Equations of the Form ∂w = aw ∂ 2 + f (x, t, w) ∂w + g(x, t, w)

1.6.9. Equations of the Form ∂w = aw ∂ 2 + f (x, t, w) ∂w + g(x, t, w)

∂t

∂x

∂x

∂w

1. = aw

+ f (x)w + bx + c.

∂t

∂x 2

Generalized separable solution:

w(x, t) = (bx + c)t + Ax + B −

( x − ξ)f (ξ) dξ,

where

A, B, and x 0 are arbitrary constants.

∂w

2. = aw ∂t

+ f (t)w + g(t).

∂x 2

1 ◦ . Degenerate solution linear in x:

Z g(t)

w(x, t) = F (t)(Ax + B) + F (t)

A and B are arbitrary constants.

2 ◦ . Generalized separable solution quadratic in x:

2 g(t)

w(x, t) = ϕ(t)(x + Ax + B) + ϕ(t)

f (t) dt , where

ϕ(t) = F (t)

A, B, and C are arbitrary constants. ∂w

3. = aw

+ cw 2 2 + f (t)w + g(t).

∂t ∂x This is a special case of equation 1.6.10.1 with b = 0.

∂w

4. = aw

– ak w + f (x)w + b 1 sinh( kx) + b 2 cosh( kx).

∂t

∂x 2

Generalized separable solution:

w(x, t) = t b 1 sinh( kx) + b 2 cosh( kx) + ϕ(x).

Here, the function ϕ(x) is determined by the linear nonhomogeneous ordinary differential equation with constant coefficients

aϕ 2 ′′

xx − ak ϕ + f (x) = 0,

whose general solution is given by

ϕ(x) = C 1 sinh( kx) + C 2 cosh( kx) −

f (ξ) sinh k(x − ξ) dξ,

ak x 0

where

A, B, and x 0 are arbitrary constants.

5. = aw

+ ak w + f (x)w + b 1 sin( kx) + b 2 cos( kx).

∂t

∂x 2

Generalized separable solution:

w(x, t) = t b 1 sin( kx) + b 2 cos( kx) + ϕ(x).

Here, the function ϕ(x) is determined by the linear nonhomogeneous ordinary differential equation with constant coefficients

aϕ 2

xx ′′ + ak ϕ + f (x) = 0,

whose general solution is given by

ϕ(x) = C 1 sin( kx) + C 2 cos( kx) −

A, B, and x 0 are arbitrary constants.

+ g(t)w.

The transformation

G(t) = exp g(t) dt leads to a simpler equation of the form 1.1.9.1:

w(x, t) = G(t)u(z, τ ), z=x+

f (t) dt, τ=

G(t) dt,

+ g(t)w + h(t).

∂t

∂x 2 ∂x

This is a special case of equation 1.6.10.5. ∂w

g(t)w.

The transformation

F 2 ( t)G(t) dt, where the functions

w(x, t) = G(t)u(z, τ ), z = xF (t), τ=

F (t) and G(t) are given by

g(t) dt , leads to a simpler equation of the form 1.1.9.1:

F (t) = exp

f (t) dt , G(t) = exp

+ xf (t) + g(t)

h(t)w.

The transformation

w(x, t) = H(t)u(z, τ ), 2 z = xF (t) + g(t)F (t) dt, τ= F ( t)H(t) dt, where the functions

F (t) and H(t) are given by

h(t) dt , leads to a simpler equation of the form 1.1.9.1:

F (t) = exp

f (t) dt , H(t) = exp

∂u

= au

∂z 2

10. = aw ∂t

+ f (x)w

+ g(t)w + h(t).

∂x 2

∂x

Generalized separable solution:

w(x, t) = ϕ(t)Θ(x) + ψ(t),

where the functions ϕ(t), ψ(t), and Θ(x) are determined by the system of ordinary differential equations

t ′ = Cϕ + g(t)ϕ, ψ ′ t = Cϕ + g(t) ψ + h(t), aΘ ′′ xx + f (x)Θ ′ x = C,

where

C is an arbitrary constant. Integrating successively, one obtains

ϕ(t) = G(t) A 1 − C G(t) dt , G(t) = exp

g(t) dt ,

Z h(t)

ψ(t) = A 2 ϕ(t) + ϕ(t)

where A 1 , A 2 , B 1 , and B 2 are arbitrary constants.

+ f (x)w

+ g(x)w 2 + h(t)w.

∂t

∂x 2 ∂x

Multiplicative separable solution:

h(t) dt . Here,

w(x, t) = ϕ(x)H(t) A−B

H(t) dt , H(t) = exp

A and B are arbitrary constants, and the function ϕ(x) is determined by the second-order linear ordinary differential equation

aϕ ′′ xx + f (x)ϕ x ′ + g(x)ϕ = B.

For exact solutions of this equation with various f (x) and g(x), see Kamke (1977) and Polyanin and Zaitsev (2003).