of

III. An Illustrative Example

Consider the system (1) with n = 2 and the rate expressions &1(c) = MC, - as), k,, k,, a,, a2 = positive constants, for subject to ciO)[O,a] = cq is given by Cl(O) =

a,

+

(01~ -al)

cos

4?,(e) = ~,@I - Cl)> which

the

general

(11) solution

(3)

wt + (kl/k2)*(a2 - u2) sin wt, (12)

(O)= a2 + (01~- a2) cos ot - (k,/k,)*(or, - a,) sin ot, I 52 where w = (k, k2)* = 2nlT to). By differentiating the functions respect to 01~and 01s,we obtain the associated quantities (&A =

to

cos cot

(k,/k,)+ sin wt

- (k,/k,)* sin wt

cos wt

(12)

with

(13)

and the inverse matrix cos wt

(“” Thus, the quantities

=

(14)

(k2/kl)*sin wt

(7) are evaluated by integration

as

qk(x) = frT’O’ V2 Ed

(15)

for both values of k = 1,2, and hence the necessary and sufficient conditions (10) b ecome 2(& k&fW&(x))

where

= V2 4x),

A = - (k, k,)* 2 T’p’ Dk/(D1 + D2) T(O) k=l

Vol. 301, No. 3, Ifarch

1976

(16)

(17)

309

Gerald Rosen is a disposable constant. A particular solution to (16) with (17) assumed fixed and positive and the rate expressions given by (11) is al(x) = a,+fikfexp(h*u*x)cos(X*u.x+8), cab

= a,-fiE~exp(X+u*x)sin(h+u.x+6),

(13)

1

where u is a constant unit vector, u. u = 1, and & 6 are additional (unconstrained) constants of integration, the general solution to (16) for fixed (eigenvalue) h being obtained by linear superposition of solutions of the form (18). The period of the associated solution follows from (4) and (17) as T = T(O)[l - (k, k,)+(D,

+ D,) A] + O(D2).

(19)

The latter formula also holds if X is negative, with X* replaced by 1h I* in both members and - j3 replaced by + /3 in the second member of (18). For validity of the weak diffusion approximation theory over a time duration large compared to w-l, X must be chosen such that (1) (12,k,)-*(II, + D,) 1h 1< 1. That the initial values (18) do indeed yield periodic solutions to the system (1) with (11) to first order in D, and D, can be checked by evaluating the integral terms in (2) with (12), (14) and (18). Alternatively, it can be noted that an exact solution to (1) with (11) is given by cl(x,t)

= a,+/3,kfexp(X*u.x)cos(X*u.x+6+&),

c,(x,t)

= a,-/3,kaexp(hfu*x)sin(X*u*x+S+Qt),

I

(20)

with and /32//3l = [I + (k, k,)-l(D,

- DJ2h2]* + (k, ks)-“(02 -01)

x.

Initial values of the form (18) obtain if t is put equal to zero and terms of order (k, k,)-+DiX are neglected in the /$‘s in the solution (20). The period of the exact solution (20) is T = 27r/Q = 2n(E1 Ic,)-*[l

+ (k, k,)-+(D, +D,) h+ O(D2)]-l,

in agreement with (19). For facility and transparency of calculation, we have illustrated the application of conditions (10) with the linear example (11). Systems of reaction-diffusion equations with rate expressions that are essentially nonlinear in the ci’s are amenable to treatment if the associated ordinary differential equations (3) can be solved analytically. In this category we find many nonlinear forms, as exemplified by sets of rate expressions that relate algebraically to the solvable linear forms (11) (1). The necessary and sufficient conditions for periodic solutions, given here in (lo), can be developed explicitly and solved immediately for such nonlinear rate expressions, as well as for other forms which admit analytical solutions to (3).

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Journal of The Franklin Institute

Periodic Xolutions to Xystems of Reaction-Diffusion IV.

Formal General

Statement Case

of the Necessary

and Su.cient

Conditions

Equations in the more

If the magnitude of one or more of the diffusivity constants in (1) is such that the assumption of weak diffusion does not hold for the dissipative structure solutions of interest, then the conditions (10) for periodic solutions must be superseded by more general equations in ai( A statement of such necessary and sufficient conditions for periodic solutions in the general ca,se is obtainable from the formal solut’ion to the initial value problem (6) associated with t)he reaction-diffusion equations (l), ci(x, t) = exp (tA) oli(x),

where the functional

differential operator

A=

2

(“1)

(7)

{DiVe.(y)+Q,(a(y))}~dS.

(22)

i=l s

For a periodic solution with ci(x, t + T) = ci(x, t), it follows from (21) that the ai must satisfy the partial differential equations [exp (Th) - l] cq(x) = 0,

(23)

where the constant T is a disposable (eigenvalue) parameter. In order to develop the latter equations explicitly for a prescribed set of rate expressions, one must use specialized exponential operator computationa, techniques (6). V.

Summary

Necessary and sufficient conditions have been derived for the existence of structure” solutions in cases of weak temporally periodic “dissipative diffusion with the reaction rate terms dominant in a generic system of reaction-diffusion equations &,/at = Di V2 ci + QJc), where the enumerator index i runs 1 to n, ci = ci(x, t) denotes the concentration or density of the ith participating molecular or biological species, Di is the diffusivity constant for the ith species, and Qi(c), an algebraic function of the n-tuple c = ($3 . . . . cm), expresses the local rate of production of the ith species due to chemical reactions or biological interactions. These necessary and sufficient conditions take the form of partial differential equations that must be satisfied by the initial values ci(x, 0) = a&x). Application of the conditions is illustrated by example for a specific set of rate expressions Q%(c). Finally a formal statement of the necessary and sufficient conditions is given in the more general case for which the assumption of weak diffusion does not hold. References (1) G. Rosen, “Validity of the weak diffusion expansion for solutions to systems of reaction-diffusion equations”, J. Chem. Phys., Vol. 63, pp. 417-421, 1975.

Vol.301,Xo. 3,March 1976

311

Gerald Rosen (2) G. Rosen, “First-order diffusive effects in nonlinear rate processes”, Phys. Letts., Vol. 43A, p. 450, 1973. Theory of Structure, Stability (3) P. Glansdorff and I. Prigogine, “Thermodynamic and Fluctuations”, Wiley-Interscience, New York, pp. 222-271, 1971. (4) G. Rosen, “Solutions to systems of nonlinear reaction-diffusion equations”, Bull. Math. Biol., Vol. 37, pp. 277-289, 1975. (5) H. T. Davis, “Introduction to Nonlinear Differential and Integral Equations”, Dover, New York, pp. 339-355, 1962. (6) G. Rosen, “Solution to the initial value problem for nonlinear classical field equations”, Lettere Nuovo Cim., Vol. 4, pp. 1301-1304, 1970. (7) G. Rosen, “Formulations of Classical and Quantum Dynamica. Theory”, Academic Press, New York, pp. 96-100, 1969.

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Journal of ‘IThe Franklin

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