Splitting Method. Reduction to a Functional Equation with

S.5.4. Splitting Method. Reduction to a Functional Equation with

Two Variables

S.5.4-1. Splitting method. Reduction to a standard functional equation. The procedure for constructing functional separable solutions, which is based on the splitting method,

involves several stages outlined below.

1 ◦ . Substitute expression (1) into the nonlinear partial differential equation under study. This results in a functional-differential equation with three arguments—the first two are usual, x and y, and the third is composite, z.

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

No. Right-hand side F(w)

Solution w(z)

Equations for ϕ(x) and ψ(y)

4 a sinh w + b sinh w ln tanh

( ϕ ′ w/2 2 x ) = C 1 sin 2 ϕ+C 2 cos 2 ϕ + σbϕ + a + A, 5 a sinh w + 2b sinh w arctan e

2 ◦ . Reduce the functional-differential equation to a purely functional equation with three arguments x, y, and z with the aid of elementary differential substitutions (by selecting and renaming terms with derivatives).

3 ◦ . Reduce the three-argument functional-differential equation by the differentiation method to the standard functional equation with two arguments (either x or y is eliminated) considered in Subsection S.4.2.

4 ◦ . Construct the solutions of the two-argument functional equation using the formulas given in Subsection S.4.4.

5 ◦ . Solve the (overdetermined) systems formed by the solutions of Item 4 ◦ and the differential substitutions of Item 2 ◦ .

6 ◦ . Substitute the solutions of Item 5 ◦ into the original functional-differential equation of Item 1 ◦ to establish the relations for the constants of integration and determine all unknown quantities.

7 ◦ . Consider all degenerate cases possibly arising due to the violation of assumptions adopted in the previous analysis.

Remark. Stage 3 ◦ is the most difficult here; it may not always be realizable. The splitting method reduces solving the three-argument functional-differential equation to (i) solv-

ing a purely functional equation with three arguments (by reducing it to a standard functional equation with two arguments) and (ii) solving systems of ordinary differential equations. Thus, the initial problem splits into several simpler problems. Examples of constructing functional separable solutions by the splitting method are given in Subsection S.5.5.

S.5.4-2. Three-argument functional equations of special form. The substitution of expression (1) with n = 2 into a nonlinear partial differential equation often leads

to functional-differential equations of the form Φ 1 ( x)Ψ 1 ( y, z) + · · · + Φ k ( x)Ψ k ( y, z) + Ψ k+1 ( y, z) + Ψ k+2 ( y, z) + · · · + Ψ n ( y, z) = 0,

(44) where the Φ j ( x) and Ψ j ( y, z) are functionals dependent on the variables x and y, z, respectively, Φ j ( x) ≡ Φ j x, ϕ, ϕ ′ x , ϕ ′′ xx , Ψ j ( y, z) ≡ Ψ j y, ψ, ψ ′ y , ψ ′′ yy , F,F ′ z , F zz ′′ .

(45) (These expressions apply to a second-order equation.)

0, we divide (44) by Ψ 1 and differentiate with respect to y to obtain a similar equation but with fewer terms containing Φ m :

as a purely functional equation, thus disregarding (45). Assuming that Ψ 1 ✯

2 x)Ψ k ( y, z) + Ψ k+1 ( y, z) + · · · + Ψ n ( y, z) = 0, (46) where Ψ (2) m = ∂ Ψ ∂y m /Ψ 1 + ψ y ′ ∂ Ψ ∂z m /Ψ 1 . We continue this procedure until an equation inde-

pendent of x explicitly is obtained:

(47) where Ψ ( k+1) = ∂ Ψ ( k) ( k) /Ψ + ψ ′ ∂

Ψ ( k+1) k+1 (

( y, z) + · · · + Ψ k+1)

Relation (47) can be regarded as an equation with two independent variables y and z. If Ψ ( k+1) m ( y, z) = Q m ( y)R m ( z) for all m = k + 1, . . . , n, then equation (47) can be solved using the

results of Subsections S.4.2–S.4.4.