At the second stage, we successively substitute the ΦiX and ΨjY of 15.5.1.4into all solutions 15.5.4.1 to obtain systems of ordinary differential equations* for the unknown functions ϕp
Trang 12◦ At the second stage, we successively substitute the Φi(X) and Ψj(Y ) of (15.5.1.4)
into all solutions (15.5.4.1) to obtain systems of ordinary differential equations* for the
unknown functions ϕp(x) and ψq(y) Solving these systems, we get generalized separable
solutions of the form (15.5.1.1).
Remark 1 It is important that, for fixed k, the bilinear functional equation (15.5.1.3) used in the splitting
method is the same for different classes of original nonlinear mathematical physics equations
Remark 2 For fixed m, solution (15.5.4.1) contains m(k – m) arbitrary constants C i,j Given k, the
solutions having the maximum number of arbitrary constants are defined by
Solution number Number of arbitrary constants Conditions on k
m= 12(k 1) 1
4(k2–1) if k is odd.
It is these solutions of the bilinear functional equation that most frequently result in nontrivial generalized separable solution in nonlinear partial differential equations
Remark 3 The bilinear functional equation (15.5.1.3) and its solutions (15.5.4.1) play an important role
in the method of functional separation of variables
For visualization, the main stages of constructing generalized separable solutions by the splitting method are displayed in Fig 15.2.
15.5.4-2 Solutions of simple functional equations and their application.
Below we give solutions to two simple bilinear functional equations of the form (15.5.1.3) that will be used subsequently for solving specific nonlinear partial differential equations.
1◦ The functional equation
Φ1Ψ1 + Φ2Ψ2 + Φ3Ψ3 = 0 , (15 5 4 2 ) where Φiare all functions of the same argument and Ψiare all functions of another argument,
has two solutions:
Φ1 = A1Φ3, Φ2 = A2Φ3, Ψ3 = –A1Ψ1– A2Ψ2;
Ψ1 = A1Ψ3, Ψ2 = A2Ψ3, Φ3 = –A1Φ1– A2Φ2 (15. 5 4 3 )
The arbitrary constants are renamed as follows: A1= C1,1and A2= C2,1in the first solution,
and A1= – 1/C1,2and A2= C1,1/C1,2in the second solution The functions on the right-hand
sides of the formulas in (15.5.4.3) are assumed to be arbitrary.
2◦ The functional equation
Φ1Ψ1 + Φ2Ψ2 + Φ3Ψ3 + Φ4Ψ4 = 0 , (15 5 4 4 ) where Φiare all functions of the same argument and Ψiare all functions of another argument, has a solution
Φ1 = A1Φ3+ A2Φ4, Φ2 = A3Φ3+ A4Φ4,
Ψ3 = –A1Ψ1– A3Ψ2, Ψ4 = –A2Ψ1– A4Ψ2 (15. 5 . 4 . 5 )
dependent on four arbitrary constants A1, , A4; see solution (15.5.4.1) with k = 4 , m = 2 ,
C1,1= A1, C1,2= A2, C2,1= A3, and C2,2 = A4 The functions on the right-hand sides of
the solutions in (15.5.4.3) are assumed to be arbitrary.
* Such systems are usually overdetermined
Trang 2Write out the functional differential equation
Obtain: (i) functional equation, (ii) determining system of ODEs
Solve the determining system of ordinary differential equations
Write out generalized separable solution of original equation
Search for generalized separable solutions
Substitute into original equation
Apply splitting procedure
Treat functional equation (i)
Figure 15.2 General scheme for constructing generalized separable solutions by the splitting method
Abbre-viation: ODE stands for ordinary differential equation
Equation (15.5.4.4) also has two other solutions:
Φ1 = A1Φ4, Φ2 = A2Φ4, Φ3 = A3Φ4, Ψ4 = –A1Ψ1– A2Ψ2– A3Ψ3;
Ψ1 = A1Ψ4, Ψ2 = A2Ψ4, Ψ3 = A3Ψ4, Φ4 = –A1Φ1– A2Φ2– A3Φ3 (15. 5 . 4 . 6 )
involving three arbitrary constants In the first solution, A1= C1,1, A2= C2,1, and A3= C3,1,
and in the second solution, A1 = – 1/C1,3, A2= C1,1/C1,3, and A3= C1,2/C1,3.
Solutions (15.5.4.6) will sometimes be called degenerate, to emphasize the fact that
they contain fewer arbitrary constants than solution (15.5.4.5).
3◦ Solutions of the functional equation
Φ1Ψ1 + Φ2Ψ2 + Φ3Ψ3 + Φ4Ψ4 + Φ5Ψ5 = 0 (15 5 4 7 )
can be found by formulas (15.5.4.1) with k = 5 Below is a simple technique for finding solutions, which is quite useful in practice, based on equation (15.5.4.7) itself Let us assume that Φ1 , Φ2 , and Φ3 are linear combinations of Φ4 and Φ5 :
Φ1 = A1Φ4+ B1Φ5, Φ2 = A2Φ4+ B2Φ5, Φ3 = A3Φ4+ B3Φ5, (15 5 4 8 )
Trang 3where An, Bnare arbitrary constants Let us substitute (15.5.4.8) into (15.5.4.7) and collect the terms proportional to Φ4 and Φ5 to obtain
(A1Ψ1+ A2Ψ2+ A3Ψ3+ Ψ4 ) Φ4 + (B1Ψ1+ B2Ψ2+ B3Ψ3+ Ψ5 ) Φ5 = 0
Equating the expressions in parentheses to zero, we have
Ψ4 = –A1Ψ1– A2Ψ2– A3Ψ3,
Ψ5 = –B1Ψ1– B2Ψ2– B3Ψ3 (15. 5 4 9 ) Formulas (15.5.4.8) and (15.5.4.9) give solutions to equation (15.5.4.7) Other solutions are found likewise.
Example 1 Consider the nonlinear hyperbolic equation
∂2w
∂t2 = a ∂
∂x
w ∂w
∂x
where f (t) and g(t) are arbitrary functions We look for generalized separable solutions of the form
Substituting (15.5.4.11) into (15.5.4.10) and collecting terms yield
aψ2(ϕϕ x) x + aψχϕ xx + (f ψ – ψ tt )ϕ + f χ + g – χ tt=0
This equation can be represented as a functional equation (15.5.4.4) in which
Φ1 = (ϕϕ x) x, Φ2= ϕ xx, Φ3= ϕ, Φ4=1,
Ψ1= aψ2, Ψ2= aψχ, Ψ3= f ψ – ψ tt , Ψ4= f χ + g – χ tt (15.5.4.12)
On substituting (15.5.4.12) into (15.5.4.5), we obtain the following overdetermined system of ordinary
differ-ential equations for the functions ϕ = ϕ(x), ψ = ψ(t), and χ = χ(t):
(ϕϕ x) x = A1ϕ + A2, ϕ xx = A3ϕ + A4,
f ψ – ψ tt = –A1aψ2– A3aψχ , f χ + g – χ tt = –A2aψ2– A4aψχ (15.5.4.13) The first two equations in (15.5.4.13) are compatible only if
A1=6B2, A2 = B2–4B0B2, A3=0, A4=2B2, (15.5.4.14)
where B0, B1, and B2are arbitrary constants, and the solution is given by
ϕ (x) = B2x2+ B1x + B0 (15.5.4.15)
On substituting the expressions (15.5.4.14) into the last two equations in (15.5.4.13), we obtain the following
system of equations for ψ(t) and χ(t):
ψ tt=6aB2ψ2+ f (t)ψ,
χ tt= [2aB2ψ + f (t)]χ + a(B2–4B0B2)ψ2+ g(t). (15.5.4.16) Relations (15.5.4.11), (15.5.4.15) and system (15.5.4.16) determine a generalized separable solution of
equation (15.5.4.10) The first equation in (15.5.4.16) can be solved independently; it is linear if B2=0and is
integrable by quadrature for f (t) = const The second equation in (15.5.4.16) is linear in χ (for ψ known) Equation (15.5.4.10) does not have other solutions with the form (15.5.4.11) if f and g are arbitrary functions and ϕ 0, ψ 0, and χ 0
Trang 4Remark It can be shown that equation (15.5.4.10) has a more general solution with the form
w (x, y) = ϕ1(x)ψ1(t) + ϕ2(x)ψ2(t) + ψ3(t), ϕ1(x) = x2, ϕ2(x) = x, (15.5.4.17)
where the functions ψ i = ψ i (t) are determined by the ordinary differential equations
ψ1=6aψ2+ f (t)ψ1,
ψ2= [6aψ1+ f (t)]ψ2,
ψ3= [2aψ1+ f (t)]ψ3+ aψ2+ g(t).
(15.5.4.18)
(The prime denotes a derivative with respect to t.) The second equation in (15.5.4.18) has a particular solution
ψ2= ψ1 Hence, its general solution can be represented as (see Polyanin and Zaitsev, 2003)
ψ2 = C1ψ1 + C2ψ1
dt
ψ2
The solution obtained in Example 1 corresponds to the special case C2=0
Example 2 Consider the nonlinear equation
∂2w
∂x∂t+
∂w
∂x
2
– w ∂
2w
∂x2 = a ∂
3w
which arises in hydrodynamics (see Polyanin and Zaitsev, 2004)
We look for exact solutions of the form
Substituting (15.5.4.20) into (15.5.4.19) yields
ϕ t θ x – ϕψθ xx + ϕ2
(θ x )2– θθ xx
– aϕθ xxx=0
This functional differential equation can be reduced to the functional equation (15.5.4.4) by setting
Φ1= ϕ t, Φ2= ϕψ, Φ3= ϕ2, Φ4= aϕ,
Ψ1= θ x, Ψ2= –θ xx, Ψ3= (θ x)2– θθ xx, Ψ4= –θ xxx (15.5.4.21)
On substituting these expressions into (15.5.4.5), we obtain the system of ordinary differential equations
ϕ t = A1ϕ2+ A2aϕ, ϕψ = A3ϕ2+ A4aϕ,
(θ x)2– θθ xx = –A1θ x + A3θ xx , θ xxx = A2θ x – A4θ xx (15.5.4.22)
It can be shown that the last two equations in (15.5.4.22) are compatible only if the function θ and its
derivative are linearly dependent,
The six constants B1, B2, A1, A2, A3, and A4must satisfy the three conditions
B1(A1+ B2– A3B1) =0,
B2(A1+ B2– A3B1) =0,
B2+ A4B1– A2=0
(15.5.4.24)
Integrating (15.5.4.23) yields
θ=
B3exp(B1x) – B2
B1 if B1≠ 0,
where B3is an arbitrary constant
The first two equations in (15.5.4.22) lead to the following expressions for ϕ and ψ:
ϕ=
⎧
⎪
⎪
A2a
C exp(–A2at ) – A1 if A2≠ 0,
A1t + C if A2=0, ψ = A3ϕ + A4a, (15.5.4.26)
where C is an arbitrary constant.
Trang 5Formulas (15.5.4.25), (15.5.4.26) and relations (15.5.4.24) allow us to find the following solutions of equation (15.5.4.19) with the form (15.5.4.20):
w= x + C1
t + C2 + C3 if A2= B1=0, B2 = –A1;
w= C1e
–λx+1
λt + C2 + aλ if A2=0, B1 = –A4, B2= –A1– A3A4;
w = C1e–λ(x+aβt)+ a(λ + β) if A1= A3= B2=0, A2= B2+ A4B1;
w= aβ + C1e
–λx
1+ C2e–aλβt+ a(λ – β) if A1= A3B1– B2, A2= B2+ A4B1,
where C1, C2, C3, β, and λ are arbitrary constants (these can be expressed in terms of the A k and B k) The analysis of the second solution (15.5.4.6) of the functional equation (15.5.4.4) in view of (15.5.4.21) leads to the following two more general solutions of the differential equation (15.5.4.19):
w= x
t + C1 + ψ(t),
w = ϕ(t)e–λx– ϕ
t (t)
λϕ (t) + aλ, where ϕ(t) and ψ(t) are arbitrary functions, and C1and λ are arbitrary constants.
15.5.5 Titov–Galaktionov Method
15.5.5-1 Method description Linear subspaces invariant under a nonlinear operator Consider the nonlinear evolution equation
∂w
where F [w] is a nonlinear differential operator with respect to x,
F [w] ≡ F
x , w, ∂w
∂x , , ∂mw
∂xm
(15 5 5 2 ) Definition A finite-dimensional linear subspace
n= 5
ϕ1(x), , ϕn(x) 6
(15 5 5 3 )
formed by linear combinations of linearly independent functions ϕ1(x), , ϕn(x) is called invariant under the operator F if F [n] ⊆ n This means that there exist functions
f1, , fnsuch that
F
n
i=1
Ciϕi(x)
=
n
i=1
fi(C1, , Cn)ϕi(x) (15 5 5 4 )
for arbitrary constants C1, , Cn.
Let the linear subspace (15.5.5.3) be invariant under the operator F Then equation
(15.5.5.1) possesses generalized separable solutions of the form
w (x, t) =
n
i=1
Here, the functions ψ1(t), , ψn(t) are described by the autonomous system of ordinary
differential equations
ψ
i= fi(ψ1, , ψn), i = 1 , , n, (15 5 5 6 )
where the prime denotes a derivative with respect to t.
The following example illustrates the scheme for constructing generalized separable solutions.
Trang 6Example 1 Consider the nonlinear second-order parabolic equation
∂w
∂t = a ∂
2w
∂x2 +
∂w
∂x
2
Obviously, the nonlinear differential operator
F [w] = aw xx + (w x)2+ kw2+ bw + c (15.5.5.8)
for k >0has a two-dimensional invariant subspace 2=5
1, cos(x√ k)6
Indeed, for arbitrary C1and C2we have
F
C1+ C2cos(x √
k)
= k(C2+ C2) + bC1+ c + C2(2kC1– ak + b) cos(x √
k)
Therefore, there is a generalized separable solution of the form
w (x, t) = ψ1(t) + ψ2(t) cos(x √
where the functions ψ1(t) and ψ2(t) are determined by the autonomous system of ordinary differential equations
ψ 1= k(ψ2+ ψ2) + bψ1+ c,
Remark 1 Example 3 below shows how one can find all two-dimensional linear subspaces invariant under the nonlinear differential operator (15.5.5.8)
Remark 2 For k > 0, the nonlinear differential operator (15.5.5.8) has a three-dimensional invariant subspace 3=5
1, sin(x√ k ), cos(x √
k)6
; see Example 3
Remark 3 For k < 0, the nonlinear differential operator (15.5.5.8) has a three-dimensional invariant subspace 3=5
1, sinh(x√ –k ), cosh(x √
–k )6
; see Example 3
Remark 4 A more general equation (15.5.5.7), with a = a(t), b = b(t), and c = c(t) being arbitrary functions, and k = const <0, also admits a generalized separable solution of the form (15.5.5.9), where the
functions ψ1(t) and ψ2(t) are determined by the system of ordinary differential equations (15.5.5.10).
15.5.5-2 Some generalizations.
Likewise, one can consider a more general equation of the form
L1[w] = L2[U ], U = F [w], (15 5 5 11 )
where L1[w] and L2[U ] are linear differential operators with respect to t,
L1[w] ≡ s1
i=0
ai(t) ∂
iw
∂ti , L2[U ] ≡ s2
j=0
bj(t) ∂
jU
and F [w] is a nonlinear differential operator with respect to x,
F [w] ≡ F
t , x, w, ∂w
∂x , , ∂mw
∂xm
, (15 5 5 13 )
and may depend on t as a parameter.
Trang 7Let the linear subspace (15.5.5.3) be invariant under the operator F , i.e., for arbitrary constants C1, , Cnthe following relation holds:
F
n
i=1
Ciϕi(x)
=
n
i=1
fi(t, C1, , Cn)ϕi(x). (15. 5 5 14 )
Then equation (15.5.5.11) possesses generalized separable solutions of the form (15.5.5.5),
where the functions ψ1(t), , ψn(t) are described by the system of ordinary differential
equations
L1
ψi(t)
= L2
fi(t, ψ1, , ψn)
, i = 1 , , n. (15 5 5 15 )
Example 2 Consider the equation
a2(t) ∂
2w
∂t2 + a1(t) ∂w
∂t =∂w
∂x
∂2w
which, in the special case of a2(t) = k2and a1(t) = k1/t , is used for describing transonic gas flows (where t
plays the role of a spatial variable)
Equation (15.5.5.16) is a special case of equation (15.5.5.11), where L1[w] = a2(t)w tt +a1(t)w t , L2[U ] = U , and F [w] = w x w xx It can be shown that the nonlinear differential operator F [w] = w x w xxadmits the three-dimensional invariant subspace 3 =5
1, x3/2, x36
Therefore, equation (15.5.5.16) possesses generalized separable solutions of the form
w (x, t) = ψ1(t) + ψ2(t)x3/2+ ψ3(t)x3,
where the functions ψ1(t), ψ2(t), and ψ3(t) are described by the system of ordinary differential equations
a2 (t)ψ1 + a1(t)ψ 1= 98ψ2,
a2(t)ψ2 + a1(t)ψ 2= 454ψ2ψ3,
a2(t)ψ3 + a1(t)ψ 3=18ψ2
Remark The operator F [w] = w x w xxalso has a four-dimensional invariant subspace 4=5
1, x, x2, x36
Therefore, equation (15.5.5.16) has a generalized separable solution of the form
w (x, t) = ψ1(t) + ψ2(t)x + ψ3(t)x2+ ψ4(t)x3
15.5.5-3 How to find linear subspaces invariant under a given nonlinear operator The most difficult part in using the Titov–Galaktionov method for the construction of exact solutions to specific equations is to find linear subspaces invariant under a given nonlinear operator.
In order to determine basis functions ϕi= ϕi(x), let us substitute the linear combination
n
i=1Ciϕi(x) into the differential operator (15.5.5.2) This is assumed to result in an expression
like
F
n
i=1
Ciϕi(x)
= A1(C) Φ1 (X) + A2(C) Φ2 (X) + · · · + Ak(C) Φk(X) + B1(C)ϕ1(x) + B2(C)ϕ2(x) + · · · + Bn(C)ϕn(x), (15 5 5 17 )
where Aj(C) and Bi(C) depend on C1, , Cn only, and the functionals Φj(X) depend
on x and are independent of C1, , Cn:
Aj(C) ≡ Aj(C1, , Cn), j = 1 , , k,
Bi(C) ≡ Bi(C1, , Cn), i = 1 , , n,
Φj(X) ≡ Φj x , ϕ1, ϕ1, ϕ1, , ϕn, ϕ n, ϕ n
.
(15 5 5 18 )