Equations of the Form ∂w = aw +f x, t, w, ∂w

1.1.9. Equations of the Form ∂w = aw +f x, t, w, ∂w

∂t

∂x 2 ∂x

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w −2

1 = C 1 C 2 w(C 1 x+C 3 , C 2 t+C 4 ),

where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . Multiplicative separable solution:

x 2 + Ax + B

w(x, t) =

C − 2at

where

A, B, and C are arbitrary constants.

3 ◦ . Traveling-wave solution in implicit form:

where C 1 , C 2 , k, and λ are arbitrary constants.

4 ◦ . For other exact solutions, see equation 1.1.9.18 with m = 1, Items 5 ◦ to 8 ◦ .

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

= w −1 1 C 1 w(C 1 x+C 2 , C 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solutions:

w(x, t) = Ax + B + bt,

x 2 + Ax + B

w(x, t) =

C − 2at),

C − 2at

A, B, and C are arbitrary constants. The first solution is degenerate and the second one is a generalized separable solution.

where

3 ◦ . Traveling-wave solution:

w = w(z),

z = kx + λt,

where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation

ak 2 ww zz ′′ − λw ′ z + b = 0.

4 ◦ . Self-similar solution:

w = tU (ξ),

ξ = x/t,

where the function U (ξ) is determined by the autonomous ordinary differential equation

1 ◦ . Generalized separable solutions:

w(x, t) = Ae bt x + Be bt c − ,

b b(x + A) 2 − Bce − bt −2 act + C

w(x, t) =

A, B, and C are arbitrary constants (the first solution is degenerate).

2 ◦ . Traveling-wave solution:

w = w(z),

z = kx + λt,

where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation

✳☎✴ This is a special case of equation 1.1.9.9 with b = 0.

Reference : V. A. Galaktionov and S. A. Posashkov (1989).

This is a special case of equation 1.6.9.3 with f (t) = ct + d and g(t) = st + k.

This is a special case of equation 1.6.10.2 with f (t) ≡ 0, g(t) = b, h(t) = ct + d, and s(t) = pt + k. ∂w

Generalized separable solution:

w(x, t) =

3 Ax 3 + f 2 ( t)x 2 + f 1 ( t)x + f 0 ( t) ,

where

A is an arbitrary constant and the functions f 2 ( t), f 1 ( t), and f 0 ( t) are determined by the system of ordinary differential equations

The general solution of this system with

A ≠ 0 has the form

f 2 ( t) = 3 ϕ(t) dt + 3B, f 1 ( t) =

ϕ(t) dt + B +

ϕ(t),

9 A 6 A 2 36 A 2 t where the function ϕ(t) is defined implicitly by

f 0 ( t) =

2 ϕ(t) dt + B +

ϕ(t)

ϕ(t) dt + B +

1 , and B, C C 2 are arbitrary constants.

Reference : J. R. King (1993), V. A. Galaktionov (1995).

This is a special case of equation 1.6.10.2 with f (t) = b, g(t) = c, h(t) = p, and s(t) = q. ∂w

1 ◦ . Generalized separable solutions involving an exponential of x: − 1 /2

w(x, t) = ϕ(t) + ψ(t) exp( λx), λ=

a+b

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

ϕ ′ t = cϕ 2 + kϕ + s,

(3) Integrating (2) yields

ψ ′ t =( aλ 2 ϕ + 2cϕ + k)ψ.

dϕ = t+C 1 . cϕ 2 + kϕ + s

On computing the integral, one can find ϕ = ϕ(t) in explicit form. The solution of equation (3) is expressed in terms of ϕ(t) as

ψ(t) = C 2 exp

( aλ 2 ϕ + 2cϕ + k) dt ,

where C 1 and C 2 are arbitrary constants.

tions (

A is an arbitrary constant): − 1 /2 c

w(x, t) = ϕ(t) + ψ(t) cosh(λx + A), λ=

a+b

1 − /2 c

w(x, t) = ϕ(t) + ψ(t) sinh(λx + A), λ=

a+b

1 c /2

w(x, t) = ϕ(t) + ψ(t) cos(λx + A),

a+b

The functions ϕ = ϕ(t) and ψ = ψ(t) are determined by autonomous systems of first-order ordinary differential equations (these systems can be reduced to a single first-order equation each).

For details about these solutions, see Items 2 ◦ to 4 ◦ of equation 1.6.10.1 with f (t) = k and g(t) = s. ✹☎✺

Reference : V. A. Galaktionov and S. A. Posashkov (1989).

The substitution w = 1/v leads to an equation of the form 1.1.10.3:

Therefore the solutions of the original equation are expressed via solutions of the linear heat equation

by the relations

The variable ✹☎✺ y should be eliminated to obtain w = w(x, t) in explicit form.

Reference : N. H. Ibragimov (1985).

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = 2 −1

1 w(C 1 x+C 2 , C 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . The transformation w = 1/u, τ = at leads to an equation of the form 1.1.11.2:

This is a special case of equation 1.1.9.19 with m = 2 and b = −1.

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions

w 1 = C −1

1 w( ✻ C 1 x+C 2 , C 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . Functional separable solutions:

w(x, t) = ✻

C 1 ( x+C 2 ) 2 + C 3 exp(2 aC 1 t) −

aC 1 where C 1 , C 2 , and C 3 are arbitrary constants.

13. = aw 2

+ bw + cw –1 .

∂t Functional separable solutions:

∂x 2

w(x, t) = 2 bC

where C 1 , C 2 , and C 3 are arbitrary constants.

This is a special case of equation 1.1.9.18 with m = 3. Functional separable solution:

3 2 1 w(x, t) = a /3 3 Ax + f 2 ( t)x + f 1 ( t)x + f 0 ( t) . Here,

f 2 ( t) = 3 ϕ(t) dt + 3B, f 1 ( t) =

ϕ(t) dt + B +

ϕ(t),

f 0 ( t) =

2 ϕ(t) dt + B +

2 ϕ(t)

ϕ(t) dt + B +

ϕ ′ t ( t),

9 A 6 A 36 A 2 where the function ϕ(t) is defined implicitly by

Z ( C 1 −8 ϕ 3 ) −1 /2 dϕ = ✼ t+C 2 ,

and

A, B, C 1 , and C 2 are arbitrary constants. Setting C 1 = 0 in the last relation, one obtains the ✽☎✾ function

ϕ in explicit form: ϕ = − 1

2 ( t+C 2 ) −2 .

Reference : G. A. Rudykh and E. I. Semenov (1999).

This is a special case of equation 1.1.9.19 with m = 3 and b = −2. The substitution

w=u 1 /3 leads to an equation of the form 1.1.9.7:

Therefore the equation in question has a generalized separable solution of the form

. ∂w

w(x, t) = a −1 /3

Functional separable solutions:

w(x, t) = ✼

ϕ(t)x 2 + ψ(t)x + χ(t),

where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations

t = 2 aϕ(4ϕχ − ψ )+2 bϕ, ψ ′ = 1 t 2 2 aψ(4ϕχ − ψ )+2 bψ,

χ ′ t = 1 2 aχ(4ϕχ − ψ 2 )+2 bχ + 2c.

It follows from the first two equations that ϕ = Cψ, where C is an arbitrary constant.

This is a special case of equation 1.6.11.1 with f (x) = bx m . ∂w

18. = aw m ∂t

∂x 2

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

1 = C 1 /m C 2 /m w(C 1 x+C 3 , C 2 t+C 4 ),

where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solutions: w(x) = Ax + B,

w(x, t) = ( ✿ βx + βλt + A) 1 /m

w(x, t) =

2 a(2 − m)(B − t)

( x + C) 2

1 /m

w(x, t) = A|t + B|

m(m−3)

2 a(m − 2) | ϕ(t)| 2 , ϕ(t) = C +

w(x, t) =

t,

m where

ϕ(t)

A, B, C, and λ are arbitrary constants (the first solution is degenerate).

3 ◦ . Traveling-wave solution in implicit form:

where C 1 , C 2 , β, and λ are arbitrary constants. To λ = 0 there corresponds a stationary solution, and to C 1 = 0 there corresponds the second solution in Item 2 ◦ .

4 ◦ . Multiplicative separable solution:

w(x, t) = (λt + A) −1 /m f (x),

where λ is an arbitrary constant, and the function f = f (x) is determined by the autonomous ordinary differential equation amf ′′ + λf 1− m xx = 0 (its solution can be written out in implicit form).

5 ◦ . Self-similar solution:

x w = w(z), z= √ ,

where the function w(z) is determined by the ordinary differential equation 2aw m w ′′ zz zw ′ z = 0.

6 ◦ . Self-similar solution of a more general form:

w=t β U (ζ), ζ = xt − mβ+1 2 ,

where β is an arbitrary constant, and the function U = U (ζ) is determined by the ordinary differential equation

aU 1 m U

ζζ ′′ = βU − 2 ( mβ + 1)ζU ζ ′ .

This equation is generalized homogeneous, and, hence, its order can be reduced.

w=e −2 λt ϕ(ξ), ξ = xe λmt ,

where λ is an arbitrary constant, and the function ϕ = ϕ(ξ) is determined by the ordinary differential equation

aϕ m ϕ ′′ ξξ = λmξϕ ′ ξ −2 λϕ.

This equation is generalized homogeneous, and, hence, its order can be reduced.

8 ◦ . Solution:

w = (At + B) −1 /m ψ(u), u = x + k ln(At + B),

where

A, B, and k are arbitrary constants, and the function ψ = ψ(u) is determined by the autonomous ordinary differential equation

aψ m ψ ′′ uu = Akψ ′

9 ◦ . The substitution u=w 1− m leads to the equation

which is considered in Subsection 1.1.10. ∂w

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions

w 1 = C 1 2 w( ❀ C 1 k−m−1 x+C 2 , C 2 1 k−2 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . Traveling-wave solutions:

w = w(z), z = kx + λt,

where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation

aw m w zz ′′ − λw ′ z + bw k = 0.

3 ◦ . Self-similar solution for k ≠ 1:

1 k−m−1

w=t 1− k u(ξ), ξ = xt 2(1− k) ,

where the function u(ξ) is determined by the ordinary differential equation

au m u ′′

m−k+1 ′

4 ◦ . For m ≠ 1, the substitution u = w 1− m leads to the equation

which is considered Subsection 1.1.11.

5 ◦ . For k = 1, the transformation

w(x, t) = e bt

U (x, τ ), τ=

e bmt + const,

bm

leads to an equation of the form 1.1.9.18:

This is a special case of equation 1.6.11.4 with f (t) = bt n and g(t) = ct k .

∂t

∂x

∂x

◮ Equations of this form admit traveling-wave solutions w = w(kx + λt). ∂w

This is a special case of equation 1.1.10.7 with m = 1.

1 ◦ . Solutions:

w(x, t) = C 2

1 x + aC 1 t+C 2 ,

( x+C 1 ) 2 C 3

w(x, t) = −

w(x, t) =

+ C 3 | x+C 1 | 1 /2 | C 2 −6 −5 at| /8 ,

C 2 −6 at ❁☎❂ where C 1 , C 2 , and C 3 are arbitrary constants.

References : D. Zwillinger (1989), A. D. Polyanin and V. F. Zaitsev (2002).

2 ◦ . Traveling-wave solution in implicit form:

w−C 2 ln | w+C 2 |=

1 x + aC 1 t+C 3 .

3 ◦ . Solution in parametric form:

x = (6at + C 2

1 ) ξ+C 2 ξ + C 3 ,

w = −(6at + C 1 ) ξ 2 −2 C 2 ξ 3 .

4 ◦ . Solution in parametric form:

x = tf (ξ) + g(ξ), w = tf ξ ′ ( ξ) + g ′ ξ ( ξ),

where the functions f = f (ξ) and g(ξ) are determined by the system of ordinary differential equations

( f ξ ′ ) 2 − ff ξξ ′′ = af ξξξ ′′′ ,

(2) The order of equation (1) can be reduced by two. Suppose a solution of equation (1) is known.

f ξ ′ g ′ ξ − fg ξξ ′′ = ag ′′′ ξξξ .

Equation (2) is linear in g and has two linearly independent particular solutions

g 1 = 1,

g 2 = f (ξ).

The second particular solution follows from the comparison of (1) and (2). The general solution of equation (1) can be represented in the form (see Polyanin and Zaitsev, 2003):

g(ξ) = C 1 + C 2 f+C 3 f ψ dξ −

f ψ dξ ,

f = f (ξ), ψ=

exp −

f dξ .

It is not difficult to verify that equation (1) has the following particular solutions:

f (ξ) = 6a(ξ + C) −1 ,

C and λ are arbitrary constants. One can see, taking into account (1) and (3), that the first solution in (4) leads to the solution of Item 3 ◦ . Substituting the second relation of (4) into (1), we obtain another solution.

Remark. The above solution was obtained, with the help of the Mises transformation, from a ❁☎❂ solution of the hydrodynamic boundary layer equation (see 9.3.1.1, Items 5 ◦ and 7 ◦ ).

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

5 ◦ . For other solutions, see Items 4 ◦ to 9 ◦ of equation 1.1.10.7 with m = 1.

2. = a .

∂t ∂x w ∂x This is a special case of equation 1.1.10.7 with m = −1.

Solutions:

w(x, y) = (C 1 x − aC 2 1 t+C 2 ) −1 ,

w(x, y) = (2at + C −2

1 )( x+C 2 ) ,

2 a(t + C 1 )

w(x, y) =

( x+C 2 ) 2 + C 3 ( t+C 1 ) 2

w(x, y) =

C 2 + C 3 exp( aC 2 t−C 1 x)

C 2 1 −1 C 1 x C 1 x

w(x, y) =

w(x, y) =

w(x, y) =

w(x, y) =

cos 2

( C 1 x+C 3 )

where ❃☎❄ C 1 , C 2 , and C 3 are arbitrary constants.

References : V. V. Pukhnachov (1987), S. N. Aristov (1999).

This is a special case of equation 1.1.10.7 with m = −2.

1 ◦ . Solutions:

w(x, t) = ❅ (2 C 1 x − 2aC 2 1 t+C 2 ) −1 /2 ,

2 at /2 C 1

w(x, t) =

w(x, t) =

+ C 3 exp( C 1 t)

where C 1 , C 2 , and C 3 are arbitrary constants. The first solution is of the traveling-wave type, the second is self-similar, and the third is a functional separable solution.

∂z

. Introduce a new unknown z = z(x, t) by w = and then integrate the resulting equation with

∂x

respect to x to obtain

By the hodograph transformation

(2) equation (1) can be reduced to a linear heat equation for u = u(y, t):

x = u, z = y,

∂u

(3) Transformation (2) means that the dependent variable z is taken to be the independent variable, and

∂t

∂y 2

the independent variable x, the dependent one.

equation (3) according to

∂u −1

w=

, x = u(y, t).

∂y

The variable y should be eliminated from (4) to obtain w = w(x, t) in explicit form.

3 ◦ . The transformation Z x

1 x= ¯

w(y, t) dy + a

x=x 0 w(x, t) where x 0 and t 0 are any numbers, leads to the linear equation

The inversion of transformation (5) is given by Z x ¯

1 x=

∂¯ w( ¯x, t ′ )

w(x ¯ ′ ,¯ t) dx ′ + dt ′ , t = ¯t − ¯t 0 , w(x, t) = .

References : M. L. Storm (1951), G. W. Bluman and S. Kumei (1980), A. Munier, J. R. Burgan, J. Gutierres, E. Fijalkow, and M. R. Feix (1981), N. H. Ibragimov (1985).

This is a special case of equation 1.1.10.7 with m = −4/3 (the equation admits more invariant solutions than for m ≠ −4/3).

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

where A 1 , A 2 , B 1 , B 2 , C 1 , and C 2 are arbitrary constants ( A 1 B 2 − A 2 B 1 ≠ 0), is also a solution of the equation. ❆☎❇

References : L. V. Ovsiannikov (1959, 1982).

2 ◦ . Solutions:

w(x, t) = ( ❈ 2 C 1 x − 3aC 2 1 t+C 2 ) −3 /4 ,

w(x, t) = (at + C 1 ) 3 /4 [( x+C 2 )( C 3 x+C 2 C 3 + 1)] −3 /2 ,

3 4 2 4 w(x, t) = ( −3 2 C

1 x + C 2 x −3 aC 1 x t) /4 ,

−3 ( /4 x+C

w(x, t) =

+ C 3 ( t+C 2 )

a(t + C 2 ) ( x+C 1 ) 2 −3 /4

w(x, t) =

+ C 3 ( t+C 2 ) 2 ( x+C 1 ) 4 ,

a(t + C 2 )

where C 1 , C 2 , and C 3 are arbitrary constants. The first solution is of the traveling-wave type, the second is a solution in multiplicative separable form, and the other are functional separable solutions.

3 ◦ . Functional separable solution:

4 −3 w(x, t) = /4 ϕ 4 ( t)x + ϕ 3 ( t)x 3 + ϕ 2 ( t)x 2 + ϕ 1 ( t)x + ϕ 0 ( t) ,

ϕ ′ 0 =− a 3 4 ϕ 2 1 +2 aϕ 0 ϕ 2 , ϕ ′ 1 =− aϕ 1 ϕ 2 +6 aϕ 0 ϕ 3 , ϕ ′ =− aϕ 2 + 2 3 2 2 aϕ 1 ϕ 3 + 12 aϕ 0 ϕ 4 , ϕ ′ 3 =− aϕ 2 ϕ 3 +6 aϕ 1 ϕ 4 , ϕ ′ 4 =− 3 4 2 aϕ 3 +2 aϕ 2 ϕ 4 .

❉☎❊ The prime denotes a derivative with respect to t.

References : V. A. Galaktionov (1995), G. A. Rudykh and E. I. Semenov (1998).

4 ◦ . There are exact solutions of the following forms:

w(x, t) = x

F (y), y = t − ;

x tx −3 2

w(x, t) = x G(z), z =

x + 1)

Reference : N. H. Ibragimov (1994).

5 ◦ . For other solutions, see equation 1.1.10.7 with m = −4/3. ∂w

This is a special case of equation 1.1.10.7 with m = −2/3.

1 ◦ . Solution:

2 −3 w = (C − 4at) /2 ( C − 4at) − x .

2 ◦ . The transformation

t=τ, x = v, w = 1/u,

where ∂v ∂ξ = u, leads to an equation of the form 1.1.10.4:

References : A. Munier, J. R. Burgan, J. Gutierres, E. Fijalkow, and M. R. Feix (1981), J. R. Burgan, A. Munier, M. R. Feix, and E. Fijalkow (1984), I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989), N. H. Ibragimov (1994).

1 ◦ . Functional separable solution:

3 2 −2 w(x, t) = a /3 3 Ax + f 2 ( t)x + f 1 ( t)x + f 0 ( t) . Here,

f 2 ( t) = 3 ϕ(t) dt + 3B, f 1 ( t) =

ϕ(t) dt + B +

ϕ(t),

9 A 6 A 2 36 A 2 t where the function ϕ(t) is defined implicitly by

f 0 ( t) =

ϕ(t) dt + B +

ϕ(t)

ϕ(t) dt + B +

A, B, C 1 , and C 2 are arbitrary constants. Setting C 1 = 0 in this relation, we find ϕ in explicit

form: ϕ=− 1 2 ( t+C 2 ) −2 .

2 ◦ . For other solutions, see equation 1.1.10.7 with m = −3/2.

This equation often occurs in nonlinear problems of heat and mass transfer, combustion the- ory, and flows in porous media. For example, it describes unsteady heat transfer in a qui- escent medium with the heat diffusivity being a power-law function of temperature. For m=

1, −1, −2, −4 /3, −2/3, −3/2, see also equations 1.1.7.1 to 1.1.7.6.

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

w 1 = C 1 w(C 2 x+C 3 , C 1 m C 2 2 t+C 4 ), where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solutions:*

w(x) = (Ax + B) m+1 ,

w(x, t) = ( ● kx + kλt + A) 1 /m , k = λm/a,

1 m(x − A) 2 m

w(x, t) =

2 a(m + 2)(B − t)

− m m+2

( x + C) 2 m

w(x, t) = A|t + B|

2 a(m + 2) t+B

m(x + A) 2 m

− m(2m+3) m

w(x, t) =

+ B|x + A| m+1

| ϕ(t)| 2( m+1) 2 , ϕ(t) = C − 2a(m + 2)t,

ϕ(t)

where

A, B, C, and λ are arbitrary constants. The third solution for B > 0 and the fourth solution for

B < 0 correspond to blow-up regimes (the solution increases without bound on a finite time interval).

Example. A solution satisfying the initial and boundary conditions

is given by

w(x, t) =

k(t − x/λ) 1 /m for 0 ≤ x ≤ λt,

0 for x > λt,

where λ= p ak m /m.

References : Ya. B. Zel’dovich and A. S. Kompaneets (1950), G. I. Barenblatt (1952), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), G. A. Rudykh and E. I. Semenov (1998).

3 ◦ . Traveling-wave solutions:

w = w(z), z= ● x + λt,

where the function w(z) is defined implicitly by

Z w m dw

a = C 2 + z, λw + C 1

and λ, C 1 , and C 2 are arbitrary constants. To λ = 0 there corresponds a stationary solution, and to

C 1 = 0 there corresponds the second solution in Item 2 ◦ .

* For the sake of brevity, here and henceforth, exact solutions of nonlinear equations are given only for the domain of

their spatial localization, where w ❏ 0 .

(1) where the function f = f (x) is defined implicitly by

w(x, t) = (λt + A) −1 /m f (x),

am(m + 2)

and λ, C 1 , and C 2 are arbitrary constants.

5 ◦ . Self-similar solution:

w = w(z), z= √

(0 ≤ x < ∞),

where the function w(z) is determined by the ordinary differential equation

(2) Solution of this sort usually describe situations where the unknown function assumes constant

2 aw m w ′ z ′ z + zw ′ z = 0.

values at the initial and boundary conditions. To the particular solution of equation (2) with w(z) = k 2 z 2 /m there corresponds the third solution in Item 2 ◦ . Fujita (1952) obtained the general solution of equation (2) for m = −1 and m = −2; see also the book by Lykov (1967). With the boundary conditions

w = 1 at z = 0,

w = 0 at z = ∞

the solution of equation (2) is localized and has the structure

w= (1 − Z) /m P (1 − Z, m) for 0 ≤ Z ≤ 1, P (1, m)

0 for 1 ≤ Z < ∞, where

b 0 = 1, b 1 =− 1 2 [ m(m + 1)] −1 , . . . ; see Samarskii and Sobol’ (1963).

6 ◦ . Self-similar solution:

− 1 m+2

(0 ≤ x < ∞). Here, the function

w=t

F (ξ), ξ = xt m+2

F = F (ξ) is determined by the first-order ordinary differential equation

(3) where

a(m + 2)F m F ξ ′ + ξF = C,

C is an arbitrary constant. To

C = 0 in (3) there corresponds the fourth solution in Item 2 ◦ , which describes the propagation of a thermal wave coming from a plane source. For details, see the book by Zel’dovich and Raiser (1966).

Performing the change of variable ϕ=F m in equation (3), one obtains

(4) where α= mC a(m+2) and β= m a(m+2) . The books by Polyanin and Zaitsev (1995, 2003) present general

ξ = αϕ /m − βξ,

solutions of equation (4) for m = −1 and m = 1.

w=t β g(ζ), ζ = xt − mβ+1 2 ,

β is any.

Here, the function g = g(ζ) is determined by the ordinary differential equation

G=g m+1 , (5) where A 1 = −( mβ + 1)/(2a) and A 2 = β(m + 1)/a. This equation is homogeneous, and, therefore, its

G ′′ ζζ = A 1 ζG − m m+1 G ζ ′ + A 2 G m+1 ,

order can be reduced (and then it can be transformed to an Abel equation of the second kind). Exact analytical solutions of equation (5) for various values of m can be found in Polyanin and Zaitsev (2003).

8 ◦ . Generalized self-similar solution:

w=e −2 λt ϕ(u), u = xe λmt ,

λ is any,

where the function ϕ = ϕ(u) is determined by the ordinary differential equation

(6) This equation is homogeneous, and, hence, its order can be reduced (and then it can be transformed

a(ϕ m ϕ ′ ) ′ u u = λmuϕ ′ u −2 λϕ.

to an Abel equation of the second kind). The substitution Φ = ϕ m+1 brings (6) to an equation that coincides, up to notation, with (5).

9 ◦ . Solution: w = (t + A) −1 /m ψ(u), u = x + b ln(t + A),

A, b are any,

where the function ψ = ψ(u) is determined by the autonomous ordinary differential equation

a(ψ m ψ ′ ′

u ) u = bψ u − ψ/m.

Introduce the new dependent variable p(ψ) = ψ m ψ ′ u . Taking into account the identity

b du

, we arrive at an Abel equation of the second kind:

a dψ

pp ′ ψ = p − sψ m+1 ,

s= a/(mb 2 ). The general solutions of this equation with m = −3, −2, − 3 2 , −1 can be found in Polyanin and Zaitsev

10 ◦ +. Unsteady point source solution with a = 1:

2 1  /m

At −1 /(m+2) η 2 w(x, t) = x 0 − 2 /(m+2)

for | x| ≤ η 0 t 1 /(m+2) ,

0 for | x| > η 0 t 1 /(m+2) , where

1 /m

Γ(1 /m + 3/2)

m/(m+2)

A π Γ(1/m + 1)

with Γ( z) being the gamma function. The above solution satisfies the initial condition

w(x, 0) = E 0 δ(x),

where δ(x) is the Dirac delta function, and the condition of conservation of energy

Z ∞ w(x, t) dx = E 0 > 0.

Reference : A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).

et= t − t 1

∂w

0 , x= e w(y, t) dy + a

x=x 0 w(x, t) takes a nonzero solution w(x, t) of the original equation to a solution e w(e x, e t) of a similar equation

References for equation 1.1.10.7: L. V. Ovsiannikov (1959, 1962, 1982), N. H. Ibragimov (1994), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).