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).
Parts
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