Solution of Functional Differential Equations by Splitting

S.4.4. Solution of Functional Differential Equations by Splitting

S.4.4-1. Preliminary remarks. Description of the method. As one reduces the number of terms in the functional-differential equation (21)–(22) by differentia-

tion, redundant constants of integration arise. These constants must be “removed” at the final stage. Furthermore, the resulting equation can be of a higher-order than the original equation. To avoid these difficulties, it is convenient to reduce the solution of the functional-differential equation to the solution of a bilinear functional equation of a standard form and solution of a system of ordinary differential equations. Thus, the original problem splits into two simpler problems. Below we outline the basic stages of the splitting method.

1 ◦ . At the first stage, we treat equation (21) as a purely functional equation that depends on two variables

X and Y , where Φ n =Φ n (

X) and Ψ n =Ψ n ( Y ) are unknown quantities (n = 1, . . . , k).

It can be shown that the bilinear functional equation (21) has k − 1 different solutions:

X) = C i,1 Φ m+1 (

X) + C i,2 Φ m+2 (

X) + · · · + C i,k−m Φ k ( X),

i = 1, . . . , m;

j = 1, . . . , k − m; (48) m = 1, 2, . . . , k − 1;

Ψ m+j ( Y ) = −C 1, j Ψ 1 ( Y)−C 2, j Ψ 2 ( Y)−···−C m,j Ψ m ( Y ),

where the C i,j are arbitrary constants. The functions Φ m+1 ( X), . . . , Φ k ( X), Ψ 1 ( Y ), . . . , Ψ m ( Y) on the right-hand sides of equations (48) are defined arbitrarily. It is apparent that for fixed m, solution (48) contains m(k − m) arbitrary constants.

X) and Ψ j ( Y ) of (22) into all solu- tions (48) to obtain systems of ordinary differential equations* for the unknown functions ϕ p ( x) and ψ q ( y). Solving these systems, we get generalized separable solutions of the form (19).

2 ◦ . At the second stage, we successively substitute the Φ i (

Remark 1. It is important that, for fixed k, the bilinear functional equation (21) used in the splitting method is the same for different classes of original nonlinear mathematical physics equa- tions.

* Such systems are usually overdetermined.

Original equation: Fxyww (,, , x , ww y , xx , w xy , w yy , ...) = 0

Search for generalized separable solutions

Define solution structure: w = j 1 () x y 1 ( ) + ... + y j n () x y n () y

Substitute into original equation

Write out the functional-differential equation

Apply splitting procedure

Obtain: (i) functional equation, (ii) determining system of ODEs

Treat functional equation (i)

Solve the functional equation: F 1 () x Y 1 ( ) + ... + y F k () x Y k ( y )=0

Substitute the F m and Y m

in determining system (ii)

Solve the determining system of ordinary differential equations

Find the j m and y m from the determining system of ODEs

Write out generalized separable solution of original equation

Figure 1. General scheme for constructing generalized separable solutions by the splitting method. Abbreviation: ODE stands for ordinary differential equation.

Remark 2. For fixed m, solution (48) contains m(k − m) arbitrary constants C i,j . Given k, the solutions having the maximum number of arbitrary constants are defined by

Solution number Number of arbitrary constants Conditions on k

1 1 m= 2

if k is even,

if k is odd. It is these solutions of the bilinear functional equation that most frequently result in nontrivial

m= 1 2 ( k ✗ 1)

4 ( k 2 − 1)

generalized separable solution in nonlinear partial differential equations. Remark 3. The bilinear functional equation (21) and its solutions (48) play an important role

in the method of functional separation of variables. For visualization, the main stages of constructing generalized separable solutions by the splitting

method are displayed in Fig. 1.

S.4.4-2. Solutions of simple functional equations and their application. Below we give solutions to two simple bilinear functional equations of the form (21) that will be

used subsequently for solving specific nonlinear partial differential equations.

1 ◦ . The functional equation

(49) where the Φ i are all functions of the same argument and the Ψ i are all functions of another argument,

has two solutions: Φ 1 = A 1 Φ 3 , Φ 2 = A 2 Φ 3 , Ψ 3 =− A 1 Ψ 1 − A 2 Ψ 2 ;

3 , Ψ 2 = A 2 Ψ 3 , Φ 3 =− A 1 Φ 1 − A 2 Φ 2 .

A 1 = −1 /C 1,2 and A 2 = C 1,1 /C 1,2 in the second solution. The functions on the right-hand sides of the equations in (50) are assumed to be arbitrary.

2 ◦ . The functional equation

(51) where the Φ i are all functions of the same argument and the Ψ i are all functions of another argument,

has a solution

Ψ 3 =− A 1 Ψ 1 − A 3 Ψ 2 , Ψ 4 =− A 2 Ψ 1 − A 4 Ψ 2

dependent on four arbitrary constants A 1 , ...,A 4 ; see solution (48) with k = 4, m = 2, C 1,1 = A 1 ,

C 1,2 = A 2 , C 2,1 = A 3 , and C 2,2 = A 4 . The functions on the right-hand sides of the equations in (50) are assumed to be arbitrary. Equation (51) has also two other solutions

Φ 1 = A 1 Φ 4 , Φ 2 = A 2 Φ 4 , Φ 3 = A 3 Φ 4 , Ψ 4 =− A 1 Ψ 1 − A 2 Ψ 2 − A 3 Ψ 3 ;

2 = A 2 Ψ 4 , Ψ 3 = A 3 Ψ 4 , Φ 4 =− A 1 Φ 1 − A 2 Φ 2 − A 3 Φ 3 involving three arbitrary constants. In the first solution, A 1 = C 1,1 , A 2 = C 2,1 , and A 3 = C 3,1 , and in

the second solution, A 1 = −1 /C 1,3 , A 2 = C 1,1 /C 1,3 , and A 3 = C 1,2 /C 1,3 .

Solutions (53) will sometimes be called degenerate, to emphasize the fact that they contain fewer arbitrary constants than solution (52).

Example 10. Consider the nonlinear hyperbolic equation

∂w

f (t)w + g(t),

where f (t) and g(t) are arbitrary functions. We look for generalized separable solutions of the form

w(x, t) = ϕ(x)ψ(t) + χ(t).

Substituting (55) into (54) and collecting terms yields

aψ 2 ( ϕϕ x ′ ) ′ x + aψχϕ ′′ xx +( fψ−ψ ′′ tt ) ϕ+fχ+g−χ ′′ tt = 0.

This equation can be represented as a functional equation (51) in which

3 tt . On substituting (56) into (52), we obtain the following overdetermined system of ordinary differential equations for the

aψ 2 ,

Ψ 2 = aψχ, Ψ = fψ−ψ ′′ tt , Ψ 4 = fχ+g−χ ′′

functions ϕ = ϕ(x), ψ = ψ(t), and χ = χ(t):

A 1 aψ − A 3 aψχ, fχ+g−χ tt ′′ =− A 2 aψ − A 4 aψχ. The first two equations in (57) are consistent only if

fψ−ψ tt ′′ =−

where B 0 , B 1 , and B 2 are arbitrary constants, and the solution is given by

ϕ(x) = B 2 2 x + B 1 x+B 0 .

On substituting the expressions (58) into the last two equations in (57), we obtain the following system of equations for ψ(t) and χ(t):

ψ ′′ tt =6 aB 2 ψ 2 +

f (t)ψ,

χ tt = [2 aB 2 ψ + f (t)]χ + a(B 1 −4 B 0 B 2 ) ψ + g(t).

Relations (55), (59) and system (60) determine a generalized separable solution of equation (54). The first equation in (60) can be solved independently; it is linear if B 2 = 0 and is integrable by quadrature for

f (t) = const. The second equation in (60) is linear in χ (for ψ known). Equation (54) does not have other solutions with the form (55) if

f and g are arbitrary functions and ϕ ✘ 0, ψ ✘ 0, and χ ✘ 0.

w(x, y) = ϕ 1 ( x)ψ 1 ( t) + ϕ 2 ( x)ψ 2 ( t) + ψ 3 ( t),

ϕ 1 ( x) = x 2 , ϕ 2 ( x) = x,

where the functions ψ i = ψ i ( t) are determined by the ordinary differential equations

aψ 2 1 +

f (t)ψ 1 ,

ψ ′′ 2 = [6 aψ 1 +

f (t)]ψ 2 ,

ψ 3 ′′ = [2 aψ 1 +

f (t)]ψ 2 3 + aψ 2 + g(t).

(The prime denotes a derivative with respect to t.) The second equation in (62) has a particular solution ψ 2 = ψ 1 . Hence, its general solution can be represented as (see Polyanin and Zaitsev, 2003)

dt

ψ 1 2 . The solution obtained in Example 10 corresponds to the special case C 2 = 0.

Example 11. Consider the nonlinear equation

which arises in hydrodynamics [see 9.3.3.1, equation (2) and 10.3.3.1, equation (4) with f 1 ( t) = 0]. We look for exact solutions of the form

w = ϕ(t)θ(x) + ψ(t).

Substituting (64) into (63) yields

2 ϕ 2 t ′ θ x ′ − ϕψθ ′′ xx + ϕ ( θ ′ x ) − θθ ′′ xx − νϕθ ′′′ xxx = 0.

This functional-differential equation can be reduced to the functional equation (51) by setting

On substituting these expressions into (52), we obtain the system of equations

It can be shown that the last two equations in (66) are consistent only if the function θ and its derivative are linearly dependent,

θ ′ x = B 1 θ+B 2 .

The six constants B 1 , B 2 , A 1 , A 2 , A 3 , and A 4 must satisfy the three conditions B 1 ( A 1 + B 2 − A 3 B 1 ) = 0, B 2 ( A 1 + B 2 − A 3 B 1 ) = 0,

B 1 2 + A 4 B 1 − A 2 = 0.

Integrating (67) yields

B 3 exp( B 1

where B 3 is an arbitrary constant. The first two equations in (66) lead to the following expressions for ϕ and ψ:

if A 2 ≠ 0,

C exp(−A 2 νt) − A 1

where C is an arbitrary constant. Formulas (69), (70) and relations (68) allow us to find the following solutions of equation (63) with the form (64):

w=C 1 e − λ(x+βνt) + ν(λ + β)

if A 1 = A 3 = B 2 = 0, A 2 = B 2 1 + A 4 B 1 ; w= νβ + C 1 e − − λx νλβt + ν(λ − β) if A 1 = A 3 B 1 − B 2 , A 2 = B 2 1 + A 4 ,

1+ C 2

where C 1 , C 2 , C 3 , β, and λ are arbitrary constants (these can be expressed in terms of the A k and B k ).

of the differential equation (63):

w = ϕ(t)e − λx

− ϕ t ( t) + νλ,

λϕ(t) ✙✂✚ where ϕ(t) and ψ(t) are arbitrary functions, and C 1 and λ are arbitrary constants.

References for Subsection S.4.4: E. R. Rozendorn (1984), A. D. Polyanin (2002, Supplement B), A. D. Polyanin and A. I. Zhurov (2002).