Equations Involving Second-Order Mixed Derivatives
9.5.1. Equations Involving Second-Order Mixed Derivatives
1. = aw
2 + ∂x∂t b ∂x ∂x 3 .
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = C 1 w(C 1 x + aC 1 ϕ(t), C 2 1 t+C 2 )+ ϕ ′ t ( t),
where C 1 and C 3 are arbitrary constants and ϕ(t) is an arbitrary function, is also a solution of the equation.
2 ◦ . There are exact solutions of the following forms:
w = U (z),
z = x + λt
traveling-wave solution;
w = |t| −1 /2
V (ξ), ξ = x|t| −1 /2 self-similar solution.
∂w 2 ∂ 2 w
2 = ∂x∂t ν ∂x ∂x ∂x 3 .
This equation occurs in fluid dynamics; see 9.3.3.1, equation (2) and 10.3.3.1, equation (4) with
f 1 ( t) = 0.
1 ◦ . Suppose w = w(x, t) is a solution of the equation in question. Then the functions
w 1 = w(x + ψ(t), t) + ψ ′ t ( t), w 2 = C 1 w(C 1 x+C 1 C 2 t+C 3 , C 1 2 t+C 4 )+ C 2 ,
where ψ(t) is an arbitrary function and C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Solutions:
w(x, t) =
+ ψ(t),
C 1 t+C 2
w(x, t) =
t ′ ( t), w(x, t) = C 1 exp − λx + λψ(t) − ψ t ′ ( t) + νλ,
x + ψ(t)
where ψ(t) is an arbitrary function and C 1 , C 2 , and λ are arbitrary constants. The first solution is “inviscid” (independent of ν).
3 ◦ . Traveling-wave solution ( λ is an arbitrary constant):
w = F (z),
z = x + λt,
where the function
F (z) is determined by the autonomous ordinary differential equation
4 ◦ . Self-similar solution:
1 w=t −1 /2 G(ξ) − /2
ξ = xt
where the function
G = G(z) is determined by the autonomous ordinary differential equation
The solutions of Items 3 ◦ and 4 ◦ can be generalized using the formulas of Item 1 ◦ .
References : A. D. Polyanin (2001 b, 2002).
∂w 2 ∂ 2 w
+ f (t).
∂x∂t ∂x
∂x 2 ∂x 3
This equation occurs in fluid dynamics; see 9.3.3.2, equation (3) and 10.3.3.1, equation (4).
1 ◦ . Suppose w = w(x, t) is a solution of the equation in question. Then the function
w 1 = w(x + ψ(t), t) + ψ ′ t ( t),
where ψ(t) is an arbitrary function, is also a solution of the equation.
2 ◦ . Degenerate solution (linear in x) for any f (t):
w(x, t) = ϕ(t)x + ψ(t),
where ψ(t) is an arbitrary function, and the function ϕ = ϕ(t) is described by the Riccati equation
ϕ ′ t + ϕ 2 = f (t). For exact solutions of this equation, see Polyanin and Zaitsev (2003).
3 ◦ . Generalized separable solutions for f (t) = Ae − βt ,
A > 0, β > 0:
w(x, t) = Be − 2 1 βt
sin[ λx + λψ(t)] + ψ ′ t ( t),
2 ν where ψ(t) is an arbitrary function.
− βt
w(x, t) = Be 2 cos[ λx + λψ(t)] + ψ t ′ ( t),
A > 0, β > 0:
w(x, t) = Be 1 2 βt sinh[ λx + λψ(t)] + ψ ′
2 ν where ψ(t) is an arbitrary function.
5 ◦ . Generalized separable solution for f (t) = Ae βt ,
A < 0, β > 0:
w(x, t) = Be 2 1 βt
2| A|ν
cosh[ λx + λψ(t)] + ψ ′
2 ν where ψ(t) is an arbitrary function.
6 ◦ . Generalized separable solution for f (t) = Ae βt ,
A is any, β > 0:
βt−λx
w(x, t) = ψ(t)e λx Ae ψ ′ t ( t)
2 ν where ψ(t) is an arbitrary function.
4 λ ψ(t)
λψ(t)
7 ◦ . Self-similar solution for f (t) = At −2 :
w(x, t) = t −1 /2 u(z) − 1 2 z , z = xt −1 /2 , where the function u = u(z) is determined by the autonomous ordinary differential equation
A − 2u z +( u z ) 2 − uu ′′ zz = νu ′′′ zzz ,
whose order can be reduced by one.
8 ◦ . Traveling-wave solution for f (t) = A:
w = w(ξ), ξ = x + λt,
where the function w(ξ) is determined by the autonomous ordinary differential equation
A + λw ξξ ′′ +( w ′ ξ ) 2 − ww ξξ ′′ = νw ′′′ ξξξ ,
whose order can be reduced by one. r✂s
References : V. A. Galaktionov (1995), A. D. Polyanin (2001 b, 2002).
∂x∂t ∂x
∂x 2 ∂x 3
1 ◦ . Suppose w = w(x, t) is a solution of this equation. Then the function
w 1 = w(x + ψ(t), t) + ψ t ′ ( t),
where ψ(t) is an arbitrary function, is also a solution of the equation.
2 ◦ . Generalized separable solutions:
w(x, t) =
+ ϕ(t),
C 1 t+C 2 w(x, t) = ϕ(t)e − λx ϕ ′ − t ( t) + λf (t),
λϕ(t) where ϕ(t) is an arbitrary function and C 1 , C 2 , and λ are arbitrary constants. The first solution is degenerate.
◮ For other equations involving second-order mixed derivatives, see Sections 9.3 and 9.4 .