Traveling-wave solutions 15.3.2.1 are invariant under the translation transformations where C is an arbitrary constant.. It should be observed that equations of the form 15.3.2.2 are inv
Trang 1where λ/k plays the role of the wave propagation velocity (the sign of λ can be arbitrary, the value λ = 0corresponds to a stationary solution, and the value k = 0corresponds to
a space-homogeneous solution) Traveling-wave solutions are characterized by the fact that the profiles of these solutions at different time* instants are obtained from one another
by appropriate shifts (translations) along the x-axis Consequently, a Cartesian coordinate
system moving with a constant speed can be introduced in which the profile of the desired
quantity is stationary For k >0and λ >0, the wave (15.3.2.1) travels along the x-axis to
the right (in the direction of increasing x).
A traveling-wave solution is found by directly substituting the representation (15.3.2.1)
into the original equation and taking into account the relations w x = kW , w t = –λW , etc
(the prime denotes a derivative with respect to z).
Traveling-wave solutions occur for equations that do not explicitly involve independent variables,
F
w,∂w
∂x,∂w
∂t ,∂
2w
∂x2,
∂2w
∂x∂t, ∂
2w
∂t2 ,
Substituting (15.3.2.1) into (15.3.2.2), we obtain an autonomous ordinary differential
equa-tion for the funcequa-tion W (z):
F (W , kW , –λW , k2W , –kλW , λ2W , ) =0,
where k and λ are arbitrary constants.
Example 1 The nonlinear heat equation
∂w
∂t = ∂
∂x
f (w) ∂w
∂x
(15 3 2 3 ) admits a traveling-wave solution Substituting (15.3.2.1) into (15.3.2.3), we arrive at the ordinary differential equation
k2[f (W )W ] + λW = 0 Integrating this equation twice yields its solution in implicit form:
k2
f (W ) dW
λW + C1 = –z + C2, where C1 and C2are arbitrary constants.
Example 2 Consider the homogeneous Monge–Amp`ere equation
∂2w
∂x∂t
2
–∂
2w
∂x2
∂2w
∂t2 = 0 (15 3 2 4 ) Inserting (15.3.2.1) into this equation, we obtain an identity Therefore, equation (15.3.2.4) admits solutions of the form
w = W (kx – λt), where W (z) is an arbitrary function and k and λ are arbitrary constants.
15.3.2-2 Invariance of solutions and equations under translation transformations Traveling-wave solutions (15.3.2.1) are invariant under the translation transformations
where C is an arbitrary constant.
* We also use the term traveling-wave solution in the cases where the variable t plays the role of a spatial
coordinate.
Trang 2It should be observed that equations of the form (15.3.2.2) are invariant (i.e., preserve their form) under transformation (15.3.2.5); furthermore, these equations are also invariant under general translations in both independent variables:
x= ¯x + C1, t = ¯t+ C2, (15.3.2.6)
where C1and C2are arbitrary constants The property of the invariance of specific equations under translation transformations (15.3.2.5) or (15.3.2.6) is inseparably linked with the existence of traveling-wave solutions to such equations (the former implies the latter) Remark 1 Traveling-wave solutions, which stem from the invariance of equations under translations, are
simplest invariant solutions.
Remark 2 The condition of invariance of equations under translations is not a necessary condition for the existence of traveling-wave solutions It can be verified directly that the second-order equation
F w , w x , w t , xw x + tw t , w xx , w xt , w tt
= 0
does not admit transformations of the form (15.3.2.5) and (15.3.2.6) but has an exact traveling-wave solution (15.3.2.1) described by the ordinary differential equation
F (W , kW , –λW , zW , k2W , –kλW , λ2W
= 0
15.3.2-3 Functional equation describing traveling-wave solutions
Let us demonstrate that traveling-wave solutions can be defined as solutions of the functional equation
where k and λ are some constants and C is an arbitrary constant Equation (15.3.2.7) states
that the unknown function does not change under increasing both arguments by proportional
quantities, with C being the coefficient of proportionality.
For C = 0, equation (15.3.2.7) turns into an identity Let us expand (15.3.2.7) into a
series in powers of C about C =0, then divide the resulting expression by C, and proceed
to the limit as C →0to obtain the linear first-order partial differential equation
λ ∂w
∂x + k ∂w
∂t =0
The general solution to this equation is constructed by the method of characteristics (see Paragraph 13.1.1-1) and has the form (15.3.2.1), which was to be proved
15.3.3 Self-Similar Solutions Invariance of Equations Under
Scaling Transformations
15.3.3-1 General form of self-similar solutions Similarity method
By definition, a self-similar solution is a solution of the form
The profiles of these solutions at different time instants are obtained from one another by a similarity transformation (like scaling)
Trang 3Self-similar solutions exist if the scaling of the independent and dependent variables,
t = C¯t, x = C k ¯x, w = C m w¯, where C ≠ 0is an arbitrary constant, (15.3.3.2)
for some k and m (| k|+|m| ≠ 0), is equivalent to the identical transformation This means that the original equation
F (x, t, w, w x , w t , w xx , w xt , w tt , ) =0, (15.3.3.3) when subjected to transformation (15.3.3.2), turns into the same equation in the new vari-ables,
F(¯x, ¯t, ¯ w,w¯¯x,w¯¯t,w¯¯x¯x,w¯¯x¯t,w¯¯t¯t , ) =0 (15.3.3.4)
Here, the function F is the same as in the original equation (15.3.3.3); it is assumed that equation (15.3.3.3) is independent of the parameter C.
Let us find the connection between the parameters α, β in solution (15.3.3.1) and the parameters k, m in the scaling transformation (15.3.3.2) Suppose
is a solution of equation (15.3.3.3) Then the function
¯
is a solution of equation (15.3.3.4)
In view of the explicit form of solution (15.3.3.1), if follows from (15.3.3.6) that
¯
Using (15.3.3.2) to return to the new variables in (15.3.3.7), we get
w = C m–α t α U C–k–β xt β
By construction, this function satisfies equation (15.3.3.3) and hence is its solution Let
us require that solution (15.3.3.8) coincide with (15.3.3.1), so that the condition for the
uniqueness of the solution holds for any C ≠ 0 To this end, we must set
In practice, the above existence criterion is checked: if a pair of k and m in (15.3.3.2)
has been found, then a self-similar solution is defined by formulas (15.3.3.1) with parame-ters (15.3.3.9)
The method for constructing self-similar solutions on the basis of scaling transformations
(15.3.3.2) is called the similarity method It is significant that these transformations involve the arbitrary constant C as a parameter.
To make easier to understand, Fig 15.1 depicts the basic stages for constructing self-similar solutions
Trang 4Here is a free parameter
C
k m
Look for a self-similar solution
Substitute into the original equation
Figure 15.1 A simple scheme that is often used in practice for constructing self-similar solutions.
15.3.3-2 Examples of self-similar solutions to mathematical physics equations
Example 1 Consider the heat equation with a nonlinear power-law source term
∂w
∂t = a ∂
2w
∂x2 + bw n (15 3 3 10 ) The scaling transformation (15.3.3.2) converts equation (15.3.3.10) into
C m–1 ∂ w¯
∂¯ t = aC m–2k ∂
2w¯
∂ ¯x2 + bC mn w¯n.
Equating the powers of C yields the following system of linear algebraic equations for the constants k and m:
m– 1= m –2k= mn.
This system admits a unique solution: k = 12, m = 1–n1 Using this solution together with relations (15.3.3.1) and (15.3.3.9), we obtain self-similar variables in the form
w = t1/(1–n) U (ζ), ζ = xt–1/2.
Inserting these into (15.3.3.10), we arrive at the following ordinary differential equation for the function U (ζ):
aU ζζ + 1
2ζU ζ + 1
n– 1U + bU n=0.
Trang 5Example 2 Consider the nonlinear equation
∂2w
∂t2 = a ∂
∂x
w n ∂w
∂x
which occurs in problems of wave and gas dynamics Inserting (15.3.3.2) into (15.3.3.11) yields
C m–2 ∂
2w¯
∂¯ t2 = aC mn+m–2k ∂
∂ ¯x
¯
w n ∂ w¯
∂ ¯x
.
Equating the powers of C results in a single linear equation, m –2= mn + m –2k Hence, we obtain
k= 12mn+ 1, where m is arbitrary Further, using (15.3.3.1) and (15.3.3.9), we find self-similar variables:
w = t m U (ζ), ζ = xt–1mn–1 (m is arbitrary).
Substituting these into (15.3.3.11), one obtains an ordinary differential equation for the function U (ζ).
Table 15.1 gives examples of self-similar solutions to some other nonlinear equations
of mathematical physics
TABLE 15.1 Some nonlinear equations of mathematical physics that admit self-similar solutions
Equation Equation name Form of solutions Determining equation
∂w
∂t =∂x ∂
f (w) ∂w ∂x Unsteady
heat equation w = w(z), z = xt–1/2 [f (w)w ]+12zw = 0
∂w
∂t = a ∂x ∂ w n ∂w ∂x
+ bw k Heat equation
with source
w = t p u (z), z = xt q,
p=1–k1 , q = k–n–12(1–k)
a (u n u ) – qzu + bu k – pu =0
∂w
∂t = a ∂ ∂x2w2+ bw ∂w ∂x equationBurgers w = t–1/2u (z), z = xt–1/2 au + buu +12zu +12u= 0
∂w
∂t = a ∂ ∂x2w2+ b ∂w ∂x 2 Potential Burgers
equation w = w(z), z = xt–1/2 aw + b(w )2+12zw = 0
∂w
∂t = a ∂w ∂x k ∂2w
∂x2
Filtration equation
w = t p u (z), z = xt q,
p= –(k+2)q+1k , q is any a (u )k u = qzu + pu
∂w
∂t = f ∂w ∂x ∂2w
∂x2
Filtration equation w = t1/2u (z), z = xt–1/2 2f(u )u + zu – u =0
∂2w
∂t2 =∂x ∂
f (w) ∂w ∂x
Wave equation w = w(z), z = x/t (z2w ) = [f (w)w ]
∂2w
∂t2 = a ∂x ∂ w n ∂w ∂x
Wave equation w = t2ku (z), z = xt–nk–1,
kis any
2k(2k–1) (nk+1) 2u+nk–4k+2 nk+1 zu
+ z2u =(nk+1) a 2(u n u )
∂2w
∂x2 +∂ ∂y2w2 = aw n Heat equation
with source w = x
2 1–nu (z), z = y/x
( 1+ z2)u –2(1+n)1–n zu
+2(1+n)(1–n)2u – au n= 0
∂2w
∂x2+ a ∂w ∂y ∂ ∂y2w2 = 0 Equation of steadytransonic gas flow w = x–3k–2u (z), z = x k y,
kis any
a k+1 u u +k+1 k2 z2u
– 5kzu+ 3 ( 3k + 2)u =0
∂w
∂t = a ∂ ∂x3w3+ bw ∂w ∂x Korteweg–de Vriesequation w = t–2/3u (z), z = xt–1/3 au + buu +13zu +23u= 0
∂w
∂y ∂
2w
∂x∂y–∂w ∂x ∂ ∂y2w2 = a ∂ ∂y3w3 Boundary-layerequation w = x λ+1 u (z), z = x λ ,
λis any
( 2λ + 1)(u )2– (λ +1)uu
= au
The above method for constructing self-similar solutions is also applicable to systems
of partial differential equations Let us illustrate this by a specific example
Trang 6Example 3 Consider the system of equations of a steady-state laminar boundary hydrodynamic boundary
layer at a flat plate (see Schlichting, 1981)
u ∂u
∂x + v ∂u
∂y = a ∂
2u
∂y2,
∂u
∂x +∂v
∂y = 0
(15 3 3 12 )
Let us scale the independent and dependent variables in (15.3.3.12) according to
x = C ¯x, y = C k ¯y, u = C m ¯u, v = C n ¯v. (15 3 3 13 ) Multiplying these relations by appropriate constant factors, we have
¯u ∂ ¯u
∂ ¯x + C n–m–k+1 ¯v
∂ ¯u
∂ ¯y = C–m–2k+1a
∂2¯u
∂ ¯y2 ,
∂ ¯u
∂ ¯x + C n–m–k+1
∂ ¯v
∂ ¯y=0.
(15 3 3 14 )
Let us require that the form of the equations of the transformed system (15.3.3.14) coincide with that of the
original system (15.3.3.12) This condition results in two linear algebraic equations, n – m – k +1 = 0 and – 2k– m +1 = 0 On solving them for m and n, we obtain
where the exponent k can be chosen arbitrarily To find a self-similar solution, let us use the procedure outlined
in Fig 15.1 The following renaming should be done: x → y, t → x, w → u (for u) and x → y, t → x,
w → v, m → n (for v) This results in
u (x, y) = x1–2kU (ζ), v (x, y) = x–k V (ζ), ζ = yx–k, (15 3 3 16 )
where k is an arbitrary constant Inserting (15.3.3.16) into the original system (15.3.3.12), we arrive at a system
of ordinary differential equations for U = U (ζ) and V = V (ζ):
U
( 1 – 2k)U – kζU ζ
+ V U ζ = aU ζζ , ( 1 – 2k)U – kζU ζ + V ζ = 0
15.3.3-3 More general approach based on solving a functional equation
The algorithm for the construction of a self-similar solution, presented in Paragraph 15.3.3-1, relies on representing this solution in the form (15.3.3.1) explicitly However, there is a more general approach that allows the derivation of relation (15.3.3.1) directly from the condition of the invariance of equation (15.3.3.3) under transformations (15.3.3.2)
Indeed, let us assume that transformations (15.3.3.2) convert equation (15.3.3.3) into the same equation (15.3.3.4) Let (15.3.3.5) be a solution of equation (15.3.3.3) Then (15.3.3.6) will be a solution of equation (15.3.3.4) Switching back to the original variables (15.3.3.2) in (15.3.3.6),we obtain
w = C mΦ C–k x , C– 1t
By construction, this function satisfies equation (15.3.3.3) and hence is its solution Let us require that solution (15.3.3.17) coincide with (15.3.3.5), so that the uniqueness condition
for the solution is met for any C ≠ 0 This results in the functional equation
Φ(x, t) = C mΦ C–k x , C– 1t
For C =1, equation (15.3.3.18) is satisfied identically Let us expand (15.3.3.18) in a
power series in C about C =1, then divide the resulting expression by (C –1), and proceed
to the limit as C →1 This results in a linear first-order partial differential equation forΦ:
kx ∂Φ
∂x + t ∂Φ
Trang 7The associated characteristic system of ordinary differential equations (see Paragraph 13.1.1-1) has the form
dx
kx = dt
t = dΦ
mΦ.
Its first integrals are
xt–k = A1, t–m Φ = A2,
where A1 and A2 are arbitrary constants The general solution of the partial differential
equation (15.3.3.19) is sought in the form A2= U (A1), where U (A) is an arbitrary function
(see Paragraph 13.1.1-1) As a result, one obtains a solution of the functional equation (15.3.3.18) in the form
Φ(x, t) = t m U (ζ), ζ = xt–k (15.3.3.20) Substituting (15.3.3.20) into (15.3.3.5) yields the self-similar solution (15.3.3.1) with parameters (15.3.3.9)
15.3.3-4 Some remarks
Remark 1. Self-similar solutions (15.3.3.1) with α =0 arise in problems with simple initial and boundary conditions of the form
w = w1 at t= 0 (x >0 ), w = w2 at x =0 (t >0 ),
where w1and w2are some constants.
Remark 2 Self-similar solutions, which stem from the invariance of equations under scaling
transforma-tions, are considered among the simplest invariant solutions.
The condition for the existence of a transformation (15.3.3.2) preserving the form of the given equation
is sufficient for the existence of a self-similar solution However, this condition is not necessary: there are equations that do not admit transformations of the form (15.3.3.2) but have self-similar solutions.
For example, the equation
a
2w
∂x2 + b ∂
2w
∂t2 = (bx2+ at2)f (w)
has a self-similar solution
w = w(z), z = xt =⇒ w – f (w) =0 ,
but does not admit transformations of the form (15.3.3.2) In this equation, a and b can be arbitrary functions
of the arguments x, t, w, w x , w t , w xx ,
Remark 3 Traveling-wave solutions are closely related to self-similar solutions Indeed, setting
t = ln τ , x = ln y
in (15.3.2.1), we obtain a self-similar representation of a traveling wave:
w = W k ln(yτ–λ/k)
= U (yτ–λ/k),
where U (z) = W (k ln z).
15.3.4 Equations Invariant Under Combinations of Translation and
Scaling Transformations, and Their Solutions
15.3.4-1 Exponential self-similar (limiting self-similar) solutions
Exponential self-similar solutions are solutions of the form