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

8.1.1. Equations of the Form ∂w =F w, ∂w ,∂ w

∂t

∂x ∂x 2

Preliminary remarks. Consider the equation

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

2 ◦ . In the general case, equation (1) admits traveling-wave solution

(2) where k and λ are arbitrary constants and the function w(ξ) is determined by the ordinary differential

w = w(ξ),

This subsection presents special cases where equation (1) admits exact solutions other than traveling wave (2). ∂w 2 ∂ w

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

1 C 1 w(C 1 x+C 2 , C 1 t+C 3 )+ C 4 x+C 5 ,

where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation.

2 ◦ . Additive separable solution:

1 w(x, t) = F (A)t + 2

2 Ax + Bx + C,

where

A, B, and C are arbitrary constants.

3 ◦ . Generalized separable solution:

w(x, t) = (Ax + B)t + C + ϕ(x),

where the function ϕ(x) is determined by the ordinary differential equation

Fϕ ′′ xx

4 ◦ . Solution:

w(x, t) = At + B + ψ(ξ),

ξ = kx + λt,

where

A, B, k, and λ are arbitrary constants and the function ψ(ξ) is determined by the autonomous ordinary differential equation

Fk 2 ψ ′′ ξξ

ξ ′ + A.

ξ = kx + λt, where

w(x, t) = 1 2 Ax 2 + Bx + C + U (ξ),

A, B, k, and λ are arbitrary constants and the function U (ξ) is determined by the autonomous ordinary differential equation

Fk 2 U ξξ ′′ + A ξ ′ .

6 ◦ . Self-similar solution:

w(x, t) = t Θ(ζ), ζ= √ , t

where the function Θ( ζ) is determined by the ordinary differential equation

. The substitution u(x, t) = brings the original equation to an equation of the form 1.6.18.3:

8 ◦ . The transformation

where α, the β i , and the γ i are arbitrary constants ( α ≠ 0, β 1 β 4 − β 2 β 3 ≠ 0) and the subscripts x and x denote the corresponding partial derivatives, takes the equation in question to an equation of the ¯ same form. The right-hand side of the equation becomes

References : I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989), N. H. Ibragimov (1994). Special case 1. Equation:

1 ◦ . Additive separable solution:

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

where C 1 , C 2 , and C 3 are arbitrary constants. 2 ◦ . Solution:

w(x, t) =

1 1− k u(x) + C 2 ,

where the function u(x) is determined by the autonomous ordinary differential equation (u xx ′′ ) k − u = 0, whose general solution can be written out in implicit form:

Special case 2. Equation:

Generalized separable solution:

w(x, t) = U (x) − 1 ( 2 +

λ where A 1 , A 2 , B 1 , B 2 , C 1 , and C 2 are arbitrary constants, and the function U (x) is determined by the ordinary differential equation

2 aλ exp(λU ′′ xx )+ B 1 ( x 2 + A 1 x+A 2 ) = 0,

which is easy to integrate; to this end, the equation should first be solved for U xx ′′ .

Generalized separable solution:

w(x, t) = (at + C) ln A at + C) −1 + D, cos 2 ( Ax + B)

where A, B, C, and D are arbitrary constants.

Apart from a traveling-wave solution, this equation has a more complicated exact solution of the form

w(x, t) = At + B + ϕ(ξ),

ξ = kx + λt,

where

A, B, k, and λ are arbitrary constants, and the function ϕ(ξ) is determined by the autonomous ordinary differential equation

Special case. Equation:

1 ◦ . Generalized separable solution:

w(x, t) = ϕ 1 ( t) + ϕ 2 ( t)x 3 /2 + ϕ 3 ( t)x 3 ,

where the functions ϕ k = ϕ k ( t) are determined by the autonomous system of ordinary differential equations

1 ′ = 8 aϕ 2 ,

ϕ 2 ′ = 45 4 aϕ 2 ϕ 3 , ϕ ′ 3 = 18 aϕ 2 3 .

The prime denotes a derivative with respect to t. 2 ◦ . Generalized separable solution cubic in x:

w(x, t) = ψ 1 ( t) + ψ 2 ( t)x + ψ 3 ( t)x 2 + ψ 4 ( t)x 3 ,

where the functions ψ k = ψ k ( t) are determined by the autonomous system of ordinary differential equations

ψ ′ 1 =2 aψ 2 ψ 3 , ψ ′ 2 =2 a(2ψ 2 3 +3 ψ 2 ψ 4 ), ψ ′ 3 = 18 aψ 3 ψ 4 , ψ ′ 4 = 18 aψ 4 2 .

3 ◦ . Generalized separable solution:

w(x, t) = θ(x) + C 3 + C 4 , C 1 t+C 2

where C 1 , ...,C 4 are arbitrary constants and the function θ = θ(x) is determined by the autonomous ordinary differential equation

aθ ′ x θ xx ′′ + C 1 θ+C 1 C 3 =0 ,

whose solution can be written out in implicit form.

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

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

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

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

e − at

F (C 1 e at , 0) dt.

3 ◦ . Traveling-wave solution:

w = w(z), z = x + λt,

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

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

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

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

2 ◦ . Degenerate solution:

x+C Z 1 1 1

w(x, t) = −

τF −

,0 dτ ,

τ=t+C 2 .

w(x, t) = U (ζ) + 2C 2

1 t,

ζ = x + aC 1 t + C 2 t,

where C 1 and C 2 are arbitrary constants and the function U (ζ) is determined by the autonomous ordinary differential equation

In the special case C 1 = 0, the above solution converts to a traveling-wave solution. ∂w

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

w 1 = w(x + aC 1 e bt + C 2 , t+C 3 )+ C 1 be bt ,

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

2 ◦ . Degenerate solution:

w(x, t) = ϕ(t)x + ψ(t),

where the functions ϕ(t) and ψ(t) are determined by the system of ordinary differential equations

ϕ ′ t = aϕ 2 + bϕ, ψ ′ t = aϕψ + bψ + F (ϕ, 0),

which is easy to integrate (the first equation is a Bernoulli equation and the second one is linear in ψ).

3 ◦ . Traveling-wave solution:

w = w(z), z = x + λt,

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

Fw z ′ , w zz ′′

′ z − λw ′ z + bw = 0.

1 ◦ . The transformation

¯t = t − t 1

0 , x=− ¯

w(y, t) dy −

F w(x 0 , τ ), w x ( x 0 , τ)

w( ¯x, ¯t) = ¯ x 0 t 0 w(x, t)

converts a (nonzero) solution w(x, t) of the original equation to a solution ¯ w( ¯x, ¯t) of a similar equation:

2 ◦ . In the special case

F w, w

it follows from (1) that

¯ F w, w x

g(w) = w ¯ 1−3 k g(w −1 ).

References : W. Strampp (1982), J. R. Burgan, A. Munier, M. R. Feix, and E. Fijalkow (1984), N. H. Ibragimov (1994).

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

w −1

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

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

2 ◦ . Solution:

w(x, t) = tϕ(z), z = kx + λ ln |t|,

where k and λ are arbitrary constants and the function ϕ(z) is determined by the autonomous ordinary differential equation

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

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

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

2 ◦ . Multiplicative separable solution:

w(x, t) = Ce λt ϕ(x),

where

C and λ are arbitrary constants and the function ϕ(x) is determined by the autonomous ordinary differential equation

ϕ xx ′′

This equation has particular solutions of the form ϕ(x) = e αx , where α is a root of the algebraic (or transcendental) equation

F (α, α 2 )− λ = 0.

w(x, t) = Ce λt ψ(ξ), ξ = kx + βt,

where

C, k, λ, and β are arbitrary constants, and the function ψ(ξ) is determined by the autonomous ordinary differential equation

This equation has particular solutions of the form ψ(ξ) = e µξ . ∂w

For the cases β = 0 and β = 1, see equations 8.1.1.7 and 8.1.1.8, respectively.

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

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

where C 1 , C 2 , and C 3 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 autonomous ordinary differential equation

w(z, t) = (t + C) 1− β Θ( z), z = kx + λ ln(t + C), where

C, k, and λ are arbitrary constants, and the function Θ(z) is determined by the autonomous ordinary differential equation

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

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

ln C 2 ,

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

2 ◦ . Additive separable solution:

1 w(x, t) = − ln( Aβt + B) + ϕ(x),

where

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

e βϕ Fϕ ′ x , ϕ ′′ xx

3 ◦ . Solution:

1 w(x, t) = − ln( t + C) + Θ(ξ), ξ = kx + λ ln(t + C),

where

C, k, and λ are arbitrary constants, and the function Θ(ξ) is determined by the autonomous ordinary differential equation

e βΘ

F kΘ

ξ ′ , k Θ ′′ ξξ

This is a special case of equation 8.1.1.2.

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

w −1

1 = C 1 w(x + C 2 , C 1 t+C 3 )+ C 4 ,

where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solution:

w(x, t) = At + B + ϕ(ξ),

ξ = kx + λt,

where

A, B, k, and λ are arbitrary constants, and the function ϕ(ξ) is determined by the autonomous ordinary differential equation

F kϕ ′′ ξξ /ϕ ′ ξ

′ ξ + A.

If A = 0, the equation has a traveling-wave solution.

3 ◦ . Solution:

w(x, t) = tΘ(z) + C, z = kx + λ ln |t|

where

C, k, β, and λ are arbitrary constants, and the function Θ(z) is determined by the autonomous ordinary differential equation

This is a special case of equation 8.1.1.2.

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

w 1 = C 1 w(x + C 2 , t+C 3 )+ C 4 , where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solution:

w(x, t) = At + B + ϕ(z),

z = kx + λt,

where

A, B, k, and λ are arbitrary constants, and the function ϕ(z) is determined by the autonomous ordinary differential equation

w(x, t) = Ae βt Θ( ξ) + B,

ξ = kx + λt,

where

A, B, k, β, and λ are arbitrary constants, and the function Θ(ξ) is determined by the autonomous ordinary differential equation

This is a special case of equation 8.1.1.2. For the cases β = 0 and β = 1, see equations 8.1.1.11 and

8.1.1.12, respectively.

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 2 , C 1 t+C 3 )+ C 4 ,

where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

w(x, t) =

1− β ϕ(x) + C,

where

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

w(x, t) = (t + A) 1− β Θ( z) + B, z = kx + λ ln(t + A), where

A, B, k, and λ are arbitrary constants, and the function Θ(z) is determined by the autonomous ordinary differential equation

This is a special case of equation 8.1.2.11. ∂w

This is a special case of equation 8.1.2.12. ∂w

∂x This is a special case of equation 8.1.2.13.

∂x

∂x