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2DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 2, 2024 Electronic Journal, reg. N &C77-39410 at 15.04.2010 ISSN 1817-2172
http://diffjournalspbu.ru/ e-mail: [email protected]
Nonlinear partial differential equations Group analysis of differential equations
Solution of the Carleman system using group analysis
Dukhnovsky S. A. Moscow State University of Civil Engineering [email protected]
Abstract. In this paper, we consider the discrete kinetic Carleman system. The Carleman system is the Boltzmann kinetic equation, and for this model momentum and energy are not conserved. Using group analysis methods, we obtain a solution representing the density of gas particles in a certain area. This limitation is due to the non-negativity of solution. Similarly, it is possible to find exact solutions for other kinetic models.
Keywords: Carleman system, group analysis, invariant solution 1 Introduction
We consider the one-dimensional Carleman system [9, 15]:
dtu + dxu = -(v2 — u2), x e R, t> 0,
£ 1 (1)
dtv — dx v = — (v2 — u2).
£
Here u = u(x, t),v = v(x, t) are the densities of two groups of particles with velocities c = 1, — 1, £ is the Knudsen parameter from the kinetic theory of gases. This system describes a monatomic rarefied gas consisting of two groups of particles. The Carleman system is a non-integrable system, i.e. the Painleve
test is not applicable. The interaction is as follows. The Carleman system describes particles of two groups, namely, the first group of particles moves at a unit speed along the axis Ox, and the second group moves at a unit speed in the opposite direction. It is worth noting that only particles within one group interact, that is, only with themselves, changing the direction of motion.
The method of group analysis is a well-known method for finding solutions, in particular invariant solutions of equations of mathematical physics. This method is described in detail in [1, 2, 3, 5]. A general description of the Boltzmann equation is described in the article [4]. In this works [6, 7], Ilyin O. V. obtained an optimal system of one-dimensional subalgebras and classes of invariant solutions for the stationary kinetic Broadwell model and the one-dimensional Boltzmann integro-differential equation for Maxwellian particles with inelastic collisions. In [8] the results of group analysis of the multidimensional Boltzmann and Vlasov equations are presented. Also, solutions of kinetic systems using Painleve series were found in [9, 10, 11, 15, 16]. Asymptotic stability for Boltzmann models as well as the numerical study are presented in [14, 17, 18, 19]. The analysis described in this paper provides solutions of the Carleman system that have not yet been found in the literature.
2 Method of group analysis
We consider a system of two second order partial differential equations
F1(u,ut,ux,utt,uxt,uxx,...) = 0, (2)
Vt ,Vx,Vtt, Vxt,Vxx,...) = 0, (3)
where u = u(x,t),v = v(x,t) are unknown functions. According to the group analysis methods, we will look for the prolonged operator in the form
^ d d , d 3,3,3 d d
X = + n^r + Z^- + x^- + — + (2^- + + X2-
1 dx dt du dv dux dut dvx dvt' where £ = £(x,t,u,v), n = £(x,t,u,v), Z = Z(x,t,u,v), x = x(x,t,u,v). Here
d d d d £ dx + n dt + Z du + x dv ( )
is called the infinitesimal operator of the group. A universal invariant of the group and the operator (4) is a function I(x,t,u,v)
xi = + n + Z + x = 0. (5)
dx dt du dv
The coordinates of the first prolongation are given by
Ci = Dx(Z) - uxDx(€) - UfDx(n). C2 = Dt(() - uxDt(0 - UtDt(n)
and
Xi = Dx(x) - VxDx(£) - VtDx(n), X2 = Dt(x) - VxDt(C) - VtDt(n),
where Dx, Dt are the operators of total differentiation with respect to x and t:
d
+ Vxt+----,
x dvt
d d d d d d
dx + ux^~ du + Vx^~ dv 1 uxx r\ dux + uxt 0 dut + VxxTT
d d d d d d
= dt + ut— du + Vt — dv + uxt 0 dux + utt— dut + Vxt 0 dvx
d
t = — + Ut— + Vt~ + uxt---h Utt---h Vxt---h Vit---1----.
dv dux dut dvx dvt After calculations, we have
Ci = Zx + (uUx + Cvv x ux (Cx + CuUx + Cv Vx) - ut (nx + Vuux + nvVx), (6)
Z2 = Zt + Zuut + ZvVt - ux(Ct + Cuut + CvVt) - ut (nx + nuut + nvVt) (7)
and
Xi = Xx + uxXu + V xXv Vx(C,x + uxCu + VxCv ) - Vt (nx + uxnu + Vxnv ) , (8)
X2 = Ct + Cuut + Cv Vt - Vx(Ct + Cuut + Cv Vt) - Vt(nt + nuut + nv Vt). (9) We require that
X =o = 0,X f2|f1=F2=o = 0. (10)
Relations (10) is called the invariance conditions.
We illustrate the above procedure for the Carleman system.
3 Application of the method
Let us insert 1
Fi = ut + ux--(v2 - u2),
£
F2 = Vt - vx + -(v2 - u2),
£
into the invariance conditions (10)
XF1\Fi=0f2=0 = (C2 + Ci - -2vx + -2uZ)|f1=g,f2=0, (11)
1 £ £
Xf2\fi=0,f2=0 = (X2 - Xi + ~2vX - -2uC)|fi=g,f2=g- (12)
1 £ £
Now, replacing ut,vt by -ux + i(v2 — u2),vx — i(v2 — u2) and taking into account the expressions (6)-(7) and (8)-(9) for the coordinates of the first prolongation from the first equation (11), we have:
Ux : -it - 1(v2 - u2)^u + iv 1(v2 - u2) + nt + nu 1(v2 - u2)-
£ £ £
-1 nv(v2 - u2) - ix + nx = 0,
£
ux : iu - nu - iu + nu = 0,
vx : Zv - 1(v2 - u2)nv + Zv - 1(v2 - u2)nv = 0,
££
uxvx : Cv + nv Cv + nv 0,
1 : it + Zu 1(v2 - u2) - Zv 1(v2 - u2) - V - u2)nt-
£ £ £
--(v2 - u2)nu 1(v2 - u2) + -(v2 - u2)nv1 (v2 - u2) + Zx-
£ £ £ £
1 . 2 2 s 2 2
— (v - u )nx--xv + -uZ = 0.
£ £ £
From the second equation (12):
1/ 2 2 \ 1 / 2 2 \
ux : Xu - -(v - u )nu - Xu--(v - u )nu = 0,
££
uxvx : iu + nu + iu + n - u = 0,
vx : Xv - it - 1(v2 - u2)iu + iv 1(v2 - u2) - nt-
££
-nu 1(v2 - u2) + nv 1(v2 - u2) + -(v2 - u2)nv - Xv + ix+
£ £ £
1 2 2 +nx--(v - u )nv = 0,
£
vX : -Zv - nv + Zv + nv = 0,
1 : Xt + Xu-(v2 - u2) - Xv-(v2 - u2) + -(v2 - u2)nt+
£ £ £
- 2 2 - 2 2 - 2 2 - 2 2 +-(v - u )nu-(v - u ) --(v - u )nv-(v - u ) - Xx-
£ £ £ £
1/2 2 \ 2 2 — (v - u )nx + -Xv--Zu = 0.
£ £ £
Rewrite the system in a more compact form
nv Zv 5
-
2-(v - u )nu - Zt + nt - nx - Zx = 0,
£
Zv - -(v2 - u2)nv = 0,
£
-(v2 - u2)(Zu - Zv - nt - nu-(v2 - u2) + nv-(v2 - u2) - nx) +
£ £ £
22
+Zx + Zt — xv + - uZ = 0
££
and
nu Zu5
xu + -(v2 - u2)nu = 0,
£
1
2-(v - u )nv - nt - £t + nx + £x = 0,
£
-(v2 - u2)(xu - xv + nt + -(v2 - u2)nu - -(v2 - u2)nv - nx) +
£ £ £
22
+xt - xx + -xv - "Zu = 0.
££
Integrating the system, we get
n(t) = at + 3, £(x) = ax + y, Z(u) = -au, x(v) = -av,
where a,3,Y are arbitrary constants. The corresponding characteristic system of ordinary differential equations for (5)
dt dx du dv
n £ Z x
or
dt dx du dv
at + ß ax + y -au - av
We obtain
ax + y
U = -7T ,
at + p
where a,Y,p is an integration constants. Invariant solution are sought in the form
$(u) W(u)
a(at + p)' a(at + /3)' where W unknown functions of similarity variable that must be defined.
-(at + p)$ + (at - ax + p - y)$' W2 - $2
(at + 3 )3 a2 (at + 3 )V
-(at + p)W - (at + ax + p + y)W'_ $2 - W2
(at + p )3 = a2 (at + p )2e
(13)
(14)
or we rewrite
a2e($(1 - u))' = -$2 + W2,
a2e(W(1 + u))' = -$2 + W2. (15)
$(u) = W(u) + (1 - u)C, 1 — u
where C is an arbitrary constant of integration.
For finding function W, we have the Riccati equation
—■4u t2 (—¿c—ea + 2(ea — c )u — ea u \
w =-w2 + ----- W —
ea2(1 - u)2(1 + u) V ea2(1 - u)2(1 + u) j
C2
ea2(1 - u)2(1 + u)' We can write the particular solution at C = 0
-ea(ea(t + x) + p + y )
u=
-¿(eat + p)2 + Ci((eat + p)2 - (eat + y)2)'
=_-ea(ea(t - x) + p - y)__(16)
V = -¿(eat + p)2 + Ci((eat + p)2 - (eat + y)2)' ( )
where Ci is an arbitrary constant of integration. The non-negativity of the solution can be chosen by choosing a constant Ci in the region of the plane x, t.
4 Conclusion
In this article, the solution was found using group analysis methods. We determined three different infinitesimal groups of transformations leaving the invariant condition. A well-known and very important special class of invariant solutions was represented by self similar solution which was constructed on the basis of invariants of extension groups. Future work will present solutions to the remaining kinetic systems.
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