June 15-19, 2014, Larnaca, Cyprus

JWorkshop in Honor of Michail Ivanov ’ 70 th Birthday

The equation studiedThe Fourier image of the spatially homogeneous and isotropicBoltzmann equation with a source term has the form:

ϕt (x , t) + ϕ(x , t)ϕ(0, t) =

∫ 1

0ϕ(xs, t)ϕ(x(1− s), t) ds + q̂(x , t).

Here the function ϕ(x , t) is

ϕ(x , t) ≡ ϕ(k2/2, t) = ϕ̃(k , t) =4πk

∫ ∞0

v sin(kv)f (v , t) dv .

where f (v , t) is the distribution function of isotropic in the3D-space of molecular velocitiesSimilarly, the transform of the isotropic source function q(v , t) is

˜̂q(k , t) =4πk

∫ ∞0

v sin(kv)q(v , t) dv ,

Determining equation

A generator of the admitted Lie group is sought in the form

X = ξ(x , t , ϕ)∂x + η(x , t , ϕ)∂t + ζ(x , t , ϕ)∂ϕ.

The determining equation for the considered equation is

Dtψ(x , t)+ψ(0, t)ϕ(x , t)+ψ(x , t)ϕ(0, t)−2∫ 1

0ϕ(x(1−s), t)ψ(xs, t)ds = 0,

where Dt is the total derivative with respect to t , and thefunction ψ(x , t) is

ψ(x , t) = ζ(x , t , ϕ(x , t))−ξ(x , t , ϕ(x , t))ϕx (x , t)−η(x , t , ϕ(x , t))ϕt (x , t).

Assume that the coefficients of the infinitesimal generator X arerepresented by the formal Taylor series with respect to ϕ:

ξ(x , t , ϕ) =∑l≥0

ql(x , t)ϕl(x , t),

η(x , t , ϕ) =∑l≥0

rl(x , t)ϕl(x , t),

ζ(x , t , ϕ) =∑l≥0

pl(x , t)ϕl(x , t).

A particular class of solutions is considered. This class isdefined by the initial conditions

ϕ0(x , t) = bxn

at a given (arbitrary) time t = t0. Here, n = 0,1,2, ....

The coefficients of the generator X are

ξ(x , t , ϕ) = c0x , η(x , t , ϕ) = −c2t+c3, ζ(x , t , ϕ) = (c2+c1x)ϕ

where c0, c1, c2and c3 are arbitrary constant.Thus, each admitted generator has the form

X = c0X0 + c1X1 + c2X2 + c3X3,

where

X0 = x∂x , X1 = xϕ∂ϕ, X2 = ϕ∂ϕ − t∂t , X3 = ∂t .

The remaining part of the determining equation becomes

(c2t − c3)q̂t − c0xq̂x + (c1x + 2c2)q̂ = 0.

Equivalence TransformationsLet us introduce the operator L:

Lϕ = ϕt (x , t) + ϕ(x , t)ϕ(0, t)−∫ 1

0ϕ(xs, t)ϕ(x(1− s), t) ds.

Equivalence transformations corresponding to the generatorsX0, X1, X2 and X3, are obtained as follows.For example, the transformations corresponding to thegenerator X0 = x∂x map a function ϕ(x , t) into the functionϕ̄(x̄ , t̄) = ϕ(x̄e−a, t̄), where a is the group parameter. Thetransformed expression becomes

L̄ϕ̄ = ϕ̄t̄ (x̄ , t̄) + ϕ̄(x̄ , t̄)ϕ̄(0, t̄)−∫ 1

0 ϕ̄(x̄s, t̄)ϕ̄(x̄(1− s), t̄) ds= ϕt̄ (x̄e−a, t̄) + ϕ(x̄e−a, t̄)ϕ(0, t̄)−

∫ 10 ϕ(x̄e−as, t̄)ϕ(x̄e−a(1− s), t̄) ds

= ϕt (x , t) + ϕ(x , t)ϕ(0, t)−∫ 1

0 ϕ(xs, t)ϕ(x(1− s), t) ds= Lϕ.

This defines the Lie group of equivalence transformations ofthe equation

x̄ = xea, t̄ = t , ϕ̄ = ϕ, ¯̂q = q̂.

Similarly, the transformations corresponding:to the generator X3 = ∂t

x̄ = x , t̄ = t + a, ϕ̄ = ϕ, ¯̂q = q̂.

to the generator X2 = ϕ∂ϕ − t∂t

x̄ = x , t̄ = te−a, ϕ̄ = ϕea, ¯̂q = q̂e2a

to the generator X1 = xϕ∂ϕ

x̄ = x , t̄ = t , ϕ̄ = ϕexa, ¯̂q = q̂exa.

Thus the generators defining an equivalence Lie group of theconsidered equation are

X e0 = x∂x , X e

1 = xϕ∂ϕ+xq̂∂q̂, X e2 = ϕ∂ϕ−t∂t +xq̂∂q̂, X e

3 = ∂t .

Let us study the change of a generator

X = x0X0 + x1X1 + x2X2 + x3X3

under these equivalence transformations. After the change onegets the generator

X = x̂0X̂0 + x̂1X̂1 + x̂2X̂2 + x̂3X̂3,

where

X̂0 = x̄∂x̄ , X̂1 = x̄ϕ̄∂ϕ̄, X̂2 = ϕ̄∂ϕ̄ − t̄∂t̄ , X̂3 = ∂t̄ .

The corresponding transformations of the basis generators are

X e0 : X0 = X̂0,X1 = e−aX̂1,X2 = X̂2,X3 = X̂3;

X e1 : X0 = X̂0 + aX̂1,X1 = X̂1,X2 = X̂2,X3 = X̂3;

X e2 : X0 = X̂0,X1 = X̂1,X2 = X̂2,X3 = e−aX̂3;

X e3 : X0 = X̂0,X1 = X̂1,X2 = X̂2 + aX̂3,X3 = X̂3.

Any generator X can be expressed as a linear combination ofthe basis generators:

X = x̂0X̂0 + x̂1X̂1 + x̂2X̂2 + x̂3X̂3 = x0X0 + x1X1 + x2X2 + x3X3

Using the invariance of a generator with respect to a change ofthe variables, the basis generators Xi (i = 0,1,2,3) and X̂j(j = 0,1,2,3) in corresponding equivalence transformations arerelated as follows:

X e0 : x̂1 = x1e−a,

X e1 : x̂1 = x1 + ax0,

X e2 : x̂3 = x3ea,

X e3 : x̂3 = x3 + ax2.

For classification an algebraic algorithm was applied,

I first we study the Lie algebra L4 composed by thegenerators X0,X1,X2,X3. The commutator table is

X0 X1 X2 X3X0 0 X1 0 0X1 −X1 0 0 0X2 0 0 0 −X3X3 0 0 X3 0

The inner automorphisms are

A0 : x̂1 = x1ea,A1 : x̂1 = x1 + ax0,A2 : x̂3 = x3ea,A3 : x̂3 = x3 + ax2,

I Second, one can notice that the results of using theequivalence transformations corresponding to thegenerators X e

0 ,Xe1 ,X

e2 ,X

e3 are similar to changing

coordinates of a generator X with regards to the basischange.These changes are similar to the inner automorphisms.

Really

A0 : x̂1 = x1ea, X e0 : x̂1 = x1e−a,

A1 : x̂1 = x1 + ax0, X e1 : x̂1 = x1 + ax0,

A2 : x̂3 = x3ea, X e2 : x̂3 = x3ea,

A3 : x̂3 = x3 + ax2, X e3 : x̂3 = x3 + ax2.

Optimal system of subalgebras

Construction of an optimal system of subalgebras of the Liealgebra L4

I It is simplified if one notices that L4 = F1 ⊕ F2,where F1 = {X0,X1} and F2 = {X2,X3} are ideals of theLie algebra L4.

I This decomposition gives a possibility to apply a two-stepalgorithm (Ovsiannikov, 1993 and 1994).

The result of construction of an optimal system of subalgebrasis presented in Table 1.

Optimal system of subalgebras

No. Basis No. Basis1. X0, X1, X2, X3 13. X0 + X3, X12. γX0 + X2, X1, X3 14. X1, X33. X0, X1, X3 15. X0, X34. X0, X1, X2 16. X0, X15. X0, X2, X3 17. γX0 + X26. X2, X3 18. X1 + X27. X2 + X0, X1 + X3 19. X1 − X28. X2 + γX0, X3 20. X0 + X39. X1 + X2, X3 21. X1 + X3

10. X1 − X2, X3 22. X011. X0, X2 23. X112. γX0 + X2, X1 24. X3

Obtaining the Function q̂

Example: Lie algebra {γX2 + 2X0, X3}For this Lie algebra there are two sets of the coefficients ci ,(i = 0,1,2,3):

γX2 + 2X0 : c0 = 2 c1 = 0 c2 = γ c3 = 0;X3 : c0 = 0 c1 = 0 c2 = 0 c3 = 1.

These sets define the system of equations by substituting thecoefficients ci into the remaining equation:

γ(12

t q̂t + q̂)− xq̂x = 0, q̂t = 0.

The general solution of these equations is q̂ = βxγ , where β isconstant.

Group Classification

No. q̂(t , x) Generators1. 0 X0, X1, X2 X32. kx2etx X2 + X0, X1 + X33. kxγ γX2 + 2X0, X34. kt−2 X0, X25. t−2Φ(xtα) αX0 + X26. t−(x+2)Φ(x) X1 + X27. tx−2Φ(x) X1 − X28. Φ(xe−t ) X0 + X39. ext Φ(x) X1 + X310. Φ(t) X011. Φ(x) X3

where α, β, γ and k are constant

Representation of invariant solution for q̂ = kx2ext

Equation

ϕt (x , t) + ϕ(x , t)ϕ(0, t) =

∫ 1

0ϕ(xs, t)ϕ(x(1− s), t) ds + kx2ext .

The corresponding admitted Lie algebra of the equation is{X2 + X0, X1 + X3}.An optimal system of subalgebras of this Lie algebra is:

{X2 + X0}, {X1 + X3}, {X2 + X0, X1 + X3}

For the subalgebra {X2 + X0} corresponding invariant solutionhas a representation

ϕ = t−1r(z), z = xt

Substituting this representation of invariant solution, we obtainthe reduced equation:

zr ′(z)− r(z) + r(z)r(0)−∫ 1

0r(zs)r(z(1− s)) ds = kz2ez .

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