Other Equations with Three Space Variables

2.5.4. Other Equations with Three Space Variables

◮ Throughout this subsection, the symbols div, ∇, and ∆ stand for the divergence operator, gradient operator, and Laplace operator in Cartesian coordinates x, y, z; cylindrical, spherical, and other three-dimensional orthogonal systems of coordinates can be used instead of the Cartesian coordinates.

∂w

a∆w + f(t)|∇w| 2 + g(t)w + h(t). ∂t There is a generalized separable solution of the form

ψ k ( t)x k + χ(t).

k,l=1

k=1

Remark. The more general equation

3 2 3 2 ∂w 3

X X ∂w =

+ g(t)w + h(t) ∂t

∂x n has solutions of the same form.

∂x n ∂x m n=1

∂x n

n,m=1

n=1

∂w

2. = ∆w + f (w)|∇w| 2 .

∂t The substitution

f (w) dw , leads to the linear heat equation

For solutions of this equation, see the books by Tikhonov and Samarskii (1990) and Polyanin (2002). ∂w

αw∆w – α|∇w| 2 – β. ∂t

1 ◦ . Solutions:

2 2 w(x, y, z, t) = Ax + By + Cz − [α(A 2 + B + C )+ β]t + D,

2 , where

2 2 2 1 w(x, y, z, t) = A − βt + B exp 2 + µ + ν ) At − βt

κx+µy+νz

A, B, C, D, κ, µ, and ν are arbitrary constants.

2 ◦ . See 2.5.4.4 with f (t) = −α, g(t) = 0, and h(t) = −β. ∂w

αw∆w + f(t)|∇w| 2 + g(t)w + h(t). ∂t There are generalized separable solutions of the form

ψ k ( t)x k + χ(t).

k,l=1

k=1

Remark. The more general equation

3 2 3 2 ∂w 3

X X ∂w =

s n ( t) + g(t)w + h(t) ∂t

nm ( t)w + b nm ( t)

c n ( t)

∂x n has solutions of the same form.

∂x n ∂x m

∂x n

n,m=1

n=1

n=1

∂t aw + f (t)]∆w + bw + g(t)w + h(t).

Here, f (t), g(t), and h(t) are arbitrary functions; a and b are arbitrary parameters (a ≠ 0). This is a special case of equation 2.5.4.6 with L [ w] ≡ ∆w.

f (t) ≡ const, g(t) ≡ const, and h(t) ≡ const, this equation was studied in Galaktionov and Posashkov (1989) and Ibragimov (1994).

Note that, with

∂w

6. =[ aw + f (t)]L [w] + bw 2 + g(t)w + h(t).

∂t Here, f (t), g(t), and h(t) are arbitrary functions; a and b are arbitrary parameters (a ≠ 0); L [w] is an arbitrary linear differential operator of the second (or any) order that depends on the space variables

x 1 = x, x 2 = y, x 3 = z only and satisfies the condition L [const] ≡ 0:

n,m=1

n=1

There is a generalized separable solution of the form

w(x 1 , x 2 , x 3 , t) = ϕ(t) + ψ(t)Θ(x 1 , x 2 , x 3 ),

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

t = bϕ + g(t)ϕ + h(t),

(2) and the function Θ( x 1 , x 2 , x 3 ) is a solution of the linear stationary equation

β = b/a,

(3) Equation (1) is independent of ψ and represents a Riccati equation for ϕ. A large number of

L [Θ] + βΘ = 0.

exact solutions to equation (1) for various g(t) and h(t) can be found in Polyanin and Zaitsev (2003). On solving (1) and substituting the resulting ϕ = ϕ(t) into (2), one obtains a linear equation for ψ = ψ(t), which is easy to integrate.

In the special case b = 0, the solution of system (1), (2) is given by

Z ϕ(t) = exp

A+

h(t) exp

, G(t) =

g(t) dt,

ψ(t) = B exp

F (t) =

f (t) dt,

A and B are arbitrary constants. In the special case L ≡ ∆, see Tikhonov and Samarskii (1990) and Polyanin (2002) for solutions of the linear stationary equation (3).

where

∂w

7. = f (t)N β [ w] + g(t)w. ∂t

Here, N β [ w] is an arbitrary homogeneous nonlinear differential operator of degree β with respect to w (i.e., N β [ αw] = α β N β [ w], α = const) that depends on the space variables x, y, z only (and is independent of t).

Using the transformation

f (t)G β−1 ( t) dt, G(t) = exp g(t) dt , one arrives at the simpler equation

w(x, y, z, t) = G(t)U (x, y, z, τ ), τ=

which has a multiplicative separable solution U = ϕ(τ )Θ(x, y, z).

number of the space variables in the original equation can be any. The coefficients of N β can be dependent on the space variables.

Remark 2. If N β is independent explicitly of the space variables, then equation (1) has also a traveling-wave solution, U = U (ξ), where ξ = k 1 x+k 2 y+k 3 z + λτ . Below are two examples of such operators:

N β [ w] = a div(w β−1 ∇ w) + b|∇w| β + cw β , N β [ w] = a div(|∇w| β−1 ∇ w) + bw µ |∇ w| β−µ ,

where

a, b, c, and µ are some constants. ∂w

v ⋅ ∇)w = ∆w + f(w)|∇w| 2 .

∂t This is a special case of equation 2.5.5.8 with n = 3.

∂w

9. +( ~ v ⋅ ∇)w = a∆w + a|∇w| 2 + f (~ x, t). ∂t

This is a special case of equation 2.5.5.9 with n = 3. ∂~ w

10. +( ~ w ⋅ ∇)~ ∂t

w = a∆~ w.

Vector Burgers equation ; w = {w ~ 1 , w 2 , w 3 } and w n = w n ( x 1 , x 2 , x 3 ). The Hamilton operator ∇ and Laplace operator ∆ can be represented in any orthogonal system of coordinates. Solution:

w=− ~

where θ is a solution of the linear heat equation

∂θ = a∆θ. ∂t

For solutions of this equation, see the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

Reference : S. Nerney, E. J. Schmahl, and Z. E. Musielak (1996).