Some Modifications and Generalizations

S.6.3. Some Modifications and Generalizations

S.6.3-1. Similarity reductions based on the ideas of the generalized separation of variables.

1 ◦ . The Clarkson–Kruskal direct method based on the representation of solutions in the forms (17) and (26) attaches particular significance to the function u = u(z), because the choice of the other functions is meant to ensure a single ordinary differential equation for u(z). However, in some cases it is reasonable to combine these methods with the ideas of the generalized and functional separation of variables, with all determining functions being regarded as equally important. Then, the function u(z) is described by an overdetermined system of equations.

2 ◦ . Exact solutions of nonlinear partial differential equations with quadratic or power nonlinearities may be sought in the form (1) with g(x, t) = g 1 ( t)x + g 0 ( t). Substituting (1) into an equation under consideration, we replace x by the expression x = [z − ψ(t)]/ϕ(t). As a result, we obtain a functional-differential equation with two arguments, t and z. Its solution can sometimes be obtained by the differentiation and splitting methods outlined in Subsections S.4.2–S.4.4.

Example 6. Consider the equation of an axisymmetric steady hydrodynamic boundary layer

∂y ∂x∂y

2 + F(x),

which, obviously, coincides with equation 9.3.1.3 in suitable notation. Its solution is sought in the form (for convenience, we introduce a coefficient a)

w(x, y) = af (x)u(z) + ag(x),

z = ϕ(x)y + ψ(x).

Let us substitute this expression into equation (31) and eliminate y, using the relation ϕ(x)y = z − ψ(x). After the division by a 2 ϕ 2 f , we arrive at the functional-differential equation

General methods for solving such equations are outlined in Section S.4. Here we use a simplified scheme for the construction of exact solutions. Assume that the functional coefficients of uu ′′ zz , u ′′ zz ,( u ′ z ) 2 , and 1 are linear combinations of the coefficients 1 and ψ of the highest-order terms (zu ′′ zz ) ′ z and u zzz ′′′ , respectively. We have

+ B 3 ψ, F/(a 2 fϕ 2 )= A 4 + B 4 ψ, + B 3 ψ, F/(a 2 fϕ 2 )= A 4 + B 4 ψ,

Case 1 . Let

A 1 = A 3 = A 4 = 0, A 2 =− n.

Then, the solution of equation (35) has the form

u(z) =

where C 1 , C 2 , and C 3 are integration constants. The solution (38) of equation (35) can be a solution of equation (36) only if the following conditions are satisfied:

n = −2, 2 B 1 = B 3 , C 1 = −4 /B 1 , C 2 =− B 4 /B 1 , C 3 =− B 2 /B 1 .

Let us insert the coefficients (37), (39) into system (34). Integrating yields

F = −(aC 2 C 4 ) x f ′ 4 f 3 ,

g(x) = 2x − C 3

where f = f (x) is an arbitrary function. Formulas (32), (38), (40) define an exact solution of the axisymmetric boundary layer equation (31).

Case 2 . For

B 1 = B 3 = B 4 = 0, B 2 =− λ, A 2 = 0, A 3 =− A 1 ,

A 4 = λ 2 /A 1 ( 41 )

a common solution of system (35), (36) can be written in the form

1 u(z) = − (

1 C 1 e λz + λz − 3).

A solution of system (34) with coefficients (41) is described by the formulas

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

g = g(x) is an arbitrary function.

Formulas (32), (42), (43) define an exact solution of the axisymmetric boundary layer equation (31). Case 3 . System (35)–(36) also admits solutions of the form

u(z) = C 2 1 z + C 2 z+C 3 ,

✸✂✹ with constants C 1 , C 2 , and C 3 related to the A n and B n . For the corresponding solutions of equation (31), see 9.3.1.3. References : G. I. Burde (1994, 1995), A. D. Polyanin and V. F. Zaitsev (2002).

Example 7. Consider the equation with a cubic nonlinearity

Let us seek its solution in the form

w(x, t) = f (x, t)u(z) + λ, z = z(x, t),

where the functions f = f (x, t), z = z(x, t), and u = u(z), as well as the constant λ, are to be determined. Substituting (45) into the equation, we obtain

From the overdetermined system of ordinary differential equations resulting from the condition of proportionality of the three

functions u ′′ , uu ′ , and u 3 and that of the two functions u ′ and u 2 , it follows that u(z) = 1/z,

where the constant factor is taken equal to unity [this factor can be included in f , since formula (45) contains the product of u and f ]. Let us substitute (47) into (46) and represent the resulting expression as a finite expansion in negative powers of z. Equating the functional coefficient of z −3 to zero, we obtain

f = βz x ,

Equating the functional coefficients of the other powers of z to zero and taking into account (48), we find that

z t − (3 a + βσ)z xx +( σλ + βb 2 +3 βb 3 λ)z x =0

(coefficient of −2 z ),

z xt − az xxx + σλz xx −( b 1 +2 λb 2 +3 b 3 λ 2 ) z x =0

(coefficient of z −1 ),

(coefficient of z 0 ). Here, the first two linear partial differential equations form an overdetermined system for the function z(x, t), while the last

b 3 λ 3 + b 2 λ 2 + b 1 λ+b 0 =0

cubic equation serves for the determination of the constant λ. Using (45), (47), and (48), we can write out a solution of equation (44) in the form

β ∂z

w(x, t) =

Let β be a root of the quadratic equation (49), and λ be a root of the last (cubic) equation in (50). According to the

value of the constant b 3 , one should consider two cases. 1 ◦ . Case b 3 ≠ 0. From the first two equations in (50), one obtains

z t + p 1 z xx + p 2 z x = 0, z xxx + q 1 z xx + q 2 z x = 0,

where βb 2 +3 βλb 3 3 b 3 λ 2 +2 b 2 λ+b 1 p 1 =− βσ − 3a,

βσ + 2a Four situations are possible.

βσ + 2a

1.1. For q 2 ≠ 0 and q 2 1 ≠4 q 2 , we have z(x, t) = C 1 exp( k 1 x+s 1 t) + C 2 exp( k 2 x+s 2 t) + C 3 ,

k n =− 2 q 1 ✺ 2 q 2 1 −4 q 2 ,

s n =− 2 k n p 1 − k n p 2 ,

where C 1 , C 2 , and C 3 are arbitrary constants; 1.2. For q 2 ≠ 0 and q 2

n = 1, 2.

1 =4 q 2 ,

z(x, t) = C 1 exp( kx + s 1 t) + C 2 ( kx + s 2 t) exp(kx + s 1 t) + C 3 ,

k=− 1

2 q 1 , s 1 =− 1 2 1 1 2 4 1 p 1 q 1 + 2 p 2 q 1 , s 2 =− 2 p 1 q 1 + 2 p 2 q 1 .

1.3. For q 2 = 0 and q 1 ≠ 0,

z(x, t) = C 1 ( x−p 2 t) + C 2 exp[− q 1 x+q 1 ( p 2 − p 1 q 1 ) t] + C 3 . 1.4. For q 2 = q 1 = 0,

z(x, t) = C 1 ( x−p 2 t) 2 + C 2 ( x−p 2 t) − 2C 1 p 1 t+C 3 .

2 ◦ . Case b 3 = 0, b 2 ≠ 0. The solutions are determined by (51), where

2 ab 2 σ(b 1 +2 β=− 2

z(x, t) = C 1 + C 2 exp Ax + A

and ✻✂✼ λ=λ 1,2 are roots of the quadratic equation b 2 λ 2 + b 1 λ+b 0 = 0.

References : M. C. Nucci and P. A. Clarkson (1992), N. A. Kudryashov (1993).

S.6.3-2. Similarity reductions in equations with three or more independent variables. The procedure of the construction of exact solutions to nonlinear equations with three or more inde-

pendent variables sometimes involves (at intermediate stages) the solution of functional-differential equations considered in Subsections S.4.2–S.4.4.

Example 8. Consider the nonlinear nonstationary wave equation anisotropic in one of the directions

Let us seek its solution in the form

w = U (z) + f (x, t),

z = y + g(x, t).

Substituting (53) into equation (52), we get

bU + ag 2 x − 2 g t +

bf + c)U z ′ ] ′ z +( ag xx − g tt ) U z ′ + af xx − f tt = 0.

where C 1 , C 2 , and C 3 are arbitrary constants. Then the function U (z) is determined by the autonomous ordinary differential equation

[( bU + c + C 3 ) U z ′ ] ′ z + C 2 U ′ z + C 1 = 0.

The general solutions of equations (54)–(55) are expressed as

f=ϕ 1 ( ξ) + ψ 1 (

g=ϕ 2 ( ξ) + ψ 2 ( η) − 1 2 C 2 t 2 , ξ=x+t √ a, η=x−t √ a.

Let us insert these expressions into equation (56) and then eliminate ξ−η

t with the help of the formula t = . After simple 2 √ a transformations, we obtain a functional-differential equation with two arguments,

bϕ 1 ( ξ) + C 2 ξϕ ′ 2 ( ξ) − kξ 2 − C 3 + bψ 1 ( η) + C 2 ηψ ′ 2 ( η) − kη 2 + ψ 2 ′ ( η)[4aϕ ′ 2 ( ξ) − C 2 ξ] + η[2kξ − C 2 ϕ ′ 2 ( ξ)] = 0,

Equation (58) can be solved by the splitting method described in Section S.4. According to the simplified scheme, set

bϕ 1 ( ξ) + C 2 ξϕ ′ 2 ( ξ) − kξ 2 − C 3 = A 1 , 4 aϕ ′ 2 ( ξ) − C 2 ξ=A 2 ,

2 kξ − C 2 ϕ ′ 2 ( ξ) = A 3 , where A 1 , A 2 , and A 3 are constants. The common solution of system (59) has the form

ϕ 1 ( ξ) = − C 2 2 2 BC 2 A 1 + 3 ξ 2 − ξ+ C , ϕ 2 ( ξ) = C ξ 2 + Bξ

( 8 60 ab b b 8 a )

and corresponds to the following values of the constants:

A 1 is arbitrary, A 2 =4 aB, A 3 =− BC 2 ,

B is arbitrary, C 1 =− C 2 , C 2 and C 3 are arbitrary, k= C 2 .

b 8 a From (58) and (59) we obtain an equation that establishes a relation between the functions ψ 1 and ψ 2 ,

Hence, taking into account (61), we get

ψ 1 ( 1 η) = − ( C 2 η + 4aB)ψ ′ 2 ( η) + C 2 η 2 + BC 2 η−A 1 ,

ψ 2 ( η) is an arbitrary function.

Ultimately, we find the functions that determine solution (53):

C η + 4aB)ψ η),

ab 2 b b b b

g(x, t) = 2 C x 2

+2 √ a xt − 3at 2 + B(x + √ a t) + ψ 2 ( η).

Remark 1. For other solutions of this equation, see 4.1.3.1. Remark 2. In the special case of

a = 1, b < 0, and c > 0, equation (52) describes spatial transonic flows of an ideal polytropic gas (Pokhozhaev, 1989).