Special Functional Separable Solutions

S.5.2. Special Functional Separable Solutions

S.5.2-1. Generalized traveling-wave solutions. Examples. To simplify the analysis, some of the functions in (1) can be specified a priori and the other functions

will be defined in the analysis. We call such solutions special functional separable solutions.

Consider functional separable solutions of the form (1) in the special case where the composite argument z is linear in one of the independent variables (e.g., in x). We substitute (1) into the equation under study and eliminate x using the expression of z to obtain a functional-differential equation with two arguments. In many cases, this equation can be solved by the methods outlined in Subsections S.4.2–S.4.4.

Below are the simplest functional separable solutions of special forms ( x and y can be swapped):

w = F (z), z = ψ 1 ( y)x + ψ 2 ( y)

( z is linear in x);

w = F (z), z = ψ 2

1 ( y)x + ψ 2 ( y)

( z is quadratic in x); w = F (z), z = ψ 1 ( y)e λx + ψ 2 ( y) (z contains an exponential of x).

The first solution will be called a generalized traveling-wave solution. In the last formula, e λx can

be replaced by cosh( ax + b), sinh(ax + b), or sin(ax + b) to obtain another three modifications. After substituting any of the above expressions into the original equation, one should eliminate x with the help of the expression for z. This will result in a functional-differential equation with two arguments, y and z. Its solution may be obtained in some cases with the methods outlined in Subsections S.4.2–S.4.4.

Original equation: w t = H (, tww , x , w xx , ..., w n x () )

Search for generalized traveling-wave solutions

Define solution structure: w = Fz ( ), where = z j () tx + y () t

Substitute into original equation and replace x by ( - )/ z yj

Write out the functional-differential equation in two arguments

Apply splitting procedure

Obtain: (i) functional equation, (ii) determining system of ODEs

Treat functional equation (i)

Solve the functional equation: F 1 () z Y 1 ( ) + ... + t F k () z Y k ( t )=0

Substitute the F m and Y m in determining system (ii)

Solve the determining system of ordinary differential equations

Find the functions , j y and F

Write out generalized traveling-wave solution of original equation

Figure 2. Algorithm for constructing generalized traveling-wave solutions for evolution equations. Abbreviation: ODE stands for ordinary differential equation.

For visualization, the general scheme for constructing generalized traveling-wave solutions for evolution equations is displayed in Fig. 2.

Remark 1. The algorithm presented in Fig. 2 can also be used for finding exact solutions of the more general form w = σ(t)F (z) + ϕ 1 ( t)x + ψ 2 ( t), where z = ϕ 1 ( t)x + ψ 2 ( t). For an example of this sort of solution, see Subsection S.6.3 (Example 6).

Remark 2. A generalized separable solution (see Section S.4) is a functional separable solution of the special form corresponding to

F (z) = z.

We consider below examples of nonlinear equations that admit functional separable solutions of the special form where the argument z is linear or quadratic in one of the independent variables.

Example 1. Consider the nonstationary heat equation with a nonlinear source

∂w = ∂ 2 w

2 + F(w).

∂t

∂x

We look for functional separable solutions of the special form

w = w(z), z = ϕ(t)x + ψ(t).

The functions w(z), ϕ(t), ψ(t), and F(w) are to be determined. On substituting (3) into (2) and on dividing by w z ′ , we have

x+ψ 2 ′ = ϕ w zz ′′

F(w)

We express x from (3) in terms of z and substitute into (4) to obtain a functional-differential equation with two variables, t and z,

t z+ϕ

2 w ′′

zz + F(w) = 0,

, Ψ 4 = F(w)

w ′ . z Substituting these expressions into relations (52) of Subsection S.4.4 yields the system of ordinary differential equations

where A 1 , ...,A 4 are arbitrary constants. Case 1 . For A 4 ≠ 0, the solution of system (5) is given by

ψ(t) = −ϕ(t) A 1 ϕ(t) dt + A 2 dt + C 2 ,

ϕ(t)

Z 3 exp − 1 2 A 3 w(z) = C 2 z − A 1 z dz + C 4 ,

3 ( A 4 2 ) exp − F(w) = −C 1 z+A 2 A 3 z 2 − A 1 z ,

where C 1 , ...,C 4 are arbitrary constants. The dependence F = F(w) is defined by the last two relations in parametric form ( z is considered the parameter). If A 3 ≠ 0 in (6), the source function is expressed in terms of elementary functions and the

inverse of the error function. In the special case A 3 = C 4 = 0, A 1 = −1, and C 3 = 1, the source function can be represented in explicit form as

F(w) = −w(A 4 ln w+A 2 ).

Solutions of equation (2) in this case were obtained by Dorodnitsyn (1982) with group-theoretic methods. Case 2 . For A 4 = 0, the solution to the first two equations in (5) is given by

1 C 2 A ϕ(t) = 1

ψ(t) = √

A t+C 1 2 − A 3 t+C − 3 3 A 3 (2 A t+C ), and the solutions to the other equations are determined by the last two formulas in (6) where A 4 = 0.

Example 2. Consider the more general equation

∂w

2 = a(t) ∂ w

∂x 2 + b(t) ∂w

∂x + c(t)F(w).

∂t

We look for solutions in the form (3). In this case, only the first two equations in system (5) will change, and the functions w(z) and F(w) will be given by (6).

Example 3. The nonlinear heat equation

∂x G(w) ∂x

F(w)

has also solutions of the form (3). The unknown quantities are governed by system (5) in which w ′′ zz must be replaced by [ G(w)w ′ z ] z ′ . The functions ϕ(t) and ψ(t) are determined by the first two formulas in (6). One of the two functions G(w) and F(w) can be assumed arbitrary and the other is identified in the course of the solution. The special case F(w) = const yields

G(w) = C 1 e 2 ke +( C 2 w+C 3 ) e kw .

Functional separable solutions (3) of the given equation are discussed in more detail in 1.6.15.2, Items 3 ◦ and 4 ◦ ; some other solutions are also specified there.

Example 4. We can treat the nth-order nonlinear equation

likewise. As before, we look for solutions in the form (3). In this case, the quantities ϕ 2 and w zz ′′ in (5) must be replaced by ϕ n and w ( z n) , respectively. In particular, for A 3 = 0, apart from equations with logarithmic nonlinearities of the form (7), we obtain other equations.

the search for exact solutions of the form (3) leads to the following system of equations for ϕ(t), ψ(t), w(z), and F(w):

ϕ t ′ = A 1 ϕ n + A 2 ϕ, − t = A 3 ϕ n + A 4 ϕ,

w z ( n)

F(w) = −A 2 − A 4 w z,

=− A 1 − A 3 z,

where A 1 , ...,A 4 are arbitrary constants. In the case n = 3, we assume A 3 = 0 and A 1 > 0 to find in particular that F(w) = −A 2 − A 4 arcsin( kw). Some functional separable solutions (3) of the given equation can be found in Subsection 11.1.3.

Example 6. In addition, searching for solutions of equation (2) with z quadratically dependent on x,

w = w(z), 2 z = ϕ(t)x + ψ(t),

also makes sense here. Indeed, on substituting (8) into (2), we arrive at an equation that contains terms with x 2 and does not contain terms linear in x. Eliminating x 2 from the resulting equation with the aid of (8), we obtain

w ϕψ zz ′ ′′ −4 + F(w) = 0.

To solve this functional-differential equation with two arguments, we apply the splitting method outlined in Subsection S.4.4. It can be shown that, for equations (2), this equation has a solution with a logarithmic nonlinearity of the form (7).

Example 7. Consider the mth-order nonlinear equation

∂y ∂x∂y

∂x ∂y

which, in the special case of f (x) = const and m = 3, 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 = 1 corresponds to a Newtonian fluid). Searching for solutions in the form

w = w(z),

z = ϕ(x)y + ψ(x),

f (x)ϕ 2 n+m−3 ( w ′′ zz ) n−1 w ( z m) , which is independent of ψ. Separating the variables and integrating yields

leads to the equation ϕ ′ x ( w ′ z ) 2 =

4−2 ϕ(x) = n−m f (x) dx + C , ψ(x) is arbitrary,

and the function w = w(z) is determined by solving the ordinary differential equation (w ′ z ) 2 = (4 − 2 n − m)(w ′′ zz ) n−1 w z ( m) . Example 8. Consider the equation

∂ n+1 w ∂x n

f (w).

∂y

We look for functional separable solutions of the special form

w = w(z),

z = ϕ(y)x + ψ(y).

We substitute (10) in (9), eliminate x with the expression for z, divide the resulting equation by w z ( n+1) , and rearrange terms to obtain the functional-differential equation with two arguments

w z ( n+1) z n+1)

It is reduced to a three-term bilinear functional equation, which has two solutions (see Subsection S.4.4). Accordingly, we consider two cases.

1 ◦ . First, we set the expression in parentheses and the last fraction in (11) equal to constants. On rearranging terms, we obtain

( z−C 1 ) w ( z n+1) + nw ( z n) = 0,

C 2 w ( z n+1) −

f (w) = 0,

ϕ n ψ ′ y − ϕ n−1 ψϕ ′ y + C 1 ϕ n−1 ϕ ′ y − C 2 = 0, ϕ n ψ ′ y − ϕ n−1 ψϕ ′ y + C 1 ϕ n−1 ϕ ′ y − C 2 = 0,

w = A ln |z| + B n−1 z n−1 + ···+B 1 z+B 0 ,

f (w) = AC 2 n! (−1) n z − n−1 ,

dy

ψ(y) = C 2 ϕ(y)

[ ϕ(y)] n+1 + C 3 ϕ(y),

where

A, the B m , and C 3 are arbitrary constants and ϕ(y) is an arbitrary function.

The first two formulas in (12) give a parametric representation of f (w). In the special case of B n−1 = ···=B 0 = 0, on eliminating z, we arrive at the exponential dependence

f (w) = αe βw ,

α = AC 2 n! (−1) n ,

β = −(n + 1)/A.

By virtue of (12), the corresponding solution of equation (9) will have functional arbitrariness. 2 ◦ . In the second case, (11) splits into three ordinary differential equations:

ϕ n−1 ϕ ′ y = C 1 , ϕ n ψ ′ y − ϕ n−1 ψϕ ′ y = C 2 ,

( C 1 z+C 2 ) w ( n+1)

where C 1 and C 2 are arbitrary constants. The solutions of the first two equations are given by

★✂✩ Together with the last equation in (13), these formulas define a self-similar solution of the form (10). References for Subsection S.5.2-1: A. D. Polyanin (2002, Supplement B), A. D. Polyanin and A. I. Zhurov (2002), A. D. Polyanin and V. F. Zaitsev (2002).

S.5.2-2. Solution by reduction to equations with quadratic (or power) nonlinearities. In some cases, solutions of the form (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 S.4.2–S.4.4 are used for finding solutions of the resulting equation.

Sometimes, quadratically nonlinear equations can be obtained using the substitutions

w(z) = z λ (for equations with power nonlinearities), w(z) = λ ln z (for equations with exponential nonlinearities), w(z) = e λz

(for equations with logarithmic nonlinearities), where λ is a constant to be determined. This approach is equivalent to specifying the form of the

function

F (z) in (1) a priori. Galaktionov and Posashkov (1989, 1994) and Galaktionov (1995) describe a large number of nonlinear equations of different type that can be reduced with similar transformations to equations with quadratic nonlinearities.

Example 9. The nonlinear heat equation with a logarithmic source

∂w = ∂ 2 w

∂x 2 +