Simplified Scheme for Constructing Generalized Separable

S.4.5. Simplified Scheme for Constructing Generalized Separable

Solutions

S.4.5-1. Description of the simplified scheme. To construct exact solutions of equations (20) with quadratic or power nonlinearities that do not

depend explicitly on x (all f i constant), it is reasonable to use the following simplified approach. As before, we seek solutions in the form of finite sums (19). We assume that the system of coordinate functions { ϕ i ( x)} is governed by linear differential equations with constant coefficients. The most common solutions of such equations are of the forms

ϕ i ( x) = x i , ϕ i ( x) = e λ i x , ϕ i ( x) = sin(α i x), ϕ i ( x) = cos(β i x). (71) Finite chains of these functions (in various combinations) can be used to search for separable

solutions (19), where the quantities λ i , α i , and β i are regarded as free parameters. The other system of functions { ψ i ( y)} is determined by solving the nonlinear equations resulting from substituting (19) into the equation under consideration.

This simplified approach lacks the generality of the methods outlined in Subsections S.4.2–S.4.4. However, specifying one of the systems of coordinate functions, { ϕ i ( x)}, simplifies the procedure of finding exact solutions substantially. The drawback of this approach is that some solutions of the form (19) 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 (71) (usually with n = 2).

S.4.5-2. 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 12. The equations of a laminar boundary layer on a flat plate are reduced to a single third-order nonlinear equation for the stream function (see Schlichting, 1981, and Loitsyanskiy, 1996):

∂y ∂x∂y

∂y 3 .

We look for generalized separable solutions with the form

w(x, y) = xψ(y) + θ(y),

which corresponds to the simplest set of functions ϕ 1 ( x) = x, ϕ 2 ( x) = 1 with n = 2 in formula (19). On substituting (73) into (72) and collecting terms, we obtain

x[(ψ ′ ) 2 − ψψ ′′ − νψ ′′′ ]+[ ψ ′ θ ′ − ψθ ′′ − νθ ′′′ ] = 0.

(The prime denotes a derivative with respect to y.) To meet this equation for any x, one should equate both expressions in square brackets to zero. This results in a system of ordinary differential equations for ψ = ψ(y) and θ = θ(y):

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

Other generalized separable solutions of equation (72) can be found in Subsection 9.3.1; see also Example 7 with n=3 and

f (x) = ν. Example 13. Consider the nth-order nonlinear equation

∂y ∂x∂y

∂x ∂y 2 ∂y n

where f (x) is an arbitrary function. In the special case n = 3 with f (x) = ν = const, this equation coincides with the boundary layer equation (72). We look for generalized separable solutions of the form

w(x, y) = ϕ(x)e λy + θ(x),

which correspond to the set of functions ψ 1 ( y) = e λy , ψ 2 ( y) = 1 in (19). On substituting (75) into (74) and rearranging terms, we obtain

λ 2 e λy ϕ[θ ′ x + λ n−2 f (x)] = 0.

This equation is met if

θ(x) = −λ n−2

f (x) dx + C, ϕ(x) is any,

where C is an arbitrary constant. (The other case, ϕ = 0 and θ is any, is of little interest.) Formulas (75) and (76) define an exact solution of equation (74),

w(x, y) = ϕ(x)e λy − λ n−2

f (x) dx + C,

which involves an arbitrary function ϕ(x) and two arbitrary constants C and λ. Note that solution (77) with n = 3 and f (x) = const was obtained by Ignatovich (1993) by a more complicated approach.

Example 14. Consider the nth-order nonlinear equation

where f (t) is an arbitrary function. In the special case n = 3 and f (t) = const, it coincides with equation (63). We look for exact solutions of the form

w = ϕ(t)e λx + ψ(t).

On substituting (79) into (78), we have

ϕ t ′ − λϕψ = λ n−1 f (t)ϕ.

We now solve this equation for ψ and substitute the resulting expression into (79) to obtain a solution of equation (78) in the form

ϕ t t) − λ n−2 f (t), λ ϕ(t)

w = ϕ(t)e ′ λx 1 (

✛✂✜ where ϕ(t) is an arbitrary function and λ is an arbitrary constant. References for Subsection S.4.5: A. D. Polyanin (2002, Supplement B), A. D. Polyanin and V. F. Zaitsev (2002).