Equations of the Form ∂w =a ∂ w ∂w +f (x, t) ∂w +g(x, t, w)
1.6.12. Equations of the Form ∂w =a ∂ w ∂w +f (x, t) ∂w +g(x, t, w)
∂t
∂x
∂x
∂x
+ f (t).
∂t ∂x
∂x
Generalized separable solutions linear and quadratic in x:
w(x, t) = C 2
w(x, t) = −
6 a(t + C 1 ) where C 1 , C 2 , and C 3 are arbitrary constants. The first solution is degenerate.
2. = a w
+ f (t)w + g(t).
∂t ∂x
∂x
This is a special case of equation 1.6.13.4 with m = 1. ∂w
∂w
3. = a w
+ bw 2 + f (t)w + g(t).
∂t ∂x
∂x
This is a special case of equation 1.6.13.5 with m = 1. ∂w
∂w
4. = a w
+ f (x)w 2 .
∂t ∂x
∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(x, C 1 t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solution:
w = (λt + C) −1 ϕ(x),
where λ and C are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
a(ϕϕ ′ x ) ′ x + f (x)ϕ 2 + λϕ = 0.
+ g(t)w.
This is a special case of equation 1.6.13.8 with m = 1. The transformation
G(t) dt, G(t) = exp g(t) dt , leads to a simpler equation of the form 1.1.10.1:
w(x, t) = G(t)u(z, τ ), z=x+
+ g(t)w.
The transformation
w(t, x) = u(z, τ )G(t), 2 z = xF (t), τ= F ( t)G(t) dt, where the functions
F (t) and G(t) are given by
g(t) dt , leads to a simpler equation of the form 1.1.10.1:
F (t) = exp
f (t) dt , G(t) = exp
+ xf (t) + g(t)
+ h(t)w.
This is a special case of equation 1.6.13.10 with m = 1.
∂t
∂x
∂x
∂x
∂w ∂
–4 /3 1. ∂w = a w + f (x)w –1 /3 .
∂t ∂x
∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w −4
C /3
1 w(x, C 1 t+C 2 ),
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . The substitution −3 w=v leads to an equation of the form 1.6.11.1:
∂v
= 4 ∂ 2 v − 1 av 5 2 3 f (x)v .
∂t
∂x
3 ◦ . Suppose u = u(x) is any nontrivial particular solution of the second-order linear ordinary differential equation
au ′′ xx − 1 3 f (x)u = 0.
The transformation
simplifies the original equation, bringing it to equation 1.1.10.4:
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
∂w ∂
m 2. ∂w = a w + f (t)w.
∂t ∂x
∂x
The transformation
f (t) dt , leads to a simpler equation of the form 1.1.10.7:
w(x, t) = u(x, τ )F (t), τ=
If m = −1 or m = −2, see 1.1.10.2 or 1.1.10.3 for solutions of this equation. ∂w
3. = a w m
∂w
+ f (t)w 1– m .
∂t ∂x
∂x
The substitution u=w m leads to an equation of the form 1.6.10.2:
∂u 2 ∂ 2 u a ∂u
= au
+ mf (t),
which admits a generalized separable solution of the form u = ϕ(t)x 2 + ψ(t)x + χ(t). ∂w
4. = a w m ∂w + f (t)w + g(t)w 1– m .
∂t ∂x
∂x
The substitution u=w m leads to an equation of the form 1.6.10.2:
+ mf (t)u + mg(t),
which admits a generalized separable solution of the form u = ϕ(t)x 2 ➷☎➬ + ψ(t)x + χ(t).
References : V. A. Galaktionov (1995), V. F. Zaitsev and A. D. Polyanin (1996).
5. = a w m
+ bw 1+ m + f (t)w + g(t)w 1– m .
∂t ∂x
∂x
For b = 0, see equation 1.6.13.4: The substitution u=w m leads to an equation of the form 1.6.10.1:
∂u 2 ∂ 2 u a ∂u
= au
+ bmu + mf (t)u + mg(t),
which admits generalized separable solutions of the following forms:
u(x, t) = ϕ(t) + ψ(t) exp( ➮ λx), u(x, t) = ϕ(t) + ψ(t) cosh(λx + C), u(x, t) = ϕ(t) + ψ(t) sinh(λx + C), u(x, t) = ϕ(t) + ψ(t) cos(λx + C),
where the functions ϕ(t) and ψ(t) are determined by systems of appropriate first-order ordinary differential equations, the parameter λ is a root of a quadratic equation, and C is an arbitrary ➱☎✃ constant.
References : V. A. Galaktionov (1995), V. F. Zaitsev and A. D. Polyanin (1996).
∂w ∂
6. = a w m ∂w + f (x)w 1+ m .
∂t ∂x
∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(x, C 1 m t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solution:
w = (λmt + C) −1 /m ϕ(x), where λ and C are arbitrary constants, and the function ϕ(x) is determined by the equation
ψ=ϕ m+1 . The book by Polyanin and Zaitsev (2003) presents exact solutions of this equation for various f (x). ∂w
aψ xx ′′ +( m + 1)f (x)ψ + λ(m + 1)ψ m+1 = 0,
m 7. ∂w = a w + g(x)w m+1 + f (t)w.
∂t ∂x
∂x
Multiplicative separable solution:
w = ϕ(x)ψ(t),
where the functions ϕ = ϕ(x) and ψ = ψ(t) are determined by the ordinary differential equations ( C 1 is an arbitrary constant)
a(ϕ m ϕ ′ x ) ′ x + g(x)ϕ m+1 + C 1 ϕ = 0, ψ ′ − f (t)ψ + C 1 ψ t m+1 = 0.
The general solution of the second equation is given by ( C 2 is an arbitrary constant)
−1 /m
ψ(t) = e F C 2 + mC 1 e mF dt
F=
f (t) dt.
8. = a w m
+ f (t)
+ g(t)w.
The transformation
G m ( t) dt, G(t) = exp g(t)dt , leads to a simpler equation of the form 1.1.10.7:
w(x, t) = u(z, τ )G(t), z=x+
+ g(t)w.
The transformation
w(t, x) = u(z, τ )G(t), 2 z = xF (t), τ= F ( t)G m ( t) dt, where the functions
F (t) and G(t) are given by
g(t) dt , leads to a simpler equation of the form 1.1.10.7:
F (t) = exp
f (t) dt , G(t) = exp
In the special case ❐☎❒ m = −2, this equation can be transformed to the linear heat equation (see 1.1.10.3).
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
+ xf (t) + g(t)
+ h(t)w.
The transformation
F 2 ( t)H m ( 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 a simpler equation of the form 1.1.10.7:
F (t) = exp
f (t) dt , H(t) = exp
❐☎❒ If m = −1 or m = −2, see 1.1.10.2 or 1.1.10.3 for solutions of this equation.
Reference : V. F. Zaitsev and A. D. Polyanin (1996).