Relation 15.8.3.5 is the basis for the construction of invariant solutions: solving 15.8.3.5 for w and substituting the resulting expression into 15.8.2.1, we arrive at an ordinary diffe
Trang 1Similarly, it can be established that the following special forms of f result in additional operators:
1 f = ew: X4= x∂ x+2∂ w;
2 f = wk , k≠ 0, –4/3, –4: X4= kx∂ x+2w∂ w;
3 f = w– 4/3: X4=2x∂ x–3w∂ w, X5= x2∂ x–3xw∂ w;
4 f = w– 4: X4=2x∂ x – w∂ w, X5= t2∂ t + tw∂ w
The symmetries obtained with the procedure presented can be used to find exact solutions
of the differential equations considered (see below).
15.8.3 Using Symmetries of Equations for Finding Exact Solutions.
Invariant Solutions
15.8.3-1 Using symmetries of equations for constructing one-parameter solutions Suppose a particular solution,
of a given equation is known Let us show that any symmetry of the equation defined
by a transformation of the form (15.8.1.1) generates a one-parameter family of solutions (except for the cases where the solution is not mapped into itself by the transformations; see Paragraph 15.8.3-2).
Indeed, since equation (15.8.2.1) converted to the new variables (15.8.1.1) acquires the same form (15.8.2.2), then the transformed equation (15.8.2.2) has a solution
¯
In (15.8.3.2), going back to the old variables by formulas (15.8.1.1), we obtain a one-parameter solution of the original equation (15.8.2.1).
Example 1 The two-dimensional heat equation with an exponential source
∂2w
∂x2 + ∂
2w
has a one-dimensional solution
w= ln 2
Equation (15.8.3.3) admits the operator X3= y∂ x –x∂ y(see Example 1 in Subsection 15.8.2), which defines
rotation in the plane The corresponding transformation is given in Table 15.7 Replacing x in (15.8.3.4) by ¯x
(from Table 15.7), we obtain a one-parameter solution of equation (15.8.3.3):
(x cos ε + y sin ε)2, where ε is a free parameter.
15.8.3-2 Procedure for constructing invariant solutions.
Solution (15.8.3.1) of equation (15.8.2.1) is called invariant under transformations (15.8.1.1)
if it coincides with solution (15.8.3.2), which must be rewritten in terms of the old variables using formulas (15.8.1.1) This means that an invariant solution is converted to itself under the given transformation The basic stages of constructing invariant solutions are outlined below.
Trang 2Invariant solutions of equation (15.8.2.1) are sought in the implicit form
I (x, y, w) = 0
Then I( ¯x, ¯y, ¯ w ) = 0 Let us find a one-parameter transformation with operator (15.8.1.3) whose coordinates are determined from the invariance condition (15.8.2.3) following the procedure described in Subsection 15.8.2 Find two functionally independent integrals (15.8.1.6) of the characteristic system of ordinary differential equations (15.8.1.5) The general solution of the partial differential equation (15.8.1.4) is determined by formula
(15.8.1.7) Setting in this formula I = 0 and solving for the invariant I2, we obtain
I2 = Φ(I1), (15 8 3 5 )
where the functions I1= I1(x, y, w) and I2= I2(x, y, w) are known,* and the function Φ is
to be determined Relation (15.8.3.5) is the basis for the construction of invariant solutions:
solving (15.8.3.5) for w and substituting the resulting expression into (15.8.2.1), we arrive
at an ordinary differential equation for Φ.
Example 2 A well-known and very important special case of invariant solutions is the self-similar
solu-tions (see Subsection 15.3.3); they are based on invariants of scaling groups The corresponding infinitesimal operator and its invariants are
X = ax ∂
∂x + by ∂
∂y + cw ∂
∂w; I1=|y|a|x|–b, I2=|w|a|x|–c Substituting the invariants into (15.8.3.5) gives|w|a|x|–c=Φ |y|a|x|–b
On solving this equation for w, we obtain the form of the desired solution, w =|x|c/aΨ y|x|–b/a
, whereΨ(z) is an unknown function.
To make it clearer, the general scheme for constructing invariant solutions for evolution second-order equations is depicted in Fig 15.4 The first-order partial differential equation (15.8.1.4) for finding group invariants is omitted, since the corresponding characteristic system of ordinary differential equations (15.8.1.5) can be immediately used.
15.8.3-3 Examples of constructing invariant solutions to nonlinear equations.
Example 3 Consider once again the stationary heat equation with nonlinear source
∂2w
∂x2 +∂
2w
∂y2 = f (w).
1◦ Let us dwell on the case f (w) = w k, where the equation admits an additional operator (see Example 1 from Subsection 15.8.2):
X4= x∂ x + y∂ y+ 2
1– k w∂ w.
In order to find invariants of this operator, we have to consider the linear first-order partial differential equation
X4I=0, or, in detailed form,
x ∂I
∂x + y ∂I
∂y + 2
1– k w
∂I
∂w =0
The corresponding characteristic system of ordinary differential equations,
dx
x = dy
y = 1– k
2
dw
w , has the first integrals
y/x = C1, x2/(k–1)w = C2,
* Usually, the invariant that is independent of w is taken to be I1
Trang 3Calculate the coordinates of the prolonged operator
Derive the determining system of PDEs
Solve the characteristic system
Figure 15.4 An algorithm for constructing invariant solutions for evolution second-order equations Notation:
ODE stands for ordinary differential equation and PDE stands for partial differential equation; ξ = ξ(x, t, w),
η = η(x, t, w), ζ = ζ(x, t, w); ζ1, ζ2, and ζ11are the coordinates of the prolonged operator, which are defined by
formulas (15.8.1.9) and (15.8.1.14) with y = t.
where C1, C2are arbitrary constants Therefore, the functions I1= y/x and I2= x2/(k–1)ware invariants of the operator X4
Assuming that I2=Φ(I1 ) and expressing w, we find the form of the invariant (self-similar) solution:
Substituting (15.8.3.6) into the original equation (15.8.2.6) yields a second-order ordinary differential equation forΦ(z):
(k –1)2(z2+1)Φ
zz+2(k2–1)zΦ
z+2(k +1)Φ – (k –1)2Φk=0,
where z = y/x.
2◦ The functions u = x2+ y2and w are invariants of the operator X3for the nonlinear heat equation concerned
The substitutions w = w(u) and u = x2+ y2lead to an ordinary differential equation describing solutions of the original equation which are invariant under rotation:
uw uu + w u = 14f (w).
Remark In applications, the polar radius r =
x2+ y2 is normally used as an invariant instead of
u = x2+ y2
Trang 4Example 4 Consider the nonlinear nonstationary heat equation (15.8.2.14).
1◦ For arbitrary f (w), the equation admits the operator (see Example 2 from Subsection 15.8.2)
X3=2t∂ t + x∂ x Invariants of X3are found for the linear first-order partial differential equation X3I=0, or
2t ∂I
∂t + x ∂I
∂x +0∂I
∂w =0
The associated characteristic system of ordinary differential equations,
dx
x = dt
2t = dw
0 , has the first integrals
xt–1/2= C1, w = C2,
where C1and C2are arbitrary constants Therefore, the functions I1 = xt– 1/2and I2 = w are invariants of the
operator X3
Assuming I2=Φ(I1), we get
whereΦ(z) is to be determined in the subsequent analysis Substituting (15.8.3.7) in the original equation
(15.8.2.14) yields the second-order ordinary differential equation
2[f (Φ)Φ
z] z + zΦ
z=0, which describes an invariant (self-similar) solution
2◦ Let us dwell on the case f (w) = w k, where the equation admits the operator
X4= kx∂ x+2w∂ w The invariants are described by the first-order partial differential equation X4I=0, or
0∂I
∂t + kx ∂I
∂x +2w ∂I
∂w =0
The associated characteristic system of ordinary differential equations,
dt
0 =
dx
kx = dw2w, has the first integrals
t = C1, x–2/k w = C2,
where C1, C2are arbitrary constants Therefore, I1= t and I2= x– 2/k ware invariants of the operator X4
Assuming I2= θ(I1) and expressing w, we get
where θ(t) is to be determined in the subsequent analysis Substituting (15.8.3.8) in the original equation (15.8.2.14) with f (w) = w kgives the first-order ordinary differential equation
2kθ t=2(k +2)θk+1 Integrating yields
θ (t) =
A– 2(k +2)
– 1/k
,
where A is an arbitrary constant Hence, the solution of equation (15.8.2.14) with f (w) = w k, which is invariant under scaling, has the from
w (x, t) = x2/k
A–2(k +2)
– 1/k
Trang 5
TABLE 15.8 Operators, invariants, and solution structures admitted by the nonlinear nonstationary heat equation (15.8.2.14)
Function f (w) Operators Invariants Solution structure
Arbitrary
X1= ∂ x,
X2= ∂ t,
X3=2t∂ t + x∂ x
I1= t, I2 = w,
I1= x, I2= w,
I1= x2/t , I2= w
w = w(t) = const,
w = w(x),
w = w(z), z = x2/t
e w X4= x∂ x+2∂ w I1= t, I2= w –2ln|x| w=2ln|x|+ θ(t)
w k (k≠ 0, –43) X4= kx∂ x+2w∂ w I1= t, I2= w|x|–k/2 w=|x|k/2θ (t)
X5= x2∂ x–3xw∂ w
I1= t, I2 = wx2/3,
I1= t, I2 = wx3
w = x– 2/3θ (t),
w = x–3θ (t)
Table 15.8 summarizes the symmetries of equation (15.8.2.14) (see Example 2 from Subsection 15.8.2 and Example 4 from Subsection 15.8.3)
Example 5 Consider the nonlinear wave equation (15.8.2.15) For arbitrary f (w), this equation admits
the following operator (see Example 3 from Subsection 15.8.2):
X3= t∂ t + x∂ x The invariants are found from the linear first-order partial differential equation X3I1=0, or
t ∂I
∂t + x ∂I
∂x +0∂I
∂w =0
The associated characteristic system of ordinary differential equations
dx
x = dt
t = dw0 admits the first integrals
xt–1= C1, w = C2,
where C1, C2are arbitrary constants Therefore, I1= xt– 1and I2= w are invariants of the operator X3
Taking I2=Φ(I1), we get
The functionΦ(y) is found by substituting (15.8.3.9) in the original equation (15.8.2.15) This results in the
ordinary differential equation
[f (Φ)Φ
y] y = (y2Φ
y) y,
which defines an invariant (self-similar) solution This equation has the obvious first integral f (Φ)Φ
y = y2Φ
y +C.
Table 15.9 summarizes the symmetries of equation (15.8.2.15) (see Example 3 from Subsection 15.8.2 and Example 5 from Subsection 15.8.3)
15.8.3-4 Solutions induced by linear combinations of admissible operators.
If a given equation admits N operators, then we have N associated different invariant
solutions However, when dealing with operators individually, one may overlook solutions that are invariant under a linear superposition of the operators; such solutions may have
a significantly different form In order to find all types of invariant solutions, one should study all possible linear combinations of the admissible operators.
Example 6 Consider once again the nonlinear nonstationary heat equation (15.8.2.14).
Trang 6TABLE 15.9 Operators, invariants, and solution structures admitted by the nonlinear wave equation (15.8.2.15)
Functions f (w) Operators Invariants Solution structure
Arbitrary
X1= ∂ x,
X2= ∂ t,
X3= t∂ t + x∂ x
I1= t, I2 = w,
I1= x, I2= w,
I1= x/t, I2 = w
w = w(t),
w = w(x),
w = w(z), z = x/t
e w X4= x∂ x+2∂ w I1= t, I2= w –2ln|x| w=2ln|x|+ θ(t)
w k (k≠ 0, –43, –4) X4= kx∂ x+2w∂ w I1= t, I2= w|x|–k/2 w=|x|k/2θ (t)
X5= x2∂ x–3xw∂ w
I1 = t, I2= wx2/3,
I1 = t, I2= wx3
w = x– 2/3θ (t),
w = x–3θ (t)
X5= t2∂ t + tw∂ w
I1 = t, I2= w|x|1/2,
I1 = x, I2= w/t
w=|x|– 1/2θ (t),
w = tθ(x)
1◦ For arbitrary f (w), this equation admits three operators (see Table 15.8):
X1= ∂ t, X2 = ∂ x, X3=2t∂ t + x∂ x The respective invariant solutions are
w=Φ(x), w = Φ(t), w = Φ(x2/t)
However, various linear combinations give another operator,
X1,2= X1+ aX2= ∂ t + a∂ x,
where a≠ 0is an arbitrary constant The solution invariant under this operator is written as
w=Φ(x – at).
It is apparent that solutions of this type (traveling waves) are not contained in the invariant solutions associated
with the individual operators X1, X2, and X3
2◦ If f (w) = e w , apart from the above three operators, there is another one, X4= x∂ x+2∂ w(see Table 15.8)
In this case, the linear combination
X3,4= X3+ aX4=2t∂ t + (a +1)x∂x+2a∂ w
gives another invariant solution,
w=Φ(ξ) + a ln t, ξ = xt a+21, where the functionΦ = Φ(ξ) satisfies the ordinary differential equation
(eΦΦ
ξ) ξ+12(a +1)ξΦ
ξ = a.
3◦ If f (w) = w k (k≠ 0, –4/3), apart from the three operators from1◦ , there is another one X4= kx∂ x+2w∂ w The linear combination
X3 , 4 = X3+ aX4=2t∂ t + (ak +1)x∂x+2aw∂ w
generates the invariant (self-similar) solution
w = t a Φ(ζ), ζ = xt ak+21, where the functionΦ = Φ(ζ) satisfies the ordinary differential equation
(ΦkΦ ζ) ζ+12(ak +1)ζΦ ζ = aΦ
The invariant solutions presented in Items1◦–3◦ are not listed in Table 15.8 It is clearly important to consider solutions induced by linear combinations of admissible operators
Trang 715.8.4 Some Generalizations Higher-Order Equations
15.8.4-1 One-parameter Lie groups of point transformations Group generator.
Here we will be considering functions dependent on n + 1 variables, x1, , xn, w The brief notation x = (x1, , xn) will be used.
The set of invertible transformations of the form
Tε=
¯xi = ϕi(x, w, ε), ¯xi|ε=0= xi
¯
w = ψ(x, w, ε), w ¯ |ε=0= w, (15. 8 4 1 )
where ϕiand ψ are sufficiently smooth functions of their arguments (i = 1 , , n) and ε is
a real parameter, is called a one-parameter continuous point group of transformations G if for any ε1 and ε2 the relation Tε1 ◦ Tε2 = Tε1+ε2 holds, that is, the successive application
(composition) of two transformations of the form (15.8.4.1) with parameters ε1 and ε2 is
equivalent to a single transformation of the same form with parameter ε1+ ε2.
Further on, we consider local one-parameter continuous Lie groups of point transfor-mations (or, for short, point groups), corresponding to the infinitesimal transformation
(15.8.4.1) as ε → 0 The expansion of (15.8.4.1) into Taylor series in the parameter ε about
ε = 0 to the first order gives
¯xi xi+ εξi(x, w), w ¯ w + εζ(x, w), (15 8 4 2 ) where
ξi(x, w) = ∂ϕi(x, w, ε) ∂ε
ε=0, ζ (x, w) =
∂ψ (x, w, ε)
∂ε
ε=0. The linear first-order differential operator
X = ξi(x, w) ∂
∂xi + ζ(x, w)
∂
∂w (15 8 4 3 )
corresponding to the infinitesimal transformation (15.8.4.2) is called a group generator (or
an infinitesimal operator) In formula (15.8.4.3), summation is assumed over the repeated index i.
THEOREM(LIE) Suppose the coordinates ξi(x, w) and ζ(x, w) of the group generator
(15.8.4.3) are known Then the one-parameter group of transformations (15.8.4.1) can be completely recovered by solving the Lie equations
dϕi
dε = ξi(ϕ, ψ), dψ
dε = ζ(ϕ, ψ) (i = 1 , , n)
with the initial conditions
ϕi|ε=0= xi ψ |ε=0= w.
Here, the short notation ϕ = (ϕ1, , ϕn) has been used.
Remark The widely known terms “Lie group analysis of differential equations,” “group-theoretic meth-ods,” and others are due to the prevailing concept of a local one-parameter Lie group of point transformations However, in this book, we prefer to use the term “method of symmetry analysis of differential equations.”
...(composition) of two transformations of the form (15.8.4.1) with parameters ε1 and ε2 is
equivalent to a single transformation of the... ϕiand ψ are sufficiently smooth functions of their arguments (i = , , n) and ε is
a real parameter, is called a one-parameter continuous point group of transformations... “Lie group analysis of differential equations,” “group-theoretic meth-ods,” and others are due to the prevailing concept of a local one-parameter Lie group of point transformations However, in