Third-Order Nonlinear Equations

B.6. Third-Order Nonlinear Equations

B.6.1. Stationary Hydrodynamic Boundary Layer Equations

The system of equations of stationary laminar boundary layer on a flat plate (Schlichting 1981, Loitsyanskiy 1996), ✗

can be reduced to this equation by introducing the stream function ✢ in accordance with the relations ✙ 1 = ✣✥✤ and ✙ 2 =− ✣✥✦ ✣✥✤ ✣✥✧

( ✚ and ✛ are the longitudinal and transverse coordinates, ✙

are the longitudinal and transverse components of the fluid velocity, and ✜

and ✙

is the kinematic fluid viscosity). Ý

1 ✁ . Suppose = ( Þ , ★ ) is a solution of the stationary hydrodynamic boundary layer equation. Then the function

1 = ✂ 1 ✂ 2 Þ + ✂ 3 , ✂ 1 ✂ 2 ★ + ú ( Þ ) ã + ✂ 4 , where ú ( Þ ) is an arbitrary function and ✂ 1 , ✂ 2 , ✂ 3 , and ✂ 4 are arbitrary constants, is also a solution

of the equation. ☎✝✆

Reference : Yu. N. Pavlovskii (1961), L. V. Ovsyannikov (1978).

2 ✁ . Exact solutions involving arbitrary functions:

+ ú ( ✬ ), where ✂ 1 , ✂ 2 , ✂ 3 , and ✂ 4 are arbitrary constants and ú ( ✬ ) is an arbitrary function. The second

( ✬ , ★ ) = −6 ✩ ✂ 1 ✬ 1 ✮ 3 tan ✯

solution is specified in Zwillinger (1998) and the fourth and fifth were obtained by Ignatovich (1993).

3 ✁ . Exact solution:

(1) where the functions ø = ø ( ★ ) and ù = ù ( ★ ) are determined by the system of ordinary differential

(3) The order of equation (2) can be reduced by two. Assume that a solution = ( ★ ø ø ) of equation (2) is

known. Then equation (3), which is linear in ù , has two linearly independent particular solutions

The second particular solution is apparent from comparing equations (2) and (3). The general solution of equation (2) can be represented in the form (see Zaitsev and Polyanin 1995):

It is not difficult to check that equation (2) has the particular solutions

where ✂ and ✵ are arbitrary constants. With reference to (1) and (4), one can see that the first solution in (5) leads to the third solution in Item 2 ✁ with ú ( ✬ ) = const. Substituting the second ☎✝✆ expression in (5) into (1) and (4) yields another solution.

Reference : A. D. Polyanin (2001b, 2001c).

The equation of laminar boundary layer with pressure gradient. ✗

1 ✭ . Suppose = ( ✬ , ★ ) is a solution of the equation in question. Then the functions (Pavlovskii 1961)

where ú ( ✬ ) is an arbitrary function and ✂ is an arbitrary constant, are also solutions of the equation.

2 ✁ . Nonviscous solutions (independent of the viscosity ✩ ):

where ú ( ✬ ) is an arbitrary function and ✂ 1 and ✂ 2 are arbitrary constants.

3 ✁ . Exact solution for ø ( ✬ )= ✺ ✬ + ✑ :

where the functions ✻ = ✻ ( ★ ) and ✓ = ✓ ( ★ ) are determined by the system of ordinary differential equations

(2) The order of the autonomous equation (1) can be reduced by one. If a particular solution ✻ ( ★ )

of equation (1) is known, the corresponding equation (2) can be reduced to a second-order linear equation by the change of variable ✼ ( ★ )= ✓ þ ✰ . If ✻ ( ★ )= ✶ ✽ ✺ ★ + ✂ , equation (2) can be integrated

in quadrature, because its two particular solutions are known if ✑ = 0, namely, ✓ 1 = 1 and ✓ 2 =

4 ✁ . Exact solution for ( ✬ )= ✺ ✳ ✾ ✿ ø :

ln | ( )|, where ú ☎✝✆ ( ✬ ) is an arbitrary function and ✵ is an arbitrary constant.

Reference : A. D. Polyanin (2001b, 2001c).

This equation can be used to model turbulent boundary layer. Exact solution:

where the functions ☛ ( ☞ ) and ✍ ( ☞ ) are determined by the system of ordinary differential equations

B.6.2. Nonstationary Hydrodynamic Boundary Layer Equations

This is the equation of nonstationary laminar boundary layer on a flat plate; ✡ ✌ and ☞ are the longitudinal and transverse coordinates and is the stream function (Schlichting 1981, Loitsyanskiy 1996).

. Suppose = ( ✌ , ☞ , ✚ ) is a solution of the equation is question. Then the function (see Vereshchagina 1973)

( ✌ , ✚ ) ✥ ✚ ✌ ✤ + ✍ ( ✚ ), where ✣

( ✌ , ✚ ) and ✍ ( ✚ ) are arbitrary functions, is also a solution of the equation.

2 ✙ . Exact solutions: ✡

( ✌ , ✚ ) is an arbitrary function of two arguments and ✦ 1 , ✦ 2 , ✦ 3 , and ✦ 4 are arbitrary constants.

3 ✙ . Exact solution:

(3) where the functions ✬ = ✬ ( ☞ , ✚ ) and ✭ = ✭ ( ☞ , ✚ ) are determined by the simpler equations with two

variables

TABLE B4 Exact solutions of equation (4)

Function ✬ = ✬ ( ☞ , ✚ ) No

Remarks (or determining equation) (or general form of solution)

( ✚ ) is an arbitrary function

( = ✚ + ( ) is an arbitrary function,

+ ✦ 1 ✦ 1 is any number

( ) is an arbitrary function

( ✚ ) is an arbitrary function,

4 = 1 exp − +

1 , ✱ are any numbers

1 exp[− ✱ ☞ + ✱ ✍ ( ✚ )] + 1

( ) is an arbitrary function,

2 ✦ 1 , ✦ 2 , ✱ are any numbers + ✦ 1 exp[− ✱ ☞

( ✚ ) is an arbitrary function,

1+ 2 exp(− ✲ )

1 , ✦ 2 , ✲ , ✱ are any numbers

Equation (4) is independent of (5). If a particular solution ✬ = ✬ ( ☞ , ✚ ) of equation (4) is known, then the corresponding equation (5) can be reduced by the change of variable ✷ = ✸ ✹ ✒ ✸ to the second-order linear equation

Exact solutions of equation (4) are listed in Table B4 . The ordinary differential equations in the last two rows are autonomous and, therefore, admit reduction of order. Table B5 presents solutions of equation (6) that correspond to the solutions of equation (4) specified in Table B4 . One can see that in the first three cases the solutions of equation (6) are expressed in terms of solutions to the classical heat equation with constant coefficients. There are other three cases where equation (6) is reduced to a separable equation.

4 ✙ . Exact solution:

where ☛ ( ✚ ) is an arbitrary function and ✦ 1 , ✦ 2 , ❀ , ❁ 1 , ❁ 2 , and ✱ are arbitrary parameters.

TABLE B5 Transformations of equation (6) for the corresponding exact solutions of equation (4) [the number in the first column corresponds to the number of the exact solution ✬ = ✬ ( ☞ , ✚ ) in Table B4 ]

No Transformations of equation (6) Resulting equation

5 ✙ . Exact solution: ✡

( ✌ , ☞ , ✚ )= ✺ ( ✚ ) exp( ❁ ✌ ☎ + ✱ ☞ )+ ✾ ( ✚ ) exp( ❁ ✌ + ✱ ☞ ✲ ✲ )+ ☛ ( ✚ ) ✌ + ❀ ☞ ,

( ✚ )= ✦ 1 exp ( ✧ ✱ 2 − ❀ ❁ ) ✚ + ✱

( ✚ )= ✦ 2 exp ( ✧ 2 ✱ ✲ 2 − ❀ ❁ ✲ ) ✚ + ✱

where ☛ ( ✚ ) is an arbitrary function and ✦ 1 , ✦ 2 , ❀ , ❁ , ✲ , and ✱ are arbitrary parameters.

6 ✙ . Exact solution: ✡

+ , where ☛ ( ✚ ) and ✍ ( ✚ ) are arbitrary functions, ❁ and ✱ are arbitrary parameters, and ❂ ( ❈ , ✚ ) is a function

satisfying the second-order linear parabolic equation

The transformation ✣

takes the last equation to the customary heat equation

. Exact solutions: ✡

= ✻ − ▼ ✿ ★ ✦ 1 sin( ✱ ❈ )+ ✦ 2 cos( ✱ ❈ ) ✩ + ✣

sin( ✱ ❈ −2 ✧ ✱ ✚ + ✦ 2 )+ ✣

Reference : A. D. Polyanin (2001b). 2 ✁

The equation of nonstationary laminar boundary layer with pressure gradient. ✡

1 ✙ . Suppose ( ✌ , ☞ , ✚ ) is a solution of the equation in question. Then the functions (see Vereshchag- ina 1973)

( ✌ , ✚ ) ✥ ✌ + ✍ ( ), where ✣

( ✌ , ✚ ) and ✍ ( ✚ ) are arbitrary functions, are also solutions of the equation.

2 ✙ . Nonviscous solution for any ✎ ( ✌ , ✚ ) (independent of the viscosity ✧ ):

where ☛ ( ✌ , ✚ ) and ✍ ( ✚ ) are arbitrary functions and ❀ is an arbitrary constant. Another nonviscous solution for any ✎ ( ✌ , ✚ ):

where ☛ ( ✌ , ✚ ) is an arbitrary function, and the function ✍ = ✍ ( ✌ , ✚ ) is determined by the first-order equation

Nonviscous solutions for ✣

3 ✙ . Exact solutions for ✎ ( ✌ , ✚ )= ✎ 1 ( ✚ ) ✌ + ✎ 2 ( ✚ ):

where the functions ✬ = ✬ ( ☞ , ✚ ) and ✭ = ✭ ( ☞ , ✚ ) are determined from the simpler equations with two variables

Equation (1) is independent of (2). If ✣

= ✬ ( ☞ , ✚ ) is a solution of equation (1), then the function

1 = ✬ ☞ + ✍ ( ✚ ), ✚❘✜ + ✍ ✰ ✏ ( ✚ ) with arbitrary ✍ ( ✚ ) is also a solution of equation (1). Table B6 presents

exact solutions of equation (1) for various ✎ 1 = ✎ 1 ( ✚ ).

The change of variable ✷ = ✸ ✹ ✒ ✸ brings equation (2) to the second-order linear equation

Let us dwell on the first solution of equation (1) in ✣ Table B6 :

where

TABLE B6

Exact solutions of equation (1) for various ✎ 1 ( ✚ ); ✍ ( ✚ ) is an arbitrary function Function

Function ✬ = ✬ ( ☞ , ✚ )

Determining equation

1 = ✎ 1 ( ✚ ) (or general form of solution) (or determining coefficients) Any

is any, ❚ >0 4

Exact solutions of the Riccati equation for ❀ = ❀ ( ✚ ) with various ✎ 1 ( ✚ ) can be found in Polyanin and Zaitsev (1995). The substitution ❀ = ✕ ✰✴❱ ✏ ✕ brings this equation to a second-order linear equation for

( ✚ ): ✕ ✰❲✰ ✏✑✏ − ✎ 1 ( ✚ ) ✕ = 0. In particular, if ✎ 1 ( ✚ ) = const, we have

( ✚ )= ❁ ✦ for ✎ =− ❁ 2 1 < 0,

1 cos( ❁ ✚ )− ✦ 2 sin( ❁ ✚ )

1 sin( ❁ ✚ )+ ✦ 2 cos( ❁ ✚ )

1 sinh( ❁ ✚ )+ ✦ 2 cosh( ❁ ✚ )

On substituting solution (4) with arbitrary ✎ 1 ( ✚ ) into equation (3), we obtain

The transformation ✣

( ✚ ) ✥ ✚ ✞ , takes (5) to the classical constant coefficient heat equation

( ✚ ) = exp

The ordinary differential equations in the last two rows in ✣ Table B6 (see the last column) are autonomous and, hence, can be reduced in order.

, ☞ , ✚ ) is a solution of the nonstationary hydrodynamic boundary layer

equation with ✎ ( ✌ , ✚ ✡ )= ✎ ✡ 1 ( ✚ ) ✌ + ✎ 2 ( ✚ ). Then the function

1 = ( ✌ + ✕ ( ✚ ), ☞ , ✚ )− ✕ ✏ ✰ ( ✚ ) ☞ , where ✕ ✏✑✏ ✰❲✰ + ✎ 1 ( ✚ ) ✕ = 0, is also a solution of this equation.

. Exact solution for ❙

where ❢ ( ✌ , ❣ ) is an arbitrary function of two arguments.

5 ❤ . Exact solutions for ✎ ( ✌ , ❣ )= ✐ ( ✌ ) ❥ ❦ ❧ , ❡ > 0:

exp q 1 2 ❣✴r s t ( ✉ ) sinh[ ♥ + ❢ ( ✉ , ❣ )] + ✈

exp q 1 2 ❣✴r s ❡ t ( ✉ ) cosh[ ♥ + ❢ ( ✉ , ❣ )] + ✈

where ❢ ( ✉ , ❣ ) is an arbitrary function of two arguments.

6 ❤ . Exact solution for ③ ( ✉ , ❣ )= ✐ ( ✉ ) ❥ − ❦ ❧ , ❡ < 0:

exp q − 1 2 ❣✴r s t ( ✉ ) sin[ ♥ + ❢ ( ✉ , ❣

exp q − 1 2 ❣✴r s ❡ t ( ✉ ) cos[ ♥ + ❢ ( ✉ , ❣ )] + ✈

where ❢ ( ✉ , ❣ ) is an arbitrary function of two arguments.

7 ❤ . Exact solution for ③ ( ✉ , ❣ )= ④ ❥ ❦ ⑤ − ⑥ ❧ :

( ✉ , ❣ ) is an arbitrary function of two arguments and is an arbitrary constant.

8 ❤ . Exact solution for ③ ( ✉ , ❣ )= ③ ( ❣ ):

+ , where ❢ ( ❣ ) and t ( ❣ ) are arbitrary functions, ❸ and are arbitrary parameters, and ( ❷ ❶ , ❣ ) is a function

satisfying the second-order linear parabolic equation

+ ★ ❸ ❢ ( ❣ )− t ( ❣ ✩ ✈ ❶

The transformation ✈

( ) brings it to the customary heat equation

. Exact solution for ③ ( ✉ , ❣ )= ③ ( ❣ ):

( ✉ , ❣ ) is an arbitrary function of two arguments and ② 1 , ② 2 , and are arbitrary constants.

10 ❤ . Exact solution for ③ ( ✉ , ❣ )= ③ ( ❣ ):

, ♥ , ❣ )= ❢ ( ✉ , ❣ ) ❥ ⑦ + t ( ✉ , ❣ ) ❥ − ⑦ + ❼ ( ✉ , ❣ )+ ④ ( ❣ ) ♥ ,

where is any number, ❢ ( ✉ , ❣ ) is an arbitrary function of two arguments, and the other functions are defined by

. Exact solutions for ③ ( ✉ , ❣ )= ③ ( ❣ ):

1 sin( ❷ )+ ② 2 cos( ❷ ) ✩ + ✈

where ❢ ( ✉ , ❣ ) is an arbitrary function of two arguments and ② 1 , ② 2 , and are arbitrary constants.

12 ❤ . Exact solutions for ③ ( ✉ , ❣ )= ➂ :

where ❢ ( ✉ , ❣ ) is an arbitrary function of two arguments and ② 1 , ➃P➄ ② 2 , and ❸ are arbitrary constants.

Reference : A. D. Polyanin (2001b). 2 ➆

This equation describes the flow of a non-Newtonian fluid in a two-dimensional nonstationary ➈ boundary layer with a pressure gradient. Here, ♠ is the stream function and the function ③ = ③ (

) depends on the rheological properties of the fluid. For power-law fluids, ③ = ❸ | ❶ | ➌ −1 ❶ .

1 ❤ . The assertion of Item 1 ❤ , equation 1 in Subsection B.6.2, remains valid for this equation.

2 ❤ . The equation admits the nonviscous solutions presented in Item 2 ❤ , equation 2 in Subsec- tion B.6.2, where ③ ( ✉ , ❣ ) must be replaced by ✐ ( ✉ , ❣ ).

3 ❤ . Exact solution for ✐ ( ✉ , ❣ )= ✐ ( ❣ ):

where ✷ = ✷ ( ♥ , ❣ ) is a function satisfying the second-order equation

The transformation

Equation (2) admits exact solutions with the forms [for any ✈

equation 1 2 ➒ ➎ − 1 2 ➏ ➎ ➒ ❹ =[ ③ ( ➎ ❹ )] ➒ ❶ ❹ , where ④ , ❸ , ② , and ➣ are arbitrary constants. The first two equations for ➎ = ➎ ( ➏ ) can be solved in

parametric form.

4 ↔ . Exact solution for ↕ ( ➙ , ➓ )= ↕ ( ➓ ):

= ➜ + ❸ ➙ , where ➞ ( ➓ ) and ➟ ( ➓ ) are arbitrary functions, ❸ is an arbitrary parameter, and ➍ ( ➝ , ➓ ) is a function

satisfying the second-order nonlinear parabolic equation

The transformation ⑨

leads to the simpler equation

For exact solutions of this equation, see Item 3 ➺ .

5 ➺ . Exact solution for ➻ ( ➼ , ➓ )= ➽ ( ➓ ) ➼ + ➾ ( ➓ ):

( ➼ , ➪ , ➓ )= ➶P➹ ( ➓ ) ➪ + ➘ ( ➓ ) ➴▲➼ + ➧ ➷ ( ➪ , ➓ ) ➨ ➪ ,

where ➘ ( ➓ ) is an arbitrary function, ➹ = ➹ ( ➓ ) is determined by the Riccati equation