Equations of the Form ∂w m =a∂ w ∂w + bw k
1.1.11. Equations of the Form ∂w m =a∂ w ∂w + bw k
∂t
∂x
∂x
◮ Equations of this form admit traveling-wave solutions w = w(kx + λt). ∂w
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions
= w −1 1 C 1 w( P C 1 x+C 2 , C 1 t+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized separable solutions linear and quadratic in x:
w(x, t) = C 1 x + (aC 2 1 + b)t + C 2 ,
( x+C 2 ) 2
w(x, t) = − −1 C
3 | t+C 1 | /3 3 + b(t + C 1 ),
6 a(t + C 1 )
where C 1 , C 2 , and C 3 are arbitrary constants. The first solution is degenerate.
3 ◦ . Traveling-wave solution in implicit form: Z
4 ◦ . For other solutions, see equation 1.1.11.11 with m = 1 and k = 0. ∂w
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions
w −1
1 = C 1 w( C 1 x+C 2 , C 1 t+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.
2 ◦ . The transformation
2 −1 ∂u b ∂ 1 ∂u
x=−
, w(x, t) = −
leads to the equation ∂ −1
u ∂t ∂y 2 It follows that any solution u = u(x, t) of the linear heat equation
(2) generates a solution (1) of the original nonlinear equation. ◆☎❖
Reference : V. A. Dorodnitsyn and S. R. Svirshchevskii (1983).
Functional separable solutions: −1 b /2
w(x, t) = ◗ C 1 ( x+C 2 ) 2 + C 3 exp(2 aC 1 t) −
aC 1 where C 1 , C 2 , and C 3 are arbitrary constants.
ab > 0, the transformation
w(x, t) = exp( ◗
3 λx)z(ξ, t), ξ=
exp( ◗ 2 λx),
3 a leads to a simpler equation of the form 1.1.10.4:
References : N. H. Ibragimov (1994), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).
2 ◦ . For
ab < 0, the transformation z(ξ, t) 1 1 b /2
w(x, t) =
tan( λx),
cos ( λx)
also leads to equation 1.1.10.4.
3 ◦ . Multiplicative separable solution:
w(x, t) = (t + C) 3 /4 u(x),
where
C is an arbitrary constant, and the function u = u(x) is determined by the autonomous ordinary differential equation
−4 /3 ′ ) ′ + a(u −1 u
bu /3 − 3 4 u = 0.
4 ◦ . See also equation 1.1.12.6 with b = c = 0. ∂w
1 ◦ . Functional separable solution: w(x, t) = a 2 /3 3 Ax 3 −2 + /3 f 2 ( t)x 2 + f 1 ( t)x + f 0 ( t) .
Here,
f 2 ( t) = 3 ϕ(t) dt + 3B, f 1 ( t) =
ϕ(t) dt + B +
ϕ(t),
2 ϕ t ( 9 t), A 6 A 36 A where the function ϕ(t) is defined implicitly by
f 0 ( t) =
2 ϕ(t) dt + B +
2 ϕ(t)
ϕ(t) dt + B +
Z ( C 1 − 108 A 2 abϕ − 8ϕ 3 ) −1 /2 dϕ = t + C 2 ,
and
A, B, C 1 , and C 2 are arbitrary constants.
2 ◦ . For other solutions, see equations 1.1.11.9 and 1.1.11.11 with m = −3/2.
3 ◦ . The substitution w=u −2 /3 leads to an equation of the form 1.1.9.7:
∂u 2 ∂ 2 u 2 ∂u 3 = au 2 − a − b.
∂t
∂x
3 ∂x
This is a special case of equation 1.1.12.5 with b = 0. See also equation 1.1.11.11 with m = −4/3 and k = 7/3.
w(x, t) = e bt
w(x, t) = e
x+
a ab
w(x, t) = e bt
bm ( x − A)
2 a(m + 2)(B − e bmt
2 bmt
bm 2 ( x + B) 2 m
w(x, t) =
w(x, t) = e A|e
2 a(m + 2) e bmt + B where
A, B, C, and λ are arbitrary constants.
2 ◦ . By the transformation
w(x, t) = e bt
v(x, τ ), τ=
e bmt + const
bm
the original equation can be reduced to an equation of the form 1.1.10.7:
❯☎❱ 3 ◦ . See also equation 1.1.11.11 with k = 1.
Reference : L. K. Martinson and K. B. Pavlov (1972).
1 ◦ . Multiplicative separable solution ( a = b = 1, m > 0):
2( m + 1) cos 2 (
πx/L) /m
for | x| ≤ ,
w(x, t) =
2 where L = 2π(m + 1) 1 /2 /m. Solution (1) describes a blow-up regime that exists on a limited time ❯☎❱ interval t ❲ [0, t 0 ). The solution is localized in the interval | x| < L/2.
References : N. V. Zmitrenko, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii (1976), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).
2 ◦ . Multiplicative separable solution:
Ae µx + Be − µx
+ 1 /m D
w(x, t) =
a(m + 1) where
A(m + 2) 2 b(m + 2)
A, C, and λ are arbitrary constants, ab(m + 1) < 0.
w(x, t) = (mλt + C) −1 /m ϕ(x),
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
(2) Equation (2) has the following solution in implicit form:
A and B are arbitrary constants.
4 ◦ . Functional separable solution [it is assumed that ab(m + 1) < 0]:
w(x, t) =
f (t) + g(t)e λx
where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations
bmf 2 ,
bm(m + 2)
f g.
m+1
Integrating yields
f (t) = (C 1 − bmt) −1 , g(t) = C 2 ( C 1 − bmt) − m+1 , where C 1 and C 2 are arbitrary constants.
m+2
5 ◦ . Functional separable solution (
A and B are arbitrary constants):
− b w(x, t) =
f (t) + g(t)(Ae λx + Be λx )
where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations
2 4 bmAB 2 bm(m + 2)
On eliminating t from this system, one obtains a homogeneous first-order equation:
The substitution ζ = f /g leads to a separable equation. Integrating yields the solution of equation (5) in the form
f= ❳
g 4AB + C 1 g − m+2 2 ,
C 1 is any.
Substituting this expression into the second equation of system (4), one obtains a separable equation for g = g(t).
6 ◦ . The functional separable solutions
1 w(x, t) = /m f (t) + g(t) cosh(λx) ,
1 w(x, t) = /m f (t) + g(t) sinh(λx)
are special cases of formula (3) with A= 1 1 1 2 1 , B= 2 and A= 2 , B=− 2 , respectively.
b w(x, t) =
f (t) + g(t) cos(λx + C)
where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations
2 = bm 2 bm(m + 2)
❨☎❩ which coincides with system (4) for AB = 1 4 .
References for equation 1.1.11.8: M. Bertsch, R. Kersner, and L. A. Peletier (1985), V. A. Galaktionov and S. A. Posash- kov (1989), V. F. Zaitsev and A. D. Polyanin (1996).
This is a special case of equation 1.1.11.11 with k = 1 − m. Functional separable solution:
w(x, t) =
A, B, and C are arbitrary constants.
Reference : R. Kersner (1978).
This is a special case of equation 1.1.11.11 with m = 2n and k = 1 − n. Generalized traveling-wave solution:
1 bn 2 2 a(n + 1) w(x, t) =
n where C 1 and C 2 are arbitrary constants.
This is a special case of equation 1.6.15.2 with f (w) = aw m and g(w) = bw k . For b = 0, see Subsection 1.1.10.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions ❬
w 1 = 2 C 2 1 w( C 1 k−m−1 x+C 2 , C 1 k−2 t+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.
2 ◦ . A space-homogeneous solution and a stationary solution are given by (the latter is written out in implicit form):
w(t) =
(1 − k)bt + C 1− k if k ≠ 1,
w m+k+1
A, B, and C are arbitrary constants.
w = w(z), z= ❭ x + λt,
where the function w(z) is determined by the autonomous ordinary differential equation
(1) The substitution
a(w m w ′ z ) ′ z − λw z ′ + bw k = 0.
a u(w) = w m w ′
brings (1) to the Abel equation
(2) The book by Polyanin and Zaitsev (2003) presents exact solutions of equation (2) with m+k=
uu ′ w − u = −abλ −2 w m+k .
4 ◦ . Self-similar solution for k ≠ 1:
1 k−m−1
w=t 1− k u(ξ), ξ = xt 2(1− k) ,
where the function u(ξ) is determined by the ordinary differential equation
a(u m u ′ ξ )
m−k+1
Reference : V. A. Dorodnitsyn (1982).