Equations with n Space Variables

2.5.5. Equations with n Space Variables

2 P n ∂w 2 P n ∂w

Notation: x=( x 1 , ...,x n ), ∆ w=

1. = ∆w + f (w)|∇w| 2 .

∂t

The substitution U=

F (w) dw, where F (w) = exp

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

∂U =∆ U. ∂t

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

f (t)∆w + g(t)w ln w + h(t)w. There is a functional separable solution of the form

∂t

w(x 1 , ...,x n , t) = exp

ϕ ij ( t)x i x j +

ψ i ( t)x i + χ(t) .

i,j=1

i=1

Example 1. Let f (t) = 1, g(t) = 1, and h(t) = 0. Solutions in the radially symmetric case:

r + Be t , r=

w = exp 1 2 q

···+x 2 n ,

where A and B are arbitrary constants, A < 1. The first solution is a special case of the second solution as A → 0. Example 2. Let

f (t) = 1, g(t) = −1, and h(t) = 0. Solution in radially symmetric case:

− r 2 ( Ae t − 1) −1 + e − t B− n ln( Ae 4 t 2 − 1) A , where ✶☎✷

w = exp 1

A and B are arbitrary constants, A > 1. Reference : A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).

∂w

h i ( t)x i + p(t). ∂t

3. = f 1 ( t)∆w + f 2 ( t)|∇w| 2 + f 3 ( t)w +

g ij ( t)x i x j +

i,j=1

i=1

There are exact solutions of the following forms:

w(x 1 , ...,x n , t) =

ϕ ij ( t)x i x j +

ψ i ( t)x i + χ(t).

i,j=1

i=1

∂w

h i ( t)x i + p(t). ∂t

4. = f 1 ( t)w∆w + f 2 ( t)|∇w| 2 + f 3 ( t)w +

g ij ( t)x i x j +

i,j=1

i=1

There are exact solutions of the following forms:

w(x 1 , ...,x n , t) =

ϕ ij ( t)x i x j +

ψ i ( t)x i + χ(t).

i,j=1

For m > 1, this equation describes the flow of a polytropic gas through a homogeneous porous medium ( w is the gas density).

1 ◦ . In the radially symmetric case the equation is written as

∂w

r n−1 w m ∂w ,

r= x 1 + ···+x n .

Its exact solutions are given in 1.1.15.9, where n should be substituted by n − 1.

2 ◦ . Solution of the instantaneous source type for a = 1: 

2 1 /m

w= t

 − n/(nm+2)

if r≤K 0 t /(nm+2) , 

2( nm + 2)

t 2 /(nm+2)

0 if

r>K 1 0 t /(nm+2) ,

1 /m

m/(nm+2)

− n/2

Γ( n/m + 1 + 1/m)

E 0 = const .

Γ(1 /m + 1)

This is the solution of the initial-value problem with initial function

w(x, 0) = E 0 δ(x),

satisfying the condition of constant energy:

w(x, t) dx = E 0 = const > 0.

✹☎✺ 3 ◦ . See also equation 2.5.5.6, in which m should be substituted by m + 1.

References : Ya. B. Zel’dovich and A. S. Kompaneets (1950), G. I. Barenblatt (1952).

∂w

6. = ∆(w m ).

∂t For m > 1, this equation describes the flow of a polytropic gas through a homogeneous porous medium ( w is the gas density). It can be rewritten in the form of equation 2.5.5.5:

1 ◦ . Solution for m > 1:

A is an arbitrary constant (A > 0), and the functions ϕ k = ϕ k ( t) are determined by the system of ordinary differential equations

System (1) admits n − 1 first integrals:

(2) where the C j are arbitrary constants.

j = 1, 2, . . . , n − 1,

The function ϕ n = ϕ n ( t) is defined in implicit form by (the C j are assumed to be positive)

B is an arbitrary constant, and the remaining ϕ j ( t) are determined by the positive roots of the ✹☎✺ quadratic equations (2).

References : S. S. Titov and V. A. Ustinov (1985), J. R. King (1993), V. V. Pukhnachov (1995).

2 ◦ . Solution for 0 < m < 1:

A is an arbitrary constant, and the functions ϕ k = ϕ k ( t) are determined by the system of ordinary differential equations (1). ✹☎✺

References : J. R. King (1993), V. V. Pukhnachov (1995).

3 ◦ . There is an exact solution of the form

b i ( t)x i + c(t)

i,j=1

i=1

Reference : G. A. Rudykh and E. I. Semenov (2000); other exact solutions are also given there.

∂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). There is a generalized separable solution of the form

w(x 1 , ...,x n , t) = ϕ(t) + ψ(t)Θ(x 1 , ...,x n ), where the functions ϕ(t), ψ(t) are determined by the system of ordinary differential equations

t = bϕ + g(t)ϕ + h(t),

β = b/a,

and the function Θ( x 1 , ...,x n ) is a solution of the Helmholtz equation

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

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.

For solutions of the linear stationary equation (3), see the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

∂w

∂t v ⋅ ∇)w = ∆w + f(w)|∇w|

Here, ~v is a prescribed vector function dependent on the space coordinates and time (but independent of w).

The substitution

f (w) dw , leads a linear convective heat and mass transfer equation for Θ = Θ( x 1 , ...,x n , t):

∂Θ +( ~v ⋅ ∇)Θ = ∆Θ. ∂t

∂w

∂t v ⋅ ∇)w = a∆w + a|∇w| + f (x, t).

Here, ~v is a prescribed vector function dependent on the space coordinates and time (but independent of w).

The substitution Θ = e w leads to the linear equation ∂Θ

+( ~v ⋅ ∇)Θ = a∆Θ + f (x, t)Θ. ∂t

∂w

10. = α∇ ⋅ w m ∇ w ∂t

1 ◦ . Multiplicative separable solution:

(1) where the function Θ(x) satisfies the Laplace equation

w(x, t) = exp

f (t) dt

m+1 ,

∆Θ = 0. For solutions of this linear equation, see Tikhonov and Samarskii (1990) and Polyanin (2002).

(2) where the function ϕ(t) is determined by the Bernoulli equation

w(x, t) = ϕ(t)

m+1 ,

(3) Here,

ϕ ′ t − f (t)ϕ + Aαϕ m+1 = 0.

A is an arbitrary constant and the function Θ(x) satisfies the stationary equation

∆Θ + A(m + 1)Θ m+1 = 0.

The general solution of equation (3) is given by

Z ϕ(t) = exp

−1 /m

f (t) dt, where

Aαm exp

F (t) =

B is an arbitrary constant.

3 ◦ . The transformation

w(x, t) = F (t)U (x, τ ), τ=

leads to a simpler equation:

= α∇ ⋅ (U m ∇ w).

Example. For α = 1, f (t) = −β < 0, we have

∂w =∇⋅( w m ∇ w) − βw. ∂t

Solution in the radially symmetric case: 

2 r 2  1 /m e w= βt/m [ g(t)] n/(nm+2)

if ∗ ( 

2( nm + 2) η

[ g(t)] 2 /(nm+2)

r≤r t), 0 if r>r ∗ ( t),

where q

1− 1 e 1− e /(nm+2) r=

− βmt

− βmt

x 1 2 + ···+x 2 n ,

g(t) = 1 +

The diameter of the support of this solution is monotonically increasing but is bounded now by the constant

The perturbation is localized in a ball of radius ✻☎✼ L. References : L. K. Martinson and K. B. Pavlov (1972), A. D. Polyanin and V. F. Zaitsev (2002).

∂w

11. =∇⋅( w m ∇ w) – w 1– m .

∂t Solution in the radially symmetric case for 0 < m < 1:

nm + 1 1 ···+x n ;

r= x +

A is an arbitrary constant (A > 0). The solution has a compact support. The diameter of the support increases with t on the time interval (0, t ∗ ), where

nm+2 nm+1

t ∗ = ( nm + 2) 3

and decreases on the interval ( t ∗ , T 0 ), where

nm+2 2( nm+1)

( nm + 2) 2

The solution vanishes at ✻☎✼ t=T 0 .

References : R. Kersner (1978), L. K. Martinson (1979).

12. = a∇ ⋅ (e λw ∇ w) + be λw + f (t) + g(t)e – λw .

∂t Functional separable solution:

w(x, t) =

ln

ψ(t) = exp λ

where the function ϕ(t) is determined by the Riccati equation

(1) and the function Θ = Θ(x) is a solution of the Helmholtz equation

t = bλϕ + λf (t)ϕ + λg(t),

(2) For details about the Riccati equation (1), see Kamke (1977) and Polyanin and Zaitsev (2003). For

a∆Θ + bλΘ = 0.

solutions of the linear equation (2), see Tikhonov and Samarskii (1990) and Polyanin (2002).

∂w

13. =∇⋅[ f (w)∇w] +

Solution in implicit form:

f (w) dw = at + U (x),

where the function U (x) is determined by the Poisson equation

∆ U + b = 0.

For details about this equation, see the books by Tikhonov and Samarskii (1990) and Polyanin (2002). ✽☎✾

Reference : V. A. Galaktionov (1994).

∂w

g(t)

14. =∇⋅[ f (w)∇w] +

+ h(x).

∂t

f (w)

Solution in implicit form:

f (w) dw =

g(t) dt + U (x),

where the function U (x) is determined by the Poisson equation

∆ U + h(x) = 0.

+ c[af (w) + b].

( w)

Solution in implicit form:

f (w) = e at

b U (x) − ,

where the function U (x) is determined by the Helmholtz equation

∆ U + acU = 0.

For details about this equation, see the books by Tikhonov and Samarskii (1990) and Polyanin (2002). ✽☎✾

Reference : V. A. Galaktionov (1994).

16. =L[ f (w)] + ′

+ h(x).

∂t

f ( w)

Here, L is an arbitrary linear differential operator of the second (or any) order with respect to the space variables with coefficients independent of t; the operator satisfies the condition L [const] = 0.

Solution in implicit form:

f (w) =

g(t) dt + U (x),

where the function U (x) is determined by the linear equation

L[ U ] + h(x) = 0.

∂w

af (w) + b

17. =L[ f (w)] + ∂t

+ g(x)[af (w) + b].

( w)

Here, L is an arbitrary linear differential operator of the second (or any) order with respect to the space variables with coefficients independent of t; the operator satisfies the condition L [const] = 0.

Solution in implicit form:

f (w) = e at U (x) − ,

where the function U (x) is determined by the linear equation

L[ U ] + ag(x)U = 0.

+ h(x).

∂t

f w ( x, w)

Here, L is an arbitrary linear differential operator of the second (or any) order with respect to the space variables with coefficients independent of t; the operator satisfies the condition L [const] = 0;

and f w stands for the partial derivative of f with respect to w. Solution in implicit form:

f (x, w) =

g(t) dt + U (x),

where the function U (x) is determined by the linear equation