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).