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).