1 It is noteworthy that many nonlinear partial differential equations that are not reducible to linear equations have exact solutions of the form 15.6.1.1 as well.. Main idea: The funct
Trang 1Here, for simplicity, the formulas are written out for the case of a second-order differential operator For higher-order operators, the right-hand sides of relations (15.5.5.18) will
contain higher-order derivatives of ϕi The functionals and functions Φ1(X), , Φk(X),
ϕ1(x), , ϕn(x) together are assumed to be linearly independent, and the Aj(C) are
linearly independent functions of C1, , Cn.
The basis functions are determined by solving the (usually overdetermined) system of ordinary differential equations
Φj x , ϕ1, ϕ1, ϕ1, , ϕn, ϕ n, ϕ n
= pj,1ϕ1+ pj,2ϕ2+ · · · + pj,nϕn, j = 1 , , k,
(15 5 5 19 )
where pj,i are some constants independent of the parameters C1, , Cn If for some
collection of the constants pi,j, system (15.5.5.19) is solvable (in practice, it suffices to find
a particular solution), then the functions ϕi= ϕi(x) define a linear subspace invariant under
the nonlinear differential operator (15.5.5.2) In this case, the functions appearing on the right-hand side of (15.5.5.4) are given by
fi(C1, , Cn) = p1 ,iA1(C1, , Cn) + p2 ,iA2(C1, , Cn) + · · ·
+ pk,iAk(C1, , Cn) + Bi(C1, , Cn).
Remark The analysis of nonlinear differential operators is useful to begin with looking for two-dimensional invariant subspaces of the form 2={1, ϕ(x)}
Proposition 1 Let a nonlinear differential operator F [w] admit a two-dimensional
invariant subspace of the form 2= {1 , ϕ(x) } , where ϕ(x) = pϕ1(x) + qϕ2(x), p and q are
arbitrary constants, and the functions 1 , ϕ1(x), ϕ2(x) are linearly independent Then the operator F [w] also admits a three-dimensional invariant subspace 2= {1 , ϕ1(x), ϕ2(x) }
Proposition 2 Let two nonlinear differential operators F1[w] and F2[w] admit one
and the same invariant subspace n = { ϕ1(x), , ϕn(x) } Then the nonlinear operator
pF1[w] + qF2[w], where p and q are arbitrary constants, also admits the same invariant
subspace.
Example 3 Consider the nonlinear differential operator (15.5.5.8) We look for its invariant subspaces of
the form 2={1, ϕ(x)} We have
F [C1 + C2 ϕ (x)] = C2[(ϕ x)2+ kϕ2] + C2 aϕ xx + kC2+ bC1 + c + (bC2+2kC1C2)ϕ.
Here, Φ1(X) = (ϕ
x)2 + kϕ2 and Φ2(X) = aϕ
xx Hence, the basis function ϕ(x) is determined by the
overdetermined system of ordinary differential equations
(ϕ x)2+ kϕ2= p1 + p2 ϕ,
ϕ xx = p3+ p4ϕ, (15.5.5.20)
where p1 = p1, 1, p2= p1, 2, p3= p2, 1/a , and p4= p2, 2/a Let us investigate system (15.5.5.20) for consistency
To this end, we differentiate the first equation with respect to x and then divide by ϕ x to obtain ϕ xx =
–kϕ + p2/2 Using this relation to eliminate the second derivative from the second equation in (15.5.5.20), we
get (p4 + k)ϕ + p3–12p2=0 For this equation to be satisfied, the following identities must hold:
p4= –k, p3= 12p2 (15.5.5.21) The simultaneous solution of system (15.5.5.20) under condition (15.5.5.21) is given by
ϕ (x) = px2+ qx if k =0 (p1= q2, p2=4p),
ϕ (x) = p sin x √
k
+ q cos x √
k
if k >0 (p1 = kp2+ kq2, p2=0),
ϕ (x) = p sinh x √
–k
+ q cosh x √
–k
if k <0 (p1= –kp2+ kq2, p2=0),
(15.5.5.22)
where p and q are arbitrary constants.
Since formulas (15.5.5.22) involve two arbitrary parameters p and q, it follows from Proposition 1 that the
nonlinear differential operator (15.5.5.8) admits the following invariant subspaces:
3=5
1, x, x26
if k =0,
3=5
1, sin(x√ k ), cos(x √
k)6
if k >0,
3=5
1, sinh(x√ –k ), cosh(x √
–k )6
if k <0
Trang 215.6 Method of Functional Separation of Variables
15.6.1 Structure of Functional Separable Solutions Solution by
Reduction to Equations with Quadratic Nonlinearities
15.6.1-1 Structure of functional separable solutions.
Suppose a nonlinear equation for w = w(x, y) is obtained from a linear mathematical physics equation for z = z(x, y) by a nonlinear change of variable w = F (z) Then, if the linear equation for z admits separable solutions, the transformed nonlinear equation for w will
have exact solutions of the form
w (x, y) = F (z), where z =
n
m=1
ϕm(x)ψm(y). (15. 6 1 1 )
It is noteworthy that many nonlinear partial differential equations that are not reducible
to linear equations have exact solutions of the form (15.6.1.1) as well We will call such
solutions functional separable solutions In general, the functions ϕm(x), ψm(y), and F (z)
in (15.6.1.1) are not known in advance and are to be identified.
Main idea: The functional differential equation resulting from the substitution of
(15.6.1.1) in the original partial differential equation should be reduced to the standard bilinear functional equation (15.5.1.3) or to a functional differential equation of the form (15.5.1.3)–(15.5.1.4), and then the results of Subsections 15.5.3–15.5.5 should be used.
Remark The function F (z) can be determined by a single ordinary differential equation or by an
overde-termined system of equations; both possibilities must be taken into account
15.6.1-2 Solution by reduction to equations with quadratic (or power) nonlinearities.
In some cases, solutions of the form (15.6.1.1) can be searched for in two stages First, one looks for a transformation that would reduce the original equation to an equation with a quadratic (or power) nonlinearity Then the methods outlined in Subsections 15.5.3–15.5.5 are used for finding solutions of the resulting equation.
Table 15.5 gives examples of nonlinear heat equations with power, exponential, and
logarithmic nonlinearities reducible, by simple substitutions of the form w = F (z), to
quadratically nonlinear equations For these equations, it can be assumed that the form of
the function F (z) in solution (15.6.1.1) is given a priori.
15.6.2 Special Functional Separable Solutions Generalized
Traveling-Wave Solutions
15.6.2-1 Special functional separable and generalized traveling-wave solutions.
To simplify the analysis, some of the functions in (15.6.1.1) can be specified a priori and
the other functions will be defined in the analysis We call such solutions special functional separable solutions.
A generalized separable solution (see Section 15.5) is a functional separable solution of
the special form corresponding to F (z) = z.
Below we consider two simplest functional separable solutions of special forms:
w = F (z), z = ϕ1(x)y + ϕ2(x);
w = F (z), z = ϕ(x) + ψ(y). (15. 6 2 1 )
Trang 3TABLE 15.5 Some nonlinear heat equations reducible to quadratically nonlinear equations by a transformation of the form
w = F (z); the constant σ is expressed in terms of the coefficients of the transformed equation
Original equation Transformation Transformed equation Form of solutions
∂w
∂t = a ∂x ∂ w n ∂w ∂x
+ bw + cw1–n w = z1/n ∂z ∂t = az ∂ ∂x2z2 +a n ∂z ∂x 2
+ bnz + cn z = ϕ(t)x
2+ ψ(t)x + χ(t)
∂w
∂t = a ∂x ∂ w n ∂w ∂x
+ bw n+1+ cw w = z
1/n ∂z ∂t = az ∂ ∂x2z2 +a n ∂z ∂x 2
+ bnz2+ cnz
z = ϕ(t)e σx + ψ(t)e–σx + χ(t)
z = ϕ(t) sin(σx) + ψ(t) cos(σx) + χ(t)
∂w
∂t = a ∂x ∂ e λw ∂w ∂x
+ b + ce–λw w= λ1 ln z ∂z ∂t = az ∂ ∂x2z2 + bλz + cλ z = ϕ(t)x2+ ψ(t)x + χ(t)
∂w
∂t = a ∂x ∂ e λw ∂w ∂x
+ be λw + c w=
1
λ ln z ∂z
∂t = az ∂ ∂x2z2 + bz2+ cλz z = ϕ(t)e σx + ψ(t)e
–σx + χ(t)
z = ϕ(t) sin(σx) + ψ(t) cos(σx) + χ(t)
∂w
∂t = a ∂ ∂x2w2
+ bw ln w + cw w = e
z ∂z ∂t = a ∂ ∂x2z2 + a ∂z ∂x 2
+ bz + c z = ϕ(t)x
2+ ψ(t)x + χ(t)
∂w
∂t = a ∂ ∂x2w2
+ bw ln2w + cw w = e
z ∂z ∂t = a ∂ ∂x2z2 + a ∂z ∂x 2
+ bz2+ c
z = ϕ(t)e σx + ψ(t)e–σx + χ(t)
z = ϕ(t) sin(σx) + ψ(t) cos(σx) + χ(t)
The first solution (15.6.2.1) will be called a generalized traveling-wave solution (x and y
can be swapped) After substituting this solution into the original equation, one should
eliminate y with the help of the expression for z This will result in a functional differential equation with two arguments, x and z Its solution may be obtained with the methods
outlined in Subsections 15.5.3–15.5.5.
Remark 1 In functional separation of variables, searching for solutions in the forms w = F ϕ (x) + ψ(y)
[it is the second solution in (15.6.2.1)] and w = F ϕ (x)ψ(y)
leads to equivalent results because the two
forms are functionally equivalent Indeed, we have F ϕ (x)ψ(y)
= F1 ϕ1(x) + ψ1(y)
, where F1(z) = F (e z),
ϕ1(x) = ln ϕ(x), and ψ1(y) = ln ψ(y).
Remark 2 In constructing functional separable solutions with the form w = F ϕ (x) + ψ(y)
[it is the
second solution in (15.6.2.1)], it is assumed that ϕconst and ψconst
Example 1 Consider the third-order nonlinear equation
∂w
∂y
∂2w
∂x∂y –∂w
∂x
∂2w
∂y2 = a
∂2w
∂y2
n–1
∂3w
∂y3,
which describes a boundary layer of a power-law fluid on a flat plate; w is the stream function, x and y are coordinates along and normal to the plate, and n is a rheological parameter (the value n =1corresponds to a Newtonian fluid) Searching for solutions in the form
w = w(z), z = ϕ(x)y + ψ(x) leads to the equation ϕ x (w z )2= aϕ2n (w zz)n–1w zzz , which is independent of ψ Separating the variables and
integrating yields
ϕ (x) = (ax + C)1/(1–2n), ψ (x) is arbitrary, and w = w(z) is determined by solving the ordinary differential equation (w z )2= (1–2n )(w zz)n–1w zzz
15.6.2-2 General scheme for constructing generalized traveling-wave solutions.
For visualization, the general scheme for constructing generalized traveling-wave solutions for evolution equations is displayed in Fig 15.3.
Trang 4Write out the functional differential equation in two arguments
Obtain (i) functional equation, (ii) determining system of ODEs
Solve the determining system of ordinary differential equations
Write out generalized traveling-wave solution of original equation
Search for generalizedtraveling-wavesolutions
Apply splitting procedure
Treat functional equation (i)
Figure 15.3 Algorithm for constructing generalized traveling-wave solutions for evolution equations
Abbre-viation: ODE stands for ordinary differential equation
Example 2 Consider the nonstationary heat equation with a nonlinear source
∂w
∂t = ∂
2w
We look for functional separable solutions of the special form
w = w(z), z = ϕ(t)x + ψ(t). (15.6.2.3)
The functions w(z), ϕ(t), ψ(t), and F(w) are to be determined.
On substituting (15.6.2.3) into (15.6.2.2) and on dividing by w z , we have
ϕ t x + ψ t = ϕ2w
zz
w z +F(w)
We express x from (15.6.2.3) in terms of z and substitute into (15.6.2.4) to obtain a functional differential equation with two variables, t and z:
–ψ t +ψ
ϕ ϕ
t–ϕ
t
ϕ z + ϕ2w
zz
w z +F(w)
w z =0, which can be treated as the functional equation (15.5.4.4), where
Φ1 = –ψ t+ψ
ϕ ϕ
t, Φ2= –ϕ
t
ϕ, Φ3= ϕ2, Φ4 =1, Ψ1 =1, Ψ2= z, Ψ3= w
zz
w z , Ψ4= F(w)
w z
Trang 5Substituting these expressions into relations (15.5.4.5) yields the system of ordinary differential equations
–ψ t +ψ
ϕ ϕ
t = A1ϕ2+ A2, –ϕ t
ϕ = A3ϕ2+ A4,
w zz
w z = –A1– A3z, F(w)
w z = –A2– A4z,
(15.6.2.5)
where A1 , , A4are arbitrary constants
ϕ (t) =
C1e2A4t– A3
A4
– 1/2 ,
ψ (t) = –ϕ(t)
A1
ϕ (t) dt + A2
dt
ϕ (t) + C2
,
w (z) = C3
exp –12A3z2– A1 z
dz + C4,
F(w) = –C3(A4z + A2) exp –12A3z2– A1 z
,
(15.6.2.6)
where C1, , C4are arbitrary constants The dependenceF = F(w) is defined by the last two relations in
parametric form (z is considered the parameter) If A3 ≠ 0in (15.6.2.6), the source function is expressed in terms of elementary functions and the inverse of the error function
In the special case A3= C4=0, A1 = –1, and C3 =1, the source function can be represented in explicit form as
F(w) = –w(A4ln w + A2)
ϕ (t) = 1
√2
A3t + C1, ψ (t) =
C2
√2
A3t + C1 –
A1
A3 –3A A23(2A3t + C1), and the solutions to the other equations are determined by the last two formulas in (15.6.2.6) where A4=0 Remark The algorithm presented in Fig 15.3 can also be used for finding exact solutions of the more
general form w = σ(t)F (z) + ϕ1 (t)x + ψ2(t), where z = ϕ1 (t)x + ψ2(t) For an example of this sort of solution,
see Subsection 15.7.2 (Example 1)
15.6.3 Differentiation Method
15.6.3-1 Basic ideas of the method Reduction to a standard equation.
In general, the substitution of expression (15.6.1.1) into the nonlinear partial differential equation under study leads to a functional differential equation with three arguments—two
arguments are usual, x and y, and the third is composite, z In some cases, the resulting
equation can be reduced by differentiation to a standard functional differential equation
with two arguments (either x or y is eliminated) To solve the two-argument equation, one
can use the methods outlined in Subsections 15.5.3–15.5.5.
15.6.3-2 Examples of constructing functional separable solutions.
Below we consider specific examples illustrating the application of the differentiation method for constructing functional separable solutions of nonlinear equations.
Example 1 Consider the nonlinear heat equation
∂w
∂t = ∂
∂x
F(w) ∂w
∂x
We look for exact solutions with the form
w = w(z), z = ϕ(x) + ψ(t). (15.6.3.2)
Trang 6On substituting (15.6.3.2) into (15.6.3.1) and dividing by w z , we obtain the functional differential equation with three variables
ψ t = ϕ xx F(w) + (ϕ
where
H (z) = F(w) w zz
w z +F
z (w), w = w(z). (15.6.3.4)
Differentiating (15.6.3.3) with respect to x yields
ϕ xxx F(w) + ϕ
x ϕ xx[F
z (w) +2H (z)] + (ϕ x)3H z =0 (15.6.3.5) This functional differential equation with two variables can be treated as the functional equation (15.5.4.2) This three-term functional equation has two different solutions Accordingly, we consider two cases
Case 1 The solutions of the functional differential equation (15.6.3.5) are determined from the system of
ordinary differential equations
F z +2H=2A1F, H z = A2F,
ϕ xxx+2A1ϕ x ϕ xx + A2(ϕ x)3=0, (15.6.3.6)
where A1 and A2are arbitrary constants
The first two equations (15.6.3.6) are linear and independent of the third equation Their general solution
is given by
F =
⎧
⎨
⎩
e A1z (B1 e kz + B2 e–kz) if A2>2A2,
e A1z (B1+ B2z) if A2=2A2,
e A1z [B1sin(kz) + B2cos(kz)] if A2<2A2,
H = A1F–1
2F
z, k=
|A2–2A2| (15.6.3.7)
Substituting H of (15.6.3.7) into (15.6.3.4) yields an ordinary differential equation for w = w(z) On
integrating this equation, we obtain
w = C1
e A1z|F(z)|– 3/2dz + C2, (15.6.3.8)
where C1 and C2are arbitrary constants The expression ofF in (15.6.3.7) together with expression (15.6.3.8)
defines the functionF = F(w) in parametric form.
Without full analysis, we will study the case A2=0(k = A1 ) and A1 ≠ 0in more detail It follows from (15.6.3.7) and (15.6.3.8) that
F(z) = B1e2A1z + B2, H = A1 B2, w (z) = C3(B1 + B2 e–2A1z)–1/2+ C2 (C1 = A1 B2C3). (15.6.3.9)
Eliminating z yields
F(w) = B2C2
The last equation in (15.6.3.6) with A2 =0has the first integral ϕ xx + A1(ϕ x)2 = const The corresponding general solution is given by
ϕ (x) = – 1
2A1 ln
D2
D1
1 sinh2 A1√
D2x + D3
for D1 >0 and D2>0,
ϕ (x) = – 1
2A1 ln
–D2
D1
1 cos2 A1√
–D2x + D3
for D1 >0 and D2<0,
ϕ (x) = – 1
2A1 ln
–D2
D1
1 cosh2 A1√
D2x + D3
for D1 <0 and D2>0,
(15.6.3.11)
where D1, D2, and D3are constants of integration In all three cases, the following relations hold:
(ϕ x)2= D1e–2A1ϕ + D2, ϕ xx = –A1D1e–2A1ϕ (15.6.3.12)
We substitute (15.6.3.9) and (15.6.3.12) into the original functional differential equation (15.6.3.3) With
reference to the expression of z in (15.6.3.2), we obtain the following equation for ψ = ψ(t):
ψ t = –A1B1D1e2A1ψ + A1B2D2 Its general solution is given by
ψ (t) = 1
2A1 ln B2D2
D4exp(–2A2B2D2t ) + B1D1, (15.6.3.13)
where D4is an arbitrary constant
Formulas (15.6.3.2), (15.6.3.9) for w, (15.6.3.11), and (15.6.3.13) define three solutions of the nonlinear
equation (15.6.3.1) withF(w) of the form (15.6.3.10) [recall that these solutions correspond to the special case
A2=0in (15.6.3.7) and (15.6.3.8)]
Trang 7Case 2 The solutions of the functional differential equation (15.6.3.5) are determined from the system of
ordinary differential equations
ϕ xxx = A1 (ϕ x)3, ϕ x ϕ xx = A2 (ϕ x)3,
A1F + A2(F
z+2H ) + H z =0 (15.6.3.14) The first two equations in (15.6.3.14) are compatible in the two cases
A1= A2=0 =⇒ ϕ(x) = B1 x + B2,
A1=2A2 =⇒ ϕ(x) = – 1
A2 ln|B1x + B2|. (15.6.3.15)
The first solution in (15.6.3.15) eventually leads to the traveling-wave solution w = w(B1 x + B2 t) of
equa-tion (15.6.3.1) and the second soluequa-tion to the self-similar soluequa-tion of the form w = w2(x2/t) In both cases, the functionF(w) in (15.6.3.1) is arbitrary.
Example 2 Consider the nonlinear Klein–Gordon equation
∂2w
∂t2 –∂
2w
We look for functional separable solutions in additive form:
w = w(z), z = ϕ(x) + ψ(t). (15.6.3.17) Substituting (15.6.3.17) into (15.6.3.16) yields
ψ tt – ϕ xx+
(ψ t )2– (ϕ x)2
g (z) = h(z), (15.6.3.18) where
g (z) = w zz /w z , h (z) = F w (z)
/w z (15.6.3.19)
On differentiating (15.6.3.18) first with respect to t and then with respect to x and on dividing by ψ t ϕ x, we have
2(ψ
tt – ϕ xx ) g z +
(ψ t)2– (ϕ x)2
g zz = h zz
Eliminating ψ tt – ϕ xxfrom this equation with the aid of (15.6.3.18), we obtain
(ψ t )2– (ϕ x)2
(g zz–2gg z ) = h zz–2g z h (15.6.3.20) This relation holds in the following cases:
g zz–2gg z=0, h zz–2g z h=0 (case 1),
(ψ t)2= Aψ + B, (ϕ x)2= –Aϕ + B – C, h zz–2g z h = (Az + C)(g zz –2gg z ) (case 2), (15.6.3.21)
where A, B, and C are arbitrary constants We consider both cases.
Case 1 The first two equations in (15.6.3.21) enable one to determine g(z) and h(z) Integrating the first equation once yields g z = g2+ const Further, the following cases are possible:
g= –1/ (z + C1), (15.6.3.22b)
g = –k tanh(kz + C1), (15.6.3.22c)
g = –k coth(kz + C1), (15.6.3.22d)
where C1and k are arbitrary constants.
The second equation in (15.6.3.21) has a particular solution h = g(z) Hence, its general solution is
expressed by [e.g., see Polyanin and Zaitsev (2003)]:
h = C2 g (z) + C3 g (z)
dz
where C2 and C3are arbitrary constants
The functions w(z) and F(w) are found from (15.6.3.19) as
w (z) = B1
G (z) dz + B2, F(w) = B1h (z)G(z), where G (z) = exp
g (z) dz
, (15.6.3.24)
and B1and B2are arbitrary constants (F is defined parametrically).
... constantFormulas (15.6.3.2), (15.6.3.9) for w, (15.6.3.11), and (15.6.3.13) define three solutions of the nonlinear
equation (15.6.3.1) withF(w) of the form (15.6.3.10)... D3
for D1 <0 and D2>0,
(15.6.3.11)
where D1, D2, and D3are constants of integration In all... traveling-wave solution w = w(B1 x + B2 t) of
equa-tion (15.6.3.1) and the second soluequa-tion to the self-similar soluequa-tion of the form w = w2(x2/t)