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The designations Ci and F i mean constants C i and functions F i , U # means

?

U ? ? 1 . Taking into account formula (12) we see that the 1st order LB symmetries

appear to be point ones. Considering the coefficients at C i and using (12) it is easy

to obtain the symmetry operators

1

XU ?U ? T ?X ,

e1 = ?X , e2 = e3 = X?X + 2T ?T ,

2

1 1

T + X 2 ?U ? XT ?X ? T 2 ?T ,

e4 = U e5 = ?T , e6 = U ?U .

2 4

As is the case for every linear partial differential equation, there is also infinite-

dimensional subalgebra e? = F 1 (X, T )?U , where F 1 is an arbitrary solution of

equation (14). This example takes 47 seconds of CPU time and 260 Kbytes memory

Computer algebra application for determining symmetries 501

on the ES 1045 computer (under the OS/VS) with the internal performance of about

700 000 op/sec.

The LBF program consists of two main parts; the computation of determining

system, and the solution of that system. The 1st part is the improved and generalised

version of the program described in [2].

We use the sequencing of u? ···il and derivatives of functions F i , included in

i1

symmetry generators, in the following way: sets of lower indices are ordered lexico-

graphically; after that the items are ordered in accordance with their upper indices.

The position of the item in such a row we shall call “ordinal”. This ordering permits to

perform the considerable part of calculations using the fast and non-wasting memory

of PL/I integer arithmetic only.

The program executes sequentially the following steps.

(1) Reading the input data and transforming them into internal form introducing

the ordering mentioned above.

(2) Computation of the differential consequences up to the required order and elimi-

nation of dependences thereof. The program tries to solve the system together

with differential consequences with respect to derivatives of the highest ordi-

nals which are included linearly in relations. To do this, the program uses the

Gauss excluding method. Note that differential consequences depend on the hi-

ghest order derivatives only linearly. Some equations may not contain linearly

included derivatives. The program marks such equations.

If the problem of classification is considered, then the system of equations

contains arbitrary parameters or functions. Some combinations of these items

may be used as denominators during the excluding process. Everywhere in the

program in similar situations such different denominators are memorised to be

printed at the end of program execution. It is necessary to consider separately

the cases when such combinations are zeros.

(3) The determination of symmetry variables, i.e. variables which the generators

depend on. (The generators do not depend on derivatives of an order higher than

the symmetry order and on derivatives excluded in the previous step.)

(4) Computation of canonical LB operator (9) and the result of its action on system

(1), i.e. computation of invariance conditions.

(5) Transition to manifold. The program eliminates the derivatives, excluded in step

(2), in the invariance conditions. If there are equations unresolved in step (2) the

program adds them to invariance conditions (11) having preliminarily multiplied

them by indefinite factors.

(6) Separation of the determining system with respect to free variables, i.e. deriva-

tives of an order higher than the symmetry order. The program separates the

equations not only with respect to different powers of the free variables but also

with respect to arbitrary different independent functions of such variables. It is

possible that the functions independent in general may be dependent in some

particular cases. For example, if uxx is a free variable, then from A sin(uxx ) +

B cos(uxx ) = 0 it follows that A = 0 and B = 0, but from Au2 + Buk = 0

xx xx

it follows that A = 0 and B = 0 if k = 0, but A + B = 0 otherwise. Another

obvious example Af (uxx )+Bf (uxx ) = 0 requires to consider the particular case

502 W.I. Fushchych, V.V. Kornyak

f = euxx . For subsequent consideration of similar particular cases the program

memorises all different separation factors containing arbitrary functions and

parameters. For the sake of economy of memory, during the separation process,

zeros are deleted immediately, i.e. when a one-term determining equation arises,

it and its differential consequences are substituted at once into all remaining

expressions.

(7) Exclusion of indefinite factors. Using the Gauss method the program excludes

indefinite factors if they were introduced at step (5).

At the end of the first part of the program the state of the determining

system (i.e. expressions for generators, dependences of functions, equations)

considered is for the example:

?

u = F 1; F 1 = F 1 (x, t, u, ux , ut );

Ft1 ? Fxx ? 2ut ux Fuux ? 2ut Fxux ? 2ux Fxu ? u2 Fux ux ? u2 Fuu = 0,

1 1 1 1 1 1

t x

1 1 1 1

Fxut + ut Fux ut + ux Fuut = 0, Fut ut = 0.

(8) Simplification of the determining system. The fragment of the program respon-

sible for this step is

BC = ‘1’B;

DO WHILE(BC);

BB = ‘1’B;

DO WHILE(BB);

BA = ‘1’B;

DO WHILE (BA);

CALL REDSYS(1); CALL ORTSYS(?); BA = SMONINT;

END;

BB = INVOL;

END;

BC = RESTINT;

END;

Here, BA, BB, BC are the control variables. The very inner loop contains

the calls of the most efficient procedures. The REDSYS procedure is auxiliary.

It reduces the determining system to some canonical form. Argument “1” means

that the reduction begins from the first equation. (There are calls of REDSYS

from other procedures with different values of argument.) The call of the

ORTSYS(K) procedure reduces the determining system to the orthonomic form

[4], i.e. the derivatives of the highest ordinals (leading derivatives) are singled

out and substituted (with theirs differential consequences of the order up to K),

equations are ordered in accordance with the increase of ordinals of leading deri-

vatives. ORTSYS deletes the equations not containing derivatives after having

performed the corresponding substitutions. The SMONINT procedure integrates

the systems of monomial equations, i.e. systems of the form {Fij1 ···ik = 0}. Here,

Computer algebra application for determining symmetries 503

the lower index il means the differentiation with respect to the il th symmetry

variable. The procedure yields the expression for F j simultaneously consideri-

ng all monomials for the given j. It allows the number of iterations to be

reduced. SMONINT substitutes also F j and its derivatives in other expressi-

ons and separates the determining system after that. If there are no monomial

equations in the determining system, then BA takes the value ‘?’B and the loop

is completed.

After this loop the state of the determining system for the heat equation is

?

u = F 3 + F 4 u + F 2 ux + F 1 ut ;

F 1 = F 1 (t), F 2 = F 2 (x, t), F 3 = F 3 (x, t), F 4 = F 4 (x, t);

2Fx ? Ft1 = 0, 2Fx ? Ft2 + Fxx = 0, Ft3 ? Fxx = 0, Ft4 ? Fxx = 0.

2 4 2 3 4

The INVOL procedure performs the further integrations or reduces the

remaining part of the determining system to the system in involution by Riquier–

Janet method [4]. The method consists in calculating differential consequences

of equations and in excluding dependences out of them. INVOL tries to separate

and integrate the relations arising during this process. If it fails in integration,

then the system is reduced to the one in involution, and BB takes the value

‘?’B; if otherwise, BB = ‘1’B, and the whole process is repeated. The INVOL

procedure integrates the monomials and often arising equations of the form

Fij1 ···ik = P , where P is the polynomial of the variables of differentiations.

As a rule, the combination of the above-mentioned operations is sufficient to

reveal the major part of dependences of the symmetry generators, because these

dependences are often polynomial. For instance, our example is completed by

the INVOL procedure. In some cases it is possible to go further in integration of

the remaining part of the determining system. This is effected by the RESTINT

procedure. In gaining experience, it is possible to add in this procedure some

particular and rare methods of integration. Now RESTINT contains the pro-

cedure for solving the ordinary differential equations or systems with constant

(with respect to differentiation variable) coefficients up to the 4th order, and the

procedure for solving differential equations of the 1st order with variable coeffi-

cients. The last procedure yields in non-polynomial cases the formal expressions

for indefinite integrals. BC takes the value ‘1’B if further simplification after

RESTINT is possible, and ‘?’B, if otherwise.

(9) The last step of the LBF program is output. The internal designations are

replaced by more expressive ones. The expressions for generators are reduced to

some form simplifying the extraction of different one-dimensional subalgebras,

as in the example above. The program removes also superfluous functions or

constants if they arise during integration. For example, if F 1 (X, Y ) and F 2 (X)

are arbitrary functions, then F 1 (X, Y ) + F 2 (X) is equivalent to F 1 (X, Y ). In

general, the output may contain two more items, in addition to those presented

in the example: the list of different denominators containing arbitrary functions

or parameters, and the analogous list of separation factors.

504 W.I. Fushchych, V.V. Kornyak

Let us demonstrate the result of the program application in obtaining the 1st order

symmetries of a more complicated non-linear equation [5]

?u ?u ?u

? 2u ? ?u µ (15)

= 0,

?? ?x ?xµ

where

?2 ?2 ?2 ?2 ?u ?u

2= ? ? 2 ? 2, = u2 t ? u2 ? u2 ? u2 .

? x y z

??2 ?x2 µ ?x

?y ?z ?x µ

t

The LBF program gives the general solution having in terms of operators the form:

e1 = ?x , e2 = ?y , e3 = ?z , e4 = ??t , e5 = ?x + x??t ,

e6 = ?t ?y + y??t , e7 = ?t ?z + z??t , e8 = y?x ? x?y , e9 = x?z ? z?x ,

e10 = z?y ? y?z , e11 = ?(u)?u , e12 = ?? ,

e13 = x?x + y?y + z?z + t?t + 2? ?? ,

x2 + y 2 + z 2 ? t2

? 2? ?(u)?u ,

2

e14 = x? ?x + y? ?y + z? ?z + t? ??t + ? ?? +

4

x y z

e15 = ? ?x + ?(u)?u , e16 = ? ?y + ?(u)?u , e17 = ? ?z + ?(u)?u ,

2 2 2

2

t ?u

e18 = ? ??t ? ?(u)?u , e? = ?(x, y, z, t, ? ) exp ? ?u ,

2 2

where

?u2 ?u2

?(u) = exp ? exp du,

2 2

?(x, y, z, t, ? ) is an arbitrary solution of equation

??

? 2u = 0. (16)

??

This example takes 5 min 18 s and 320 Kbytes on ES 1045. Generators e1 –e10 create

the Poincar? algebra. As it follows from the above operators, equation (15) turned

e

out to be automorphic, i.e. all its solutions lie on one group orbit. It allows to reduce

non-linear equation (15) to linear one (16) using standard techniques of symmetry

analysis.

4. Conclusion

There are some problems in connection with the considered one where the com-

puter algebra may be successfully applied. For example, to complete the symmetry

analysis of the system of differential equations it is important to learn the subgroup

structure of the symmetry group, i.e. to classify the subalgebras of Lie-algebra of

symmetries into conjugacy classes. This problem is also important in many other

fields. In [9] the algorithm for classification of subalgebras of finite-dimensional Lie-

algebras and its computer implementation were described.

Modern development of symmetry analysis includes several approaches considering

non-local symmetries, i.e. symmetries depending on integrals or even more general

operators acting in the functional space of the dependent functions [7]. Some class of

such symmetries and corresponding algorithms and programs were considered in [6].

Computer algebra application for determining symmetries 505

We are grateful to Professors B. Buchberger, B. F. Caviness and J. A. van Hulzen

for their interest in this work.

1. Eliseev V.P., Fedorova R.N., Kornyak V.V., A REDUCE program for determining point and contact

Lie symmetries of differential equations, Comput. Phys. Commun., 1985, 36, 383.

2. Fedorova R.N., Kornyak V.V., Determination of Lie–B?cklund symmetries of differential equations

a

using FORMAC, Comput. Phys. Commun., 1986, 39, 93.

3. Fedorova R.N., Kornyak V.V., A REDUCE program for computing determining equations of Lie–

B?cklund symmetries of differential equations, Dubna, JINR, 1987, R11-87-19 (in Russian).

a

4. Finikov S.P., Carton’s exterior forms method, Moscow–Leningrad, Gostechizdat, 1948, 432 p. (in

Russian).

5. Fushchych W.I., Symmetry in the problems of mathematical physics, in Algebraic-Theoretical Studi-

es in Mathematical Physics, Kyiv, Inst. of Math., 1981, 6–28 (in Russian).

6. Fushchych W.I., Kornyak V.V., Computation on a computer of non-local symmetries of linear

systems in mathematical physics, In International Conference on Computer Algebra and its Appli-

cations in Theoretical Physics, Dubna, JINR, 1987, D11-85-791, 345–350 (in Russian).

7. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, D. Reidel, 1987,

217 p.

8. Ibragimov N.H., Transformation groups in mathematical physics, Moscow, Nauka, 1983, 286 p. (in

Russian).

9. Kornyak V.V., Classification of subalgebras for finite-dimensional Lie-algebra using computer, In

international Conference on Computer Algebra and its Applications in Theoretical Physics, Dubna,

JINR, D11-85-791, 339–344 (in Russian).

10. Ovsiannikov L.V., Group analysis of differential equations, Moscow, Nauka, 400 p. (in Russian).

11. Schwarz F.A., A REDUCE package for determining Lie symmetries of ordinary and partial di-

fferential equations, Comput. Phys. Commun., 1982, 27, 179.

12. Schwarz F.A., Automatically determining symmetries of partial differential equations, Computing,

1985, 34, 91.

13. Wolf T., Analytic solutions of differential equations with computer algebra systems, Preprint N 87/5,

Friedrich Schiller Universitat, Jena, 1987.

14. Zhdanov R.Z., On application of Lie–B?cklund method to study of symmetries of the Dirac equati-

a

ons, in Group-Theoretical Studies of Mathematical Physics Equations, Kyiv, Inst. of Math., 1985,

70–73 (in Russian).

W.I. Fushchych, Scientific Works 2001, Vol. 3, 506–510.

On vector and pseudovector Lagrangians

for electromagnetic field

W.I. FUSHCHYCH, I.Yu. KRIVSKY, V.M. SIMULIK

A Lagrange function in terms of electromagnetic field strengths is constructed which is

?

a 4-vector with respect to the total Poincar? group P (1, 3) and whose Euler–Lagrange

e

equivalent to the Maxwell equations. The advantages of the known pseudovector with

?

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