[PPT] - Application of the method of differential constraints to PowerPoint Presentation

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Application of the method of differential constraints to constructing exact solutions

f the gas dynamics equations

S.V. Meleshko

School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, Thailand

Abrau-Durso, Russia September 14-19, 2015 1 / 38

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Outline

1

Introduction Equations describing motion of fluids with internal inertia Examples Group classification

2

Conservation law Noether’s theorem Shmyglevskii’s approach Ibragimov’s approach

3

Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

4

Method of differential constraints Generalized simple waves

5

Conclusion

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Introduction Equations describing motion of fluids with internal inertia

Equations describing motion of fluids with internal inertia S.L.Gavrilyuk, V.M.Teshukov (2001) ˙ ρ + ρdiv(u) = 0, ρ ˙ u + ∇p = 0, ˙ S = 0

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Introduction Equations describing motion of fluids with internal inertia

Equations describing motion of fluids with internal inertia S.L.Gavrilyuk, V.M.Teshukov (2001) ˙ ρ + ρdiv(u) = 0, ρ ˙ u + ∇p = 0, ˙ S = 0 p = ρδW δρ − W, W = W(ρ, ˙ ρ, S)

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Introduction Equations describing motion of fluids with internal inertia

Equations describing motion of fluids with internal inertia S.L.Gavrilyuk, V.M.Teshukov (2001) ˙ ρ + ρdiv(u) = 0, ρ ˙ u + ∇p = 0, ˙ S = 0 p = ρδW δρ − W, W = W(ρ, ˙ ρ, S) p = ρ (Wρ − (W ˙

ρ − div (uW ˙ ρ)) − W,

t is time, ∇ is the gradient operator with respect to space variables, ρ is the fluid density, u is the velocity field, W (ρ, ˙ ρ, S) is a given potential, ’dot’ denotes the material time derivative: ˙ f = d f dt = ft + u∇f, δW δρ denotes the variational derivative of W with respect to ρ at a fixed value of u. Abrau-Durso, Russia September 14-19, 2015 3 / 38

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Introduction Examples

Example 1: Iordanski-Kogarko-Wingaarden model (1960, 1961, 1968)

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Introduction Examples

Example 2: Green-Naghdi equations (1975)

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Introduction Examples

Example 2: Green-Naghdi equations (1975) Yu.Yu.Bagderina, A.P.Chupakhin (2005) Invariant and partially invariant solutions of the Green–Naghdi equations

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Introduction Group classification

Isentropic flows

Abrau-Durso, Russia September 14-19, 2015 6 / 38

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Introduction Group classification

Isentropic flows

A.Hematulin, S.V.Meleshko and S.L.Gavrilyuk (2007)

Group classification of one-dimensional equations of fluids with internal inertia Abrau-Durso, Russia September 14-19, 2015 6 / 38

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Introduction Group classification

Nonisentropic flows

P.Siriwat, C.Kaewmanee and S. V. Meleshko (2015) Group classification of one-dimensional nonisentropic equations of fluids with internal inertia II. General case Abrau-Durso, Russia September 14-19, 2015 7 / 38

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Introduction Group classification

Nonisentropic flows

P.Siriwat, C.Kaewmanee and S. V. Meleshko (2015) Group classification of one-dimensional nonisentropic equations of fluids with internal inertia II. General case Abrau-Durso, Russia September 14-19, 2015 8 / 38

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Conservation law Noether’s theorem

Noether’s theorem XF + FDiξi = W k δF δuk + Di(N iF)

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Conservation law Noether’s theorem

Noether’s theorem XF + FDiξi = W k δF δuk + Di(N iF) N iF = ξiF +W k δF δuk

i

+

s=1

Di1...Dis(W k) δF δuk

ii1i2...is

, (i = 1, 2, ..., n).

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Conservation law Noether’s theorem

Noether’s theorem XF + FDiξi = W k δF δuk + Di(N iF) N iF = ξiF +W k δF δuk

i

+

s=1

Di1...Dis(W k) δF δuk

ii1i2...is

, (i = 1, 2, ..., n). W k = ηk − ξiuk

i , (k = 1, 2, ..., m),

δ δuk = ∂ ∂uk +

s=1

(−1)sDi1...Dis ∂ ∂uk

i1i2...is

, (k = 1, 2, ..., m)

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Conservation law Noether’s theorem

Noether’s theorem δ δuj

XF + FDiξi

= X( δF δuj ) + δF δuk ∂ηk ∂uj − ∂ξi ∂uj uk

i + δkjDiξi

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Conservation law Noether’s theorem

Noether’s theorem δ δuj

XF + FDiξi

= X( δF δuj ) + δF δuk ∂ηk ∂uj − ∂ξi ∂uj uk

i + δkjDiξi

If

XF + FDiξi = DiBi, δF δuj = 0 then X δF δuj

| δF

δu =0

= 0 Variational (divergent) symmetry is a symmetry

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Conservation law Shmyglevskii’s approach

Shmyglevskii’s approach. Gas dynamics Terent’ev&Shmyglevskii (1980)

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Conservation law Shmyglevskii’s approach

Shmyglevskii’s approach. Gas dynamics Terent’ev&Shmyglevskii (1980) Bateman (1929), Ito (1955), Shmyglevskii (1980) L = ρ

u2

2 + ˙ ϕ + S ˙ µ

− ρU(ρ, S)

where ϕ and µ play the roles of Lagrange’s multipliers. S is the entropy , U(ρ, S) is the internal energy

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Conservation law Shmyglevskii’s approach

Shmyglevskii’s approach. Fluids with internal inertia L = ρ

u2

2 + ˙ ϕ + S ˙ µ

− W(ρ, ˙

ρ, S). Euler–Lagrange equations u = −∇ϕ − S∇µ + ρ−1W ˙

ρ∇ρ,

˙ µ = ρ−1WS, u2 2 + ˙ ϕ + S ˙ µ = Wρ − ∂W ˙

ρ

∂t − div(W ˙

ρu),

∂ρ ∂t + div (ρu) = 0, ∂(ρS) ∂t + div (ρSu) = 0,

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Conservation law Shmyglevskii’s approach

Shmyglevskii’s approach. Fluids with internal inertia L = ρ

u2

2 + ˙ ϕ + S ˙ µ

− W(ρ, ˙

ρ, S). Euler–Lagrange equations u = −∇ϕ − S∇µ + ρ−1W ˙

ρ∇ρ,

˙ µ = ρ−1WS, u2 2 + ˙ ϕ + S ˙ µ = Wρ − ∂W ˙

ρ

∂t − div(W ˙

ρu),

∂ρ ∂t + div (ρu) = 0, ∂(ρS) ∂t + div (ρSu) = 0, ˙ ρ + ρux = 0, ρ ˙ u + px = 0, ˙ S = 0, p = ρδW δρ − W = ρ (Wρ − (W ˙

ρ − div (uW ˙ ρ)) − W,

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Conservation law Shmyglevskii’s approach

Shmyglevskii’s approach. Example of a conservation law W = ρ−3 ˙ ρ2η, X = t∂t − u∂u ∂ ∂tC1 + ∂ ∂xC2 = 0,

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Conservation law Shmyglevskii’s approach

Shmyglevskii’s approach. Example of a conservation law W = ρ−3 ˙ ρ2η, X = t∂t − u∂u ∂ ∂tC1 + ∂ ∂xC2 = 0, C1 = tρu2 2 − tρ−3η ˙ ρ2 + ρ(ϕ + ηµ), C2 = −tρu3 2 − tuηρ−3 ˙ ρ2 − 4tu2ηρ−3 ˙ ρρx + 4tu2ηρ−2ρtx − 2tu2ρ−2 ˙ ρηx +2tu2ρ−2 ˙ ρηηx + 2tuρ−2 ˙ ρuxη − 2tu2ρ−2uxηρx + 2tuρ−2ηρtt +2tuρ−2ηutρx + 2tu3ρ−3ηρxx + ρu(ϕ + ηµ).

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Conservation law Ibragimov’s approach

Ibragimov’s approach Ibragimov Conservation law’s Theorem Consider a system of m equations Fα(x, u, u(1), u2, . . . , u(s)) = 0, α = 1, . . . , m (1) with n independent variables x = (x1, x2, . . . , xn) and m dependent variables u = (u1, u2, . . . , um). The adjoint system F ∗

α(x, u, v, u(1), v(1), u(2), v(2), . . . , u(s), u(s)) ≡ δL

δuα = 0 (2) inherits the symmetries of the system (1), where L = vβFβ(x, u, u(1), . . . , u(s))

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Conservation law Ibragimov’s approach

Ibragimov Conservation law’s Theorem (continue) If system (1) admits a point transformation group with a generator X = ξi(x, u) ∂ ∂xi + ηα(x, u) ∂ ∂uα (3) then also the adjoint system (2) admits the operator (3). Then the quantities Ci = vβ

ξiFβ + (ηα − ξjuα

j )∂Fβ

∂uα

i

, i = 1, . . . , n

(4) furnish a conserved vector C = (C1, . . . , Cn) for the system (1).

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Conservation law Ibragimov’s approach

Ibragimov’s approach. Example of a conservation law The formal Lagrangian L = (R + u2 2 )( ˙ ρ + ρux) + U(ut + uux + ρ−1px) + P ˙ η

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Conservation law Ibragimov’s approach

Ibragimov’s approach. Example of a conservation law The formal Lagrangian L = (R + u2 2 )( ˙ ρ + ρux) + U(ut + uux + ρ−1px) + P ˙ η Example of multipliers U = ρu, P = Wη − W ˙

ρη( ˙

ρ + ρux), R = δW δρ + W ˙

ρη ˙

η

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Conservation law Ibragimov’s approach

Ibragimov’s approach. Example of a conservation law The formal Lagrangian L = (R + u2 2 )( ˙ ρ + ρux) + U(ut + uux + ρ−1px) + P ˙ η Example of multipliers U = ρu, P = Wη − W ˙

ρη( ˙

ρ + ρux), R = δW δρ + W ˙

ρη ˙

η W = ρ−3 ˙ ρ2η, X = t∂t − u∂u

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Conservation law Ibragimov’s approach

Ibragimov’s approach. Example of a conservation law The formal Lagrangian L = (R + u2 2 )( ˙ ρ + ρux) + U(ut + uux + ρ−1px) + P ˙ η Example of multipliers U = ρu, P = Wη − W ˙

ρη( ˙

ρ + ρux), R = δW δρ + W ˙

ρη ˙

η W = ρ−3 ˙ ρ2η, X = t∂t − u∂u

C1 = −tuρut − ρu2 − 1 2tu2 ˙ ρ + 1 2tu3ρx + 11tηρ−4 ˙ ρ3 − 11tuηρ−4 ˙ ρ2ρx +8tuηρ−3 ˙ ρρtx + tuηxρ−3 ˙ ρ2 C2 = −3 2tu2ρut − 3 2u2ρ − 1 2tu3 ˙ ρ + 1 2tu4ρx − 5tηρ−3 ˙ ρ2ut + 23tuηρ−4 ˙ ρ3 +8tuηρ−3 ˙ ρutρx − 47tu2ηρ−4 ˙ ρ2ρx + 8tu2ηρ−3ρ2

xut + 11uηρ−3 ˙

ρ2 +5tu2ρ−3 ˙ ρ2ηx − 8tu3ρ−3 ˙ ρρxηx + 8tu2ηρ−3ρx(ρtt + uρtx) + 24tu3ηρ−4 ˙ ρρ2

x

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Equations in Lagrangian variables

Equations in Lagrangian variables x = Φ(t, X) Here x ∈ D(t) is called a trajectory of the point X ∈ D(t0), the deformation gradient F = ∂x ∂X = ∂Φ(t, X) ∂X The functions u(t, x), ρ(t, x), and S(t, x) in the Eulerian coordinate system are written through the Lagrangian coordinates ρ(t, Φ(t, X))ΦX(t, X) = ρ0(X), S(t, Φ(t, X)) = S0(X), u(t, Φ(t, X)) = Φt(t, X)

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Equations in Lagrangian variables

Lagrangian In Eulerian variables LE(ρ, u, ˙ ρ, S) = ρu2 2 − W(ρ, ˙ ρ, S)

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Equations in Lagrangian variables

Lagrangian In Eulerian variables LE(ρ, u, ˙ ρ, S) = ρu2 2 − W(ρ, ˙ ρ, S) In Lagrangian variables LL(ρ0, Φt, ΦX, Φtx, S0) = ρ0 Φ2

t

2 − ΦXW ρ0 ΦX , −Φtx Φ2

x

, S0

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September 14-19, 2015 18 / 38

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Equations in Lagrangian variables

Euler-Lagrange equations The Euler-Lagrange equation δ δΦLL = 0 δ δΦ = ∂ ∂Φ − Dt ∂ ∂Φt − DX ∂ ∂ΦX + D2

t

∂ ∂Φtt + DtDX ∂ ∂ΦtX + D2

X

∂ ∂ΦXX reduces to the moment equation in Eulerian coordinates ρ ˙ u + px = 0, p = ρδW δρ − W,

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Equations in Lagrangian variables

Euler-Lagrange equations The Euler-Lagrange equation δ δΦLL = 0 δ δΦ = ∂ ∂Φ − Dt ∂ ∂Φt − DX ∂ ∂ΦX + D2

t

∂ ∂Φtt + DtDX ∂ ∂ΦtX + D2

X

∂ ∂ΦXX reduces to the moment equation in Eulerian coordinates ρ ˙ u + px = 0, p = ρδW δρ − W, The constraints ρ0 = ρ0(X), S0 = S0(X) become the equations ˙ ρ + ρux = 0, ˙ S = 0

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

The case W ˙

ρ = 0

Φ3

XΦtt − Wρρρ0ΦXX − WS

Φ3

XS′

ρ0 + Wρρρ′

0ΦX + WρSρ0Φ2 XS′ 0 = 0,

Equivalence transformation ˆ X = g(X), ˆ ρ0 = α(X)ρ0(X), where α(X) = h′(g(X)), X = h( ˆ X) is the inverse function of g(X): h(g(X)) = X.

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Gas dynamics equations in Lagrangian mass coordinate In particular ρ0(X)α(X) = 1 gas dynamics equations become Φtt + pX = 0, p = ρδW δρ − W

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Some review of group properties utt = ϕx, ϕ = ϕ(x, u, ux)

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Some review of group properties utt = ϕx, ϕ = ϕ(x, u, ux) ϕ = f(u)ux Ames, Lohner, Adams (1981)

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Some review of group properties utt = ϕx, ϕ = ϕ(x, u, ux) ϕ = f(u)ux Ames, Lohner, Adams (1981) ϕ = f(ux) Baikov & Gazizov (1989), Suhubi & Bakkaloglu (1991) Conservation laws (Vinokurov & Nurgalieva (1985))

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Some review of group properties utt = ϕx, ϕ = ϕ(x, u, ux) ϕ = f(u)ux Ames, Lohner, Adams (1981) ϕ = f(ux) Baikov & Gazizov (1989), Suhubi & Bakkaloglu (1991) Conservation laws (Vinokurov & Nurgalieva (1985)) ϕ = −b(x)uγ

x

Polytropic gas Andreev, Kaptsov, Rodionov, Pukhnachev (1998)

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Some review of group properties utt = ϕx, ϕ = ϕ(x, u, ux) ϕ = f(u)ux Ames, Lohner, Adams (1981) ϕ = f(ux) Baikov & Gazizov (1989), Suhubi & Bakkaloglu (1991) Conservation laws (Vinokurov & Nurgalieva (1985)) ϕ = −b(x)uγ

x

Polytropic gas Andreev, Kaptsov, Rodionov, Pukhnachev (1998) ϕ = c(x)ux Bluman & Kumei (1987), Grimshaw, Pelinovskii, Pelinovskii (2011)

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Group classification utt = Dxϕ, ϕ = ϕ(x, ux) The objective is to construct conservation laws using Noether’s theorem.

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Group classification utt = Dxϕ, ϕ = ϕ(x, ux) The objective is to construct conservation laws using Noether’s theorem. Webb (2015)

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Equations in Lagrangian variables Gas dynamics equations in Lagrangian coordinates

Group classification utt = Dxϕ, ϕ = ϕ(x, ux) The objective is to construct conservation laws using Noether’s theorem. Webb (2015) ϕ = −b(x)uγ

x,

γ > 1 (Polytropic gas: Andreev, Kaptsov, Rodionov, Pukhnachev (1998)) ϕ = c(x)ux, (γ = 1) (Grimshaw, Pelinovskii, Pelinovskii (2011)) ϕ = ϕ(x, ux) (C.Kaewmanee, S.V.Meleshko, S.G.Gavrilyuk (2015))

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Method of differential constraints

Invariant solutions Solutions of partial differential equations with two independent variables (t, x) F(t, x, u, ut, ux, utt, utx, uxx) = 0 invariant with respect to 1-dimensional Lie algebra are reduced to a system of ordinary differential equations.

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Method of differential constraints

Methods for constructing exact solutions utt = uxx ⇐ ⇒ ut = vx, vt = ux, u = f(x − t) + g(x + t), v = −f(x − t) + g(x + t)

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Method of differential constraints

Methods for constructing exact solutions utt = uxx ⇐ ⇒ ut = vx, vt = ux, u = f(x − t) + g(x + t), v = −f(x − t) + g(x + t) Assumptions: a) Method of generate hodograph Φ(u, v) = 0: v = φ(u) vt = φ′ut, vx = φ′ux, ut = uxφ′, utφ′ = ux, = ⇒ (φ′)2 = 1 φ′ = −1 = ⇒ ut = −ux, = ⇒ u = f(x − t)

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Method of differential constraints

Methods for constructing exact solutions utt = uxx ⇐ ⇒ ut = vx, vt = ux, u = f(x − t) + g(x + t), v = −f(x − t) + g(x + t) Assumptions: a) Method of generate hodograph Φ(u, v) = 0: v = φ(u) vt = φ′ut, vx = φ′ux, ut = uxφ′, utφ′ = ux, = ⇒ (φ′)2 = 1 φ′ = −1 = ⇒ ut = −ux, = ⇒ u = f(x − t) b) Method of differential constraints Φ(u, v, ut, ux, vt, vx, x, t) = 0: ux + vx = 0 = ⇒ u = f(x − t), v = −f(x − t).

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Method of differential constraints

Methods for constructing exact solutions utt = uxx ⇐ ⇒ ut = vx, vt = ux, u = f(x − t) + g(x + t), v = −f(x − t) + g(x + t) Assumptions: a) Method of generate hodograph Φ(u, v) = 0: v = φ(u) vt = φ′ut, vx = φ′ux, ut = uxφ′, utφ′ = ux, = ⇒ (φ′)2 = 1 φ′ = −1 = ⇒ ut = −ux, = ⇒ u = f(x − t) b) Method of differential constraints Φ(u, v, ut, ux, vt, vx, x, t) = 0: ux + vx = 0 = ⇒ u = f(x − t), v = −f(x − t). c) Group analysis: t∂t + x∂x : xux + tut = 0 D∂x + ∂t : Dux + ut = 0 ∂x : ux = 0

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Method of differential constraints

Method of differential constraints N.N. Yanenko (1921-1984)

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Method of differential constraints

Method of differential constraints N.N. Yanenko (1921-1984) Si(x, u, p) = 0, (i = 1, s). (5) Φk(x, u, p) = 0, (k = 1, q). (6)

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Method of differential constraints

Method of differential constraints A solution of system Si(x, u, p) = 0, (i = 1, s), satisfying differential constraints Φk(x, u, p) = 0, (k = 1, q) is called a solution characterized by the differential constraints.

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Method of differential constraints

Method of differential constraints A solution of system Si(x, u, p) = 0, (i = 1, s), satisfying differential constraints Φk(x, u, p) = 0, (k = 1, q) is called a solution characterized by the differential constraints. Compatibility of system (5), (6).

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SLIDE 54

Method of differential constraints

Method of differential constraints A solution of system Si(x, u, p) = 0, (i = 1, s), satisfying differential constraints Φk(x, u, p) = 0, (k = 1, q) is called a solution characterized by the differential constraints. Compatibility of system (5), (6). Involutiveness of system (5), (6).

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Method of differential constraints

Method of differential constraints A solution of system Si(x, u, p) = 0, (i = 1, s), satisfying differential constraints Φk(x, u, p) = 0, (k = 1, q) is called a solution characterized by the differential constraints. Compatibility of system (5), (6). Involutiveness of system (5), (6). Invariant solutions and Lie-B¨ acklund symmetries (involutive), Nonclassical, weak and conditional symmetries

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Method of differential constraints

Method of differential constraints A solution of system Si(x, u, p) = 0, (i = 1, s), satisfying differential constraints Φk(x, u, p) = 0, (k = 1, q) is called a solution characterized by the differential constraints. Compatibility of system (5), (6). Involutiveness of system (5), (6). Invariant solutions and Lie-B¨ acklund symmetries (involutive), Nonclassical, weak and conditional symmetries

A.F.Sidorov, V.P.Shapeev and N.N.Yanenko ”Methods of differential constraints and its applications in gas dynamics”, (Nauka, 2005) S.V.Meleshko ”Methods for constructing exact solutions of PDEs”, (Springer, 2005) Abrau-Durso, Russia September 14-19, 2015 27 / 38

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Method of differential constraints

One–dimensional gas flows    ut + uux + ρ−1px = 0, ρt + uρx + ρux = 0, pt + upx + A(ρ, p)ux = 0. τ = 1/ρ, c2 = Aτ, α = ±c, A = γp, γ > 1 S ≡ Lut + ΛLux = 0, L =   −A ρ ρc 1 −ρc 1   , Λ =   u u + c u − c   . Riemann (simple) waves: u = u(ρ), p = p(ρ) = ⇒ entropy is constant px − ρα2ρx = 0, ρux + αρx = 0,

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Method of differential constraints Generalized simple waves

Solutions with functional arbitrariness

Theorem. The general solution for differential constraints

px − α2ρx = ψ, ux + ρ−1αux = φ

1. ψ = 0 (isentropic flow)

1.a Riemann wave φ = 0 1.b γ = 3 1.c γ = 5/3 (one-atomic gas)

2. ψ = 0 (nonisentropic flow)

ψ = kρβ1pβ2, φ = − 3γ (3γ − 1)αρψ β1 = 1 − γ (3γ − 1), β2 = 1 + 1 (3γ − 1)

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SLIDE 59

Method of differential constraints Generalized simple waves

Integration of generalized simple waves      dx dt = u − α dρ dt = −3γkpβ1−1/2ρβ2+1/2, dp dρ = p 3ρ, du dρ = α 3γρ Rarefaction waves (p, u)–diagram: pξ − ρα2ρξ = 0, ρuξ + αρξ = 0, This gives nonisentropic rarefaction waves.

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Method of differential constraints Generalized simple waves

Interaction of two generalized simple waves

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Method of differential constraints Generalized simple waves

Interaction of two generalized simple waves Integration along a shock wave. Integration along characteristics.

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SLIDE 62

Method of differential constraints Generalized simple waves

The equation utt = Dxϕ, (ϕ = ϕ(x, ux)) Potential form ut = hx, ht = ϕ

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SLIDE 63

Method of differential constraints Generalized simple waves

The equation utt = Dxϕ, (ϕ = ϕ(x, ux)) Potential form ut = hx, ht = ϕ Differential constraint hx = g(t, x, u, h, ux)

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SLIDE 64

Method of differential constraints Generalized simple waves

The equation utt = Dxϕ, (ϕ = ϕ(x, ux)) Potential form ut = hx, ht = ϕ Differential constraint hx = g(t, x, u, h, ux) Determining equation uxx(ϕux − g2

ux) = gt + guxgx + uxgugux − ϕx + ghguxg + ghϕ + gug

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Method of differential constraints Generalized simple waves

The equation utt = Dxϕ, (ϕ = ϕ(x, ux)) Potential form ut = hx, ht = ϕ Differential constraint hx = g(t, x, u, h, ux) Determining equation uxx(ϕux − g2

ux) = gt + guxgx + uxgugux − ϕx + ghguxg + ghϕ + gug

gux = G, gt = −gh(ϕ + gG) − gu(g + Gux) − gxG + ϕx

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Method of differential constraints Generalized simple waves

Polytropic gas ϕ(x, ux) = −b(x)u−γ

x ,

G2 = bγu−(γ+1)

x

gux = G, gu = 0, gt = 0, gx = gh(u−γ

x bG−1 − g) − u−γ x bxG−1

(3γ − 1)bbxx − 3γb2

x = 0

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SLIDE 67

Method of differential constraints Generalized simple waves

Polytropic gas ϕ(x, ux) = −b(x)u−γ

x ,

G2 = bγu−(γ+1)

x

gux = G, gu = 0, gt = 0, gx = gh(u−γ

x bG−1 − g) − u−γ x bxG−1

(3γ − 1)bbxx − 3γb2

x = 0 ⇒ b = (k1x + k0)−

1 3γ−1 Abrau-Durso, Russia September 14-19, 2015 33 / 38

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Method of differential constraints Generalized simple waves

Polytropic gas ϕ(x, ux) = −b(x)u−γ

x ,

G2 = bγu−(γ+1)

x

gux = G, gu = 0, gt = 0, gx = gh(u−γ

x bG−1 − g) − u−γ x bxG−1

(3γ − 1)bbxx − 3γb2

x = 0 ⇒ b = (k1x + k0)−

1 3γ−1

g = − b′ b(3ga − 1)h + g1, g1 = kb

1 3γ−1 − 2α (γb)1/2

(γ − 1)u(1−γ)/2

x

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Method of differential constraints Generalized simple waves

Polytropic gas ut = hx, ht = −bu−γ

x , hx = −

b′ b(3γ − 1)h + g1, g1 = kb

1 3γ−1 − 2α (γb)1/2

(γ − 1)u(1−γ)/2

x

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Method of differential constraints Generalized simple waves

Polytropic gas. Finding a solution of the overdetermined system

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Method of differential constraints Generalized simple waves

Polytropic gas. Finding a solution of the overdetermined system Set k0, k1 and initial data for u: u = u0(x),

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Method of differential constraints Generalized simple waves

Polytropic gas. Finding a solution of the overdetermined system Set k0, k1 and initial data for u: u = u0(x), Find initial data for h by solving the equation hx = − b′ b(3γ − 1)h + g1, b = (k1x + k0)−

1 3γ−1 Abrau-Durso, Russia September 14-19, 2015 35 / 38

SLIDE 73

Method of differential constraints Generalized simple waves

Polytropic gas. Finding a solution of the overdetermined system Set k0, k1 and initial data for u: u = u0(x), Find initial data for h by solving the equation hx = − b′ b(3γ − 1)h + g1, b = (k1x + k0)−

1 3γ−1

Solve the system of ordinary differential equations (v = ux) dx dt = −G, dv dt = −3 b′G b(3γ − 1)v, du dt = − b′ b(3γ − 1)h+g1−Gv, dh dt = G( b′ b(3γ − 1)h−g1)−bv−γ

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SLIDE 74

Method of differential constraints Generalized simple waves

Polytropic gas. Finding a solution of the overdetermined system Set k0, k1 and initial data for u: u = u0(x), Find initial data for h by solving the equation hx = − b′ b(3γ − 1)h + g1, b = (k1x + k0)−

1 3γ−1

Solve the system of ordinary differential equations (v = ux) dx dt = −G, dv dt = −3 b′G b(3γ − 1)v, du dt = − b′ b(3γ − 1)h+g1−Gv, dh dt = G( b′ b(3γ − 1)h−g1)−bv−γ G = α (bγ)1/2 v− (γ+1)

2 ,

α = ±1, g1 = kb

1 3γ−1 − 2α (γb)1/2

(γ − 1)v(1−γ)/2

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Conclusion

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Conclusion

Conclusion We realized that for constructing conservation laws of equations with internal inertia by using Noether’s theorem we need to consider them in Lagrangian variables.

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Conclusion

Conclusion We realized that for constructing conservation laws of equations with internal inertia by using Noether’s theorem we need to consider them in Lagrangian variables. The first step is group classification of these equations. It is completed for W ˙

ρ = 0

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Conclusion

Conclusion We realized that for constructing conservation laws of equations with internal inertia by using Noether’s theorem we need to consider them in Lagrangian variables. The first step is group classification of these equations. It is completed for W ˙

ρ = 0

Attract attention to the method of differential constraints

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Conclusion

Acknowledgements Thanks to

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Conclusion

Acknowledgements Thanks to Organising Committee for the invitation

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Conclusion

Thank you very much for your attention!

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