Equations of the Form ∂w =F t, w, ∂w ,∂ w
8.1.2. Equations of the Form ∂w =F t, w, ∂w ,∂ w
∂t
∂x ∂x 2
Suppose w(x, t) is a solution of this equation. Then the function
w 1 = w(x, t) + Ce at ,
where
C are arbitrary constants, is also a solution of the equation. ∂w
∂w
∂w ∂ 2 w
2. + f (t)w
=F t,
+ g(t)w.
Suppose w(x, t) is a solution of this equation. Then the function Z
f (t)ϕ(t) dt, where
w 1 = w x + ψ(t), t
ϕ(t) = C exp
g(t) dt , ψ(t) = −
C is an arbitrary constant, is also a solution of the equation.
Multiplicative separable solutions:
w(x, t) = A exp 2 λx + F (t, λ ) dt ,
w(x, t) = 2 F (t, λ ) dt ,
w(x, t) = 2 F (t, −λ ) dt , where
A, B, and λ are arbitrary constants. ∂w
Multiplicative separable solution:
w(x, t) = A exp λx +
F (t, λ, λ 2 ) dt ,
where
A and λ are arbitrary constants. ∂w 2 w
1 ◦ . Multiplicative separable solution for k ≠ −1:
w(x, t) =
1 ( k + 1)x + C 2 k+1 ϕ(t),
where the function ϕ = ϕ(t) is determined by the first-order ordinary differential equation ϕ ′ = ϕF t, C 1 ϕ t k+1 ,− kC 2 1 ϕ 2 k+2
2 ◦ . For k = −1, see equation 8.1.2.4. ∂w
∂w 1 ∂ 2 w
6. = f (t)w β
+ g(t)w.
∂t
w ∂x w ∂x 2
The transformation
G(t) = exp g(t) dt , leads to a simpler equation of the form 8.1.1.9:
w(x, t) = G(t)u(x, τ ), τ=
f (t)G β−1 ( t) dt,
which has a traveling-wave solution u = u(Ax + Bτ ) and a solution in the multiplicative form u = ϕ(x)ψ(τ ).
∂w
7. = f (t)
+ g(t)w + h(t).
Generalized separable solution:
w(x, t) = ϕ(t)Θ(x) + ψ(t), w(x, t) = ϕ(t)Θ(x) + ψ(t),
ϕ ′ t = Af (t)ϕ k + g(t)ϕ,
ψ ′ t = g(t)ψ + Bf (t)ϕ k + h(t),
C is an arbitrary constant, and the function Θ(x) is determined by the second-order ordinary differential equation
(3) The general solution of system (1), (2) is expressed as
Θ ′ k x ΦΘ ′′ xx /Θ ′ x
1 1− k
ϕ(t) = G(t)
C − kA
f (t)G k−1 ( t) dt
, G(t) = exp
g(t) dt ,
dt
ψ(t) = DG(t) + G(t)
( t) + h(t)
G(t)
where
A, B, C, and D are arbitrary constants. For k = 1 and Φ(x, y) = Φ(y), a solution to equation (3) is given by
Θ( x) = αe λx − B/A,
where α is an arbitrary constant and λ is determined from the algebraic (or transcendental) equation λΦ(λ) = A.
∂w
∂w ∂ 2 w
8. = f (t)e βw Φ
+ g(t).
∂t
∂x ∂x 2
The transformation
Z w(x, t) = u(x, τ ) + G(t), τ=
G(t) = g(t) dt, leads to a simpler equation of the form 8.1.1.10:
which has a traveling-wave solution u = u(Ax+Bτ ) and an additive separable solution u = ϕ(x)+ψ(τ ). ∂w 2 ∂w ∂ w
∂w
9. = f (t) Φ w,
2 + g(t)
With the transformation
f (t) dt one arrives at the simpler equation
w = U (z, τ ),
z=x+
g(t) dt,
which has a traveling-wave solution U = U (kz + λτ ). ∂w
Multiplicative separable solutions:
w(x, t) = 1 sin x a 2 cos x a F (t, 0) dt
if
a > 0,
w(x, t) = p
a < 0, where C 1 and C 2 are arbitrary constants.
1 sinh x | a|
2 cosh x | a|
F (t, 0) dt if
1 ◦ . Multiplicative separable solution for a > 0:
w(x, t) =
1 sin x a 2 cos x a
where C 1 and C 2 are arbitrary constants, and the function ϕ = ϕ(t) is determined by the ordinary differential equation ϕ ′
2 t = ϕF t, a(C 2 1 + C 2 2 ) ϕ ,− a
2 ◦ . Multiplicative separable solution for a < 0:
√ w(x, t) = C 1 e | a| x + C 2 e − | a| x
where C 1 and C 2 are arbitrary constants, and the function ϕ = ϕ(t) is determined by the ordinary
differential equation ϕ ′ = ϕF t, 4C 1 C 2 aϕ t 2 ,− a
Example. For C 1 C 2 =0 , a solution is given by
w(x, t) = C exp
| a| x +
F (t, 0, −a) dt ,
where C is an arbitrary constant.
1 ◦ . Multiplicative separable solution:
w(x, t) = C exp 2 λx + F (t, λ , 0) dt ,
where
C and λ are arbitrary constants.
2 ◦ . Multiplicative separable solution:
w(x, t) = (Ae λx + Be − λx ) ϕ(t),
where
A, B, and λ are arbitrary constants, and the function ϕ = ϕ(t) is determined by the ordinary
differential equation ϕ ′ = ϕF t, λ 2 t ,4 ABλ 2 ϕ 2
3 ◦ . Multiplicative separable solution:
w(x, t) = [A sin(λx) + B cos(λx)]ϕ(t),
where
A, B, and λ are arbitrary constants, and the function ϕ = ϕ(t) is determined by the ordinary
differential equation ϕ ′ = ϕF t, −λ 2 ,− λ 2 2 2 ✞✂✟ 2 t ( A + B ) ϕ
Reference : Ph. W. Doyle (1996), the case ∂ t