Differentiation Method

B.3.3. Differentiation Method

B.3.3-1. Basic ideas of the method. Reduction to a standard equation. In general, the substitution of expression (1) into the nonlinear partial differential equation under

study leads to a functional differential equation with three arguments—two arguments are usual,

and ✎ , and the third is composite, ❊ . In many cases, the resulting equation can be reduced by differentiation to a standard functional differential equation with two arguments (either ▲ or ✎ is

eliminated). Two solve the two-argument equation, one can use the methods outlined in Section B.2.

B.3.3-2. 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 7. Consider the nonlinear heat equation

(12) We look for exact solutions with the form

On substituting (13) into (12) and dividing by ✥ ✦ , we obtain the functional differential equation

(15) Differentiating (14) with respect to ✢ yields

(16) This functional differential equation with two variables can be treated as the functional equation (23) of Section B.2. This

three-term functional equation has two different solutions. Accordingly, we consider two cases. Case 1. The solutions of the functional differential equation (16) are determined from the system of ordinary differential

where ✬ 1 and ✬ 2 are arbitrary constants. The first two equations (17) are linear and independent of the third equation. Their general solution is given by

1 [ ❯ 1 sin( ❈ ✤ )+ ❯ 2 cos( ❈ ✤ )] if ✬ 1 <2 ✬ 2 ,

Substituting P

of (18) into (15) yields a differential equation for ✛ = ( ✤ ). On integrating this equation, we obtain

(19) where ✛ ✯ 1 and ✯ 2 are arbitrary constants. The expression of ❏ in (18) together with expression (19) define the function

= ❏ ( ) in parametric form. Without full analysis, we will study the case ✬ 2 =0( ❈ = ✬ 1 ) and ✬ 1 ≠ 0 in more detail. It follows from (18) and (19) that

−2 ✱ 1 ) −1 ✳ 2 + ✯ 2 ( ✯ 1 = ✬ 1 ❯ 2 ✯ 3 ). (20) Eliminating ✤ yields

+ ✬ 1 ( ✷ ◆ ) 2 = const. The corresponding general solution is given by

The last equation in (17) with ✥ ✬ 2 = 0 has the first integral

where ❲ 1 , ❲ 2 , and ❲ 3 are constants of integration. In all three cases, the following relations hold:

(23) We substitute (20) and (23) into the original functional differential equation (14). With reference to the expression of ✜ ✤

in (13), we obtain the following equation for ❂ = ❂ ( ):

Its general solution is given by

where ❲ 4 is an arbitrary constant. ✛

Formulas (13), (20) for , (22), and (24) define three solutions of the nonlinear equation (12) with ❏ ✛ ( ) of the form (21)

[recall that these solutions correspond to the special case ✬ 2 = 0 in (18) and (19)].

Case 2. The solutions of the functional differential equation (16) are determined from the system of ordinary differential equations

The first two equations in (25) are consistent in the two cases

The first solution in (26) eventually leads to the traveling wave solution ✜

= ( ❯ 1 + ❯ 2 ) of equation (12) and the second solution to the self-similar solution of the form

( ✛ ). In both cases, the function ❏ ( ) in (12) is arbitrary. A more detailed analysis of functional separable solutions (13) of equation (12) can be found in the reference cited

below. ❴❛❵

Reference : P. W. Doyle and P. J. Vassiliou (1998). Example 8. One can look for more complicated functional separable solutions of equation (12) with the form

We substitute this into (12), divide the resulting functional differential equation by ✢ ✦ , and differentiate with respect to to obtain

[ ❏ ✦ ( )+2 P ( ✤ )] + ( ✷ ❝ ) 3 P ✥ ✦ = 0, where the function P = P ( ✤ ) is defined by (15). This functional differential equation with two variables ❜ and ✤ can be

treated as the functional equation (25) of Section B.2. The solution of (25) is given by relations (26), thus representing a system of ordinary differential equations for ❏ , P , and ✷ .

Example 9. Consider the nonlinear Klein–Gordon equation

(27) We look for functional separable solutions in additive form:

(28) Substituting (28) into (27) yields

On differentiating (29) first with respect to ✥ and then with respect to and on dividing by ❂

, we have

◆❖◆ Eliminating ✥★✥ − ✷ from this equation with the aid of (29), we obtain

(31) This relation holds in the following cases:

−2 ❑ ✦ ❣ =( ✬ ✤ + ✯ )( ❑ ✦✩✦ −2 ❑✼❑ ✦ ) (case 2 ❤ ), where ✬ , ❯ , and ✯ are arbitrary constants. We consider both cases.

Case 1 . The first two equations in (32) enable one to determine ❑ 2 ( ✤ ) and ❣ ( ✤ ). Integrating the first equation once yields

= ❑ + const. Further, the following cases are possible:

(33 ✰ ) where ✯ 1 and ❈ are arbitrary constants.

= ❈ tan( ❈ ✤ + ✯ 1 ),

The second equation in (32) has the particular solution ❣ = ❑ ( ✤ ). Hence, its general solution in expressed by (e.g., see Polyanin and Zaitsev 1995)

where ✯ 2 and ✯ 3 are arbitrary constants. ✛ ✛

The functions ( ✤ ) and ✣ ( ) are found from (30) as

( ✤ ) ✸✼✤ ✹ , (35) and ❯ 1 and ❯ 2 are arbitrary constants ( ✣ is defined parametrically).

( )= ❯ 1 ❣ ( ✤ ) ✐ ( ✤ ), where ✐ ( ✤ ) = exp ✴ ✶

Let us dwell on the case (33b). According to (34),

where ✬ 1 =− ✯ 3 ❫ 3 and ✬ 2 =− ✯ 2 are any numbers. Substituting (33b) and (36) into (35) yields

1 ln | + 1 |+ 2 ,

Eliminating ✤ , we arrive at the explicit form of the right-hand side of equation (27):

(37) For simplicity, we set ✯ 1 = 0, ❯ 1 = 1, and ❯ 2 = 0 and denote ✬ 1 = ✽ and ✬ 2 = ✾ . Thus, we have

( ✤ ) = −1 ❫ ✤ , ❣ ( ✤ )= ✽❧✤ 2 + ✾ ❫ ✤ . (38) * In case 2, equation (31) can be represented as the functional equation considered in Paragraph B.3.5-1.

) = ln | ✤ |, ✣ ( )= ✽ ✰ ❅ + ✾ ✰ −2 ❅ ,

TABLE B1 ♥

Nonlinear Klein–Gordon equations ♥

= ✣ ( ) admitting functional separable solutions of the form = ( ✤ ),

= ✷ ( )+ ❂ ( ♣ ). Notation: ✬ , ✯ 1 , and ✯ 2 are arbitrary constants; q = 1 for ✤ > 0 and q = −1 for ✤ <0

No ♥ Right-hand side

Solution ( ✤ )

Equations for ❂ ( ♣ ) and ✷ ( r )

+ , ln tanh

( ❂ ) 2 = ✯ 1 ✰ 2 ❩ + ✯ 2 ✰ −2 ❩ − q ✾❞❂ + ✽

4 ✬ sinh + sinh + 2 sinh 2 ln

5 ✽ sinh +2 ✾ ✮ sinh arctan ✰

It remains to determine ❂ ( ♣ ) and ✷ ( r ). We substitute (38) into the functional differential equation (29). Taking into account (28), we find ✥★✥ ✧✵✧

−3 ✽✼❂ 2 ) ✷ − ❂ ( ✷ ◆❖◆ +3 ✽ ✷ 2 ) = 0. (39) Differentiating (39) with respect to ♣ and ✥★✥★✥ ✧✵✧✵✧ r yields the separable equation* ✧ ✥ ✥ ✥★✥★✥ ✥

whose solution is determined by the ordinary differential equations ✧✵✧✵✧ ✥★✥★✥ ✥ ✧

where ✬ is the separation constant. Each equation can be integrated twice, thus resulting in

where ✯ 1 , ✯ 2 , ✯ 3 , and ✯ 4 are arbitrary constants. Eliminating the derivatives from (39) with the aid of (40), we find that the arbitrary constants are related by ✯ 3 =− ✯ 1 and ✯ 4 = ✯ 2 + ✾ . So, the functions ❂ ( ♣ ) and ✷ ( r ) are determined by the first-order nonlinear autonomous equations

The solutions of these equations are expressed in terms of elliptic functions. For the other cases in (33), the analysis is performed in a similar way. Table B1 presents the final results for the cases (33a)–(33e).

Case 2 . Integrating the third and fourth equations in (32) yields

1 and ❲ 2 are arbitrary constants. In both cases, the function ♥ ♥ ✣ ( ) in equation (27) is arbitrary. The first row in (41) ♥ ♥

❴❛❵ corresponds to the traveling wave solution = ( ❈ r + t ♣ ). The second row leads to a solution of the form = ( r 2 − ♣ 2 ). References : A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V. K. Andreev,

O. V. Kaptsov, V. V. Pukhnachev, and A. A. Rodionov (1994). Example 10. The nonlinear stationary heat (diffusion) equation

is analyzed just as the nonlinear Klein–Gordon equation considered in Example 9. The final results are listed in ♥ ♥ ♥ ♥ ♥ Table B2 ; the traveling wave solutions ❴❛❵

= ( ❈ r + t ♣ ) and solutions of the form = ( r 2 + ✉ 2 ), existing for any ✈ ( ), are omitted. References : A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V. K. Andreev,

O. V. Kaptsov, V. V. Pukhnachev, and A. A. Rodionov (1994). * To solve equation (39), one can use the solution of equation (25) in Section B.2 [see (26a)].

TABLE B2

Nonlinear equations ♥ +

= ✈ ( ) admitting functional separable solutions of the form = ( ① ),

= ✷ ( r )+ ❂ ( ✉ ). Notation: ② , ✯ 1 , and ✯ 2 are arbitrary constants; q = 1 for ①

> 0, q = −1 for ① <0

No ♥ Right-hand side

Solution ( ① )

Equations for ✷ ( r ) and ❂ ( ✉ )

4 sinh + sinh ln tanh

+ 2 sinh

2 ln coth

5 ✽ sinh +2 ✾ ✮ sinh arctan ✰