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Pergamon Mathl. Comput. Modelling Vol. 25, No. 819, pp. 181-193, 1997

Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

PII: SO8957177(97)00068-X 0895-7177/97 $17.00 + 0.00

The Role of Symmetries in Solving Differential Equations

M. C. NUCCI Dipartimento di Matematica, Universitk di Perugia

06123 Perugia, Italy nucci@gauss.dipmat.unipg.it

Abstract-A review of the role of symmetries in solving differential equations is presented. Af- ter showing some recent results on the application of classical Lie point symmetries to problems in fluid draining, meteorology, and epidemiology of AIDS, the nonclassical symmetries method is presented. Finally, it is shown that iterations of the nonclassical symmetries method yield new non- linear equations, which inherit the Lie symmetry algebra of the given equation. Invariant solutions of these equations supply new solutions of the original equation. Furthermore, the equations yield both partial symmetries 8s given by Vorobev, and differential constraints as given by Vorobev and by Olver. Some examples are given. The importance of using ad hoc interactive REDUCE programs is underlined.

Keywords-Lie symmetries, Computer algebra, Fluid mechanics, Medical epidemiology.

1. INTRODUCTION

The most famous and established method for finding exact solutions of differential equations is

the classical symmetries method (CSM), also called group analysis, which originated in 1881 from

the pioneering work of S. Lie [l]. Many good books have been dedicated to this subject and its

generalizations [2-l 11.

Although Lie point symmetries represent a very powerful tool, they can yield very lengthy

calculations. In fact, interest in them and their generalizations has increased during the last

twenty years because of the availability of symbolic computation packages.

In this paper, we review some of our recent work in the area. First, we illustrate Lie group

analysis with some examples of ordinary differential equations. We show a third-order ordinary

differential equation admitting a three-dimensional nonsolvable Lie symmetry algebra [12], a

third-order ordinary differential equation admitting a two-dimensional Lie symmetry algebra, and

a system of three first-order ordinary differential equations admitting a three-dimensional solvable

Lie symmetry algebra [13]. Each of these represents mathematical models of fluid draining [14],

meteorology [15], and epidemiology of AIDS [16], respectively.

Second, we illustrate the nonclassical symmetries method introduced by Bluman and Cole in

1969 [17] d h an s ow its application to the modified Korteweg-deVries equation [18].

Finally, we illustrate how to iterate the nonclassical symmetries method [19,20] for the purpose

of obtaining new solutions and show some examples.

A reliable and user-friendly symbolic computation program is the key to the widespread and

successful application of Lies method and its generalizations. Each given example has been

derived by using our own interactive REDUCE programs [21], which calculate the Lie point

symmetries, and the nonclassical symmetries.

Supported in part by Fondi M.U.R.S.T. 60% and 40%.

181

Typeset by &,+5-&X

182 M. C. NUCCI

2. APPLICATIONS OF LIE GROUP ANALYSIS TO ORDINARY DIFFERENTIAL EQUATIONS

2.1. Fluid Draining

The equation Wnt

-2 =w , w = w(s), (1)

represents the small-w limit of an equation which is relevant to fluid draining problems on a dry wall, and the large-w limit to draining over a wet wall [14]. Its general solution was given in 1221 by a lucky guess method. In [12] Lies method was applied to (1) and led to a three-dimensional nonsolvable Lie algebra LB with basis

Xl = 33, x.2 = sa, +w&, x, = s=iJ, + 2swa,. (2)

Let us consider the two-dimensional subalgebra spanned by X1, X2. A basis of its differential invariants of order I 2 is given by

u = w, v = wwl. (3)

Then, equation (1) is reduced to the following first-order equation:

dv

OdZL = 2Lv+ 1,

which admits the nonlocal operator Xs in the space of variables u, v, i.e.,

x3 = w(8, + ?&). (5)

We put (5) into its semicanonical form [12], i.e., X3 = wd,, by introducing the new variable

2L= z=qJ--- 2

which is obtained by solving

;++0.

Then, equation (4) becomes a Riccati equation in the variables z, 26, i.e.,

dzl U=

dx=-2-+e* (6)

Finally, the general solution of (6) is easily found in terms of Airy functions [22].

2.2. Meteorology

Let us consider the Lorenz system [15]

5 = o(y - x), (7) y=-0z-+Tz-~, (3)

L = sy - bz, (9)

where o, b, and T are parameters. This system can be reduced to a single third-order ordinary differential equation for 2 (231, which admits a two-dimensional Lie symmetry algebra if q = l/2, b = 1, and T = 0. Then, this third-order equation becomes

2xxett - 222 + 5x2 - 3xn + 2x32' + 32x1 + x4 + x2 = 0, 00)

Solving Differential Equations 183

and admits a two-dimensional Lie symmetry algebra Lz with basis

Xl = 4, x2=et/z(at-%a,>. 01)

A basis of its differential invariants of order I 2 is given by

9=(X+-2, ?f!?= X+qXf+;)X-3. (

(12)

Then, equation (10) is reduced to the following first-order equation:

(It, - 24) $ = -2? - I# > (13)

which can be easily integrated [24] to give

with an arbitrary constant cl ordinary differential equation

1+ 4ti - 4+2 = c1

Cl+ 2$)2 (14)

Substituting I and its derivatives into (14) yields a second-order

1 + 4 (2 + (312)s + z/2) x-~ - 4 (s -t ~/2)~ X-* =I c1

(X3 + 2x + 3x + x)2 5-s 1

which admits the Lie symmetry algebra Lg. Lies classification of two-dimensional algebras into four canonical types [25] allows us to integrate (15) by quadrature if we introduce the canonical variables

2/ =I --2eeti2, eTt/2

u=-, X

(16)

transform equation into

l+4(~~z-4~~

2u$$ - (F&)2 - = C1

and the (11) into = a,, X2=va,+uau. (18)

Then, the general solution of (17) can be easily derived [25] to be

J F 2c2?.? c2u > 4 -112dU=f2)+Cg (19) with arbitrary constants ~2, cg. This solution has already been obtained by Sen and Tabor [23], ~though they used a lengthier analysis.

2.3. Epidemiology of AIDS

In [13], Lie group analysis has been applied to a model formulated by Anderson (161, which describes HIV transmission in male homosexual/bisexual cohorts:

dul --P~1~2 -L= dt Ul -i-u2 +u3

-WI, (20)

(21)

cw dw dt = vu2 - Qu3*

184 M. C. NUCCI

This compartmental model divides the population at time t into susceptibles (HIV negatives), infecteds (HIV positives), and AIDS patients, represented by al(t), zlz(t), and ~sft), respectively. HIV infecteds are individuals who test positive for specific antibodies to the virus [26]. AIDS patients are persons exhibiting characteristic clinical manifestations of full-blown AIDS, the end- stage of the disease [27]. The parameter /.J is the per capita natural death rate (nonAIDS related) of both susceptibles and infecteds, and cr is the AIDS-related death rate. The parameter @ is the average probability that an infected individual will infect a susceptible partner over the duration of their relationship [16,28,29], and c is the effective rate of partner change within the specified risk category [16,29]. In the model, all infecteds are supposed to develop AIDS with an average incubation period l/v [16,27].

It is well known that a first-order system of ordinary differential equations admits an infinite- dimensional Lie symmetry algebra [25]. Lies theorem allows us to integrate the given system by quadrature, if we find a thr~dimension~ solvable Lie algebra. Let the operator r be the

generator of a symmetry group

r =

(26)

if the death rate of AIDS patients is the sum of the death rate of HIV infecteds plus the probability of transmi~ion per partner contact multiplied by the effective rate of partner change, i.e., ~3: = p + PC. It is easy to show that X2 and Xs span a two-dimensional ideal Ls of La. If L1 denotes the ideal spanned by X3, we get the following chain of inclusions:

L3 2 L2 3 Ll,

which means that the algebra we have found is solvable. By using a basis of differential invariants of L~L, we can reduce the system to a first-order ordinary differential equation which can be integrated by quadrature, because it admits the Lie algebra Ls/Ls spanned by Xr in the new variables. A basis of differential invariants of Ls of order 0 is

t, +-Jc u1+ 213

(27)

Then, the system (20)-(22) can be easily reduced to the following first-order equation:

which admits the operator Xl = &, i.e., the operator Xr in the variables (t, 6). Therefore, (28) can be easily integrated by quadrature. Its general solution is

c e@ t (PC - u) = eBC % + eYt,i3c ct. - evtclu'

Solving Differential Equations 185

where cl is an arbitrary constant. From (29) and (27) we obtain

u3 = ePct (211 + ~2) v - /3cu~ + evt (/3c - v) qu2

eflct (PC - v) ,

which substituted into the system (20)-(22) and after some easy calculations yields the following general solution of the system (20)-(22):

eVtc2

e@ [evt (0 c - u) cl + eflctp C]

ept+vt ,

[ev(/?c-u)cl+e pctpc] [evt (p c - u) cl + eflctu]

eflct+pt+vt [evt (Pc - u) cl + ePctPc] (PC - v)

w-a ,p Ct+Zt dtPcc2 + c3 +

(eP=t~C+eLBCC1-eYtC~~)~

eo Ct+pt+Yt [eVt (/3 c - v) cl + eP ctp c] (p c