The path consists of con-tinuous segments representing solutions of the ordinary differential equations 15.14.4.65 rarefaction waves, line segments that connect two points u–and u+satisf
Trang 1Hence, the velocity vector for the continuous solution u(ξ) is a right eigenvector of
the matrix 2 G for any point u, and the corresponding eigenvalue equals the self-similar
coordinate:
ξ = λk, u ξ = αrk (15 14 4 66 )
Here, λk= λk(u) is a root of the algebraic equation det | G – λI 2 | = 0 , rk= rk(u) is a solution
of the corresponding degenerate linear system of equations G – λI
r = 0 , and α = α(u) is
a positive function that will be defined below.
Differentiating both sides of the first equation (15.14.4.66) with respect to ξ yields
〈∇ λk, rk〉 , 〈∇ λk, rk〉 =
∂λk
∂u1r
k
1 + · · · + ∂λk
∂unr
k
n.
Any n × n hyperbolic system allows for n continuous solutions [of system (15.14.4.65)] corresponding to n characteristic velocities λ = λk The continuous solutions are
deter-mined by n systems of ordinary differential equations Each system is represented by a
phase portrait in the n-dimensional u-space A solution/trajectory that corresponds to a
characteristic velocity λkis called a kth rarefaction wave.
3◦ A trajectory of solution (15.14.4.47) in the space u = (u1, , u
n)Tis called a solution
path The path is parametrized by the self-similar coordinate ξ The path connects the
point u = uLwith the point u = uR The self-similar coordinate ξ monotonically increases
along the path varying from – ∞ at u = uL to + ∞ at u = uR The path consists of con-tinuous segments representing solutions of the ordinary differential equations (15.14.4.65)
(rarefaction waves), line segments that connect two points u–and u+satisfying the Rankine– Hugoniot conditions (15.14.4.54) and evolutionary conditions (15.14.4.59), and rest points
u(ξ) = const.
Example 8 Consider a solution consisting of two shocks and one rarefaction The structural formula* for the solution path is uL→1—2→ uR; specifically,
u(x, t) =
⎧
⎪
⎨
⎪
⎩
uL if –∞ < x/t < D1,
u1 if D1< x/t < λ2(u1),
u( 2 )(ξ) if λ2(u1) < x/t < λ2(u2),
u2 if λ2(u2) < x/t < D2,
uR if D2< x/t < ∞.
The shock speed D1(resp., D2) can be found from the Hugoniot condition by setting u–= uL and u+= u1
(resp., u–= u2and u+= uR) Points 1 and 2 are located on the same rarefaction curve The vector u(2)(ξ) is a
second-family rarefaction wave that is described by the system of ordinary differential equations (15.14.4.65)
with ξ = λ2(u).
Figure 15.10a depicts a sequence of rarefactions and shocks in the plane (x, t) Figure 15.10b shows the profile of the solution component u i along the x-axis The self-similar curves u = u(ξ) coincide with the profiles u(x, t =1) For t >1, the graphs of u(x, t) are obtained from the self-similar curves by extending them
along the axis x by a factor of t.
Example 9 Let us discuss an adiabatic gas flow in a tube in front of an impermeable piston moving with a
velocity vL(see Fig 15.11) The initial state is defined by prescribing initial values of the velocity and density:
v=0, ρ = ρR at ξ = ∞ (ξ = x/t). (15.14.4.67) The piston is impermeable; therefore, the gas velocity in front of the shock is equal to the piston velocity
(Fig 15.12a):
The gas density in front of the piston is unknown in this problem
* In structural formulas like uL→1—2→ uR, the symbol “→” stands for a shock wave and “—” stands
for a rarefaction
Trang 21 2
t
R
ui L
x
x
0
0
( )a
( )b
Figure 15.10 Solution for the Riemann problem: (a) centered waves in the (x, t) plane; (b) the u iprofile
vR= 0, rR
vL
vL
Figure 15.11 Constant velocity piston motion in a tube.
L
x =vL
x =D
0
r
vR= 0, rR
t
-R
vL, rL
vL v
Figure 15.12 Solution of the piston problem: (a) shock wave in the (x, t) plane; (b) shock locus in the phase
plane
Figure 15.12b shows the locus of points that can be connected by a shock to the point (vR=0, ρR) This
locus is a first-family shock The intersection of the locus with the line v = vL defines the value ρL Hence, ρL
can be found from the equation
vL2= [p(ρL ) – p(ρR)](ρL – ρR)
ρLρR
which has been obtained by taking the square of equation (15.14.4.56) There exists a root ρLof (15.14.4.69)
such that ρL > ρR Hence, the gas is compressed ahead of the piston (Fig 15.12a) The expression for the
shock speed can be found from the first Hugoniot condition (15.14.4.55):
D= ρLvL
The shock speed exceeds the piston velocity of (15.14.4.70) for ρL > ρR Both characteristics of the first family
as well as the characteristic ahead of the shock from the second family arrive at the shock, so that the Lax condition is satisfied It can be proved that there are no other configurations that satisfy the initial-boundary conditions (15.14.4.67), (15.14.4.68)
Trang 315.14.5 Systems of Second-Order Equations of Reaction-Diffusion
Type
15.14.5-1 Traveling-wave solutions and some other invariant solutions.
In this subsection, we consider systems of nonlinear second-order equations
∂u
∂t = a1∂
2u
∂x2 + F1(u, w),
∂w
∂t = a2∂
2w
∂x2 + F2(u, w),
(15 14 5 1 )
which often arise in the theory of mass exchange of reactive media, combustion theory, mathematical biology, and biophysics.
It is apparent that systems (15.14.5.1) admit exact traveling-wave solutions
u = u(z), w = w(z), z = kx – λt, where k and λ are arbitrary constants, and the functions u(z) and w(z) are determined by
the autonomous system of ordinary differential equations
a1k2u
zz+ λu z+ F1(u, w) = 0 ,
a2k2w
zz+ λwz + F2(u, w) = 0 Table 15.13 lists some systems of the form (15.14.5.1) admitting other invariant solu-tions that are found using combinasolu-tions of translation and scaling transformasolu-tions in the independent and dependent variables; these transformations preserve the form of the equa-tions (i.e., a given system of equaequa-tions is converted into an identical one) The procedure for constructing such solutions for a single equation is detailed in Section 15.3.
TABLE 15.13 Systems of equations of the form (15.14.5.1) admitting invariant solutions that are found using
combinations of translation and scaling transformations (C is an arbitrary constant)
Systems of equations Invariant transformations Form of invariant solutions
∂u
∂t = a ∂ ∂x2u2+e λu f (λu–σw),
∂w
∂t = b ∂ ∂x2w2+e σw g (λu–σw)
t = C2λσ ¯t, x=C λσ ¯x,
u=¯u–2σ ln C,
w=w¯–2λ ln C
u = y(ξ)– λ1 ln t, w = z(ξ)– σ1 ln t, ξ = √ x
t
∂u
∂t = a ∂ ∂x2u2+u1+kn f u n w m
,
∂w
∂t = b ∂ ∂x2w2+w1–km g u n w m
t = C–kmn ¯t,
x = C–1kmn ¯x,
u = C m ¯u, w=C–n w¯
u = t– kn1 y (ξ), w = t km1 z (ξ), ξ = √ x
t
∂u
∂t = a ∂ ∂x2u2+uf u n w m
,
∂w
∂t = b ∂ ∂x2w2+wg u n w m
t =¯t+ln C,
x=¯x+λ ln C,
u = C m ¯u, w=C–n w¯
u = e mt y (ξ), w = e–nt z (ξ), ξ = x–λt
15.14.5-2 Generalized separable solutions.
In some cases, nonlinear systems admit generalized separable solutions of the form
u = ϕ1(t)θ(x, t) + ψ1(t),
w = ϕ2(t)θ(x, t) + ψ2(t), (15. 14 5 2 )
where the functions ϕ1(t), ϕ2(t), ψ1(t), and ψ2(t) are selected so that both equations of system (15.14.5.1) are reduced to one and the same equation for θ(x, t).
Trang 4Example 1 Consider the system
∂u
∂t = a ∂
2u
∂x2 + uf (bu – cw) + g1(bu – cw),
∂w
∂t = a ∂
2w
∂x2 + wf (bu – cw) + g2(bu – cw),
(15.14.5.3)
where f (z), g(z), and h(z) are arbitrary functions.
Exact solutions are sought in the form (15.14.5.2) Let us require that the arguments of the functions
appearing in the system must depend on time t only, so that
∂
∂x (bu – cw) = [bϕ1(t) – cϕ2(t)] ∂
∂x θ (x, t) =0
It follows that
ϕ1(t) = cϕ(t), ϕ2(t) = bϕ(t). (15.14.5.4) Substituting (15.14.5.2), in view of (15.14.5.4), into (15.14.5.3) after elementary rearrangements, we obtain
∂θ
∂t = a ∂
2θ
∂x2 +
f (bψ1– cψ2) – ϕ
ϕ
θ+ 1
cϕ
ψ1f (bψ1– cψ2) + g1(bψ1– cψ2) – ψ 1
,
∂θ
∂t = a ∂
2θ
∂x2 +
f (bψ1– cψ2) – ϕ
ϕ
θ+ 1
bϕ
ψ2f (bψ1– cψ2) + g2(bψ1– cψ2) – ψ2
(15.14.5.5)
For the equations of (15.14.5.5) to coincide, let us equate the expressions in square brackets to zero As a result,
we obtain an autonomous system of ordinary differential equations for ϕ = ϕ(t), ψ1= ψ1(t), and ψ2= ψ2(t):
ϕ = ϕf (bψ1– cψ2),
ψ1 = ψ1f (bψ1– cψ2) + g1(bψ1– cψ2),
ψ2 = ψ2f (bψ1– cψ2) + g2(bψ1– cψ2)
(15.14.5.6)
From (15.14.5.5)–(15.14.5.6) it follows that θ = θ(x, t) satisfies the linear heat equation
∂θ
∂t = a ∂
2θ
From the first equation in (15.14.5.6) we can express the function ϕ via the other two:
ϕ = C1exp
f (bψ1– cψ2) dt
(The constant of integration C1 can be set equal to1, since ϕ and θ appear in the solution in the form of a product.) As a result, we obtain an exact solution of system (15.14.5.2) in the form
u = ψ1(t) + c exp
f (bψ1– cψ2) dt
θ (x, t),
w = ψ2(t) + b exp
f (bψ1– cψ2) dt
θ (x, t),
(15.14.5.9)
where the functions ψ1= ψ1(t) and ψ2= ψ2(t) are described by the last two equations of system (15.14.5.6), and the function θ(x, t) is a solution of the linear heat equation (15.14.5.7).
Remark Now we show how the functions ϕ(t) and ψ(t), appearing in (15.14.5.9), can be found Multiply the second equation (15.14.5.6) by b and the third by –c and add up to obtain a separable first-order ordinary
differential equation for the linear combination of the unknowns:
z t = zf (z) + bg1(z) – cg2(z), z = bψ1– cψ2 (15.14.5.10) Its general solution is expressed in implicit form as
dz
zf (z) + bg1(z) – cg2(z) = t + C2. (15.14.5.11)
For given functions f (z), g(z), and h(z), from (15.14.5.11) we find z = z(t) Substituting it into the second equation (15.14.5.6), we get a linear equation for ψ1 Solving this equation and taking into account the relation
for z, ψ1, and ψ2in (15.14.5.10), we obtain
ψ1 = C3F (t) + F (t)
g1(z)
F (t) dt, ψ2=
b
c ψ1–1
c , F (t) = exp
f (z) dt
, z = z(t).
Trang 5Example 2 Consider now the system
∂u
∂t = a ∂
2u
∂x2 + uf
u w
,
∂w
∂t = a ∂
2w
∂x2 + wg
u w
An exact solution is sought in the form (15.14.5.2) with ϕ2(t) = 1and ψ1(t) = ψ2(t) = 0 After simple rearrangements we get
u = ϕ(t) exp
g (ϕ(t)) dt
θ (x, t), w= exp
g (ϕ(t)) dt
θ (x, t), where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary differential equation
ϕ t = [f (ϕ) – g(ϕ)]ϕ, (15.14.5.12)
and the function θ = θ(x, t) satisfies the linear heat equation (15.14.5.7).
The general solution of equation (15.14.5.12) is written out in implicit form as
dϕ
[f (ϕ) – g(ϕ)]ϕ = t + C.
Exact solutions to some other second-order nonlinear systems of the form (15.14.5.1) can be found in Subsection T10.3.1.
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