Equations of the Form ∂w =F x, t, w, ∂w ,∂ w
8.1.4. Equations of the Form ∂w =F x, t, w, ∂w ,∂ w
∂t
∂x ∂x 2
+ h(t)w.
The transformation
F m+2n ( t)H m+n−1 ( t) dt, where the functions
w(x, t) = u(z, τ )H(t), z = xF (t) +
g(t)F (t) dt, τ=
F (t) and H(t) are given by
h(t) dt , leads to the simpler equation
F (t) = exp
f (t) dt , H(t) = exp
The last equation admits a traveling-wave solution, a self-similar solution, and a multiplicative separable solution.
The transformation
G(t) = exp g(t) dt , leads to the simpler equation
z = xG(t) + h(t)G(t) dt, τ=
The last equation admits a traveling-wave solution and a self-similar solution. ∂w
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 2 ∂w ∂ w
w = w(ξ), ξ = ax + bt,
where the function w(ξ) is determined by the ordinary differential equation
5. = f (t)x k Φ w, x
,x
+ xg(t)
Passing to the new independent variables
g(t) dt , we obtain a simpler equation of the form 8.1.3.6:
z = xG(t), τ=
f (t)G − k ( t) dt,
G(t) = exp
∂w
∂w
w, z
∂z
∂z 2
6. = wF t, ∂t
w ∂x 2
Multiplicative separable solution:
w(x, t) = ϕ(x) exp
F (t, λ) dt ,
where the function ϕ = ϕ(x) satisfies the linear ordinary differential equation f (x)ϕ ′′ xx = λϕ.
∂w
7. =w t, + f (t)e Φ λx ∂t
w ∂x 2
Generalized separable solution:
w(x, t) = e λx E(t) A+
f (t)
dt + Be − λx E(t), E(t) = exp
A, B, and λ are arbitrary constants.
∂w 2 1 ∂ w
8. – =w Φ t, + f (t)e λx + g(t)e λx .
∂t
w ∂x 2
Generalized separable solution:
λx Z f (t) − λx g(t) w(x, t) = e E(t) A+
dt + e E(t) B+
E(t) = exp
Φ( t, λ 2 ) dt ,
where
A, B, and λ are arbitrary constants.
There is a generalized separable solution of the form
w(x, t) = e λx ϕ(t) + e − λx ψ(t).
∂w
10. =w Φ t,
+ f (t) cosh(λx) + g(t) sinh(λx).
∂t
w ∂x 2
Generalized separable solution:
f (t) Z g(t) w(x, t) = cosh(λx)E(t) A+
dt + sinh( λx)E(t) B+ dt ,
E(t)
E(t)
E(t) = exp
Φ( t, λ 2 ) dt ,
where
A, B, and λ are arbitrary constants.
∂w 2 1 ∂ w
11. =w Φ t,
+ f (t) cos(λx).
∂t
w ∂x 2
Generalized separable solution:
f (t)
w(x, t) = cos(λx)E(t) A+
dt +
B sin(λx)E(t),
E(t)
E(t) = exp
Φ( t, −λ 2 ) dt ,
where
A, B, and λ are arbitrary constants.
2 ∂t + f (t) cos(λx) + g(t) sin(λx). w ∂x Generalized separable solution:
12. =w Φ t,
f (t) Z g(t) w(x, t) = cos(λx)E(t) A+
dt + sin( λx)E(t) B+
E(t) = exp
Φ( t, −λ 2 ) dt ,
where
A, B, and λ are arbitrary constants. ∂w
13. = wF 1 t,
2 + cos(λx)F 2 t,
2 + sin(λx)F 3 t,
There is a generalized separable solution of the form
w(x, t) = cos(λx)ϕ(t) + sin(λx)ψ(t).
+ f (t)e λx .
∂t
w ∂x w ∂x 2
Multiplicative separable solution:
w(x, t) = e λx E(t) A+
f (t)
dt , E(t) = exp
A, B, and λ are arbitrary constants. ∂w
1 ∂w
15. = f (t)w β
Φ x,
+ 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 8.1.3.10:
w(x, t) = G(t)u(x, τ ), τ=
f (t)G β−1 ( t) dt,
which has a multiplicative separable solution u = ϕ(x)ψ(τ ). ∂w
+ g(t)w + h(t).
Generalized separable solution:
w(x, t) = ϕ(t)Θ(x) + ψ(t),
where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations
ϕ ′ = Af (t)ϕ k t + 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
Θ ′ k x Φ x, Θ ′′ 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 of equation (3) is given by
Θ( x) = αe λx − B/A,
where α is an arbitrary constant, and λ is found from the algebraic (or transcendental) equation λΦ(λ) = A.
2 +g 1 (t)w + g 0 ∂t (t). ∂x ∂x ∂x Generalized separable solution:
17. = 1 (t)w + f 0 (t)
Φ x,
w(x, t) = ϕ(t)Θ(x) + ψ(t),
where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations (
C is an arbitrary constant):
ϕ ′ = Cf 1 ( t)ϕ t k+1 + g 1 ( t)ϕ,
(2) and the function Θ( x) is determined by the second-order ordinary differential equation
1 ( t)ϕ k + g 1 ( t)
0 ( t)ϕ k + g 0 ( t),
(3) The general solution of system (1), (2) is expressed as
Θ ′ k x Φ x, Θ ′′ xx /Θ ′ x
−1 /k
ϕ(t) = G(t)
A − kC
f 1 ( t)G k ( t) dt
, G(t) = exp
g 1 ( t) dt ,
dt
ψ(t) = Bϕ(t) + ϕ(t)
A, B, and C are arbitrary constants. Further, we assume that Φ is independent of x explicitly, i.e., Φ(x, y) = Φ(y). For Φ(0) ≠ 0 and Φ(0) ≠ ∞, particular solution to equation (3) has the form Θ(x) = αx + β, where α k Φ(0) =
C and β is an arbitrary constant. For k = 0, the general solution of equation (3) is expressed as
Θ( x) = αe λx + β,
where α and β are arbitrary constants, and λ is determined from the algebraic (transcendental) equation Φ( λ) = C.
∂w 2 ∂w ∂
18. = f (t)e βw
Φ x,
+ 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 8.1.3.11:
which has an additive separable solution u = ϕ(x) + ψ(τ ).
Multiplicative separable solution:
w(x, t) = (C 2
2 x + C 1 x+C 0 ) ϕ(t),
where C 0 , C 1 , and C 2 are arbitrary constants, and the function ϕ = ϕ(t) is determined by the ordinary
differential equation ✡✂☛ ϕ ′ t = ϕF t, 2C 2 ϕ, C 1 ϕ, 2C 0 ϕ
Reference : Ph. W. Doyle (1996), the case ∂ t
F ≡ 0 was treated.
+ h(t).
Additive separable solution:
w(x, t) = ϕ(x) +
h(t) dt,
where the function ϕ(x) is determined by the ordinary differential equation
+ h(t)w.
Multiplicative separable solution:
w(x, t) = C exp
h(t) dt ϕ(x),
where the function ϕ(x) is determined by the ordinary differential equation
22. =g 0 (t)F 0 + xg 1 (t)F 1 +x g 2 (t)F 2
+ h(t) 2 +
0 (t) + xp 1 (t)
+ q(t)w + ☞ 0 (t) + x ☞ 1 (t) + x ☞ 2 (t).
∂x
∂x
There is a generalized separable solution of the form
w(x, t) = x 2 ϕ(t) + xψ(t) + χ(t).
Generalized separable solution quadratic in x:
f 0 ( t, 2ϕ) dt + C 1 x+C 2 , where C 1 and C 2 are arbitrary constants, and the function ϕ = ϕ(t) is determined by the first-order
w(x, t) = x 2 ϕ(t) + x
f 1 ( t, 2ϕ) dt +
ordinary differential equation
ϕ ′ t = f 2 ( t, 2ϕ).
2 +f 0 ∂t t, ∂x ∂x ∂x 2 + g(t)w.
There is a generalized separable solution of the form
w(x, t) = x 2 ϕ(t) + xψ(t) + χ(t).