Equations of the Form ∂w 2 = f (x, t, w) ∂ w +g x, t, w, ∂w
1.6.17. Equations of the Form ∂w 2 = f (x, t, w) ∂ w +g x, t, w, ∂w
∂t
∂x 2 ∂x
1 ◦ . Multiplicative separable solutions for a > 0: Ý
w(x, t) = C exp x a+a
f (t) dt ,
where
C is an arbitrary constant.
2 ◦ . Multiplicative separable solution for a > 0:
w(x, t) = C 1 e x a + C 2 e − x √ a e F C 3 +8 aC 1 C 2 e 2 F dt
where C 1 , C 2 , and C 3 are arbitrary constants.
3 ◦ . Multiplicative separable solution for a < 0:
Z p −1 /2
2 2 w(x, t) = 2 C
where F=a
f (t) dt; C 1 , C 2 , and C 3 are arbitrary constants.
+ g(t)w + h 2 ( t)x 2 + h 1 ( t)x + h 0 ( t).
∂t ∂x
∂x
Generalized separable solution quadratic in x:
w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t),
where the functions ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are determined by the system of ordinary differential equations
ϕ ′ t =6 f (t)ϕ 2 + g(t)ϕ + h 2 ( t), ψ t ′ =6 f (t)ϕψ + g(t)ψ + h 1 ( t),
χ ′ t =2
f (t)ϕχ + f (t)ψ 2 + g(t)χ + h
0 ( t).
3. + f (t)w
= g(t)
Degenerate solution linear in x:
1 Z g(t)
w(x, t) =
where C 1 and C 2 are arbitrary constants. ∂w
∂w
∂w
4. + f (t)w
= g(t)
Degenerate solution linear in x:
w(x, t) = (x + C 1 ) ϕ(t),
where the function ϕ = ϕ(t) is determined by the ordinary differential equation
g(t)ϕ
f (t)ϕ .
f (x)w m
∂t ∂x
∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(x, C m 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) = (mλt + C) −1 /m ϕ(x),
C and λ are arbitrary constants, and the function ϕ = ϕ(x) is determined by the ordinary differential equation
where
(1) The transformation
[ f (x)ϕ m ϕ ′ x ] ′ x + λϕ = 0.
brings (1) to the generalized Emden–Fowler equation
(2) where the function
Φ ′′ zz +
F (z)Φ m+1 = 0,
F = F (z) is defined parametrically by
Z dx
F = λ(m + 1)f (x),
z=
f (x)
The book by Polyanin and Zaitsev (2003, Sections 2.3 and 2.7) presents a large number of solutions to equation (2) for various
F = F (z).
3 ◦ . The transformation
Z w(x, t) = ψ(x) m+1 u(ξ, t), ξ=−
1 m+2
dx
ψ(x) m+1
dx,
ψ(x) = ,
f (x) leads to an equation of the similar form
∂u
F (ξ)u m
where the function
F = F (ξ) is defined parametrically by
F = f (x)[ψ(x) m+1 , ξ=−
ψ(x) m+1
dx,
ψ(x) = .
f (x)
6. = f (x)w m ∂ 2 w + g(x)w m+1 .
∂t
∂x 2
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(x, C m 1 t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solution (
C and λ are arbitrary constants): w(x, t) = (mλt + C) −1 /m ϕ(x),
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
(1) In the special case of f (x) = ax n and g(x) = bx k , equation (1) becomes
f (x)ϕ m ϕ ′′ xx + g(x)ϕ m+1 + λϕ = 0.
(2) The books by Polyanin and Zaitsev (1995, 2003) present a large number of solutions to equa-
ϕ ′′ xx +( b/a)x k−n ϕ + (λ/a)x − n ϕ 1− m = 0.
tion (2) for various values of n, m, and k. ∂w
w m ∂w + g(t)w 1– m .
Functional separable solution:
2 1 w(x, t) = /m ϕ(t)x + ψ(t) ,
where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by the system of first-order ordinary differential equations
Integrating yields
A and B are arbitrary constants.
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
f (x)w m
+ g(x)w m+1 .
∂t ∂x
∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(x, C m 1 t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solution (
C and λ are arbitrary constants): w(x, t) = (mλt + C) −1 /m ϕ(x),
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
[ f (x)ϕ m ϕ ′ x ] ′ x + g(x)ϕ m+1 + λϕ = 0.
(2) where the functions
Φ ′′ zz +
F (z)Φ m+1 + G(z)Φ = 0,
F = F (z) and G = G(z) are defined parametrically by ( F = λ(m + 1)f(x), Z
( G = (m + 1)f(x)g(x),
In the special case of f (x) = ax n and g(x) = bx k , equation (2) becomes
A = λa(m + 1) a(1 − n) 1− n and
B = ab(m + 1) a(1 − n) 1− n .
The books by Polyanin and Zaitsev (1995, 2003) present a large number of solutions to equa- tion (3) for various values of n, m, and k.
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
+ g(t)
+ h(t)w.
The transformation
Z w(x, t) = u(z, τ ) exp
f (t) exp m h(t)dt dt leads to a simpler equation of the form 1.1.10.7:
h(t) dt , z=x+
g(t) dt, τ=
+ xg(t) + h(t)
+ â ( t)w.
The transformation
f (t)G 2 ( t)S m ( t) dt, where the functions S(t) and G(t) are given by
w(x, t) = u(z, τ )S(t), z = xG(t) + h(t)G(t) dt, τ=
g(t) dt , leads to a simpler equation of the form 1.1.10.7:
S(t) = exp
s( t) dt , G(t) = exp
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
11. = x k f (t)
+ xg(t)
+ h(t)w.
The transformation
f (t)G 2− k ( t)H m ( t) dt, where the functions G(t) and H(t) are given by
w(t, x) = u(z, τ )H(t), z = xG(t), τ=
h(t) dt , leads to a simpler equation of the form 1.1.15.6:
G(t) = exp
g(t) dt , H(t) = exp
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
∂w ∂
f (x)e βw ∂w .
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
1 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 1 A − βx
w(x, t) = − ln( βt + C) +
A, B, and C are arbitrary constants. ∂w
∂w
f (x)e βw
+ g(x)e βw .
∂t ∂x
∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
1 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( βt + C) + ϕ(x), β
where β and C are arbitrary constants, and the function ϕ(x) is determined by the second-order linear ordinary differential equation
[ f (x)ψ ′
x ] ′ x + βg(x)ψ + β = 0,
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
Generalized traveling-wave solution:
1 w(x, t) = /n ϕ(t)x + ψ(t) ,
where the functions ϕ(t) and ψ(t) are determined by the system of ordinary differential equations
which is easy to integrate (the first equation is a Bernoulli equation and the second one is linear in ψ).
∂w
15. = f (w)
+ ax + g(w)
+ h(w).
∂t
∂x 2 ∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
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.
2 ◦ . Generalized traveling-wave solution:
w = w(z),
z=x+C 1 e − at ,
where the function w(z) is determined by the ordinary differential equation
f (w)w zz ′′ +[ az + g(w)]w ′ z + h(w) = 0.
Nonlinear problems of the diffusion boundary layer, defined by equation 1.6.19.2, are reducible to equations of this form. For n = 1, see equation 1.6.15.1, and for f (w) = aw m , see equation 1.1.15.6.
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = w(C 1 x, C n+1 1 t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Self-similar solution for n ≠ −1:
w = w(z), z = xt − 1 n+1
(0 ≤ x < ∞),
where the function w(z) is determined by the ordinary differential equation
(1) which is often accompanied by the boundary conditions of (3) in 1.6.15.1.
( n + 1)[f (w)w ′ ] ′ + z n z z w ′ z = 0,
The general solution of equation (1) with f (w) = a(w + b) −1 and arbitrary n can be found in Zaitsev and Polyanin (1993).
solutions. Let us integrate (1) with respect to z and then apply the hodograph transformation (with w regarded as the independent variable and z as the dependent one) to obtain
f (w) = −
z n dw + A ,
A is any. (2)
n+1
Substituting a specific z = z(w) for z on the right-hand side of (2), one obtains a one-parameter family of functions f (w) for which z = z(w) solves equation (1). An explicit form of the solution, w = w(z), is determined by the inversion of z = z(w).
For example, setting z = (1 − w) k , one obtains from (2) the corresponding f (w):
f (w) = A(1 − w) k−1
(1 − w) k(n+1) ,
A is any.
( n + 1)(nk + 1)
4 ◦ . There is another way to construct f (w) for which equation (1) admits exact solutions. It involves the following. Let ¯ w=¯ w(z) be a solution of equation (1) with some function f (w). Then ¯ w=¯ w(z) is also a solution of the more complicated equation ( n + 1)[F (w)w ′ z ] ′ z + z n w ′ z = 0 with
(3) where the function g = g(w) is defined parametrically by
F (w) = f (w) + Ag(w)
A is any),
1 g(w) = ′ , w=¯ w(z).
For example, if f (w) is a power-law function of w, f (w) = aw m
n+1
, then ¯ w = bz m is a solution of equation (1), with b being a constant. It follows from (3) and (4) that ¯ w is also a solution of
equation (1) with f (w) = aw m
m−n−1
+ Aw n+1 .
5 ◦ . For n = −1, there is an exact solution of the form
w = w(ξ),
ξ = ln |x| + λt,
where the function w(ξ) is defined implicitly by
å☎æ where λ, C 1 , and C 2 are arbitrary constants. To λ = 0 there corresponds a stationary solution.
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
+ g(w).
This is a nonlinear equation of heat and mass transfer in the radial symmetric case ( n = 1 corresponds to a plane problem and n = 2 to a spatial one).
1 ◦ . Let f (w) and g(w) be defined by
g(w) = a(n + 1)w + 2a
, ϕ ′ w ( w)
where ϕ(w) is an arbitrary function. In this case, there is a functional separable solution defined implicitly by
ϕ(w) = Ce 2 at − 1 2 ax 2 ,
where å☎æ
C is an arbitrary constant.
Reference : V. A. Galaktionov (1994).
g(w) = b ,
where ϕ = ϕ(w) is an arbitrary function. In this case, there is a functional separable solution defined implicitly by
C is an arbitrary constant. ∂w
18. = f (t)ϕ(w)
+ xg(t) + h(t)
∂t
∂x 2 ∂x
The transformation
z = xG(t) + 2 h(t)G(t) dt, τ= f (t)G ( t) dt, G(t) = exp g(t) dt , leads to a simpler equation of the form 1.6.16.3:
+ xg(t) + h(t)
The transformation
z = xG(t) + 2 h(t)G(t) dt, τ= f (t)G ( t) dt, G(t) = exp g(t) dt , leads to a simpler equation of the form 1.6.15.1:
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
∂w
∂ 2 g(t)
f (x, w) +
+ h(x).
Here, f w is the partial derivative of f with respect to w. Functional separable solution in implicit form:
f (x, w) =
g(t) dt −
( x − ξ)h(ξ) dξ + C 1 x+C 2 ,
where C 1 and C 2 are arbitrary constants, and x 0 is any number.
1.6.18. Equations of the Form ∂w =f x, w, ∂w w
+g x, t, w, ∂w
∂t
∂x ∂x 2
∂x
f (t)x + g(t)
+ h(t)w.
With the transformation
w(x, t) = u(z, τ )H(t), z = xF (t) +
g(t)F (t) dt, τ=
F k+2 ( t)H k ( t) dt, F k+2 ( t)H k ( t) dt,
F (t) = exp
f (t) dt , H(t) = exp
∂u k ∂u ∂ 2 u
∂x ∂z 2 ∂τ .
See equation 1.6.18.3, the special case 1. ∂w
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(x, C k 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 the function ϕ(x) is given by
ϕ(x) = C 1 exp A dx + C 2 if k = −1,
f (x)
A, B, C 1 , and C 2 are arbitrary constants.
3 ◦ . Solution:
w(x, t) = (Akt + B) −1 /k Θ( x) + C,
where
A, B, and C are arbitrary constants, and the function Θ(x) is determined by the second-order ordinary differential equation
This equation occurs in the nonlinear theory of flows in porous media; it governs also the motion of
a nonlinear viscoplastic medium.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C −1 w(C 1 x+C 2 , C 1 2 1 t+C 3 )+ C 4 , where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solution:
(1) where
w(x, t) = At + B + ϕ(z),
z = kx + λt,
A, B, k, and λ are arbitrary constants, and the function ϕ(z) is determined by the ordinary differential equation k 2 f kϕ ′ z ) ϕ ′′ zz = λϕ ′ z + A. (2)
The general solution of equation (2) can be rewritten in parametric form as
where C 1 and C 2 are arbitrary constants, Relations (1) and (3) define a traveling-wave solution for
A = 0 and an additive separable solution for λ = 0.
w(x, t) = t Θ(ξ), ξ= √ , t
where the function Θ( ξ) is determined by the ordinary differential equation
2 fΘ ′ ξ Θ ′′ ξξ + ξΘ ′ ξ − Θ = 0.
∂w
. The substitution u(x, t) = leads to an equation of the form 1.6.15.1:
5 ◦ . The hodograph transformation
x = w(x, t), ¯ w( ¯x, t) = x ¯
leads to an equation of the similar form
6 ◦ . The transformation ¯t = αt + γ 1 , x=β ¯ 1 x+β 2 w+γ 2 , w=β ¯ 3 x+β 4 w+γ 3 , where α, the β i , and the γ i are arbitrary constants such that α ≠ 0 and β 1 β 4 − β 2 β 3 ≠ 0, takes the
original equation to an equation with the same form. We have
f(¯ w x ¯ )= ( β 1 + β 2 w x ) f (w x ),
β 4 − β 2 w ¯ x ¯ where the subscripts x and ¯x denote the corresponding partial derivatives.
Special case 1. Equation
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(C 2 x+C 3 , C k 1 C k+2 2 t+C 4 )+ C 5 ,
where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation. 2 ◦ . Solution:
1 ) w(x, t) = (t + C −1 /k u(x) + C 2 ,
where C 1 and C 2 are arbitrary constants, and u(x) is determined by the ordinary differential equation ak(u ′ x ) k u ′′ xx + u = 0, the general solution of which can be written out in the implicit form as
C 3 − k+2 u 2 2 k+2 du = x + C 4 .
ak
Special case 2. Equation
w(x, t) = p C 1 − b 2 ( x+C 2 ) 2 −2 at + C 3 ,
where C 1 , C 2 , and C 3 are arbitrary constants. 2. Solution:
w = bx tan 1 é z − arctan ψ(z) é a
2 b 2 t+C ,
z=x 2 cos −2 é 1 2 z − arctan ψ(z) é a
t+C ,
= 1 (1 + ψ 2 z
The function z = z(x, t) in the solution is defined implicitly. 3. Solution:
C at w = bx tan ϕ(z) + 2 ln 2 , b
where C is an arbitrary constant, and the functions ϕ(z) and ψ(z) are determined by the system of ordinary differential equations
, ψ = (1 + ψ 2 ) C − z ψ − ψ 2
The function z = z(x, t) in the solution is defined implicitly. Special case 3. 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 2 , C 1 2 e C 3 t+C 4 )+ C 3 x+C 5 ,
where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation. 2 ◦ . Solution:
x+B
kt + C
A ( x + B) 2 + − 2 (2 + ln 2) A x + D, where ë☎ì
w(x, t) = 2A arctan
x + B) ln
A, B, C, and D are arbitrary constants. References for equation
1.6.18.3: E. V. Lenskii (1966), I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989), N. H. Ibragimov (1994).
∂w
∂w
4. = f (x)g(w)h(w x )
The hodograph transformation, according to which x is taken to be the independent variable and w the dependent one,
x = u, w = y,
leads to a similar equation for u = u(y, t):
∂u
= g(y)f (u)h(u
y −2 )
where h(z) = z h(1/z).
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 −1 w(C 1 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 ◦ . The Euler transformation
∂u
w(x, t) + u(ξ, η) = xξ, x=
, t=η
leads to the linear equation
for details, see Subsection S.2.3 (Example 7).