Equations of the Form ∂w =F x, w, ∂w ,∂ w

8.1.3. Equations of the Form ∂w =F x, w, ∂w ,∂ w

∂t

∂x ∂x 2

Preliminary remarks. Consider the equation

Suppose that the auxiliary ordinary differential equation

w = F (x, w, w x ′ , w xx ′′ )

is reduced, by a linear transformation

x = ϕ(z), w = ψ(z)u + χ(z)

and the subsequent division of the resulting equation by ψ(z), to the autonomous form

u = F(u, u ′ z , u ′′ zz ),

where

F = F/ψ. Then, the original equation (1) can be reduced, by the same transformation

x = ϕ(z), w(x, t) = ψ(z)u(z, t) + χ(z),

to the equation

which has a traveling-wave solution u = u(kz + λt). The above allows using various known transformations of ordinary differential equations (see Kamke, 1977, and Polyanin and Zaitsev, 2003) for constructing exact solutions to partial differential equations. If the original equation is linear, then such transformations will result in linear constant- coefficient equations.

Generalized separable solution:

w(x, t) = Axt + Bt + C + ϕ(x),

where

A, B, and C are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation

Additive separable solution:

w(x, t) = At + B + ϕ(x),

where

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

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = w(x + C 1 e − at , t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

w = w(z), z = x + Ce − at ,

where

C is an arbitrary constant and the function w(z) is determined by the ordinary differential equation

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = C 1 −1 w(C 1 x, C 1 t+C 2 )+ C 3 ,

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Additive separable solution:

w(x, t) = At + B + ϕ(x),

where

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

Fϕ ′ x , xϕ ′′ xx

3 ◦ . Solution:

w(x, t) = tΘ(ξ) + C, ξ = x/t,

where

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

The substitution x= ✠ e z leads to the equation

which has a traveling-wave solution w = w(kz + λt). ∂w 2 ∂w

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = w(C 1 x, C − k 1 t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Self-similar solution:

w(x, t) = w(z), z = xt 1 /k ,

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

kz k−1

F w, zw z ′ , 2 z w ′′ zz

′ z = 0.

Passing to the new independent variables

we obtain an equation of the form 8.1.3.6:

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w(x + C 1 1 , e λC t+C

where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solution:

w(x, t) = w(z), z = λx + ln t,

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

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = C 1 w(x, t + C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Multiplicative separable solution:

w(x, t) = e λt ϕ(x),

where λ is an arbitrary constant and the function ϕ(x) is determined by the ordinary differential equation

For β = 1, see equation 8.1.3.9.

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

w β−1

1 = C 1 w(x, C 1 t+C 2 ),

where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Multiplicative separable solution:

w(x, t) =

1− β ϕ(x),

where

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

F x, ϕ ′ x /ϕ, ϕ ′′ xx /ϕ

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = w(x, C 1 t+C 2 )+

ln C 1 ,

β where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Additive separable solution:

1 w(x, t) = − ln( Aβt + B) + ϕ(x), β

where

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

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = C 1 w(x, t + C 2 )+ C 3 ,

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Additive separable solution:

w(x, t) = At + B + ϕ(x),

where

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

F x, ϕ ′′ xx /ϕ ′ x

3 ◦ . Generalized separable solution:

w(x, t) = Ae µt Θ( x) + B

where

A, B, and µ are arbitrary constants, and the function Θ(x) is determined by the ordinary differential equation

For β = 1, see equation 8.1.3.12.

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

w β−1

1 = C 1 w(x, C 1 t+C 2 )+ C 3 ,

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.