Some Functional Equations and Their Solutions. Exact Solutions of Heat and Wave Equations

B.3.5. Some Functional Equations and Their Solutions. Exact Solutions of Heat and Wave Equations

In this subsection, we discuss several types of three-argument functional equations that arise most frequently in functional separation of variables in nonlinear equations of mathematical physics. The results are used to construct exact solutions for some classes of nonlinear heat and wave equations.

B.3.5-1. The functional equation ✥ ( ✂ )+ ✦ ( ☎ )= ✛ ( ✆ ), where ✆ = ☞ ( ✂ )+ ✏ ( ☎ ). Here, one of the two functions ✥ ( ✂ ) and ☞ ( ✂ ) is prescribed and the other is assumed unknown, also

one of the functions ✦ ( ☎ ) and ✏ ( ☎ ) is prescribed and the other is unknown, and the function ✛ ( ✆ ) is assumed unknown.*

Differentiating the equation with respect to ✂ and ☎ yields ✛ ✌✎✌ ✔✕✔ = 0. Consequently, the solution is given by

(46) where ✧ , ★ , and ✩ are arbitrary constants.

B.3.5-2. The functional equation ✥ ( ✪ )+ ✦ ( ✂ )+ ✫ ( ✂ ) ✛ ( ✆ )+ ✜ ( ✆ ) = 0, where ✆ = ☞ ( ✂ )+ ✏ ( ✪ ). Differentiating the equation with respect to ✂ yields the two-argument equation

* In similar equations with a composite argument, it is assumed that ✬ ( ✭ ) ✮ const and ✯ ( ✰ ) ✮ const.

Such equations were discussed in Section B.2. Hence, the following relations hold [see formulas (25) and (26a) in Section B.2]:

where ✧ 1 , ✧ 2 , ✧ 3 , and ✧ 4 are arbitrary constants. By integrating system (48) and substituting the resulting solutions into the original functional equation, one obtains the results given below. Case 1 . If ✧ 3 = 0 in (48), the corresponding solution of the functional equation is given by

where the ✟

and ★ are arbitrary constants and ☞ = ☞ ( ✂ ) and ✏ = ✏ ( ✪ ) are arbitrary functions. Case 2 . If ✧ 3 ≠ 0 in (48), the corresponding solution of the functional equation is

where the ✟

and ★ are arbitrary constants and ☞ = ☞ ( ✂ ) and ✏ = ✏ ( ✪ ) are arbitrary functions. Case 3 . In addition, the functional equation has the two degenerate solutions:

= ✧ 1 ✏ + ★ 1 , ✦ = ✧ 1 ☞ + ★ 2 , ✫ = ✧ 2 , ✜ =− ✧ 1 ✆ − ✧ 2 ✛ − ★ 1 − ★ 2 , (51a) where ☞ = ☞ ( ✂ ), ✏ = ✏ ( ✪ ), and ✛ = ✛ ( ✆ ) are arbitrary functions, ✧ 1 , ✧ 2 , ★ 1 , and ★ 2 are arbitrary

constants; and

= ✧ 1 ✏ + ★ 1 , ✦ = ✧ 1 ☞ + ✧ 2 ✫ + ★ 2 , ✛ =− ✧ 2 , ✜ =− ✧ 1 ✆ − ★ 1 − ★ 2 , (51b) where ☞ = ☞ ( ✂ ), ✏ = ✏ ( ✪ ), and ✫ = ✫ ( ✂ ) are arbitrary functions, ✧ 1 , ✧ 2 , ★ 1 , and ★ 2 are arbitrary

constants. The degenerate solutions (51a) and (51b) can be obtained directly from the original equation or its consequence (47) using formulas (26b) in Section B.2.

Example 11. Consider the nonstationary heat equation with a nonlinear source

We look for exact solutions of the form

Substituting (53) into (52) and dividing by ✼ ✽ yields the functional differential equation

We rewrite it as the functional equation B.3.5-2 in which

, ✸ ❅ ✸ ( ✸ ✭ )=( ✬ ❀ ) 2 , ❆ ( ✻ )= ✽❂✽✒❇ ✽ , ❈ ( ✻ )= ( ( ✻ )) ❇ ✽ . (54) We now use the solutions of equation B.3.5-2. On substituting the expressions of ❄ and ❅ of (54) into (49)–(51), we

arrive at overdetermined systems of equations for ✬ = ✬ ( ✭ ). Case 1 . The system

following from (49) and corresponding to ❉ 3 = 0 in (48) is consistent in the cases

= ❋ 1 ✭ + ❋ 2 for ❉ 2 =− ❉ 1 ❋ 1 2 , ❉ 4 = ❊ 2 = 0, ❊ 1 = ❋ 1 2 ,

+ ❋ 1 ✭ + ❋ 2 for ❉ 1 = ❉ 2 = 0, ❊ 1 = ❋ 2 1 − ❉ 4 ❋ 2 , ❊ 2 = 1 2 ❉ 4 , where ❋ 1 and ❋ 2 are arbitrary constants.

The first solution in (55) with ❉ 1 ≠ 0 leads to a right-hand side of equation (52) containing the inverse of the error function [the form of the right-hand side is identified from the last two relations in (49) and (54)]. The second solution in (55) ✸ ✸ ✸ ✸

+ ● 2 in (52). In both cases, the first relation in (49) is, taking into account that =− ✯ ✾ , a first-order linear solution with constant coefficients, whose solution is an exponential plus a constant. Case 2 . The system

corresponds to the right-hand side ❃ ✼ ✺ ( )= ● 1 ln

3 , following from (50) and corresponding to ❉ 3 ≠ 0 in (48) is consistent in the following cases:

ln ◗ ◗ sinh ❘ 1 2 ❙ − ❉ 3 ❉ 4 ✭ + ❋ 1 ❚ ◗ ◗ + ❋ 2 for

=− 2 ln ◗ ◗ cosh ❘ 1 2 ❙ − ❉ 3 ❉ 4 ✭ + ❋ 1 ❚ ◗ + ❋ 2 for

1 and ❋ 2 are arbitrary constants. The right-hand sides of equation (52) corresponding to these solutions are represented in parametric form. Case 3 . Traveling wave solutions of the nonlinear heat equation (52) and solutions of the linear equation (52) with

= const correspond to the degenerate solutions of the functional equation (51). Example 12. Likewise, one can analyze the more general equation

(56) It arises in convective heat/mass exchange problems ( ✼ ❱ = const and ❲ = const), problems of heat transfer in inhomogeneous

media ( ❲ = ❱ ❀ ≠ const), and spatial heat transfer problems with axial or central symmetry ( ❱ = const and ❲ = const ❇ ✭ ).

Searching for exact solutions of equation (56) in the form (53) leads to the functional equation B.3.5-2 in which

. Substituting these expressions into (49)–(51) yields a system of ordinary differential equations for the unknowns.

Example 13. Equation (52) also admits more complicated functional separable solutions with the form ✸ ✸ ✹ ✹

Substituting these expressions into equation (52) yields the functional equation B.3.5-2 again, in which ( ✭ must be replaced by ❳ )

)=− ✸ ( ❳ ✸ ✼ ✯ ❃ ✸ ( ✼ , )= ✬ − ❱ ✬ , ❅ ( ❳ )=( ✬ ❨ ) 2 , ❆ ( ✻ )= ✽❂✽ ❇ ✽ , ❈ ( ✻ )= ( ( ✻ )) ❇ ✽ . Further, one should follow the same procedure of constructing the solution as in Example 11.

In Examples 11–13, different equations were all reduced to the same functional equation. This demonstrates the utility of isolation and independent analysis of individual types of functional equations, as well as the expedience of developing methods for solving functional equations with a composite argument.

B.3.5-3. The functional equation ✥ ( ✪ )+ ✦ ( ✂ ) ✛ ( ✆ )+ ✫ ( ✂ ) ✜ ( ✆ ) = 0, where ✆ = ☞ ( ✂ )+ ✏ ( ✪ ). Differentiating with respect to ✂ yields the two-argument functional differential equation

(57) which coincides with equation (25) in Section B.2, up to notation.

Nondegenerate case. Equation (57) can be solved using formulas (49) in Section B.2, just as was the case for equation (25). In this way, we arrive at the system of ordinary differential equations

where ✧ 1 , ✧ 2 , ✧ 3 , and ✧ 4 are arbitrary constants.

The solution of equation (58) is given by ✟

( ✆ )=( ✣ 1 − ✧ 1 ) ★ ✱ − 1 +( ✣ 2 − ✧ 1 ) ★ 4 ✱ − 3 2 , where ★ 1 , ★ 2 , ★ 3 , and ★ 4 are arbitrary constants and ✣ 1 and ✣ 2 are roots of the quadratic equation ( ✣ − ✧ 1 )( ✣ − ✧ 4 )− ✧ 2 ✟ ✧ ✟ 3 = 0.

(60) In the degenerate case ✟ ✣ 1 = ✣ the terms ✱ 2 2 ✶ and ✱ − 2 in (59) must be replaced by ☞ ✱ 1 ✔ ✶ and

− 1 , respectively. In the case of purely imaginary or complex roots, one should extract the real (or imaginary) part of the roots in solution (59).

On substituting (59) into the original functional equation, one obtains conditions that must be met by the free coefficients and identifies the function ✥ ( ✪ ), specifically, ✟

. Solution (59), (61) involves arbitrary functions ☞ = ☞ ( ✂ ) and ✏ = ✏ ( ✪ ).

Degenerate case. In addition, the functional equation has the two degenerate solutions:

, ✫ = ★ ✱ − 1 1 ✶ , ✜ =− ★ ✱ 1 2 − ✧ 2 ✛ , where ☞ = ☞ ( ✂ ), ✏ = ✏ ( ✪ ), and ✛ = ✛ ( ✆ ) are arbitrary functions, ✧ 1 , ✧ 2 , ★ 1 , and ★ 2 are arbitrary

constants; and

1 = ✔ 1 2 , =− 1 − 2 , = 2 ★ 2 ✱ , ✜ = ★ 2 ✱ , where ☞ = ☞ ( ✂ ), ✏ = ✏ ( ✪ ), and ✦ = ✦ ( ✂ ) are arbitrary functions, ✧ 1 , ✧ 2 , ★ 1 , and ★ 2 are arbitrary

constants. The degenerate solutions can be obtained immediately from the original equation or its consequence (57) using formulas (26b) in Section B.2.

Example 14. For the first-order nonlinear equation

the search for exact solutions in the form (53) leads to the functional equation B.3.5-3 in which

B.3.5-4. Equation ✥ 1 ( ✂ )+ ✥ 2 ( ☎ )+ ✦ 1 ( ✂ ) ❡ ( ✆ )+ ✦ 2 ( ☎ ) ✛ ( ✆ )+ ✜ ( ✆ ) = 0, ✆ = ☞ ( ✂ )+ ✏ ( ☎ ). Differentiating with respect to ☎ and dividing the resulting relation by ✏ ✌ ✑ ❡ ✔ ✌ and differentiating with

respect to ☎ , one arrives at the functional equation with two arguments ☎ and ✆ that is discussed in Section B.2 [see equation (3) and its solution (22)].

Example 15. Consider the following equation of steady-state heat transfer in an anisotropic inhomogeneous medium with a nonlinear source:

The search for exact solutions in the form ✸ = ( ✻ ), ✻ = ✬ ( ✭ )+ ✯ ( ✰ ), leads to the functional equation B.3.5-4 in which

Here we confine ourselves to studying functional separable solutions existing for arbitrary right-hand side ✸

With the change of variable ✻ = ❥ 2 , we look for solutions of equation (62) in the form

Taking into account that ❦ ❀ ❧

= 2 ❧ and ❦

= 2 ❧ , we find from (62)

For this functional differential equation to be solvable we require that the expressions in square brackets be functions of ❥ :

) ❀ +( ❲❩✯ ❤ ) ❤ = t ( ❥ ), ❱ ( ✬ ❀ ) 2 + ❲ ( ✯ ❤ ) 2 = ✉ ( ❥ ).

Differentiating the first relation with respect to ✼

and ✰ yields the equation ( t

) ❧ = 0, whose general solution is

( ❥ )= ❋ 1 ❥ 2 + ❋ 2 . Likewise, we find ✉ ( ❥ )= ❋ 3 ❥ 2 + ❋ 4 . Here, ❋ 1 , ❋ 2 , ❋ 3 , and ❋ 4 are arbitrary constants. As a result, we have

1 ( ✬ + ✯ )+ ❋ 2 , ❱ ( ✬ ❀ ) 2 + ❲ ( ✯ ❤ ) 2 = ❋ 3 ( ✬ + ✯ )+ ❋ 4 . The separation of variables results in a system of ordinary differential equations for ✬ ( ✭ ), ❱ ( ✭ ), ✯ ( ✰ ), and ❲ ( ✰ ):

This system is always integrable in quadrature and can be rewritten as

(65) ) −2 . Here, the equations for ✬ and ✯ do not involve ❱ and ❲ and, hence, can be solved independently. Without full analysis of

system (65), we note a special case where the system can be solved in explicit form.

For ❋ 1 = ❋ 2 = ❋ 4 = ● 1 = ● 2 = 0 and ❋ 3 = ❋ ≠ 0, we find

where ✈ , ② , ⑤ , and ⑥ are arbitrary constants. Substituting these expressions into (64) and taking into account (63), we obtain ✸

the ordinary differential equation for ( ❥ )

System (65) has other solutions as well; these lead to various expressions of ❱ ( ✭ ) and ❲ ( ✰ ). Table B3 lists the cases where these functions can be written in explicit form (the traveling wave solution, which corresponds to ❱ = const and ❲ = const, is omitted). In general, the solution of system (64) enables one to represent ❱ ( ✭ ) and ❲ ( ✰ ) in parametric form.

Reference : V. F. Zaitsev and A. D. Polyanin (1996), A. D. Polyanin and A. I. Zhurov (1998).

TABLE B3 ✸ ✸

Functional separable solutions of the form

( ❥ ), ❥ 2 = ⑩ ( ❶ )+ ❷ ( ❸ ), for heat

equations in an anisotropic inhomogeneous medium with an arbitrary nonlinear source. Notation: ❋ , ✈ , ② , ⑤ , ⑥ , ❹ , and ● are free parameters ( ❋ ≠ 0, ⑤ ≠ 0, ⑥ ≠ 0, ❹ ≠ 2, and ● ≠ 2)

Heat equation ✸ Functions

( ❶ ) and ❷ ( ❸ )

Equation for = ( ❥ )

Equation (64); both expressions in square brackets are constant

= ⑤ ❶ , ❷ = ⑥ ln | ❸ |

Equation (64); both expressions

in square brackets are constant