Equations Involving ∂ m w and ∂ w

11.4.2. Equations Involving ∂ m w and ∂ w

∂x m ∂y

1. a n +b ∂y n

= (ay n + bx n )f (w).

∂x Solution:

w = w(z),

z = xy,

where the function w(z) is determined by the autonomous ordinary differential equation

w ( n)

= f (w).

Remark. This remains true if the constants a and b in the equation are replaced by arbitrary functions a = a(x, y, w, w x , w y , . . .) and b = b(x, y, w, w x , w y , . . .).

Multiplicative separable solution:

w(x, y) = Ae λy ϕ(x),

where

A and λ are arbitrary constants, and the function ϕ(x) is determined by the nth-order ordinary differential equation

n ; w ∂y 2 w ,..., ∂x w ∂x w ∂y 2 m

1 ◦ . Multiplicative separable solution:

w(x, y) =

A cosh(λy) + B sinh(λy) ϕ(x),

where

A, B, and λ are arbitrary constants, and the function ϕ(x) is determined by the nth-order ordinary differential equation

2 F x, ϕ 2 ′ /ϕ, . . . , ϕ ( n) /ϕ; λ , ...,λ m

2 ◦ . Multiplicative separable solution:

w(x, y) =

A cos(λy) + B sin(λy) ϕ(x),

where

A, B, and λ are arbitrary constants, and the function ϕ(x) is determined by the nth-order ordinary differential equation

2 F x, ϕ 2

x /ϕ, . . . , ϕ x /ϕ; −λ , . . . , (−1) m λ m = 0.

( n)

Additive separable solution:

w(x, y) = ϕ(x) + ψ(y).

Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations

F 1 x, ϕ ′ x , ...,ϕ ( n) x − kϕ = C,

F 2 y, ψ ′ y , ...,ψ ( m) y − kψ = −C, where

C is an arbitrary constant.

Multiplicative separable solution:

w(x, y) = ϕ(x)ψ(y).

Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations

C is an arbitrary constant. ∂w

Additive separable solution:

w(x, y) = ϕ(x) + ψ(y).

Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations

e ( λϕ F 1 x, ϕ n)

x ′ , ...,ϕ x = C,

e λψ F 2 y, ψ ′ , ...,ψ ( m) y y =− C, where

C is an arbitrary constant.

w ∂y m Multiplicative separable solution:

w(x, y) = ϕ(x)ψ(y).

Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations

F 1 x, ϕ x ′ /ϕ, . . . , ϕ ( n) x /ϕ − k ln ϕ = C,

y ′ /ψ, . . . , ψ y /ψ − k ln ψ = −C, where

F ( 2 y, ψ m)

C is an arbitrary constant.

Solution: w = w(ξ), ξ = ax + by,

where the function w(ξ) is determined by the ordinary differential equation

F ξ, w, aw ′ ξ , ...,a n ( n) w , bw ξ ′ ξ , ...,b m ( m) w ξ = 0.

w = ϕ(ξ) + Cx, ξ = ax + by,

where

C is an arbitrary constant and the function ϕ(ξ) is determined by the ordinary differential equation

ξξ ...,a n ( , ϕ n)

ξ , bϕ

m) ξ ′ , ...,b ϕ ξ

10. n a 1 x+b 1 y + f (w)

a 2 x+b 2 y + g(w)

∂y m Solutions are sought in the traveling-wave form

w = w(z),

z = Ax + By,

where the constants

A and B are evaluated from the algebraic system of equations

a 1 A n+m + a 2 B n+m = A,

b 1 A n+m + b 2 B n+m = B.

The desired function w(z) is determined by the mth-order ordinary differential equation z+A n+m f (w) + B n+m g(w) w ( m)

= C 0 + C 1 z+···+C n−1 z , where C 0 , C 1 , ...,C n−1 are arbitrary constants.

∂y ∂y Generalized traveling-wave solution:

w = w(z),

z = Ax + By,

where the constants

A and B are evaluated from the algebraic system of equations

a 1 A n + a 2 B n = A,

b 1 A n + b 2 B n = B,

and the desired function w(z) is determined by the ordinary differential equation

, Bw z ′ , ...,B w z . Remark. If the right-hand side of the equation is also dependent on mixed derivatives, solutions

zw ( n) =

F w, Aw ′ , ...,A m w ( m)

( k)

are constructed likewise.

Supplements

Exact Methods for Solving Nonlinear Partial Differential Equations