Special Functional Separable Solutions

B.3.2. Special Functional Separable Solutions

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.

B.3.2-1. Solutions of the form (1) with ÿ linear in one of the independent variables. Consider functional separable solutions of the form (1) in the special case where the composite

argument ÿ is linear in one of the independent variables (e.g., in ï ). We substitute (1) into the equation under study and eliminate ï using the expression of ÿ to obtain a functional differential equation with two arguments. In many cases, this equation can be solved by the methods outlined in Section B.2.

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

(5) We look for functional separable solutions of the special form

The functions ✛ (

), û ( ), ( ), and ✣ ( ) are to be determined. ✛ ✥

On substituting (6) into (5) and on dividing by ✦ , we have

We express ✜ from (6) in terms of

and substitute into (7) to obtain a functional differential equation with two variables and ✤ ,

which can be treated as the functional equation (25) in Section B.2 where

Substituting these expressions into relations (26a) of Section B.2 yields the system of ordinary differential equations

where ✬ 1 , ✬ 2 , ✬ 3 , and ✬ 4 are arbitrary constants. The solution of system (8) is given by

1 , ✯ 2 , ✯ 3 , and ✯ 4 are arbitrary constants. The dependence ✣ = ✣ ( ) is defined by the last two relations in parametric form ( ✤ is considered the parameter). In the special case ✬ 3 = ✯ 4 = 0, ✬ 1 = −1, and ✯ 3 = 1, the source function can be represented in explicit form as

If ✬ 3 ≠ 0 in (9), the source function is expressed in terms of elementary functions and the inverse of the error function. Example 2. Consider the more general equation

We look for solutions in the form (6). In this case, only the first two equations in system (8) will change, and the functions ✛

( ✤ ) and ✣ ( ) will be given by (9). Example 3. The nonlinear heat equation

has also solutions of the form (6). The unknown quantities are governed by system (8) in which ✥★✥

must be replaced by [ ❀ ( ✛ ) ✦ ✥ ✥

] ✛ ✦ ✜ . The functions

( ) and ❂ ( ) are determined by the first two formulas in (9). One of the two functions ✛ ❀ ( ) and

( ✛ ) can be assumed arbitrary and the other is identified in the course of the solution. The special case ✛ ✣

( ) = const yields

Example 4. Likewise, we can treat the ❆ th-order nonlinear equation

and ✦✩✦ in (8) must be replaced by ✷

As before, we look for solutions in the form (6). In this case, the quantities ✷ 2 ✛ ✥★✥

and ( ✦ ) , respectively. In particular, for ✬ 3 = 0, apart from equations with logarithmic nonlinearities of the form (10), we obtain other equations.

Example 5. For the ❆ th-order nonlinear equation

the search for exact solutions of the form (6) leads to the following system of equations for ✛

where ✬ 1 , ✬ 2 , ✬ 3 , and ✬ 4 are arbitrary constants. In the case ❆ = 3, we assume ✬

3 ✛ = 0 and ✬ 1 > 0 to find in particular that ✣ ( )=− ✬ 2 − ✬ 4 arcsin( ❈ ). Example 6. In addition, searching for solutions of equation (5) with ✢

quadratically dependent on ,

(11) also makes sense here. Indeed, on substituting (11) into (5), we arrive at an equation that contains terms with ✢ 2

and does not contain terms linear in . Eliminating 2 from the resulting equation with the aid of (11), we obtain

To solve this functional differential equation with two arguments, we apply the splitting method outlined in Subsection B.2.3. It can be shown that, for equations (5), this equation has a solution with a logarithmic nonlinearity of the form (10).

B.3.2-2. Solution by reduction to equations with quadratic 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 Section B.2 are used to find solutions of the resulting equation.

Sometimes, quadratically nonlinear equations can be obtained using the substitutions

( ❊ )= ❊ ❋ (for equations with power nonlinearities),

( ❊ )= ● ln ❊ (for equations with exponential nonlinearities),

( ❊ )= ❍ ❋ ✑ (for equations with logarithmic nonlinearities), where ● is a constant to be determined. This approach is equivalent to specifying the form of the

function ■ ( ❊ ) in (1) a priori.

Example 6. The nonlinear heat equation with a logarithmic source

= ✽ ✚ ✢ 2 + ❏ ( ) ln

can be reduced by the change of variable ✦ =

to the quadratically nonlinear equation

which admits separable solutions with the form

where ✜

1 ( )= , ✷ 2 ( )= , and the functions ❂ ❃ ( ) are determined by an appropriate system of ordinary differential equations.