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