Other Equations Explicitly Independent of x and t

1.2.4. Other Equations Explicitly Independent of x and t

On passing from t, x to the new variables t, z = x + βt, one arrives at a simpler equation of the form 1.2.1.3:

This is a special case of equation 1.6.3.7 with f (w) = be λw .

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

2 1 w 1 = w(C 1 x+C 2 , C 1 t+C 3 )+ ln C 1 , λ

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Apart from the traveling wave w = w(x + λt), there is also an exact solution of the form

This is a special case of equation 1.6.6.8 with f (w) = ae λw . The substitution

leads to the linear heat equation ∂ t u=∂ xx u. ∂w

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

C 1 a w 1 1 = 1 w e x+ C 1 e C 1 t+C 2 , e C t+C 3 − C 1 ,

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Traveling-wave solution in implicit form:

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

3 ◦ . Solution:

w(x, t) = u(z) + ln t, z=

− ln t,

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

au − z −

= b(e λ uu ′

5. = ae λw ∂ w ∂t

∂x 2

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

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

where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . Traveling-wave solution:

w(x, t) = ln C 2 exp

z = kx + βt,

ak 2

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

3 ◦ . Additive separable solutions:

1 cos 2 ( C 2 x+C 3 )

w(x, t) = ln

2 ( at + C 1 )

1 2 sinh ( C

2 x+C 3 )

w(x, t) = ln

2 ( at + C 1 )

1 cosh 2 ( C 2 x+C 3 )

w(x, t) = ln

2 ( C 1 − at)

where C 1 , C 2 , and C 3 are arbitrary constants; note that ln( A/B) = ln |A| − ln |B| for AB > 0.

4 ◦ . Self-similar solution:

w = w(y),

y = x/ t,

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

ae λw w yy ′′ + 1 2 yw ′ y = 0.

5 ◦ . Solution:

w(x, t) = U (ξ) + 2kt, − ξ = xe kλt ,

where k is an arbitrary constant, and the function U = U (ξ) is determined by the ordinary differential equation

2 k − kλξU ξ ′ = ae λU U ξξ ′′ .

6 ◦ . Solution:

1 w(x, t) = F (ζ) − ln t, ζ = x + β ln t,

where β is an arbitrary constant, and the function F = F (ζ) is determined by the autonomous ordinary differential equation

w(x, t) = G(θ) −

ln t, θ = xt b ,

where b is an arbitrary constant, and the function G = G(θ) is determined by the ordinary differential equation

2 b+1

+ bθG ′

θ = ae λG G ′′ θθ .

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

w 1 = w( ➀ C 1 λ−1 x+C 2 , C 1 2 λ t+C 3 ) + 2 ln | C 1 |,

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . Traveling-wave solution:

w = w(ξ), ξ = kx + βt,

where k and β are arbitrary constants, and the function w(ξ) is determined by the autonomous ordinary differential equation

ak 2 e w w ′′

ξξ − βw ′ ξ + be λw = 0.

3 ◦ . Solution for λ ≠ 0:

w(x, t) = u(z) − ln t,

z = 2 ln x +

ln t,

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

2 aλe u−z 2 u ′′ zz − u ′ z + bλe λu = (1 − λ)u ′ z − 1.

4 ◦ . Additive separable solution for λ = 1:

w(x, t) = − ln(kt + C) + ϕ(x),

where the function ϕ(x) is determined by the autonomous ordinary differential equation

aϕ − ′′

xx + b + ke ϕ = 0.

5 ◦ . Additive separable solutions for λ = 0: