Differentiation Method

S.5.3. Differentiation Method

S.5.3-1. Basic ideas of the method. Reduction to a standard equation. In general, the substitution of expression (1) into the nonlinear partial differential equation under

study leads to a functional-differential equation with three arguments—two arguments are usual, x and y, and the third is composite, z. In many cases, the resulting equation can be reduced by differentiation to a standard functional-differential equation with two arguments (either x or y is eliminated). To solve the two-argument equation, one can use the methods outlined in Subsec- tions S.4.2–S.4.4.

S.5.3-2. Examples of constructing functional separable solutions. Below we consider specific examples illustrating the application of the differentiation method for

constructing functional separable solutions of nonlinear equations.

Example 10. Consider the nonlinear heat equation

We look for exact solutions with the form

w = w(z),

z = ϕ(x) + ψ(t).

On substituting (15) into (14) and dividing by w ′ z , we obtain the functional-differential equation with three variables

xx 2 ′′ F(w) + (ϕ ′ x ) H(z),

H(z) = F(w) zz + w ′ F z ′ ( w),

w = w(z).

Differentiating (16) with respect to x yields

xxx 3 ′′′ F(w) + ϕ ′ x ϕ ′′ xx [ F z ′ ( w) + 2H(z)] + (ϕ ′ x ) H ′ z = 0.

This functional-differential equation with two variables can be treated as the functional equation (49) of Subsection S.4.4. This three-term functional equation has two different solutions. Accordingly, we consider two cases.

Case 1. The solutions of the functional-differential equation (18) are determined from the system of ordinary differential equations

where A 1 and A 2 are arbitrary constants. The first two equations (19) are linear and independent of the third equation. Their general solution is given by

  e A 1 z [ B 1 sin( kz) + B 2 cos( kz)] if A 2 1 <2 A 2 ,

Substituting H of (20) into (17) yields an ordinary differential equation for w = w(z). On integrating this equation, we obtain

w=C 1 e A 1 z |

F(z)| −3 /2 dz + C 2 ,

where C 1 and C 2 are arbitrary constants. The expression of F in (20) together with expression (21) define the function F = F(w) in parametric form. Without full analysis, we will study the case A 2 =0( k=A 1 ) and A 1 ≠ 0 in more detail. It follows from (20) and (21) that

F(z) = B 2 1 e A 1 z + B 2 , H=A 1 B 2 ,

w(z) = C 3 ( B 1 +

B −2 2 e A 1 z ) −1 /2 + C 2 ( C 1 = A 1 B 2 C 3 ).

Eliminating z yields

F(w) =

ln

2 A 1 D 1 sinh 2 A 1 √ D 2 x+D 3

D 1 < 0 and D 2 A > 0; cosh A 1 √ D 2 x+D 3

where D 1 , D 2 , and D 3 are constants of integration. In all three cases, the following relations hold:

We substitute (22) and (25) into the original functional-differential equation (16). With reference to the expression of z in (15), we obtain the following equation for ψ = ψ(t):

ψ ′ t =− A 1 B 1 D 1 e 2 A 1 ψ + A 1 B 2 D 2 .

Its general solution is given by

1 2 ψ(t) = 2

1 ln

2 A D 4 exp(−2 A 2 1 B 2 D 2 t) + B 1 D 1 ,

where D 4 is an arbitrary constant. Formulas (15), (22) for w, (24), and (26) define three solutions of the nonlinear equation (14) with F(w) of the form (23)

[recall that these solutions correspond to the special case A 2 = 0 in (20) and (21)].

Case 2. The solutions of the functional-differential equation (18) are determined from the system of ordinary differential equations

The first two equations in (27) are consistent in the two cases

A 1 = A 2 =0

=⇒ ϕ(x) = B 1 x+B 2 ,

1 =2 2 1 ( 28 A ) A 2 =⇒ ϕ(x) = − ln | B 1 x+B 2 |.

The first solution in (28) eventually leads to the traveling-wave solution 2 w = w(B 1 x+B 2 t) of equation (14) and the second

solution to the self-similar solution of the form ✪✂✫ w=e w(x /t). In both cases, the function F(w) in (14) is arbitrary.

References : P. W. Doyle and P. J. Vassiliou (1998), A. D. Polyanin (2002, Supplement B), A. D. Polyanin and V. F. Zaitsev (2002).

Remark. The more general nonlinear heat equation

∂w = ∂

∂w +

∂x F(w) ∂x

G(w)

∂t

has also solutions of the form (15). For the unknown functions ϕ(x) and ψ(t), we have the functional-differential equation in three variables

ψ 2 t ′ = ϕ ′′ xx F(w) + (ϕ ′ x ) H(z) + G(w)/w z ′ ,

where w = w(z) and H(z) is defined by (17). Differentiating with respect to x yields ϕ ′′′ xxx F(w) + ϕ ′ x ϕ ′′ xx [ F z ′ ( w) + 2H(z)] + (ϕ ′ x ) 3 H ′ z + ϕ ′ x [ G(w)/w ′ z ] ′ z = 0. This functional-differential equation in two variables can be treated as the bilinear functional equation (51) of Subsection S.4.4

with Φ 1 = ϕ ′′′ xxx ,Φ 2 = ϕ ′ x ϕ ′′ xx ,Φ 3 =( ϕ ′ x ) 3 , and Φ 4 = ϕ ′ x .

See also Est´evez, Qu, and Zhang (2002), where a more general equation was considered. Example 11. Consider the nonlinear Klein–Gordon equation

∂x 2 = F(w).

∂t

We look for functional separable solutions in additive form:

w = w(z),

z = ϕ(x) + ψ(t).

Substituting (30) into (29) yields

ψ ′′ tt − ϕ ′′ xx + ( ψ ′ t ) 2 −( ϕ ′ x ) 2 g(z) = h(z),

where

g(z) = w zz ′′ /w ′ z , h(z) = F w(z) /w ′ z .

ψ 2 tt ϕ xx g z ψ t ϕ x ′ ) g ′′ zz = h ′′ zz .

Eliminating ψ ′′ tt − ϕ ′′ xx from this equation with the aid of (31), we obtain

t 2 ) 2 −( ϕ x ′ ) ( g ′′ zz −2 gg ′ z )= h ′′ zz −2 g ′ z h. ( 33 )

This relation holds in the following cases:

h zz ′′ −2 g z ′ h = (Az + C)(g zz ′′ −2 gg ′ z ) (case 2), where

= Aψ + B, ( ϕ ′

x ) 2 =− Aϕ + B − C,

A, B, and C are arbitrary constants. We consider both cases. Case 1 . The first two equations in (34) enable one to determine g(z) and h(z). Integrating the first equation once yields

g ′ z = g 2 + const. Further, the following cases are possible:

g = k,

(35a)

g = −1/(z + C 1 ),

(35b)

g = −k tanh(kz + C 1 ),

(35c)

g = −k coth(kz + C 1 ),

(35d)

(35e) where C 1 and k are arbitrary constants.

g = k tan(kz + C 1 ),

The second equation in (34) has a particular solution h = g(z). Hence, its general solution in expressed by (e.g., see Polyanin and Zaitsev (2003))

2 3 h=C dz g(z) + C g(z)

z)

where C 2 and C 3 are arbitrary constants. The functions w(z) and F(w) are found from (32) as

w(z) = B 1 G(z) dz + B 2 ,

F(w) = B 1 h(z)G(z), where G(z) = exp g(z) dz ,

and B 1 and B 2 are arbitrary constants (

F is defined parametrically).

Let us dwell on the case (35b). According to (36),

where A 1 =− C 3 /3 and A 2 =− C 2 are any numbers. Substituting (35b) and (38) into (37) yields

Eliminating z, we arrive at the explicit form of the right-hand side of equation (29):

F(w) = A 1 B 1 e + A 2 B 1 e ,

where u=

b. Thus, we have w(z) = ln |z|, 2 F(w) = ae w + −2 be w , g(z) = −1/z, h(z) = az + b/z.

For simplicity, we set C 1 = 0, B 1 = 1, and B 2 = 0 and denote A 1 =

a and A 2 =

It remains to determine ψ(t) and ϕ(x). We substitute (40) into the functional-differential equation (31). Taking into account (30), we find

Differentiating (41) with respect to t and x yields the separable equation*

( ψ ttt ′′′ −6 aψψ t ′ ) ϕ ′ x −( ϕ ′′′ xxx +6 aϕϕ ′ x ) ψ ′ t = 0,

whose solution is determined by the ordinary differential equations

ψ ttt ′′′ −6 aψψ ′ t = Aψ ′ t , ϕ ′′′ xxx +6 aϕϕ ′ x = Aϕ ′ x ,

where A is the separation constant. Each equation can be integrated twice, thus resulting in

* To solve equation (41), one can use the solution of equation (51) in Subsection S.4.4 [see (52)].

Nonlinear Klein–Gordon equations ∂ tt w−∂ xx w = F(w) admitting functional separable solutions of the form w = w(z), z = ϕ(x) + ψ(t). Notation: A, C 1 , and C 2 are arbitrary constants; σ = 1 for z > 0 and σ = −1 for z < 0

No. Right-hand side F(w)

Solution w(z)

Equations for ψ(t) and ϕ(x)

4 a sinh w + b sinh w ln tanh w + 2 sinh w

2 ln

2 2 ϕ −2 x ′ ) = C 2 e ϕ + C 1 e ϕ + σbϕ + A ′ w 2 z ( ψ t ) = C 1 sin 2 ψ+C 2 cos 2 ψ + σbψ + a + A,

5 a sinh w + 2b sinh w arctan e w/2 + cosh

ϕ 2 x ′ ) =− C 1 sin 2 ϕ+C 2 cos 2 ϕ − σbϕ + A where C 1 , ...,C 4 are arbitrary constants. Eliminating the derivatives from (41) with the aid of (42), we find that the arbitrary

2 ln

b. So, the functions ψ(t) and ϕ(x) are determined by the first-order nonlinear autonomous equations

constants are related by C 3 =− C 1 and C 4 = C 2 +

( ψ ′ t ) 2 =2 aψ 3 + Aψ 2 + C 1 ψ+C 2 , ( ϕ ′ x ) 2 = −2 aϕ 3 + Aϕ 2 − C 1 ϕ+C 2 + b.

The solutions of these equations are expressed in terms of elliptic functions. For the other cases in (35), the analysis is performed in a similar way. Table 17 presents the final results for the cases (35a)–(35e).

Case 2 . Integrating the third and fourth equations in (34) yields

Bt+D 1 ,

B−Ct+D 2 if

where D 1 and D 2 are arbitrary constants. In both cases, the function F(w) in equation (29) is arbitrary. The first row in (43) corresponds to the traveling wave solution w = w(kx + λt). The second row leads to a solution of the form w = w(x 2 − t ✭✂✮ 2 ).

References : A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Rodionov (1999).

Example 12. The nonlinear stationary heat (diffusion) equation

∂ 2 w + ∂ 2 w = ∂x 2 ∂y 2 F(w)

is analyzed in much the same way as the nonlinear Klein–Gordon equation considered in Example 11. The final results are listed in Table 18; the traveling wave solutions w = w(kx + λt) and solutions of the form w = w(x 2 + y 2 ), existing for any ✭✂✮ F(w), are omitted.

References : A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Rodionov (1999).