6.15. Equations of the Form ∂w i

h 1.6.15. Equations of the Form ∂w i

f (w) ∂w +g x, t, w, ∂w

∂t

∂x

∂x

∂x

This equation is frequently encountered in nonlinear problems of heat and mass transfer (with f being the thermal diffusivity or diffusion coefficient) and the theory of flows through porous media. For

f (w) = aw m , see Subsection 1.1.10; for f (w) = e λw , see equation 1.2.2.1; and for f (w) = a ln w + b, see equation 1.4.2.7.

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

w(C 2

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 ◦ . Traveling-wave solution in implicit form:

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

3 ◦ . Self-similar solution:

w = w(z), z= √

(0 ≤ x < ∞),

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

(2) Solutions of this form usually correspond to constant w in the initial and boundary conditions

f (w)w ′ ′ z z + 1 2 zw z ′ = 0.

for the original partial differential equation ( w 0 , w 1 = const):

w=w 0 at t=0

(initial condition),

w=w 1 at x=0

(boundary condition),

w→w 0 as x → ∞ (boundary condition).

Then the boundary conditions for the ordinary differential equation (2) are as follows:

w=w 1 at z = 0,

w→w 0 as z → ∞. (3) For f (w) = aw −1 , f (w) = aw −2 , and f (w) = (αw 2 + βw + γ) −1 , the general solutions of (2) were

obtained by Fujita (1952); see also the book by Lykov (1967).

Solutions equation 1.6.15.1 for various f = f (w), where z = xt −1 /2 . No.

Function f = f (w)

Solution z = z(w) Conditions

5 1 8 sin( πw) πw + sin(πw)

cos 2 1 2 πw

16 sin 2 ( πw) 5 + cos( πw)

1 − sin( 1 2 πw)

w arccos w + 1

arccos w

2 1− w 2 2

π − 2(1 − w) arcsin(1 − w)

arcsin(1 − w)

4 2 w−w 2

w arcsin w

4 ◦ . We now describe a simple method for finding an f (w) such that equation (2) admits an exact solution. To this end, we integrate equation (2) with respect to z and then apply the hodograph transformation (where w is regarded as the independent variable and z as the dependent one) to obtain

A is an arbitrary constant. (4) Substituting a specific expression z = z(w) for z on the right-hand side of relation (4), one obtains a

f (w) = − z ′

z dw + A ,

one-parameter family of functions f (w) for which z = z(w) solves equation (2). The explicit form of w = w(z) is obtained by the inversion of z = z(w).

The method just outlined was devised by Philip (1960); he obtained a large number of exact solutions to the original equation for various f = f (w). Some of his results, those corresponding to

a problem with the initial and boundary conditions of (3) with w 0 = 0 and w 1 = 1, are listed below in Table 1. All solutions are written out in implicit form, z = z(w), and are valid within the range of their spatial localization 0 ≤ w ≤ 1.

5 ◦ . There is another way to find an f (w) for which equation (2) admits exact solutions. By direct substitution, one can verify that equation (2) is satisfied by

where φ = φ(z) is an arbitrary function, and s is an arbitrary constant. Expressions (5) define a parametric representation of f = f (w); the explicit representation is obtained by eliminating z.

For example, assuming in (5) that

φ(z) = w 0 z+ ( w 0 − w 1 ) e − λz

( λ > 0, w 1 > w 0 ), ( λ > 0, w 1 > w 0 ),

B + C ln(w − w 0 ),

w=w 0 +( w 1 − w 0 ) e − λz ,

w−w 0

1 + ln( w 1 − w 0 ) , and C=− 1 2 λ −2 . Note that this solution satisfies the boundary conditions of (3). Likewise, one can construct other f (w).

where A=− 1 2 s λ −1 , B= 1 2 λ −2

6 ◦ . Here is one more method for constructing an f (w) for which equation (2) admits exact solutions. Suppose ¯ w=¯ w(z) is a solution of equation (2) with an f (w). Then ¯ w=¯ w(z) is also a solution of the more complicated equation [

F (w)w ′ ′ z z + 1 2 zw ′ z = 0 with

A is an arbitrary constant), (6) where the function g = g(w) is defined parametrically by

F (w) = f (w) + Ag(w)

g(w) =

(7) For example, the function ¯ w = bz 2 /m , where b is some constant, is a particular solution of

, w=¯ w(z).

equation (2) if f (w) is a power-law function, f (w) = aw m . It follows from (6) and (7) that ¯ w is also

a solution of equation (2) with f (w) = aw m

m−2

+ Aw 2 .

For the first solution presented in Table 1, the method outlined gives the following one-parameter family of functions:

for which z=1−w n is a solution of equation.

7 ◦ . The transformation

w(y, t) dy +

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

In the special case of power-law dependence, f (w) = aw m , transformation (8) leads to equa- tion (9) where ¯ f (w) = aw − m−2 .

8 ◦ . The equation in question is represented in conservative form, i.e., in the form of a conservation law.

Another conservation law:

Z where

F (w) =

f (w) dw.

9 ◦ . For f (w) = a(w 2 + b) −1 , see equation 1.1.13.2 and Subsection S.5.3 (Example 10).

References for equation 1.6.15.1: L. V. Ovsiannikov (1959, 1962, 1982), V. A. Dorodnitsyn and S. R. Svirshchevskii (1983), W. Strampp (1982), J. R. Burgan, A. Munier, M. R. Feix, and E. Fijalkow (1984), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), N. H. Ibragimov (1994), V. F. Zaitsev and A. D. Polyanin (1996), P. W. Doyle and P. J. Vassiliou (1998).

f (w)

+ g(w).

∂t ∂x

∂x

This equation governs unsteady heat conduction in a quiescent medium in the case where the thermal diffusivity and the rate of reaction are arbitrary functions of temperature.

1 ◦ . Traveling-wave solutions:

w = w(z), z= Ò x + λt,

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

(1) The substitution

[ f (w)w ′ z ] ′ z − λw z ′ + g(w) = 0.

y(w) =

f (w)w z ′

brings (1) to an Abel equation of the second kind:

where ϕ(w) = −λ −2 f (w)g(w). (2) The book by Polyanin and Zaitsev (2003) present a considerable number of solutions to equation (2)

yy w ′ − y = ϕ(w),

for various ϕ = ϕ(w).

2 ◦ . Let the function f = f (w) be arbitrary and let g = g(w) be defined by

g(w) =

A and B are some numbers. In this case, there is a functional separable solution, which is defined implicitly by

f (w) dw = At − Bx + C 1 x+C 2 ,

Ó☎Ô where C 1 and C 2 are arbitrary constants.

Reference : V. A. Galaktionov (1994).

3 ◦ . Let now g = g(w) be arbitrary and let f = f (w) be defined by

where A 1 , A 2 , and A 3 are some numbers. Then there are generalized traveling-wave solutions of the form

x+C 2 A 1 A 2

w = w(Z), Z= √

(2 A 3 t+C 1 ),

2 A 3 t+C 1 A 3 3 A 3

where the function w(Z) is determined by the inversion of (4), and C 1 and C 2 are arbitrary constants.

4 ◦ . Let g = g(w) be arbitrary and let f = f (w) be defined by

A 4 g(w)

where A 1 , A 2 , A 3 , and A 4 are some numbers ( A 4 ≠ 0). In this case, there are generalized traveling- wave solutions of the form

w = w(Z), Z = ϕ(t)x + ψ(t),

where the function w(Z) is determined by the inversion of (6),

A 3 dt ϕ(t) =

ψ(t) = −ϕ(t) A 1 ϕ(t) dt + A 2 + C 2 ,

A 4 ϕ(t) and Ó☎Ô C 1 and C 2 are arbitrary constants.

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

aϕ(w) + b

f (w) = ϕ ′ ( w),

g(w) =

+ c[aϕ(w) + b],

( w)

where ϕ(w) is an arbitrary function and a, b, and c are any numbers (the prime denotes a derivative with respect to w). Then there are functional separable solutions defined implicitly by

ϕ(w) = e at C 1 cos( x

ac ) + C 2 sin( x

ϕ(w) = e at C 1 cosh( x −

ac ) + C 2 sinh( x − ac ) −

if

ac < 0,

and Õ☎Ö C 1 and C 2 are arbitrary constants.

Reference : V. A. Galaktionov (1994).

6 ◦ . Let f (w) and g(w) be as follows:

g(w) = a w+2

ϕ ′ w ( w)

where ϕ(w) is an arbitrary function and a is any number. Then there are functional separable solutions defined implicitly by

2 ϕ(w) = C 2

1 e at − 1 2 a(x + C 2 ) ,

where Õ☎Ö C 1 and C 2 are arbitrary constants.

Reference : V. A. Galaktionov (1994).

7 ◦ . Group classification of solutions to the equation in question was carried out by Dorodnitsyn (1979, 1982); see also Dorodnitsyn and Svirshchevskii (1983), Galaktionov, Dorodnitsyn, Elenin, Kurdyumov, and Samarskii (1986), and Ibragimov (1994). As a result, only a limited number of equations were extracted that possess symmetries other than translations.

8 ◦ . If f = dF (w)/dw and g = aF (w) + bw + c, where F (w) is an arbitrary function, and a, b, and c are arbitrary constants, then there is a conservation law

F (w) − p(x)(F (w)) x + ϕ(x) x = 0. Here,

e − bt p(x)w bt

t + e p(x) x

 C 1 sin( √ ax) + C 2 cos(

ax) if

a > 0,

p(x) = C 1 e − ax + C 2 e − − ax

where ϕ ′ Õ☎Ö x = cp(x); C 1 and C 2 are arbitrary constants.

References : V. A. Dorodnitsyn (1979), V. A. Galaktionov, V. A. Dorodnitsyn, G. G. Elenin, S. P. Kurdyumov, and A. A. Samarskii (1986).

9 ◦ . For specific equations of this form, see Subsections 1.1.1 to 1.1.3, 1.1.11 to 1.1.13, 1.2.1 to 1.2.3, and 1.4.1.

+ h(x).

Functional separable solution in implicit form:

f (w) dw =

g(t) dt −

( x − ξ)h(ξ) dξ + C 1 x+C 2 ,

where C 1 and C 2 are arbitrary constants, and x 0 is any number.

This equation governs unsteady heat and mass transfer in an inhomogeneous fluid flow in the cases where the thermal diffusivity is arbitrarily dependent on temperature.

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

w 1 = w(x + C 1 e at , t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Generalized traveling-wave solution:

w = w(z),

z=x+C 1 e at ,

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

[ f (w)w ′ z ] ′ z −( az + b)w z ′ = 0.

3 ◦ . Functional separable solution in implicit form:

f (w) dw = C 1 e − at ( ax + b) + C 2 .

4 ◦ . On passing from t, x to the new variables

one obtains a simpler equation of the form 1.6.15.1 for w(ζ, τ ):

+ g(w)

1 ◦ . Traveling-wave solution in implicit form:

2 f (w) dw

= kx + λt + C 2 ,

G(w) =

g(w) dw,

λw − kG(w) + C 1

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

2 ◦ . The transformation dz = w dx +

dz = z x dx + z t dt leads to an equation of the similar form

f (w)w x + G(w) dt, dτ = dt, u = 1/w

where Z

Φ( u) = 2 f , Ψ( u) = g − G , G(w) =

g(w) dw.

Example. For f (w) = a and g(w) = bw, the original equation is an unnormalized Burgers equation 1.1.5.3. The above transformation brings it to the solvable equation

Reference : A. S. Fokas and Y. C. Yortsos (1982).

g(w) = A 2 + A 4 Z, where

f (w) = Z ′ w A 1 w+A 3 Z dw ,

(1) is a prescribed function (chosen arbitrarily). Then the original equation has the following generalized

Z = Z(w)

traveling-wave solution:

w = w(Z), Z = ϕ(t)x + (A 2 t+C 1 ) ϕ(t) + A 1 ϕ(t) ϕ(t) dt, where C 1 is an arbitrary constant, the function w(Z) is determined by the inversion of (1), and the

function ϕ(t) is determined by the first-order separable ordinary differential equation

(2) whose general solution can be written out in implicit form.

In special cases, solutions of equation (2) are given by

ϕ(t) = (C 2 −2 A 3 t) −1 /2 if A 4 = 0,

ϕ(t) = (C −1

2 − A 4 t)

if A 3 = 0.

4 ◦ . Conservation law:

g(w) dw. ∂w

D t ( w) + D x − f (w)w x − G(w) = 0,

G(w) =

+ g(t)

This equation governs unsteady heat conduction in a moving medium in the case where the thermal diffusivity is arbitrarily dependent on temperature.

On passing from t, x to the new variables t, z = x + g(t) dt, one obtains a simpler equation of the form 1.6.15.1:

+ xg(t)

On passing from t, x to the new variables (A and B are arbitrary constants) Z

g(t) dt , one obtains a simpler equation of the form 1.6.15.1 for w(τ , z):

τ= 2 G ( t) dt + A, z = xG(t),

where G(t) = B exp

+ xg(t) + h(t)

The transformation

G(t) = exp g(t) dt , leads to a simpler equation of the form 1.6.15.1:

w = U (z, τ ), z = xG(t) +

h(t)G(t) dt, τ=

Reference : V. F. Zaitsev and A. D. Polyanin (1996).

f (w)

+ g(w)

+ h(w).

For g ≡ const, this equation governs unsteady heat conduction in a medium moving at a constant velocity in the case where the thermal diffusivity and the reaction rate are arbitrary functions of temperature.

Traveling-wave solution:

w = w(z), z = x + λt,

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

(1) The substitution y(w) = f (w)w ′ z brings (1) to the Abel equation

[ f (w)w ′ z ] ′ z +[ g(w) − λ]w ′ z + h(w) = 0.

(2) The books by Polyanin and Zaitsev (1995, 2003) present a large number of exact solutions to

yy ′ w +[ g(w) − λ]y + f (w)g(w) = 0.

equation (2) for various f (w), g(w), and h(w). ∂w

+ ax + g(w)

+ h(w).

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

1 = w(x + C 1 e at , t+C 2 ),

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

2 ◦ . Generalized traveling-wave solution:

w = w(z),

z=x+C 1 e − at ,

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

[ f (w)w ′ z ] ′ z +[ az + g(w)]w ′ z + h(w) = 0.