Local and nonlocal conserved vectors for the nonlinear filtration equation Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том 4. № 4 (2012). С. 179-185.

УДК 517.958: 537.84

LOCAL AND NONLOCAL CONSERVED VECTORS FOR THE NONLINEAR FILTRATION EQUATION

A.A. ALEXANDROVA, N.H. IBRAGIMOV, K.V. IMAMUTDINOVA AND

V.O. LUKASHCHUK

Abstract. It is demonstrated that the nonlinear filtration equation is nonlinarly self-adjoint. Using this property, the conserved vectors associated with Lie point and nonlocal symmetries are constructed.

Keywords: nonlinear filtration equation, nonlinear self-adjointness, Lie point and nonlocal symmetries, conserved vectors.

1. Introduction

The present paper is a continuation of the Preprint [1], where we have applied the method of nonlinear self-adjointness [2] and constructed conservation laws

Dt {C1) + Dx (C2) =0 (1.1)

for the nonlinear heat and filtration equations associated with their Lie point symmetries.

In this introduction we revise and outline the results of [1] concerning the conservation laws for the nonlinear heat conduction equation

ut = (k(u)ux)x . (1.2)

It is well known that Eq. (1.2) with an arbitrary function k(u) admits the three-dimensional Lie algebra L3 with the basis

d d d d X1 = «■ *2 = X3 = 2tm + U-3)

and that this equation has a wider symmetry Lie algebra in the following special cases (see e.g. [3]): if k(u) = eu, the admitted Lie algebra L3 extends by the operator

d d

X4 = x— + 2 — ; (1.4)

ox Ou

if k(u) = ua, where a = 0, — |, the algebra L3 extends by the operator

d d

XA = ox— +2u — ; (1.5)

ox Ou

finally, if k(u) = u 4/i, the algebra L3 extends by two operators Using the substitution

д д д d

X4 = -2x — + 3-u — , X5 = -x2— + 3xu— ■ (1.6)

Ox Ou Ox Ou

v = Ax + В, A, В = const., (1.7)

found in the [2] from the equation

F* lv=^(t,x,u) = ^ [ut — k{u)uxx — kl(u)u^\

A.A. Alexandrova, N.H. Ibragimov, K.V. Imamutdinova and V.O. Lukashchuk, Local and nonlocal conserved vectors for the nonlinear filtration equation.

© A.A. Alexandrova, N.H. Ibragimov, K.V. Imamutdinova and V.O. Lukashchuk 2012.

We acknowledge the financial support of the Government of Russian Federation through Resolution No. 220, Agreement No. 11.G34.31.0042.

Поступила 29 октября 2012 г.

that connects Eq. (1.2) with its adjoint equation

F * = vt + k(u)vxx = 0,

and applying the general procedure from [2] to the Lie point symmetries (1.3)-(1.6), we have found in [1] the following conserved vectors for the nonlinear heat equation.

In the case of the arbitrary function k(u) the symmetries X2 and X3 provide two linearly independent conserved vectors:

C1 = u, C2 = —k(u)ux (1.8)

and

C1 = xu, C2 = K(u) — xk(u)ux , (1.9)

respectively, where

K '(u) = k(u).

The time-translational symmetry X1 leads to a trivial conserved vector (the similar result is proved in [4], Section 1.3 for the multi-dimensional case). The conservation law (1.1) for the vector (1.9) coincides with Eq. (1.1), whereas the vector (1.9) satisfies the conservation law (1.1) in the following form:

Dt (C1) + Dx (C2) = x[ut — (k(u)ux)x].

In the special case k(u) = eu the additional symmetry X4 given by Eq. (1.4) does not lead to a new conservation law. Indeed, one can verify that the conserved vector provided by this symmetry X4 is equivalent to the conserved vector (1.9) with k(u) = K(u) = eu.

In the special case k(u) = ua the additional symmetry X4 given by Eq. (1.5) also does not lead to a new conservation law. Indeed, the calculation shows that the conserved vector provided by this symmetry X4 is a linear combination of the conserved vectors (1.8) and (1.9) with

k(u) = ua, K (u) = —ua+1.

Finally, in the case k(u) = u-4/3 the conserved vector provided by the operator X4 from (1.6) is a linear combination of the corresponding conserved vectors (1.8) and (1.9), whereas the operator X5 lead again to the conserved vector (1.9).

Thus, the extended symmetries (1.4)-(1.6) do not give new conservation laws.

In the rest of the paper we dwell upon the nonlinear filtration equation

ut = k(ux)uxx (1.10)

and construct the conserved vectors associated not only with its Lie point symmetries, but also with the nonlocal symmetries found in [5]. Eq. (1.10) describes, in particular, a distribution of the pressure in a porous medium.

2. Nonlinear self-adjointness of the filtration equation

2.1. The general case. We will write Eq. (1.10) in the form

F =—ut + k(ux)uxx = 0. (2.1)

Its adjoint equation has the form

F * = vt + k(ux)vxx + k'(ux)vxuxx = 0. (2.2)

Let us find a function (p(t, x, u) satisfying the nonlinear self-adjointness condition

F \v=ip(t,x,u) = ^ [Vrt k(ux)uxx]. (2.3)

The expanded form of Eq. (2.3) is

^Uut + + k(ux) \<puu XX + ^UU + 2^

XU^X + ^xx\ +

+ k (u,x) [lfuUx + <fix\ u,xx = X \ut k(ux)uxx\.

Equating the terms with ut in both sides of Eq. (2.4) we obtain

A = yu.

(2.4)

Taking this into account and equating the terms with uxx in both sides of Eq. (2.4) we arrive at the equation

pu [2k(ux) + uxk'(ux)] + pxkf(ux) = 0. (2.5)

Then Eq. (2.4) reduces to the following:

ipt + k(ux) [ifuuul + 2ipxuux + ipxx] = 0. (2.6)

In the case of an arbitrary function k(ux) the determining equations (2.5)-(2.6) for tp(t,x,u) are satisfied only if tp = const. We can let

<p = 1. (2.7)

2.2. A special case. We will find now the particular form of k(ux) when Eqs. (2.5)-(2.6) are satisfied for a non-constant function tp(t,x,u). Separating the variables in Eq. (2.5) we have:

2k(ux) + = _ ^

k'(ux) Ux <p. .

It follows that

U

2k(ux) px .

+ ux = —a,-----------= —a, a = const. (2.8)

"?u

k' (ux) ’ p.

The first equation (2.8) written in the form

dk 2k

dux ux + a

gives

k(ux) = ----------, m = const. (2.9)

(ux + a)2

The solution of the second equation (2.8), i.e. of the partial differential equation

aw — TT = 0•

Ou ox

has the form

<p = <fi(t,z), z = u + ax. (2.10)

The substitution of (2.9) and (2.10) in Eq. (2.6) yields:

$t + m0zz = 0. (2.11)

We further simplify Eqs. (2.9)-(2.11) by using the equivalence transformation

u = u + ax (2.12)

of Eq. (2.1). Applying this transformation and denoting a again by u we conclude that the nonlinear

filtration equation

Ut = 2 V'xx (2.13)

u$

satisfies the nonlinear self-adjointness condition (2.3) with the function

p = 4>(t ,u), (2.14)

where <fi(t, u) is an arbitrary solution of the equation

$t + m<puu = 0. (2.15)

3. Construction of conserved vectors

The nonlinear filtration equation (1.10) admits the four-dimensional Lie algebra L4 with the basis

d d d d d d

X1 = 7£, X2 = 7T, X3 = ^, X4 = 2t - +x— +u— • (3.1)

Ot ox Ou Ot Ox Ou

The algebra L4 extends by one additional admitted operator X5 in the following cases ([3], Sect. 10.3):

if k(ux) = eWx, then

Y , d d

X5 = t — —x— ;

Ot OU

if k(ux) =ux (n > —l,n = 0), then

9 d

X5 = nt — — u— ;

Ot OU

^(n arctan ux)

if k(ux) =------— (n > 0), then

ux + l

d d d

X5 = nt — + u---------•

Ot OX OU

Let us construct the conservation laws

Dt {C1) + Dx (C2) = 0

for the operators X1,..., X7 using the algorithm given in [2]. Namely, writing the formal Lagrangian in the form

C = v [ut — k(ux)uxx] (3.2)

we have the following expressions for the components of the conserved vectors:

o p

C1 = W— = Wv, dut

C2 = W

D

(—)

V dUxx )

+ DAW) -Iе- = <3-3>

OUXX

dux

= Wk(u,x)vx — DX(W )k(u,x)v,

where we should make the substitution v = <p(t, x, u).

In the general case we have <p = 1 (see Eq. (2.7)). One can verify that X\,X2 and X3 provide only trivial conserved vectors whereas X4 yields the following conserved vector:

C1 = u, C2 = —K(ux), (3.4)

where

K.' (Ux) = k(ux).

In the case

k(ux) = eUx

the operator X5 provides the conserved vector

C1 = —x — teUx Uxx, C2 = eUx + te2Ux (u2xx + Uxxx).

In the case

k(ux) = u™

the operator X5 yields

п.П+1

at n> —1,n = 0 C1 = —u, C2 = x

n + 1'

at n = —1 C1 = —u, C2 = lnux.

In the case

an arctan Ux

6

k(ux) = —^--------------, n > 0,

v ’ u% + 1 ’ > ’

the operator X5 yields the trivial conserved vector

C1 = -x, C2 = 0.

Remark 1. The conservation law for the conserved vector (3.4) coincides with Equation (1.10). The other conserved vectors obtained in this section can be reduced to the trivial conserved vector.

4. Conservation laws in the special case

Let us turn to Eq. (2.13). In this case <p(t,x,u) given by Eq (2.14). The symmetries of Eq.(2.13) are given by (3.1).

Let us begin with Consider the symmetry X3. We have W = 1, and Eqs. (3.3) give the infinite set of conserved vectors

C1 = fa, C2 = — fau (4.1)

Ux

involving an arbitrary solution fa = fa(t, u) of Eq. (2.15). We have:

Dt(C ^ + DX(C2) = fat + mfauu +

m

Ut------2 u

9 ^XX

uz

fav

Hence, invoking Eq. (2.15), we obtain the conservation equation

Dt(C 1) + DX(C2) = Consider the symmetry X1. Eqs.(3.3) give

m

Ut 22 UXx

UZ

fau.

„1 , „2 m , m ,

C = —<put, C =------------------fauUt +------2 fautx.

ux ux

Since

—faut = —2 0uxx = ™DX ( — ) — mfav Ux \ ux J

we can write the above conserved vector in the form

C1 = fau, C2 = -—fat. (4.2)

ux

This vector satisfies the conservation equation due Eq.(2.15) because

Dt(C1) + DX(C2) = (fa + mfauu)+ (ut — ^uxx\ fauu.

ux \ ux J

For X2 we obtain

/~i1 j r'i 2 j j

C = —faux, C = —mfau + 22 faU%x.

Ux We have

faux = Dx[$(t,u)],

where §(t,u) is defined by the equation

= fa(t,u).

Therefore the above conserved vector is equivalent to

C1 = 0, C2 = —mfau(t,u) — ®t(t,u). (4.3)

The conservation equation for this vector is satisfied due to Eq. (2.15). Namely, we have:

Dt(C1) + DX(C2) = —(fat + mfauu)ux.

For X4 we obtain

C1 = (u — 2tut + xux)fa,

^2 m / \ 1 m /

C =—(u — 2tut — xux)fau +—7r(2tutx + xux

Ux U,i

We have

^ f fa \

—2tfaut = —2tfa^uxx = Dx\ 2mt— — 2mtfav

Ux y uxJ

^ s ^

and

—xfaux = —Dx (^$) + $,

where $ = $(t,u) has been defined in the previous case. Therefore the above conserved vector is equivalent to

C1 = ufa — 2mtfau + $,

^2 ^ 2m, , , , . mfau, . (4.4)

C = —x$t +----------(fa + tfat) +---(u — xux).

ux ux

The conservation equation for this vector is satisfied in the following form:

\

u — xux — -^2uxxj (fat + mfauu) +

+ (2fa + ufau — 2mtfauu)[ Ut--------2 uxx

u^

( m \

I Ut-------2 UXX I .

V ux j

5. Nonlocal symmetries and conserved vectors

The nonlinear filtration equation (1.10) has nonlocal symmetries (see [5]) in the case when the

function k(ux) has the form

k(ux) = uax-1 (5.1)

with a = 1/?j and a = —1/3.

In the case <j = 1/3 the corresponding equation (1.10) is written

Ut = u- 2 uxx . (5.2)

It has the nonlocal symmetry

d d

Xe = w-----------u2—, (5.3)

ox Ou

(5.6)

where w is a nonlocal variable defined by the equations

1

wx = u, wt = 3(wxx)3 . (5.4)

The application of the general method to the nonlocal symmetry (5.3) gives the conserved vector

C1 = u2 + wux, C2 = —3uu]J3 — wu-223uxx . (5.5)

The conservation law for the vector (5.5) is satisfied in the following form:

Dt(C 1) + DX(C2) =

= 2u, (u,t — u-223 uxx^J + wDx (ut — u-223 uxx^J + ux (wt — 3w\2x3 j .

In the case <j = —1/3 the corresponding equation (1.10) is

ut = u-423 uxx . (5.7)

It has the nonlocal symmetry

^ 2 d d

X7 = x — + (w — xu—),

OX OU

where w solves the equations

wx = u, wt = —3(wxx)~3 . (5.8)

In this case the conserved vector has the form

_ 4

C1 = w — xu — x2ux, C2 = ux 3 (3xux + x2uxx)

and satisfies the conservation equation

_1 _4 _4

Df(C ) + DX(C ) = Wt + ^Wxx X(Uf W- 3UXx) + (^t 3^xx)x.

Remark 2. The nonlocal conserved vectors obtained in this section can be reduced to the trivial conserved vector.

СПИСОК ЛИТЕРАТУРЫ

1. A.A. Alexandrova, N.H. Ibragimov, K.V. Imamutdinova and V.O. Lukashchuk Conservation laws of nonlinear heat and filtration equations // Archives of ALGA, V. 9, 2012. P. 53-62.

2. N.H. Ibragimov Nonlinear self-adjointness in constructing conservation laws // Archives of ALGA, V. 7/8, 2010-2011. P. 1-99, See also arXiv:1109.1728v1[math-ph] (2011) P. 1-104.

3. N.H. Ibragimov, ed. CRC Handbook of Lie group analysis of differential equations. Vol. 1: Symmetries, exact solutions and conservation laws, Boca Raton, CRC Press Inc., 1994.

4. E.D. Avdonina and N.H. Ibragimov Conservation laws of anisotropic heat equations // Archives of ALGA, V. 9, 2012. P. 13-22,

5. I.Sh. Akhatov, R.K. Gazizov, and N.H. Ibragimov Quasi-local symmetries of nonlinear heat conduction type equations // Dokl. Akad. Nauk SSSR, 295, No.1, 1987. P. 75-78. (Russian).

Nail H. Ibragimov,

Laboratory Group analysis of mathematical models in natural and engineering sciences,

Ufa State Aviation Technical University,

K. Marx Str. 12,

450 000 Ufa, Russia

and Research Centre ALGA: Advances in Lie Group Analysis,

Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden E-mail: [email protected]

A.A. Alexandrova, K.V. Imamutdinova, V.O. Lukashchuk,

Laboratory Group analysis of mathematical models in natural and engineering sciences,

Ufa State Aviation Technical University,

K. Marx Str. 12,

450 000 Ufa, Russia E-mail: [email protected]