Equations of the Form ∂w n = a∂ w n + f (x, t, w) ∂w + g(x, t, w)
11.1.4. Equations of the Form ∂w n = a∂ w n + f (x, t, w) ∂w + g(x, t, w)
∂t
∂x
∂x
+ f (w).
∂x n
∂x
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = − w(x + C 1 e bt , t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized traveling-wave solution:
w = w(z), − z=x+C 1 e bt ,
where the function w(z) is determined by the ordinary differential equation
aw ( n)
+( bz + c)w ′ z + f (w) = 0.
2. =a ∂t
+ f (t)
+ g(w).
∂x n
∂x
The transformation w = u(z, t), z = x +
f (t) dt leads to the simpler equation
∂u
∂z n + g(u),
∂t
which has a traveling-wave solution u = u(kz + λt). ∂w
∂w
3. =a n + bx + f (t)]
+ g(w).
∂t ∂x
∂x
Generalized traveling-wave solution:
w = w(z), − z = x + Ce bt + e bt
e bt f (t) dt,
where
C is an arbitrary constant and the function w(z) is determined by the ordinary differential equation
aw ( n) z + bzw ′ z + g(w) = 0.
+ bw ln w + g(x) + h(t) w.
∂x n
∂x
Multiplicative separable solution:
bt
bt
w(x, t) = exp − Ce + e e bt h(t) dt ϕ(x),
where
C is an arbitrary constant and the function ϕ(t) is determined by the ordinary differential equation
aϕ ( n) + f (x)ϕ ′
x + bϕ ln ϕ + g(x)ϕ = 0.
+ f (t).
∂x n
∂x
The transformation
w = u(z, t) +
f (τ ) dτ ,
z=x+b
( t − τ )f (τ ) dτ ,
where t 0 is any, leads to an equation of the form 11.1.3.1:
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = w(x + bC 1 e ct + C 2 , t+C 3 )+ C 1 ce ct ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
where C 1 and C 2 are arbitrary constants and the function U (z) is determined by the autonomous ordinary differential equation
aU ( n) z + bU U z ′ − C 2 U z ′ + cU = 0.
For C 1 = 0, we have a traveling-wave solution.
3 ◦ . There is a degenerate solution linear in x:
w(x, t) = ϕ(t)x + ψ(t).
7. =a n + bw + f (t)
+ g(t).
∂t ∂x
∂x
The transformation
w = u(z, t) +
( t − τ )g(τ ) dτ ,
where t 0 is any, leads to an equation of the form 11.1.3.1:
+ f (t)w
+ g(t)w = 0.
∂t ∂x n
∂x
Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = wx+C 1 ψ(t) + C 2 , t − C 1 ϕ(t),
where
ϕ(t) = exp −
g(t) dt ,
ψ(t) =
f (t)ϕ(t) dt,
is also a solution of the equation ( C 1 and C 2 are arbitrary constants).
Remark. This also remains true if a in the equation is an arbitrary function of time, a = a(t). ∂w
∂w
9. =a ∂t
+ [f (t) ln w + g(t)]
∂x n
∂x
Generalized traveling-wave solution:
w(x, t) = exp[ϕ(t)x + ψ(t)],
where
ψ(t) = ϕ(t) [ g(t) + aϕ n−1 ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants.* ∂w
+ [f (t) arcsinh(kw) + g(t)]
∂x 2 n+1
∂x
Generalized traveling-wave solution:
1 w(x, t) = sinh ϕ(t)x + ψ(t) , k
where
f (t) dt + C 1 , 2 ψ(t) = ϕ(t) [ g(t) + aϕ n ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants. ∂w
+ [f (t) arccosh(kw) + g(t)]
∂x 2 n+1
∂x
Generalized traveling-wave solution:
1 w(x, t) = cosh ϕ(t)x + ψ(t) , k
where
ϕ(t) = − 2 f (t) dt + C
ψ(t) = ϕ(t) [ g(t) + aϕ n ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants.
* In equations 11.1.4.9 to 11.1.4.13 and their solutions, a can be an arbitrary function of time, a = a(t).
12. =a ∂t
+ [f (t) arcsin(kw) + g(t)]
∂x 2 n+1
∂x
Generalized traveling-wave solution:
1 w(x, t) = sin ϕ(t)x + ψ(t) , k
where
ϕ(t) = − 2 f (t) dt + C 1 , ψ(t) = ϕ(t) [ g(t) + a(−1) n ϕ n ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants.
2 n+1 + [f (t) arccos(kw) + g(t)]
∂t ∂x
∂x
Generalized traveling-wave solution:
1 w(x, t) = cos ϕ(t)x + ψ(t) , k
where
ϕ(t) = − 2 f (t) dt + C
ψ(t) = ϕ(t) [ g(t) + a(−1) n ϕ n ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants.