Equations of the Form ∂w λw =a∂ e ∂w + f (w)
1.2.2. Equations of the Form ∂w λw =a∂ e ∂w + f (w)
∂t
∂x
∂x
This equation governs unsteady heat transfer in a quiescent medium in the case where the thermal diffusivity is exponentially dependent on temperature.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. 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 ◦ . Solutions:
2 ⑨ x+A
w(x, t) =
ln √
B − 2at
w(x, t) = − ln
C − 2aλµt + ln λµx 2 + Ax + B ,
where
A, B, C, and µ are arbitrary constants. The first solution is self-similar and the second one is an additive separable solution.
3 ◦ . Traveling-wave solution in implicit form:
Z e λw dw x + βt + C 1 = a .
βw + C 2
4 ◦ . Self-similar solution:
w = w(y),
y = x/ t,
where the function w(y) is determined by the ordinary differential equation
a(e λw w ′ y ) ′ y + 1 2 yw ′ y = 0.
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 ′ = a(e λU U ξ ′ ) ′ ξ .
6 ◦ . Solution:
1 w(x, t) = F (ζ) − ln t, ζ = x + β ln t, λ
where β is an arbitrary constant, and the function F = F (ζ) is determined by the first-order ordinary differential equation (
C is an arbitrary constant)
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 ′ θ ) θ ′ .
8 ◦ . The substitution ϕ=e λw leads to an equation of the form 1.1.9.1:
References : L. V. Ovsiannikov (1959, 1982), N. H. Ibragimov (1994), A. A. Samarskii, V. A. Galaktionov, S. P. Kur- dyumov, and A. P. Mikhailov (1995).
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = w(C 1 x+C 2 , t+C 3 ) − 2 ln | C 1 |,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Additive separable solutions:
w(x, t) = ln |C 1 x+C 2 |+ bt + C 3 , w(x, t) = 2 ln | ❹ x+C 1 | − ln C 2 e − bt 2 − a .
The first solution is degenerate.
3 ◦ . The transformation
w = bt + u(x, τ ), τ= e bt + const
leads to an equation of the form 1.2.2.1:
a exp[2( ❹ ax + B)] + 2 exp( ❹ ax + B) + A
w(x, t) = ln ax − B,
2 a 2 ( t + C)
where
A, B, and C are arbitrary constants.
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions
w 1 = w( ❻ x+C 1 , C 2 t+C 3 ) + ln C 2 ,
where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.
2 ◦ . Additive separable solution:
w(x, t) = u(x) − ln(aC 1 t+C 2 ),
where C 1 and C 2 are arbitrary constants, and the function u = u(x) is determined by the ordinary differential equation
Integrating yields the general solution in implicit form:
The integral is computable, so the solution can be rewritten in explicit form (if a = 1 and b > 0, see
1.2.2.3 for a solution).
3 ◦ . The substitution u=e w leads to an equation of the form 1.1.9.9:
1 ◦ . Additive separable solution for a=k 2 > 0: w(x, t) = ln C 1 cos( kx) + C 2 sin( kx) + bt + C 3 , where C 1 , C 2 , and C 3 are arbitrary constants.
2 ◦ . Additive separable solution for a = −k 2 < 0:
w(x, t) = ln C 1 cosh( kx) + C 2 sinh( kx) + bt + C 3 .
3 ◦ . The transformation
w = bt + u(x, τ ), τ= e bt + const
leads to an equation of the form 1.2.2.4:
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions
w 1 = w( ❻ C 1 λ−1 x+C 2 , 2 C 1 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 ◦ . 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 − z 2( e u u ′
z ′ − e u ′ z + bλe = (1 − λ)u ′ z − 1.
λu
Functional separable solution:
where C 1 and C 2 are arbitrary constants. ∂w
This is a special case of equation 1.6.14.4 with f (t) = c and g(t) = s. Functional separable solutions:
w= ln e αt
C 1 cos( x β)+C 2 sin( x β) + γ if abλ > 0,
po w= ln e αt C 1 cosh( x − β)+C 2 sinh( x − β) + γ
if abλ < 0.
λ Here, C 1 and C 2 are arbitrary constants and
α = λ(bγ + c), β = bλ/a, ❽☎❾ where γ=γ 1,2 are roots of the quadratic equation bγ 2 + cγ + s = 0.
Reference : V. A. Galaktionov and S. A. Posashkov (1989).