Equations of the Form ∂w =a∂ w n ∂w w k +b∂ ∂w

2.1.2. Equations of the Form ∂w =a∂ w n ∂w w k +b∂ ∂w

∂t

∂x

∂x

∂y ∂y

This is a special case of equation 2.1.3.1 with c = 0. ∂w

Boussinesq equation. It arises in nonlinear heat conduction theory and the theory of unsteady flows through porous media with a free surface (see Polubarinova–Kochina, 1962). This is a special case of equation 2.1.2.4 with n = 1.

1 ◦ . Solution linear in all independent variables:

2 w(x, y, t) = Ax + By + (A 2 + B ) t + C,

where

A, B, and C are arbitrary constants.

2 ◦ . Traveling-wave solution ( k 1 , k 2 , and λ are arbitrary constants): w = w(ξ),

ξ=k 1 x+k 2 y + λt,

where the function w(ξ) is determined by the ordinary differential equation

2 λw 2 ′

ξ =( k 1 + k 2 )( ww ′ ′ ξ ) ξ .

The solution of this equation can be written out in implicit form:

A and B are arbitrary constants.

3 ◦ . Generalized separable solution quadratic in the space variables:

2 w(x, y, t) = f (t)x 2 + g(t)xy + h(t)y , 2 w(x, y, t) = f (t)x 2 + g(t)xy + h(t)y ,

t ′ =6 f +2 fh+g ,

g t ′ = 6( f + h)g,

(3) It follows from (1) and (3) that f t ′ − h ′ t = 6( f + h)(f − h). Further, using (2) and assuming g

t =6 h +2 fh+g .

0, we find that f = h + Ag, where A is an arbitrary constant. With this relation, we eliminate h from (2) and (3) to obtain a nonlinear ordinary differential equation for g(t):

3 gg ′′ tt −5 g t ′2

2 − 36(1 + 4 A ) g = 0.

B is an arbitrary constant)

On solving this equation with the change of variable u(g) = (g t ′ ) 2 , we obtain (

g ′ t = gΦ(g),

Φ( g) = ✁ pBg 4 /3 + 36(1 + A 2 ) g 2 , (4)

1 1 1 h= 1

12 Φ( g) − 2 Ag, f= 12 Φ( g) + 2 Ag,

where the first equation is separable, and, hence, its solution can be written out in implicit form. In the special case

B = 0, the solution can be represented in explicit form (C is an arbitrary constant):

, g(t) =

, h(t) =

4 ◦ . Generalized separable solution (generalizes the solution of Item 3 ◦ ):

2 w(x, y, t) = f (t)x 2 + g(t)xy + h(t)y + ϕ(t)x + ψ(t)y + χ(t), where the functions f (t), g(t), h(t), ϕ(t), ψ(t), and χ(t) are determined by the system of ordinary

differential equations

t =6 f +2 fh+g , ϕ t ′ = 2(3 f + h)ϕ + 2gψ,

g ′ t = 6( f + h)g,

ψ t ′ =2 gϕ + 2(f + 3h)ψ,

2 t = ϕ + ψ + 2( f + h)χ. The first three equations for f , g, and h can be solved independently (see Item 3 ◦ ).

2 2 =6 2 h +2 fh+g , χ ′

Example. Solution:

w(x, t) = −

+ Cxt −1 /3 + 3 /3

where C is an arbitrary constant.

5 ◦ . There is a “two-dimensional” solution in multiplicative separable form:

w(x, y, t) = (At + B) −1 Θ( x, y),

where

A and B are arbitrary constants, and the function Θ is determined by the stationary equation written out in Item 4 ◦ ✂☎✄ of equation 2.1.2.4 with n = α = 1.

References : S. S. Titov and V. A. Ustinov (1985), V. V. Pukhnachov (1995), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

This is a special case of equation 2.1.2.4 with n = −1.

αA(k 1 + k 2 )

where

A, B, k 1 , k 2 , and λ are arbitrary constants.

2 ◦ . Solutions:

2 αt + B

w(x, y, t) =

(sin y + Ae x

2 A 2 αt + C w(x, y, t) = 2 2 −

e x sinh Ae x sin y+B

C − 2A 2 αt

w(x, y, t) =

cosh 2 Ae − x sin y+B

2 2 A αt + C w(x, y, t) =

e 2 x cos 2 Ae − x sin y+B

where

A, B, and C are arbitrary constants.

3 ◦ . The exact solutions specified in Item 2 ◦ are special cases of a more general solution having the form of the product of two functions with different arguments:

w(x, y, t) = (Aαt + B)e Θ( x,y) ,

where

A and B are arbitrary constants, and the function Θ(x, y) is a solution of the stationary equation

which is encountered in combustion theory. For solutions of this equation, see 5.2.1.1. ✆☎✝

References : V. A. Dorodnitsyn, I. V. Knyazeva, and S. R. Svirshchevskii (1983), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

4 ◦ . Other exact solutions:

2 sinh( αt + C) cosh(αt + C)

w(x, y, t) = (

2 2 x + A) 2 sinh ( 2 ,

αt + C) + (y + B) cosh ( αt + C)

sin( µy + η 0 )

w(x, y, t) =

sin( µy + η 0 )

w(x, y, t) =

A coth θ(t) + B sinh θ(t) e µx A ,

sinh θ(t)

sin( µy + η 0 )

w(x, y, t) =

A cot θ(t) + B sin θ(t) e µx A ,

sin θ(t)

A −1

1 + sin θ(t)

1 − sin θ(t)

w(x, y, t) =

A cosh( µx + ξ 0 )+s A sin( µy + η 0 ) , cos θ(t)

2 cos θ(t)

2 cos θ(t)

1 + sin θ(t) w(x, y, t) = −

A −1

1 − sin θ(t)

A cosh( µx + ξ 0 )+s A sin( µy + η 0 ) ,

cos θ(t)

2 cos θ(t)

2 cos θ(t)

1 + cosh θ(t) w(x, y, t) =

A −1

1 − cosh θ(t)

A cosh( µx + ξ 0 )+s A sin( µy + η 0 ) , sinh θ(t)

2 sinh θ(t)

2 sinh θ(t)

1 − cosh θ(t) w(x, y, t) = −

A −1

1 + cosh θ(t)

A cosh( µx + ξ 0 )+s A sin( µy + η 0 ) ,

sinh θ(t)

2 sinh θ(t)

2 sinh θ(t)

A, B, µ, ξ 0 , η 0 , and τ 0 are arbitrary constants; and s is a parameter that admits the values 1 or −1 (the first solution was indicated by Pukhnachov, 1995). By swapping the variables, x ⇄ y, in the above relations, one can obtain another group of solutions (not written out here).

5 ◦ . Solutions with axial symmetry:

λ 2 r λ−2

w(r, t) =

+ Ce αt λϕr ϕ−2

w(r, t) =

where r= px + y ;

1 , C, C C 2 , and λ are arbitrary constants.

Reference : S. N. Aristov (1999).

6 ◦ . The transformation w = 1/U leads to an equation of the form 2.1.4.3 with β = 0:

This is a two-dimensional heat and mass transfer equation with power-law temperature-dependent thermal conductivity (diffusion coefficient), where n can be integer, fractional, and negative. This is a special case of equation 2.4.3.3 with f (w) = αw n .

1 ◦ . Suppose w(x, y, t) is a solution of the equation in question. Then the functions

w 2 = w(x cos β − y sin β, x sin β + y cos β, t), where C 1 , ...,C 5 and β are arbitrary constants, are also solutions of the equation. The plus or

minus signs can be chosen arbitrarily.

2 ◦ . Traveling-wave solution:

1 , C, k k 2 , and λ are arbitrary constants.

3 ◦ . Traveling-wave solution in implicit form (generalizes the solution of Item 2 ◦ ):

λw + C 1 where C 1 , C 2 , k 1 , k 2 , and λ are arbitrary constants.

4 ◦ . Multiplicative separable solution: w(x, y, t) = f (t)Θ(x, y), −1 f (t) = (Aαnt + B) /n .

Here,

A and B are arbitrary constants, and the function Θ(x, y) is a solution of the two-dimensional stationary equation

If n ≠ −1, this equation can be reduced to

u + A(n + 1)u n+1

2 ∂x + ∂y 2 , u=Θ n+1 .

w(x, y, t) = F (z, t),

z=k 1 x+k 2 y;

p 2 w(x, y, t) = G(r, t), 2 r= x + y ;

w(x, y, t) = H(ξ 1 , ξ 2 ),

ξ 1 = k 1 x+λ 1 t, ξ 2 = k 2 y+λ 2 t;

w(x, y, t) = t β

w(x, y, t) = e 2 βt

V (ζ − , ζ 2 ), ζ 1 = xe βnt , ζ 2 = ye βnt ,

where k 1 , k 2 , λ 1 , λ 2 , and β are arbitrary constants.

6 ☛☎☞ ◦ . See also equations 2.5.5.5 and 2.5.5.6 for the case of two space variables.

References : V. A. Dorodnitsyn, I. V. Knyazeva, and S. R. Svirshchevskii (1983), S. S. Titov and V. A. Ustinov (1985), J. R. King (1993), V. V. Pukhnachov (1995).

This is a special case of equation 2.4.3.4 with f (w) = a 1 w n 1 and g(w) = a 2 w n 2 .

1 ◦ . Suppose w(x, y, t) is a solution of the equation in question. Then the functions

= 2 A w( ✌ A n 1 Bx + C 1 , ✌ A n 2 By + C 2 , B t+C 3 ), where

A, B, C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation (the plus or minus signs can be chosen arbitrarily).

2 ◦ . Traveling-wave solution in implicit form:

dw = k 1 x+k 2 y + λt + C 2 ,

λw + C 1

☛☎☞ where C 1 , C 2 , k 1 , k 2 , and λ are arbitrary constants.

Reference : A. A. Samarskii and I. M. Sobol’ (1963).

3 ◦ . “Two-dimensional” solution:

− 1 w(x, y, t) = t +1) k U (ξ, η), ξ = xt 2 ( kn 1 , η = yt 2 ( kn 2 , where k is an arbitrary constant and the function U (ξ, η) is determined by the differential equation

4 ◦ . “Two-dimensional” solution:

w(x, y, t) = e 2 βt

1 V (z , z 2 ), z 1 = − xe βn 1 t , z 2 = − ye βn 2 t , where β is an arbitrary constant and the function V (z 1 , z 2 ) is determined by the differential equation

5 ◦ . There is a “two-dimensional” solution of the form w(x, y, t) = F (ξ 1 , ξ 2 ), ξ 1 = α 1 x+β 1 y+γ 1 t, ξ 2 = α 2 x+β 2 y+γ 2 t, ☛☎☞ where the α i , β i , and γ i are arbitrary constants.

References : V. A. Dorodnitsyn, I. V. Knyazeva, and S. R. Svirshchevskii (1983), N. H. Ibragimov (1994).

2.1.3. Equations of the Form ∂w =∂

f (w)∂w +∂ g(w)∂w +h(w)

∂t ∂x

∂x ∂y ∂y

1 ◦ . Suppose w(x, y, t) is a solution of this equation. Then the functions

where C 1 , ...,C 5 are arbitrary constants, are also solutions of the equation (the plus or minus signs can be chosen arbitrarily).

2 ◦ . Traveling-wave solution in implicit form:

2 2 2 bk 2

2 w + (ak 1 + ck 2 − C 1 bk 2 ) ln | w+C 1 |= λ(k 1 x+k 2 y + λt) + C 2 ,

where C 1 , C 2 , k 1 , k 2 , and λ are arbitrary constants.

3 ◦ . Solution:

2 2 w = u(z) − 4abC 2

z = y + bC 1 x + bC 2 x+C 3 t, where C 1 , C 2 , and C 3 are arbitrary constants and the function u(z) is determined by the first-order

1 x −4 abC 1 C 2 x,

ordinary differential equation

2 ) u z ′ + (2 abC 1 − C 3 ) u = 8a bC 1 z+C 4 . With appropriate translations in both variables, one can reduce this equation to a homogeneous one,

2 2 2 ( 2 bu + c + ab C

which can be integrated by quadrature.

4 ◦ . Generalized separable solution linear in y (a degenerate solution):

w = F (x, t)y + G(x, t),

where the functions

F and G are determined by solving the one-dimensional equations

Equation (1) is a linear homogeneous heat equation. Given

F = F (x, t), equation (2) can be treated as a linear nonhomogeneous heat equation. For these equations, see the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

5 ◦ . Generalized separable solution quadratic in y:

w = f (x, t)y 2 + g(x, t)y + h(x, t),

where the functions f = f (x, t), g = g(x, t), and h = h(x, t) are determined by the system of differential equations

Here, the subscripts denote partial derivatives.

w = |y + C| 1 /2

c θ(x, t) − ,

where the function θ(x, t) is determined by the linear heat equation

7 ◦ . “Two-dimensional” solution:

where C 1 and C 2 are arbitrary constants and the function U (ξ, t) is determined by a differential equation of the form 1.10.1.1:

8 ◦ . “Two-dimensional” solution:

2 w = V (η, t) − 4abC 2

1 x −4 abC 1 C 2 x, 2 η = y + bC 1 x + bC 2 x, where C 1 and C 2 are arbitrary constants and the function

V (η, t) is determined by the differential equation

This is a two-dimensional heat and mass transfer equation with a linear temperature-dependent thermal conductivity (diffusion coefficient).

The substitution U = αw + β leads to an equation of the form 2.1.2.2 for U = U (x, y, t). ∂w

This is a two-dimensional heat and mass transfer equation with a hyperbolic temperature-dependent thermal conductivity (diffusion coefficient).

The substitution U = αw + β leads to an equation of the form 2.1.2.3 for U = U (x, y, t). ∂w

1 ◦ . The transformation

βt

w(x, y, t) = e u(x, y, τ ), τ=C− e βt ,

where

C is an arbitrary constant, leads to a simpler equation of the form 2.1.2.3:

2 ◦ . In Zhuravlev (2000), a nonlinear superposition principle is presented that allows the construction of complicated multimodal solutions of the original equation; some exact solutions are also specified there.

The substitution ✎☎✏ w = 1/U leads to an equation of the form 2.1.4.3 for U = U (x, y, t).

References : V. A. Galaktionov and S. A. Posashkov (1989), N. H. Ibragimov (1994).

The transformation (

C is an arbitrary constant)

w(x, y, t) = e βt

leads to a simpler equation of the form 2.1.2.4:

1 ◦ . Suppose w(x, y, t) is a solution of this equation. Then the functions w 2 1 = −1 A w( ✑ A k−n 1 x+B 1 , ✑ A k−n 2 −1 y+B 2 , 2 A k−2 t+B 3 ),

where

1 , A, B B 2 , and B 3 are arbitrary constants, are also solutions of the equation (the plus or minus signs can be chosen arbitrarily).

2 ◦ . Traveling-wave solution:

w(x, y, t) = u(z), z=t−λ 1 x−λ 2 y,

where λ 1 and λ 2 are arbitrary constants, and the function u = u(z) is determined by the ordinary differential equation

u ′ = [( a 1 λ 2 1 u n 1 z 2 + a 2 λ 2 u n 2 ) u ′ z ] ′ z + bu k .

3 ◦ . “Two-dimensional” solution:

1 n 1 − k+1

n 2 − k+1

F (ξ, η), ξ = x(αt + β) 2( k−1) , η = y(αt + β) 2( k−1) , where the function

w(x, y, t) = (αt + β) 1− k

F = F (ξ, η) is determined by the differential equation

References : V. A. Dorodnitsyn, I. V. Knyazeva, and S. R. Svirshchevskii (1983), M. I. Bakirova, S. N. Dimova, V. A. Dorodnitsyn, S. P. Kurdyumov, A. A. Samarskii, and S. R. Svirshchevskii (1988), N. H. Ibragimov (1994).