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