Equations Involving Power Law Nonlinearities

4.3.1. Equations Involving Power Law Nonlinearities

1 ◦ . “Two-dimensional” generalized separable solution linear in y:

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

The first equation is a linear homogeneous telegraph equation. Given f = f (x, t), the second one represents a linear nonhomogeneous telegraph equation. For these equations, see the book by Polyanin (2002).

2 ◦ . There is a “two-dimensional” generalized separable solution quadratic in y:

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

3 ◦ . The substitution u = w + (c/b) leads to the special case of equation 4.3.1.4 with m = 1.

1 ◦ . Additive separable solution:

1 + 2 B a 2 ) t + D, where

w(x, y, t) = Akx + Bky + Ce − kt +

k(A 2 a

A, B, C, and D are arbitrary constants.

2 ◦ . Generalized separable solution linear in the space variables:

2 w(x, y, t) = (A −2 B

where A 1 , A 2 , B 1 , B 2 , C 1 , and C 2 are arbitrary constants.

3 ◦ . Traveling-wave solution in implicit form ( k ≠ 0):

2 2 2 2 2 kλ(a 2

1 β 1 a 2 β 2 ) w + [kλ(b 1 β 1 + b 2 β 2 − λ )− C 1 ( a 1 β 1 + a 2 β 2 )] ln( kλw + C 1 ) = k 2 λ 2 ( β 1 x+β 2 y + λt) + C 2 ,

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

4 ◦ . There is a generalized separable solution of the form w(x, y, t) = f (t)x 2 + g(t)xy + h(t)y 2 + ϕ(t)x + ψ(t)y + χ(t).

This is a special case of equation 4.3.1.6 with n = m = −1.

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

1 = C 1 w( ✬ C 1 x+C 2 , ✬ C 1 y+C 3 , t+C 4 ),

w 2 = w(x cos β + y sin β, −x sin β + y cos β, t), where C 1 , ...,C 4 and β are arbitrary constants, are also solutions of the equation (the plus or minus

signs are chosen arbitrarily).

2 at + A + Be − kt

w(x, y, t) =

k(sin y + Ce ) 2

1 (2 at + A + Be )

C 2 − kt

w(x, y, t) =

ke 2 x sinh 2 ( C 1 e − x sin y+C 2 )

C 1 2 (−2 at + A + Be − kt )

w(x, y, t) =

ke 2 x cosh 2 ( C 1 e − x

sin y+C 2 )

C 2 (2 at + A + Be − 1 kt )

w(x, y, t) =

A, B, C, C 1 , and C 2 are arbitrary constants.

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

w(x, y, t) = (Aat + B + Ce − kt ) Θ( e x,y) ,

where

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

✭✝✮ which occurs in combustion theory. For solutions of this equation, see 5.2.1.1.

Reference : N. H. Ibragimov (1994).

This is a special case of equation 4.3.1.6 with n = 0.

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

w 1 = −2 C 1 w( ✯ x+C 2 , ✯ C 1 m y+C 3 , t+C 4 ),

where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation (the plus or minus signs are chosen arbitrarily).

2 ◦ . “Two-dimensional” solution:

w(x, y, t) = u(x, t)y 2 /m ,

where the function u(x, t) is determined by the differential equation

For m = −2 and m = −1, this equation is linear.

3 ◦ . “Two-dimensional” multiplicative separable solution:

U (x, t)|y + C| 1 /(m+1) if m ≠ −1, w(x, y, t) =

U (x, t) exp(Cy)

if m = −1,

where

C is an arbitrary constant and the function U (x, t) is determined by the telegraph equation

For solutions of this linear equation, see the book by Polyanin (2002). ✭✝✮

Reference : N. H. Ibragimov (1994).

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

w −2

1 = C 1 w( ✰ C 1 n x+C 2 , ✰ C 1 n y+C 3 , t+C 4 ),

w 2 = w x cos β + y a/b sin β, −x b/a sin β + y cos β, t , where C 1 , ...,C 4 and β are arbitrary constants, are also solutions of the equation (the plus or minus

signs are chosen arbitrarily).

2 ◦ . Multiplicative separable solution:

w(x, y, t) = F (t)Φ(x, y),

where the function

F (t) is determined by the autonomous ordinary differential equation (C is an arbitrary constant)

(1) and the function Φ( x, y) satisfies the stationary equation

B, where A and B are arbitrary constants. For C = 0, equation (2) is reduced to the Laplace equation

Example. For

C = 0, it follows from (1) that F = Ae −kt +

where Ψ = Φ n+1 ,¯ x= √ ,¯

3 ◦ . “Two-dimensional” solution:

p w(x, y, t) = u(r, t), r= bx 2 + ay 2 ,

where the function u(r, t) is determined by the differential equation

w(x, y, t) = U (t)(bx 2 + ay 2 ) 1 /n ,

where the function U (t) is determined by the autonomous ordinary differential equation

ab(n + 1)

Reference : N. H. Ibragimov (1994).

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

w 1 = C 1 −2 w( ✰ C 1 n x+C 2 , ✰ C 1 m y+C 3 , t+C 4 ),

where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation (the plus or minus signs are chosen arbitrarily).

2 ◦ . Traveling-wave solution in implicit form:

dw = β 1 x+β 2 y + λt + C 2 ,

kλw + C 1 where C 1 , C 2 , β 1 , β 2 , and λ are arbitrary constants.

w(x, y, t) = U (ξ, t), ξ = βx + µy,

where β and µ are arbitrary constants, and the function U = U (ξ, t) is determined by the differential equation

Remark. There is a more general, “two-dimensional” solution of the form w(x, y, t) = V (ξ 1 , ξ 2 ), ξ 1 = β 1 x+µ 1 y+λ 1 t, ξ 2 = β 2 x+µ 2 y+λ 2 t, where the β i , µ i , and λ i are arbitrary constants.

4 ◦ . “Two-dimensional” solution:

w(x, y, t) = y 2 /m

u(z, t), − z = xy n/m ,

where the function u = u(z, t) is determined by the differential equation

m + ∂u nm amu + bnz u + bn(n − 3m − 4)zu +2 b(m + 2)u m+1 .

Reference : N. H. Ibragimov (1994).