ñòð. 55 |

1975, 30, 261–262 (in Russian).

17. Tau?k M.S., On maximal subalgebras in classical real Lie algebras, in Voprosy Teorii Grupp

i Gomologicheskoj Algebry, Jaroslavl Univ., 1979, 148–168 (in Russian).

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 247–252.

Lowering of order and general solutions

of some classes of partial di?erential

equations

V.M. BOYKO, W.I. FUSHCHYCH

A procedure of lowering the order and construction of general solutions for some clas-

ses of partial di?erential equations (PDEs) are proposed. Some classes of general

solutions of some linear and nonlinear equations of mathematical physics are

constructed and a series of examples is presented.

The construction of the general solution of a de?nite partial di?erential equation

is in a number cases an unsolved problem. In what follows, we propose an algorithm

of lowering the order and constructing general solutions of speci?c partial di?erential

equations.

Consider the following partial di?erential equation

(1)

L(D[u]) + F (D[u]) = 0,

where u = u(x), x = (x0 , x1 , . . . , xk ); L is a ?rst-order di?erential operator of the

form

L ? ai (x, u)?xi , (2)

i = 0, 1, . . . , k,

and ai (x, u) are arbitrary smooth functions which are not identically equal to zero

simultaneously. D[u] is an n-order di?erential expression

(3)

D[u] = D x, u, u(1) , u(2) , . . . , u(n) ,

where u(m) is the collection of m-th order derivatives, m = 1, . . . , n, and F is an

arbitrary smooth function of D[u]. As a particular case, D[u] may depend only on x

and u. In this case we say that D[u] is of order zero. In general, (1) is an (n + 1)-th

order partial di?rential equation.

For equations of the type (1), we propose a method of lowering the order and

construction of solutions based on the local change of variables which reduces opera-

tors (2) to the operator of di?erentiation with respect to one of independent variables.

We introduce the change of variables

? = f 0 (x, u), ? a = f a (x, u), a = 1, . . . , k, (4)

z = u,

where z(?, ?) is a new dependent variable, ? = (? 1 , . . . , ? k ).

We determine functions f 0 , f a from the conditions

L(f 0 ) = 1, L(f a ) = 0, (5)

a = 1, . . . , k,

Reports on Math. Phys., 1998, 41, ¹ 3, P. 311–318.

248 V.M. Boyko, W.I. Fushchych

and functions f 1 , . . . , f k and u must form a complete collection of functionally-in-

dependent invariant of operator (2). We choose f 0 as a particular solution of the

equation Ly = 1.

Relations (5) determine the change of variables (4) such that operator L is reduced

to the operator of di?erentiation with respect to the variable ? , i.e.,

L ? ?? . (6)

We obtain a new form of (3) in new variables (4) and rewrite the initial equation

(1) in the form

(7)

?? D[z] + F D[z] = 0,

where D[z] is D[u] in the new variables (4).

Relation (7) is the ?rst order ordinary di?erential equation with respect to the

variable ? . We integrate it and obtain D[z]. Thus, when we solve (7), we obtain

an n-th order partial di?erential equation with respect to z(?, ?) with one arbitrary

function depending on ? which is a “constant” of integration of Eq. (7).

Remark. This algorithm is also e?ective in the case where Eq. (1) has the form

L(D[u]) + F (D[u], f 0 , f 1 , . . . , f k ) = 0. (8)

Here, functions f 0 , . . . , f k must satisfy relations (5). In this case, integrating the

corresponding ordinary di?erential equation (an analog of equation (7)) we regard

variables ? a as parameters.

Example 1. Consider the one-dimensional wave equation

?2u ?2u

? (9)

= 0.

?t2 ?x2

Equation (9) can be written in the form (1), namely:

? ? ?u ?u

? (10)

+ = 0.

?t ?x ?t ?x

After the change of variables

? = t, ? = x + t, z = u,

Eq. (10) can be rewritten in the form

?? (z? + 2z? ) = 0.

We integrate this equation and obtain

(11)

z? + 2z? = g(?),

Since g(?) is arbitrary, we set g(?) = 2h (?). Then characteristic system of for the

inhomogeneous quasi-linear Eq. (11) has the form

d? d? dz

= = .

1 2 2h (?)

Lowering of order and general solutions of some classes of PDEs 249

We ?nd the ?rst integrals of the characteristic system and we get the following solution

of Eq. (11),

z ? h(?) = f (? ? 2? ), (12)

where h and f are arbitrary functions. Then we rewrite (12) in variables (t, x, u) and

get the following well-known general solutions of Eq. (9)

u = h(x + t) + f (x ? t).

Example 2. Consider the following equation proposed in [3] for description of motion

of a liquid,

L ? ?t + u?x . (13)

L(Lu) + ?Lu = 0,

This equation can be regards as a generalization of the one-dimensional Newton–Euler

equation (the equation of simple wave). In the explicit form, Eq. (13) has the form

2

?2u ?2u 2

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

+ 2u + +u +u +? +u = 0.

2 ?x2

?t ?t?x ?t ?x ?x ?t ?x

Since Eq. (13) belongs to the class of (1), the change of variables

? = x ? ut,

? = t, z = u,

allows us to write it as

z? z?

(14)

?? +? = 0.

1 + ? z? 1 + ? z?

Having integrated (14), e.g., for ? = 0, we obtain the parametric solution

d?

z± ? 2 ? h(?) = p, (15)

= ?(p),

h(?) + p

where p is a parameter, h and ? are arbitrary functions.

Then we return to the initial variables and obtain a solution of Eq. (13). Below, we

give several classes of solutions of Eq. (13) with one arbitrary function [1] (The fact

that we have only one arbitrary function associated with the problem of integration

of system of type (15)).

1. L(Lu) = 0:

1.1 u ± ln(x ? ut ? t) = ? t2 ? (x ? ut)2 ,

t(x ? ut)3 1

= ? t2 ?

1.2 u + ,

t2 (x ? ut)2 ? 1 (x ? ut)2

x ? ut x ? ut

? exp t2 dt.

1.3 u = ?

exp (t2 ) exp (t2 )

2. L(Lu) = a:

a C a

x ? ut + t3 + t2 = ? u ? t2 ? Ct .

3 2 2

250 V.M. Boyko, W.I. Fushchych

3. L(Lu) + Lu = a

a

x ? ut ? C(t + 1) exp(?t) + t2 = ? (u + C exp(?t) ? at) .

2

Here, C = const, ? is arbitrary function.

Example 3. The equation

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

? ? 2 +2 (16)

=0

?t2 ?x2 ?y ?x?y

can be written in the form (1) as follows:

? ? ? ?u ?u ?u

? ?

+ + = 0.

?t ?x ?y ?t ?x ?y

Using the change of variables

? 2 = t ? y,

? 1 = t + x,

? = t, z = u,

and applying the algorithm described earlier, we obtain the following solution of

Eq. (16)

u = f (t + x, t ? y) + g(t ? x, t + y),

where f and g are arbitrary functions.

It is natural to generalize the described algorithm for equations of the form (1) to

the classes of partial di?erential equations of the form

Lm (D[u]) + bm?1 Lm?1 (Du) + · · · + b1 L(D[u]) + b0 = 0, (17)

where

bj = bj (Du, f 0 , f 1 , . . . , f k ), j = 0, m ? 1; Lm = LLL · · · LL;

m

L, D[u], f 0 , f 1 , . . . , f k are determined according to the relations (2)–(6).

After the change of variables (4)–(6), the problem lowering the order of Eq. (17) is

reduced to the problem of integrating the m-th order ordinary di?erential equation.

Example 4. For

D ? xµ ?xµ ,

Dn (u) = 0, µ = 0, . . . , k,

we use the change of variables

xa

?a =

? = ln x0 , , a = 1, k, z = u,

x0

and we obtain the solution

u = Cn?1 (ln x0 )n?1 + Cn?2 (ln x0 )n?2 + · · · + C1 ln x0 + C0 ,

x1 xk

;···; , i = 0, n ? 1.

where Ci = Ci

x0 x0

Lowering of order and general solutions of some classes of PDEs 251

The obtained results can be easily generalized to the case of system of equations

L(D[u]) = F f 0 , f 1 , . . . , f k , D[u] ,

where u = (u1 (x), . . . , um (x)), x = (x0 , x1 , . . . , xk ); L, f 0 , f 1 , . . . , f k are determined

according to relations (2), (4), (5) and (6). Here, u ? u ; D[u] = (D1 , . . . , Dm ), where

Di = Di x, u, u(1) , u(2) , . . . , u(n) , i = 1, . . . , m, u(i) is a collection of i-th order

derivatives for each component of the vector u; and F = (F 1 , . . . , F m ). In particular,

the components of the vector D[u] can dependent only on x and u.

Example 5. Consider the system of Euler equations

?v ?v

+ vk (18)

= 0,

?x0 ?xk

where v = (v 1 , v 2 , v 3 ), v l = v l (x0 , x1 , x2 , x3 ), l = 1, 2, 3.

The system (18) can be written as follows:

?0 + v k ?k v l = 0, l = 1, 2, 3. (19)

After the change of variables

? = x0 ,

? a = xa ? v a x0 , a = 1, 2, 3,

z l = v l , l = 1, 2, 3

the system (19) takes the form

?? z l = 0, (20)

l = 1, 2, 3.

Then we integrate Eq. (20), apply the inverse change of variables, and obtain a solution

of system (18) in an implicit form (compare this solutions with one from [2])

v l = g l (x1 ? v 1 x0 , x2 ? v 2 x0 , x3 ? v 3 x0 ).

where g l are arbitrary functions.

Example 6. Consider the following system of equation for vector-potential Aµ ,

?Aµ

A? (21)

= 0, µ = 0, . . . , 3.

?x?

Assume that A0 = 0. By the change of variables

x0

? = 0,

A

? = xa A0 ? x0 Aa , a = 1, 2, 3,

a

Aµ = Aµ , µ = 0, 1, 2, 3

we obtain the following solutions of system (21)

Aµ = g µ (x1 A0 ? x0 A1 , x2 A0 ? x0 A2 , x3 A0 ? x0 A3 ),

ñòð. 55 |