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 Lie’s 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] Lie’s 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 = ww’l. (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 - 22’2” + 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. Lie’s 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]. Lie’s 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 = <(t,Ul,Z&,U3)& + r]l(t,%r~2,‘113)&, +7)2(t,Ul,U2,U3)& +~3(k%,U2~U3)f%a. (23)

To find a thre~dimension~ solvable Lie algebra, we make an ansatz on the form of the operator I’, and then verify if such an operator is admitted by the system. We assume that the functions 5, r]r, and ~2 are polynomials of second degree on ~1, us, and us. After some lengthy analysis, we obtain that (20)-(22) admits a three-dimensional solvable Lie algebra Ls with basis

Xl = at _ Pld,, - pu2avz - /Lu3&&, (24)

x2 = Ul&* + uza,, + u3&, (25)

x3 -_ ,-+++ a,, + I!!a+uaus ) 262 >

(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+Z”t

dtPcc2 + c3 +

(eP=t~C+e”LBCC1-eYtC~~)~

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

-evtc2

+ e@ [eyt (PC - v) cl + eflctpc] ’

(30)

(31)

(32)

where c2,cs are arbitrary constants. In [13], the behaviour of this solution is compared with

epidemiologic data on the incidence of HIV-infection, and AIDS in male homosexual/bisexual populations in U.S. areas most affected by the HIV epidemic: the Multicenter AIDS Cohort Study [30], the San Francisco Men’s Health Study [31], and the San Francisco City Clinic Co-

hort, [32].

3. NONCLASSICAL SYMMETRIES METHOD

The nonclassical symmetries method (NSM) was introduced in 1969 by Bluman and Cole [17]

to obtain new exact solutions of the linear heat equation, i.e., solutions not deducible from the

classical symmetries method (CSM). The NSM consists of adding the invariant surface condition to the given equation, and then applying the CSM. The main difficulty of this approach is that

the determining equations are no longer linear. On the other hand, the NSM may give more

solutions than the CSM. The NSM has been successfully applied to various equations [33--37]‘, for the purpose of finding new exact solutions.

3.1. Modified Korteweg-de Vries Equation

In [18], the NSM was applied to the modified Korteweg-de Vries (mKdV) equation

ut = u,,, - 6~~212,

with the invariant surface condition given by

u, = G(t, 2, u).

Then, G must satisfy

3GGzz, + 3GG,G,, + 3G,G,, + G3G,w + 3G2G,,,

+3G2G,G,, + 3GG,G,, - Gt + G,,, - 6u2G, - 12uG2 = 0.

(33)

(34)

(35)

1 Just to cite some of numerous papers on this subject.

186 M. C. NUCCI

If we apply the NSM to (35), we find that its invariant surface condition

Sl(t, 2, u, G)G. + 52(4 2, u, Wz + b(t, r, u, G>G - v(t, 5, u, G) = 0 (36)

is given by G, = -221,

which implies G = 4 - u2,

with an arbitrary function q(z,t). Then, substituting into (35) yields

(37)

(38)

qt = qxxx - Wm (39)

which is the Kortweg-deVries (KdV) equation. In fact, (34) and (33)2 yield the Biicklund trans- formation between the mKdV and KdV equations [38], i.e.,

u, =q-u2,

Ut = qzz - 292 + 2u2q - 2uq,.

(40)

(41)

This means that the Backlund tr~sformation (40),(41) between the mKdV and KdV equations can be obtained by applying the NSM “twice” to the mKdV equation.

4. ITERATING THE NONCLASSICAL SYMMETRIES METHOD

Recently, Galaktionov [39] and King [40] have found exact solutions of certain evolution equa tions which apparently do not seem to be derivable by either the CSM or NSM. However, we have shown [19] that these solutions can be obtained by iterating the NSM (INSM). Successive iterations generate new equations which, although more complex than the original equations, in- herit the same Lie symmetry algebra. Invariant solutions of these equations are just the solutions found by Galaktionov and by King.

The use of a symbolic computation program becomes imperative, because these equations can be quite long: one more independent variable is added at each iteration.

4.1. Applications to Evolution Equations

Let us consider an evolution equation in two independent variables and one dependent variable

The invariant surface condition is given by

Vi@, 2, u)ut + Vi@, 5, u)uz = G(t, z, u). (43)

Let us take the case with VI = 0 and Vs = 1, so that (43) becomes

uz = G(t, 2, u). (44)

Applying the NSM leads to an equation for G (the G-equation [18]). Its invariant surface condition is given by

&(t, 2, U, G)Gt + Ez(G G u, W, + 53@, 5, u, G)Gu = v(t, G u, G). (45)

aAfter substituting us with C, and usziF with @, respectively.

Solving Differential Equations 187

Let us consider the case 51 = 0, <2 = 1, and <s = G, so that (45) becomes

G, + GG, = q(t, x, u, G). (46)

Applying the NSM leads to an equation for 77 (the q-equation). Clearly,

G, + GG, E u,, = q. (47)

We could keep iterating to obtain the Q-equation, which corresponds to

the p-equation, which corresponds to

0, + GQ, + q% + RR, = u,,,, = p(t, x, u, G, q, O), (49)

and so on. Each of these equations inherits the symmetry algebra of the original equation, with

the appropriate prolongation: first prolongation for the G-equation, second prolongation for the

v-equation, and so on.

This iterating method is strongly connected to the definition of partial symmetries given by

Vorobev in [41]. To exemplify, let us consider the heat equation

with G-equation

and v-equation

2GG,, i- G2G,, - Gt +- G,, = o, (51)

273%G -k 2GQ’%G i- 7j2qGG + 2GqZU + qZs - qt + G2qUU = 0. (52)

The G-equation corresponds to the zeroth-order differential constraint as given by Vorobev [42,

p. 76, formula (3.5)], while the q-equation gives the partial symmetry of the heat equation as in

[41, p. 324, formula (14)], and in [42, p. 83, formula (4.10)].

Also, it, should be noticed that the uxx +Y

-equation of (42) is just one of many possible

r-extended equations as defined by Guthrie ;n [43].

In [19], it has been shown that the solutions obtained in [39,40] are just t-independent invariant

solutions, which have x as similarity variable. In this way, ordinary differential equations are

obtained and their general solution will depend on arbitrary functions oft. Substituting into the

original equation will give rise to ordinary differential equations to be satisfied by these functions

oft.

GALAKTIONOV’S EXAMPLE. [39] Let us consider the equation

Ut=uz,+U2fU2, (53)

with G-equation

2GG,, -I- G2G,, + G2G, - u2G, - Gt + G,, + 2GG, + zuG = 0, (54)

and q-equation

27777,G + 2%‘uG + q2?,GG - 2UGqG + 2GqzU + qzl

+2Gqz - qt + G2qw, + G2qu - u2q, + zq2 + suq + 2G2 = 0. (55)

188 M. C. Nuccr

The symmetry algebra of (53) is spanned by the two vector fields Xi = at and Xz = 8,. Therefore, t-independent invariant solutions of (55) are given in the form n = 71(x, u,G). A particular case is nU = 0, which implies 7) = L(z, G). Substituting this expression for q into (55) leads to L = f(z)G where

f(x) = -cl sin x + cs cos x

c2sinz + cl cosx (56)

and cl, c2 are arbitrary constants. If we take cr = 0, then

q = cot(z)G,

which is just the differential constraint for (53) given by Olver in [44], i.e.,

(57)

u IX = cot(x)?&. (58)

Integrating (58) with respect to z gives rise to

u = ml(t) cos(z) + wz(t), (59)

where 201, ws are arbitrary functions oft. Finally, the substitution of (59) into (53) leads to

$1 =w:+wg, ?iQ = 2WlW2 _ w2. (60)

This is the solution derived by Galaktionov for (53). In the same way, iterating the NSM twice will generate the solutions of the other equations in [39].

KING'S EXAMPLE. [40] Let us consider the equation

N-l ?Jt = vu,, + ml,” + -vvp.

r (61)

Its symmetry algebra is spanned by the vector fields

x1 = ra, + 2v& x2 = tat - vd,,

More vector fields exist in the following cases:

x3 =a$. (62)

4-N

a = 2(N - 3) =+ rBsNd,. + 2(3 - N)r2-Nvt3,,

N=le&., (64)

N=:,cr= -1 I rlogrd, + 2(1+ logr)v&, (65)

N = 1, cy = -; & r2& -t-47-v&,, a,.. (66)

We derive the R-equation3 of (61) and look for t-independent invariant solutions which have T

as similarity variable. Such invariant ~lutio~ can be obtained by the subalgebra spanned by (X2, X3). The third prolongation of X2 is given by

9 = tat - v& - vr&+ - %+aJ,, - vm&lprr, (67)

which implies that the corresponding invariant solutions of form:

the R-equation have the following

52 = %-r = vF(r, E4, E5), (68)

Solving Differential Equations 189

where <4 = G/u - v,./v, (5 = q/v E vTT/v, and F must satisfy a very long equation not reported here [19]. Let us assume that F is linear with respect to & and [s, i.e.,

F = S1(T)55 + S2(r)t4 + S3(T), (69)

with arbitrary functions ~1, sz, and sg. Then, we obtain five cases, two more than those considered in [40]. In particular, we show the case N = 1, cr = -3/4 with

F = sl(~)Q - S;(T) + ;s:(T) J4 - 2 1 [ s:(r) + $(r)s:(T) 1 , i.e.,

V r~r=srv,,-- (s;+~+.-2(s;+~s+,

(70)

(71)

where sl(r) must satisfy the equation

9s’l’+9sls’l+s; =o, (72)

which admits a Lie symmetry algebra of dimension eight, so it is linearizable. The linearizing

transformation [25] can be obtained by using a two-dimensional subalgebra Pi, l?z, such that

l?i V I72 = 0, e.g.4,

r3 ’ Ii = -@pi - 6)&. + 18( rsl - 6)(r2sq - 6rsi + 18)&,,

rz = r2Sl& - $(r’S; - 6~s~ -I i8)&, .

The general solution of (72) is

sr = -6(~2 + T)

6cr - c$ - 2czr - r2 ’

which transforms (71) into

V -6(~2 + r) 18(2cr - c; - 2c2r - r2)

“‘= 6cl-c;-2c2~-r2 “r + (6~ - c; - 2c2r - r2)2 ”

24(c2 + r)

+ (6cr - cf - 2c2r - r2)2v*

A particular solution of (75) is

v = a&) (6~1 - c; - 2~27’ - T2)‘.

Substituting (76) into

4.2. Application to

(73)

(74

(75)

(76)

(61) imposes q,(t) to have the form

1

e” = 24clt + c3’

Prandtl Equation

(77)

In [20], INSM has been applied to the laminar boundary layer model introduced in 1904 by

Ludwig Prandtl [45]

uus + VT& = WY, 215+vy =o, Uy = w, (78)

4The commutator of these two vector fields is [I’l, rs] = -6I’l.

190 M. C. Nuccr

which was called the Prandtl system in [46], and to an equivalent single equation for u

%%YY - u21yuzy + uuxuyy - urJa, = 2 0 ) (79)

which was called the Prandtl equation. Both the equation and the system were reduced to several ordinary differential equations, and new solutions were found. Therefore, it was shown that INSM can be applied to systems of partial differential equations, and to single equations which are not of evolutive type.

The Prandtl system (78) admits an infinite-dimensional Lie symmetry algebra generated by the operators [4]

where o(z) is an arbitrary function of 2. It is easy to verify that the NSM is equivalent to the CSM for the Prandtl system (78). Details on INSM applied to the Prandtl system (78) can be found in [20].

The Prandtl equation (79) admits the same infinite-dimensional Lie symmetry algebra of the Prandtl system, i.e.,

x1 = za, -k Il&, x2 = & - a?.&&, x3 = &-, x, = cy(z)d$/. (81)

The NSM is equivalent to the CSM for the Prandtl equation also. Application of INSM to the Prandtl equation (79) means that we consider the case with VI = 0 and Vs = 1, so that the invariant surface condition

becomes

uy = G(z, M, u). (83)

The NSM leads to an equation for G, the G-equation. Its invariant surface condition is given by

&(z:, Y, a, G)G + Cz(x, Y, utt, GE, + 53(~ Y, 21, G)Gu = q(x, Y, u, G).

If we consider the case & = 0, (2 = 1, and <s = G, then (84) becomes

G, + GG, = rl(x, 2/, 21, G),

(84)

(85)

and the NSM leads to an equation for q, the q-equation. Clearly,

G, + GG, G uyy z q. 036)

Continuing with the iterations gives rise to the Q-equation, which corresponds to

and so on. Each of these equations inherits the Lie symmetry algebra of (79) with the appropriate prolongation: first prolongation for the G-equation, second prolongation for the n-equation, third prolongation for the Q-equation, and so on.

Let us consider the R-equation and find its z-independent invariant solutions, which have y as similarity variable [20]. Such invariant solutions can be obtained by the subalgebra spanned by {Xl, X3). The third prolongation of Xr is given by

Solving Differential Equations 191

while xs zxs.

3

The corresponding invariant solution of the R-equation is

(89)

with (1 = G/u. z ~~/~, & = 771% sz ~*~/~, which substituted into the R-equation leads to an equation for @(<I, y,&). Let us assume that Qi is linear with respect to y and 53, i.e.,

* = ~I(&)53 + ~dJI)Y + S3(G), (91)

with arbitrary functions si, se, and sg. Then, we obtain that Cp must be of the following form:

Q = 4~5 + 4~2 + 3c3[4 + 4c3t3 - 4Ef - 4M3

4 (92)

with arbitrary constants cl, cs, and ca. Substituting (92) into (90) yields the following third-order equation for u. and its y-derivatives:

If ci = -6?3&1936, and c2 = 15&704, then (93) becomes

U YY?/ = (-- 7744v,,u,u + 7744u,,csu2 - 77442~; + 5808~;~~~ - 2692u&u2 + 165~;~~)

7744u2 ! (94

which admits a three-dimensional solvable Lie symmetry algebra generated by the operators

& = e-(Wll)Y~y + 5C3,-W”fVu~~, 44

22 = a,, 23 = ua, 195)

with the commutators

I&,Z2I = ~21, I&, 231 = 0, [Zz, 231 = 0. (96)

Let us show that (94) is a particular case of equation (2.7) as given in [12] with X = 0. ‘In fact, by introducing the change of variables

g=logu.-5c3 - -Try7 l1 (2CSlll)Y

U=Ge 1

equation (94) transforms into

f&j = iigj ( 32 +3%-4 , >

(98)

which corresponds to equation (2.7) in (121 with X = 0 and f(*) = 3 * +3/ * -4. Operators (95) transform into

21 = a,, z2 = a,, 23 = ,iz+j, (99)

with commutators

[G, Z2] = 0, [&, &] = 0, [.z2, .z3] = z2. (100)

Therefore, 21 and 22 span the ideal L2 = (2,) 22). A basis of differential invariants of La of order < 2 is given by

z = Gsig, v=Qj. (101)

192 M. C. Nucct

Then, equation (94) is reduced to the following first-order equation:

dv -=3;+3;-4, dz (102)

which admits the operator .!?a in the space of variables (2,~). Through this operator we obtain the general solution of (102) in the following implicit form:

(103)

with an arbitrary function hi(z). Then, by introducing the original variables, we obtain

_2JZarctan “%_VRj + log 3% - 4iigQg + 2Gs _ 410gc, + 4k = 0,

-z/Y a; (194)

which is a second-order differential equation admitting the symmetry algebra L2. This means

that (104) can be integrated by using Lie’s approach [25] to obtain

a = J F(g + hz(z), h(z)) d@ -t- h3(2), (105)

where F is a suitable function, while hsfzr) and h3(2) are arbitrary functions of 3~. Finally,

substituting (105) with (97) into (79) yields a system of equations for h;(z) (i = 1,2,3). Therefore,

(105), (97)> and the system for hi(s) give rise to a new solution of the Prandtl equation (79).

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18. 19. 20.

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