Solutions of Some Nonlinear Functional Equations and Their Applications

S.5.5. Solutions of Some Nonlinear Functional Equations and Their Applications

In this subsection, we discuss several types of three-argument functional equations that arise most frequently in the functional separation of variables in nonlinear equations of mathematical physics. The results are used for constructing exact solutions for some classes of nonlinear heat and wave equations.

S.5.5-1. The functional equation f (x) + g(y) = Q(z), where z = ϕ(x) + ψ(y). Here, one of the two functions f (x) and ϕ(x) is prescribed and the other is assumed unknown, also

one of the functions g(y) and ψ(y) is prescribed and the other is unknown, and the function Q(z) is assumed unknown.*

Differentiating the equation with respect to x and y yields Q ′′ zz = 0. Consequently, the solution is given by

f (x) = Aϕ(x) + B, g(y) = Aψ(y) − B + C, Q(z) = Az + C, (48) where

A, B, and C are arbitrary constants.

S.5.5-2. The functional equation f (t) + g(x) + h(x)Q(z) + R(z) = 0, where z = ϕ(x) + ψ(t). Differentiating the equation with respect to x yields the two-argument equation

(49) Such equations were discussed in Subsections S.4.2–S.4.4. Hence, the following relations hold [see

g ′ x + h ′ x Q + hϕ ′ x Q ′ z + ϕ ′ x R ′ z = 0.

formulas (51) and (52) in Subsection S.4.4]:

where A 1 , ...,A 4 are arbitrary constants. By integrating system (50) and substituting the resulting solutions into the original functional equation, one obtains the results given below.

* In similar equations with a composite argument, it is assumed that ϕ(x) ✰ const and ψ(y) ✰ const.

where the A k and B k are arbitrary constants and ϕ = ϕ(x) and ψ = ψ(t) are arbitrary functions. Case 2 . If A 3 ≠ 0 in (50), the corresponding solution of the functional equation is

where the A k and B k are arbitrary constants and ϕ = ϕ(x) and ψ = ψ(t) are arbitrary functions. Case 3 . In addition, the functional equation has the two degenerate solutions:

f=A 1 ψ+B 1 , g=A 1 ϕ+B 2 , h=A 2 , R = −A 1 z−A 2 Q−B 1 − B 2 , (53a) where ϕ = ϕ(x), ψ = ψ(t), and Q = Q(z) are arbitrary functions, A 1 , A 2 , B 1 , and B 2 are arbitrary

constants, and f=A 1 ψ+B 1 , g=A 1 ϕ+A 2 h+B 2 , Q = −A 2 , R = −A 1 z−B 1 − B 2 ,

(53b) where ϕ = ϕ(x), ψ = ψ(t), and h = h(x) are arbitrary functions, A 1 , A 2 , B 1 , and B 2 are arbitrary

constants. The degenerate solutions (53a) and (53b) can be obtained directly from the original equation or its consequence (49) using formulas (53) in Subsection S.4.4.

Example 13. Consider the nonstationary heat equation with a nonlinear source

∂w = ∂ 2 w

∂t

∂x 2 + F(w).

We look for exact solutions of the form

w = w(z),

z = ϕ(x) + ψ(t).

Substituting (55) into (54) and dividing by w z ′ yields the functional-differential equation

ϕ ′′ xx +( ϕ ′ x )

zz + F(w(z)) . w z ′

We rewrite it as the functional equation S.5.5-2 in which

f (t) = −ψ t ′ , g(x) = ϕ ′′ xx , h(x) = (ϕ ′ x ) 2 ,

Q(z) = w zz ′′ /w ′ z , R(z) = f (w(z))/w ′ z .

We now use the solutions of equation S.5.5-2. On substituting the expressions of g and h of (56) into (51)–(53), we arrive at overdetermined systems of equations for ϕ = ϕ(x). Case 1 . The system

ϕ 2 ′′ xx = 1

2 A 1 A 4 ϕ +( A 1 B 1 + A 2 ) ϕ+B 2 ,

( ϕ ′ x ) 2 = A 4 ϕ+B 1

ϕ= 4 A 4 x + C 1 x+C 2 for A 1 = A 2 = 0, B 1 = C 1 − A 4 C 2 , B 2 = 2 A 4 , where C 1 and C 2 are arbitrary constants.

The first solution in (57) with A 1 ≠ 0 leads to a right-hand side of equation (54) containing the inverse of the error function [the form of the right-hand side is identified from the last two relations in (51) and (56)]. The second solution in (57) corresponds to the right-hand side F(w) = k 1 w ln w + k 2 w in (54). In both cases, the first relation in (51) is, taking into account that

f = −ψ ′ t , a first-order linear solution with constant coefficients, whose solution is an exponential plus a constant. Case 2 . The system

ϕ ′′ = A 1 B 1 e 1 A 3 ϕ + A 2 − A A xx 4

ϕ+B 2 ,

following from (52) and corresponding to A 3 ≠ 0 in (50) is consistent in the following cases: ϕ= ✱ √ − A 4 /A 3 x+C 1 for A 2 = A 1 A 4 /A 3 , B 1 = B 2 = 0,

2 3 4 2 2 A 3 A 4 , B 2 = 0, A 3 A A < 0, where C 1 and C 2 are arbitrary constants. The right-hand sides of equation (54) corresponding to these solutions are

1 − A 3 A 4 x+C 1 2 for A 1 = 1 A 2 3 , A 2 = 1

represented in parametric form. Case 3 . Traveling wave solutions of the nonlinear heat equation (54) and solutions of the linear equation (54) with F w ′ = const correspond to the degenerate solutions of the functional equation (53). Remark. It may be reasonable to look for more complicated solutions of equation (54) of the form

w = w(z),

z = ϕ(ξ) + ψ(t),

ξ = x + at.

Substituting these expressions into equation (54) yields the functional equation S.5.5-2 again, in which ( x must be replaced by ξ)

R(z) = f (w(z))/w z ′ . Further, one should follow the same procedure of constructing the solution as in Example 13.

f (t) = −ψ ′ t ,

g(ξ) = ϕ ′′ ξξ − aϕ ′ ξ ,

′ ξ ) h(ξ) = (ϕ 2 , Q(z) = w ′′ zz /w ′ z ,

Example 14. Likewise, one can analyze the more general equation

∂w

2 = a(x) ∂ w + b(x) ∂w

∂t

∂x 2 ∂x + F(w).

It arises in convective heat/mass exchange problems ( a = const and b = const), problems of heat transfer in inhomogeneous media ( b=a ′ x ≠ const), and spatial heat transfer problems with axial or central symmetry ( a = const and b = const /x).

Searching for exact solutions of equation (58) in the form (55) leads to the functional equation S.5.5-2 in which f (t) = −ψ ′ t ,

g(x) = a(x)ϕ ′′ xx + b(x)ϕ ′ ( x), h(x) = a(x)(ϕ ′ x ) 2 , Q(z) = w ′′ zz /w z ′ , R(z) = f (w(z))/w z ′ .

Substituting these expressions into (51)–(53) yields a system of ordinary differential equations for the unknowns.

Remark. In Examples 13 and 14, different equations were all reduced to the same functional equation. This demonstrates the utility of the isolation and independent analysis of individual types of functional equations, as well as the expedience of developing methods for solving functional equations with a composite argument.

S.5.5-3. The functional equation f (t) + g(x)Q(z) + h(x)R(z) = 0, where z = ϕ(x) + ψ(t). Differentiating with respect to x yields the two-argument functional-differential equation

(59) which coincides with equation (51) in Subsection S.4.4, up to notation.

g ′ x Q + gϕ ′ x Q z ′ + h ′ x R + hϕ ′ x R z ′ = 0, g ′ x Q + gϕ ′ x Q z ′ + h ′ x R + hϕ ′ x R z ′ = 0,

where A 1 , ...,A 4 are arbitrary constants.

The solution of equation (60) is given by

g(x) = A 2 B 1 e k 1 ϕ +

h(x) = (k 1 1 −

A 2 1 ) B 1 e k ϕ +( k

Q(z) = A 2 3 B 3 e − k 1 z

− 1 − R(z) = (k 2

1 − A 1 ) B 3 e k z +( k 2 − A 1 ) B 4 e k z , where B 1 , ...,B 4 are arbitrary constants and k 1 and k 2 are roots of the quadratic equation

(62) In the degenerate case k 1 = k 2 , the terms 2 2 e 1 k ϕ and e − k z in (61) must be replaced by ϕe k ϕ and

( k−A 1 )( k−A 4 )− A 2 A 3 = 0.

ze − k 1 z , respectively. In the case of purely imaginary or complex roots, one should extract the real (or imaginary) part of the roots in solution (61). On substituting (61) into the original functional equation, one obtains conditions that must be met by the free coefficients and identifies the function f (t), specifically,

B 1 2 = B 4 =0 =⇒ f(t) = [A 2 A 3 +( k 1 − A 1 ) 2 ] B 1 B 3 e − k ψ ,

B 1 = B 3 =0 =⇒ f(t) = [A 2 A 3 +( k 2 − A 1 ) 2 ]

2 A 3 + k 1 2 +( 2 ) B 2 B 4 e − k ψ . Solution (61), (63) involves arbitrary functions ϕ = ϕ(x) and ψ = ψ(t).

1 A 2 =⇒ f(t) = (A 2 A 3 + k 2 ) B 1 B 3 e − k ψ A

Degenerate case . In addition, the functional equation has two degenerate solutions,

f=B 1 B 2 e A 1 ψ , g=A 2 B 1 e − A 1 ϕ

1 , 1 h=B 1 e − A ϕ , R = −B

2 e A z − A 2 Q, where ϕ = ϕ(x), ψ = ψ(t), and Q = Q(z) are arbitrary functions, A 1 , A 2 , B 1 , and B 2 are arbitrary

constants; and

f=B 1 B 2 e A 1 ψ ,

h = −B 1 e − A 1 ϕ − A 2 g, Q=A 2 B 2 e A 1 z , R=B 2 e A 1 z , where ϕ = ϕ(x), ψ = ψ(t), and g = g(x) are arbitrary functions, and A 1 , A 2 , B 1 , and B 2 are arbitrary

constants. The degenerate solutions can be obtained immediately from the original equation or its consequence (59) using formulas (53) in Subsection S.4.4.

Example 15. For the first-order nonlinear equation

∂x + G(x),

the search for exact solutions in the form (55) leads to the functional equation S.5.5-3 in which

R(z) = 1/w ′ z , w = w(z). Example 16. For the nonlinear heat equation (14) [see Example 10 in S.5.3-2] the search for exact solutions in the

f (t) = −ψ ′ t , g(x) = (ϕ ′ x ) 2 , h(x) = G(x), Q(z) = F(w)w ′ z ,

form w = w(z), where z = ϕ(x) + ψ(t), leads to the functional equation (16), which coincides with equation S.5.5-3 if

f (t) = −ψ ′

g(x) = ϕ xx ′′ ,

h(x) = (ϕ ′ x ) , Q(z) = F(w), R(z) = F(w)w z z w z ′

, w = w(z).

S.5.5-4. The equation f 1 ( x) + f 2 ( y) + g 1 ( x)P (z) + g 2 ( y)Q(z) + R(z) = 0, z = ϕ(x) + ψ(y). Differentiating with respect to y and dividing the resulting relation by ψ ′ y P z ′ and differentiating

with respect to y again, one arrives at the functional equation with two arguments, y and z, that is discussed in Subsections S.4.2–S.4.4 [see equation (21) and its solutions (48)].

Example 17. Consider the following equation of steady-state heat transfer in an anisotropic inhomogeneous medium with a nonlinear source:

F(w).

The search for exact solutions in the form w = w(z), z = ϕ(x) + ψ(y), leads to the functional equation S.5.5-4 in which

f 2 ( y) = b(y)ψ ′′ yy + b ′ y ( y)ψ ′ y , g 1 ( x) = a(x)(ϕ ′ x ) 2 , g 2 ( y) = b(y)(ψ y ′ ) 2 , P (z) = Q(z) = w zz ′′ /w ′ z , R(z) = −F(w)/w z ′ , w = w(z).

f 1 ( x) = a(x)ϕ ′′ xx + a ′ x ( x)ϕ ′ x ,

Here we confine ourselves to studying functional separable solutions existing for arbitrary right-hand side With the change of variable z=ζ 2

F(w).

, we look for solutions of equation (64) in the form

w = w(ζ), ζ 2 = ϕ(x) + ψ(y).

Taking into account that ′ ∂ζ

∂x = ϕ 2 x and ∂ζ = ψ 2 ζ y ∂y ζ , we find from (64)

4 ζ 3 F(w),

F(w) = F w(ζ)

For this functional-differential equation to be solvable we require that the expressions in square brackets be functions of ζ:

( aϕ ′

x 2 ) ′ x +( bψ ′ y ) ′

y 2 = M (ζ), a(ϕ ′ x ) + b(ψ ′ y ) = N (ζ).

Differentiating the first relation with respect to x and y yields the equation (M ζ ′ /ζ) ′ ζ = 0, whose general solution is M (ζ) = C 1 ζ 2 + C 2 . Likewise, we find N (ζ) = C 3 ζ 2 + C 4 . Here, C 1 , ...,C 4 are arbitrary constants. Consequently, we have

( aϕ ′ x ) ′ x +( bψ ′ y ) ′ y = C 1 ( ϕ + ψ) + C 2 , a(ϕ ′ x ) 2 + b(ψ y ′ ) 2 = C 3 ( ϕ + ψ) + C 4 . The separation of variables results in a system of ordinary differential equations for ϕ(x), a(x), ψ(y), and b(y):

This system is always integrable by quadrature and can be rewritten as

2 ψ −2 y ′ ) = 0, b = (C 3 ψ−k 2 )( ψ ′ y ) . Here, the equations for ϕ and ψ do not involve a and b and, hence, can be solved independently. Without full analysis of

( C 3 ψ−k 2 ) ψ ′′ yy +( C 1 ψ−k 1 − C 3 )(

system (67), we note a special case where the system can be solved in explicit form.

For C 1 = C 2 = C 4 = k 1 = k 2 = 0 and C 3 =

C ≠ 0, we find

µx − νy Ce − µx Ce νy

ψ(y) = 2 αµ , βν where α, β, µ, and ν are arbitrary constants. Substituting these expressions into (66) and taking into account (65), we obtain

a(x) = αe ,

b(y) = βe , ϕ(x) =

the ordinary differential equation for w(ζ)

′′ − 1 ′ = w 4 ζζ w

C F(w).

System (67) has other solutions as well; these lead to various expressions of a(x) and b(y). Table 19 lists the cases where these functions can be written in explicit form (the traveling-wave solution, which corresponds to

a = const and b = const, is omitted). In general, the solution of system (67) enables one to represent a(x) and b(y) in parametric form.

References for Subsection S.5.5: V. F. Zaitsev and A. D. Polyanin (1996), A. D. Polyanin and A. I. Zhurov (1998), A. D. Polyanin (2002, Supplement B), A. D. Polyanin and V. F. Zaitsev (2002).

Functional separable solutions of the form w = w(ζ), ζ 2 = ϕ(x) + ψ(y), for heat equations in an anisotropic inhomogeneous medium with an arbitrary nonlinear source. Notation:

C, α, β, µ, ν, n, and k are free parameters (C ≠ 0, µ ≠ 0, ν ≠ 0, n ≠ 2, and k ≠ 2) Heat equation

Equation for w = w(ζ) ∂

Functions ϕ(x) and ψ(y)

β(2 − n) 2 ζζ (2 − m)(2 − n) ζ ζ C F(w) ∂

∂x ∂y

∂y

= F(w)

ζ ζ C F(w) ∂

C F(w) ∂

Equation (66); both expressions ∂x

ϕ = µ ln |x|, ψ = ν ln |y|

∂x ∂y

∂y

in square brackets are constant

Equation (66); both expressions ∂x 2 ∂y

in square brackets are constant