Methods of Generalized Separation of Variables

B.2. Methods of Generalized Separation of Variables

B.2.1. Structure of Generalized Separable Solutions

B.2.1-1. General form of solutions. The classes of nonlinear equations considered. To simplify the presentation, we confine ourselves to the case of mathematical physics equations

with two independent variables ✁ , ✂ and a dependent variable ✄ (one of the independent variables can play the role of time).

Linear separable equations of mathematical physics admit exact solutions in the form

( ✁ , ✂ )= ☎ 1 ( ✁ ) ✆ 1 ( ✂ )+ ☎ 2 ( ✁ ) ✆ 2 ( ✂ )+ ❱✮❱✮❱ + ☎ ❲ ( ✁ ) ✆ ❲ ( ✂ ), (1) where the ✄ ❳ = ☎ ❳ ( ✁ ) ✆ ❳ ( ✂ ) are particular solutions; the functions ☎ ❳ ( ✁ ), as well as the functions

( ✂ ), with different numbers ❨ are not related to one another. Also having exact solutions of the form (1) are many nonlinear partial differential equations with quadratic or power nonlinearities

1 ( ✁ ) ❩ 1 ( ✂ ) ❬ 1 [ ✄ ]+ ✌ 2 ( ✁ ) ❩ 2 ( ✂ ) ❬ 2 [ ✄ ]+ ❱✮❱✮❱ + ✌ ❭ ( ✁ ) ❩ ❭ ( ✂ ) ❬ ❭ [ ✄ ] = 0, (2) where the ❬ ❳ [ ✄ ] are differential forms that are the products of nonnegative integer powers of the

function ✄ and its partial derivatives ✟ ✠ ✄ , ✟ ☛ ✄ , ✟ ✠ ✠ ✄ , ✟ ✠ ☛ ✄ , ✟ ☛☞☛ ✄ , ✟ ✠ ✠ ✠ ✄ , etc. We will refer to solutions (1) of nonlinear equations (2) as generalized separable solutions. Unlike linear equations, in nonlinear equations the functions ☎ ❳ ( ✁ ) with different subscripts ❨ are usually related to one another [and to the functions ✆ ❪ ( ✂ )]. Subsections B.1.2 and B.1.3 give examples of exact solutions (1) to

nonlinear equations (2) for some simple cases with ❫ = 1 or ❫ = 2 (for ✆ 1 = ☎ 2 = 1).

If the ✌ ❢ ( ✁ ) and ❩ ❢ ( ✂ ) in (2) are all constant, then one can seek solutions in the more general form

where ❥ 1 , ❥ 2 , ❦ 1 , and ❦ 2 are constants. Some solutions of this sort are discussed in Subsections B.7.1 and B.8.1.

B.2.1-2. General form of functional differential equations. In general, on substituting expression (1) into the differential equation (2), one arrives at a functional

differential equation ❧

( ♠ ) ♥ ( ♦ )=0 (3) for the ☎ ❳ ( ✁ ) and ✆ ❳ ( ✂ ). The functionals ❧ ❧ ❪ ( ♠ ) and ♥ ❪ ( ♦ ) depend only on ✁ and ✂ , respectively,

( ♠ )≡ ❪ qr✁ , ☎ 1 , ☎ 1 s , ☎ sts 1 , ✉✮✉✮✉ , ☎ ❲ , ☎ s ❲ , ☎ sts ❲ ✈ ,

( ♦ )≡ ♥ ❪ qr✂ , ✆ 1 , ✆ 1 s , ✆ 1 sts , ✉✮✉✮✉ , ✆ ❲ , ✆ s ❲ , ✆ sts ❲ ✈ .

Here, for simplicity, the formulas are written out for the case of a second-order equation (2); for higher-order equations, the right-hand sides of relations (4) will contain higher-order derivatives of ☎ ❳ and ✆ ❪ .

B.2.2. Solution of Functional Differential Equations by Differentiation

B.2.2-1. Description of the method. ♣

0. We divide equation (3) by ❧ ♥ ❧ and differentiate with respect to ♣ ♣ ✂ . This results in a similar equation but with fewer terms: ② ❧ ② ❧ ② ②

1 ✇ . Assume that ♥ ❧

We continue the above procedure until we obtain a separable two-term equation ④ ④ ④ ④

(5) Three cases must be considered. ④ ④

0. Then equation (5) is equivalent to the ordinary differential equations ④ ④ ❧ ④

Nondegenerate case ④ :|

where ⑤ is an arbitrary constant. The equations 2 = 0 and ♥ 1 = 0 correspond to the limit case

Two degenerate cases ❧

1 ,2 ( ) are any;

1 ,2 ( ♠ ) are any.

2 ✇ . The solutions of the two-term equation (5) should be substituted into the original function- al differential equation (3) to “remove” redundant constants of integration [these arise because ♣ ♣

equation (5) is obtained from (3) by differentiation].

3 ✇ . The case ♥ ≡ 0 should be treated separately (since we divided the equation by ♥ at the first stage). Likewise, we have to study all other cases where the functionals by which the intermediate functional differential equations were divided vanish.

❴ ❵★❛ ❜ ❝P❞ ⑨ ❡

The functional differential equation (3) can happen to have no solutions.

❴ ❵★❛ ❜ ❝P❞ ⑩ ❡

At each subsequent stage, the number of terms in the functional differential equation ♣ ① ❧ ♣

can be reduced by differentiation with respect to either ❶ or ❷ . For example, we can assume at the first stage that

0. On dividing equation (3) by and differentiating with respect to ❷ , we again obtain a similar equation that has fewer terms.

B.2.2-2. Examples of constructing exact generalized separable solutions. Below we consider specific examples illustrating the application of the above method to constructing

exact generalized separable solutions of nonlinear equations.

Example 1. The two-dimensional stationary equations of motion of a viscous incompressible fluid are reduced to a single fourth-order nonlinear equation for the stream function (see equation 1 in Subsection B.7.1), specifically, ❸ ❹ ❸ ❸ ❹ ❸ ❸ ❹ ❸ ❹

= ❸ ❻ 2 + ❸ ❺ 2 . (6) We seek exact separable solutions of equation (6) in the form ❹ ❻ ❺

(7) Substituting (7) into (6) yields

(8) Differentiating (8) with respect to and , we obtain

(9) Nondegenerate case. If ❽ ➁✷➁ ❿✫❿ ➂

0, we separate the variables in (9) to obtain the ordinary differential equations

(11) which have different solutions depending on the value of the integration constant ➃ .

1 ➄ . Solutions of equations (10) and (11) for ➃ = 0:

where ➅ ➇ and ➆ ➇ are arbitrary constants ( ➈ = 1, 2, 3, 4). On substituting (12) into (8), we evaluate the integration constants. Three cases are possible:

4 = ➆ 4 = 0, ➅ ➉ , ➆ ➉ are any numbers ( ➊ = 1, 2, 3);

are any numbers

are any numbers

The first two sets of constants determine two simple solutions (7) of equation (6):

where ➃ 1 , ➋➌➋❊➋ , ➃ 5 are arbitrary constants.

2 ➄ . Solutions of equations (10) and (11) for ➃ = ➍ 2 > 0:

Substituting (13) into (8), dividing by ➍ 3 , and collecting terms, we obtain

+ ➆ 4 ( ❼ ➍ − ➅ 2 ) ➎ − ➏ = 0. Equating the coefficients of the exponentials to zero, we find

(The other constants are arbitrary.) These sets of constants determine three solutions (7) of equation (6):

where ➃ 1 , ➃ 2 , ➃ 3 , and ➍ are arbitrary constants.

3 ➄ . Solution of equations (10) and (11) for ➃ =− ➍ 2 < 0:

Substituting (14) into (8) does not yield new real solutions. Degenerate cases. If ❽ ➁✷➁

≡ 0 or ❾ ➀★➀ ❿✫❿ ≡ 0, equation (9) becomes an identity for any ❾ = ❾ ( ) or ❽ = ❽ ( ), respectively. These cases should be treated separately from the nondegenerate case. For example, if ❽ ➁✷➁

( ❻ )= ➅ + ➆ , where ➅ and ➆ are arbitrary numbers. Substituting this ❽

≡ 0, we have ❽

. Its general solution is given by ( )= 1 exp(−

into (8), we arrive at the equation −

+ ❺ 3 + ➃ 4 . Thus, we obtain another solution (7) of equation (6):

Example 2. Consider the second-order nonlinear parabolic equation

We look for exact separable solutions of equation (15) in the form

(16) Substituting (15) into (16) and collecting terms yields

On dividing this relation by ❻

2 and differentiating with respect to and , we obtain

Separating the variables, we arrive at the ordinary differential equations

(19) where ➞ is an arbitrary constant. The general solution of equation (18) is given by

)+ ➅ 2 cos( ➍ )+ ➅ 3 if ➞

where ➅ 1 , ➅ 2 , and ➅ 3 are arbitrary constants. Integrating (19) yields

where ➆ is an arbitrary constant. On substituting solutions (20) and (21) into (17), one can “remove” the redundant constants and define the functions ↔ and ↕ . Below we summarize the results.

1 ➄ . Solution for ➒ ≠− ➓ and ➒ ≠ −2 ➓ :

(corresponds to ➞ = 0), 2( + )

where ➃ 1 , ➃ 2 , and ➃ 3 are arbitrary constants. 2 ➄ . Solution for ➓ =− ➒ :

(corresponds to ➞ = ➍ 2 > 0), where the function ➑

= ↕ ( ) is determined from the autonomous ordinary differential equation

whose solution can be found in implicit form. In the special case ➑➧➦

1 = 0 or ➅ 2 = 0, we have ↕ = ➃ 1 exp ➥ 1 ➒✮➣❊➍ 2 2 2 + ➃ 2 . 3 ➄ . Solution for ➓ =− ➒ :

(corresponds to ➒✮➍ ➞ ↕ =− ➍ 2 < 0). where the function ➑

2 + ↕ [ ➅ 1 sin( ➍ )+ ➅ 2 cos( ➍ )]

= ↕ ( ) is determined from the autonomous ordinary differential equation

whose solution can be found in implicit form.

B.2.3. Solution of Functional Differential Equations by Splitting

B.2.3-1. Preliminary remarks. Description of the method. As one reduces the number of terms in the functional differential equation (3) by differentiation,

redundant constants of integration arise. These constants must be “removed” at the final stage. Furthermore, the resulting equation can be of a higher-order than the original equation. To avoid these difficulties, it is convenient to reduce the solution of the functional differential equation to the solution of a linear functional equation of a standard form and solution of a system of ordinary differential equations. Thus, the original problem splits into two simpler problems. Below we outline the basic stages of the splitting method.

The case of even number of terms in equation (3), ➨ =2 ➩ .

1 ➫ . At the first stage, we treat equation (3) as a purely functional equation that depends on two variables ➭ and ➯ , where ➲ 1 ( ➭ ), ➳✮➳✮➳ , ➲ ➵ ( ➭ ), ➸ ➵ +1 ( ➯ ), ➳✮➳✮➳ , ➸ 2 ➵ ( ➯ ) are unknown quantities and the functions ➲ ➵ +1 ( ➭ ), ➳✮➳✮➳ , ➲ 2 ➵ ( ➭ ), ➸ 1 ( ➯ ), ➳✮➳✮➳ , ➸ ➵ ( ➯ ) are assumed to be known. It can be shown (by induction and differentiation) that the functional equation (3) has a solution depending on ➩ 2 arbitrary constants

( ➽ = 1, ➳✮➳✮➳ , ➩ ), where the ➻ ➺➚➾ are arbitrary constants. Note that there are also “degenerate” solutions depending on

fewer arbitrary constants (see Item 2 ➫ in Paragraph B.2.3-2).

2 ➫ . At the second stage, we substitute the ➲ ➺ ( ➭ ) and ➸ ➾ ( ➯ ) of (4) into (22). This results in an overdetermined system of ordinary differential equations for the unknown functions ➪ ➶ ( ❷ ) and ➹ ➘ ( ➴ ).

The case of odd number of terms in equation (3), ➨ =2 ➩ − 1.

1 ➫ . If the number or terms is odd ( ➨ =2 ➩ − 1), the functional equation (3) has two different solutions with ➩ ( ➩ − 1) arbitrary constants. One of them can be obtained from formulas (22) by setting ➲ 2 ➵ ≡0 and discarding the last term with ➸ 2 ➵ . The other solution can be obtained from the first one by renaming ➲ ➺ ( ➭ ) ➷ ➸ ➺ ( ➯ ).

2 ➫ . Further analysis for each solution should be performed following the same scheme as in the case of even number of terms in (3).

B.2.3-2. Solutions of simple functional equations and their application. Below we give solutions of two simple functional equations of the form (3) that will be used

subsequently for solving specific nonlinear partial differential equations.

1 ➫ . The functional equation

(23) where the ➲ ➺ are all functions of the same argument and the ➸ ➺ are all functions of another argument,

has two solutions:

1 = ➬ 1 ➸ 3 , ➸ 2 = ➬ 2 ➸ 3 , ➲ 3 =− ➬ 1 ➲ 1 − ➬ 2 ➲ 2 , where ➬ 1 and ➬ 2 are arbitrary constants.

2 ➫ . The functional equation

(25) where the ➲ ➺ are all functions of the same argument and the ➸ ➺ are all functions of another argument,

has a solution

depending on four arbitrary constants ➬ ➮ [see solution (22) with ➩ = 2, ➻ 11 = ➬ 1 , ➻ 12 = ➬ 2 , ➻ 21 = ➬ 3 , and ➻ 22 = ➬ 4 ]. Equation (25) has also two “degenerate” solutions

1 = ➬ 1 ➲ 4 , ➲ 2 = ➬ 2 ➲ 4 , ➲ 3 = ➬ 3 ➲ 4 , ➸ 4 =− ➬ 1 ➸ 1 − ➬ 2 ➸ 2 − ➬ 3 ➸ 3 , (26b)

involving three arbitrary constants.

Example 3. Consider the nonlinear hyperbolic equation

( ) and ❮ ( ) are arbitrary functions. We look for generalized separable solutions with the form

(28) Substituting (28) into (27) and collecting terms yield

This equation can be represented as a functional equation (25) in which

4 = ❒ ❰ + ❮ − ❰ Ï✫Ï ➛✬➛ . On substituting (29) into (26a), we obtain the following overdetermined system of ordinary differential equations for the ❐ ➑ ➑

functions ↔ = ↔ ( ), ↕ = ↕ ( ), and ❰ = ❰ ( ):

=− 1 − 3 , ❒ ❰ + ❮ − ❰ Ï✫Ï ➛✬➛ =− ➅ 2 ➒☞↕ 2 − ➅ 4 ➒☞↕ ❰ . The first two equations in (30) are consistent only if

(31) where ➆ 0 , ➆ 1 , and ➆ 2 are arbitrary constants, and the solution is given by

(32) On substituting the expressions (31) into the last two equations in (30), we obtain the following system of equations for ➑

Relations (28), (32) and system (33) determine a generalized separable solution of equation (27). The first equation in (33) can be solved independently; it is linear if ➆

2 ➑ = 0 and is integrable in quadrature for ❒ ( ) = const. The second equation in (33) is linear in ❰ (for ↕ known).

Equation (27) does not have other solutions with the form (28) if ❒ and ❮ are arbitrary function and ↔ Ó 0, ↕ Ó 0, and

Ô Õ▼Ö ×PØ❘Ù Ú

It can be shown that equation (27) has a more general solution with the form

2 ( )= , (34) where the functions ➑

= ↕ Ü ( ) are determined by the ordinary differential equations

(The prime denotes the derivative with respect to ➑ .) The second equation in (35) has a particular solution ↕ 2 = ↕ 1 . Hence, its general solution can be represented as (see Polyanin and Zaitsev, 1995)

1 2 . The solution obtained in Example 3 corresponds to the special case Ý 2 = 0.

Example 4. Consider the nonlinear equation

(36) which arises in hydrodynamics (see equations B.6.2.1, Item 3 à

and B.7.2.1, Item 2 ä ).

We look for exact solutions of the form

Substituting (37) into (36) yields

This functional differential equation can be reduced to the functional equation (25) by setting

1 Ò = , 2 =− , 3 =( ➙ Ï Ð ) 2 − ➙✷➙ Ð✷Ð Ï✫Ï , 4 =− ➙ Ï✫Ï✫Ï Ð✷Ð✷Ð . On substituting these expressions into (26a), we obtain the system of equations

It can be shown that the last two equations in (38) are consistent only if the function ➙ and its derivative are linearly dependent,

The six constants æ 1 , æ 2 , å 1 , å 2 , å 3 , and å 4 must satisfy the three conditions

Integrating (39) yields

where æ 3 is an arbitrary constant. The first two equations in (38) lead to the following expressions for ↔ and ↕ :

is an arbitrary constant. Formulas (41), (42) and relations (40) allow us to find the following solutions of equation (36) with the form (37):

where Ý 1 , Ý 2 , Ý 3 , í , and ➍ are arbitrary constants (these can be expressed in terms of the å î and æ î ). The analysis of the second degenerate solution (26b) of the functional equation (25) leads to the following two more general solutions of the differential equation (36):

( à ) where ↔ ( à ) and ↕ ( à ) are arbitrary functions, and Ý 1 and ➍ are arbitrary constants.

B.2.4. Simplified Scheme for Constructing Exact Solutions of Equations with Quadratic Nonlinearities

B.2.4-1. Description of the simplified scheme. To construct exact solutions of equations (2) with quadratic or power nonlinearities that do not

depend explicitly on ï (all ð ➺ constant), it is reasonable to use the following simplified approach. As depend explicitly on ï (all ð ➺ constant), it is reasonable to use the following simplified approach. As

(43) Finite chains of these functions (in various combinations) can be used to search for separable

( ï )= ï ➺ , ➪ ➺ ( ï )= ñ ò ó➧ô , ➪ ➺ ( ï ) = sin( õ ➺➧ï ), ➪ ➺ ( ï ) = cos( ö ➺➧ï ).

solutions (1), where the quantities ÷ ➺ , õ ➺ , and ö ➺ are regarded as free parameters. The other system of functions { ➹ ➺ ( ➴ )} is determined by solving the nonlinear equations resulting from substituting (1) into the equation under consideration.

This simplified approach lacks the generality of the methods outlined in Subsections B.2.2 and B.2.3. However, specifying one of the systems of coordinate functions, { ➪ ➺ ( ï )}, simplifies the procedure of finding exact solutions substantially. The drawback of this approach is that some solutions of the form (1) can be overlooked. It is significant that the overwhelming majority of generalized separable solutions known to date, for partial differential equations with quadratic nonlinearities, are determined by coordinate functions (36) (usually with ø = 2).

B.2.4-2. Examples of constructing exact solutions of higher-order equations. Below we consider specific examples that illustrate the application of the above simplified scheme

to constructing generalized separable solutions of higher-order nonlinear equations.

Example 5. The equations of laminar boundary layer on a flat plate are reduced to a single third-order nonlinear equation for the stream function (see Schlichting 1981, Loitsyanskiy 1996): ➱

(44) We look for generalized separable solutions with the form

(45) which corresponds to the simplest set of functions û 1 ( )= , û 2 ( ) = 1 with ➊ = 2 in formula (1). On substituting (45) into

(44) and collecting terms, we obtain ❐ ù

(The prime denotes the derivative with respect to Û .) To meet this equation for any , one should equate both expressions in ù ù

square brackets to zero. This results in a system of ordinary differential equations for = ( Û ) and ú = ú ( Û ):

For example, this system has an exact solution