Connection Between the Differential Constraints Method and Other Methods

S.8.4. Connection Between the Differential Constraints Method and Other Methods

The differential constraints method is one of the most general methods for the construction of exact solutions to nonlinear partial differential equations. Many other methods can be treated as its particular cases.*

S.8.4-1. Generalized and functional separation of variables versus differential constraints. Table 20 lists examples of second-order differential constraints which are essentially equivalent to

most common forms of separable solutions. For functional separable solutions (rows 5 and 6), the function g can be expressed through f .

Table 21 lists examples of third-order differential constraints which may be regarded as equivalent to direct specification of most common forms of functional separable solutions.

* The basic difficulty of applying the differential constraints method is due to the great generality of its statements and the necessity of selecting differential constraints suitable for specific classes of equations. This is why for the construction of exact solutions of nonlinear equations, it is often preferable to use more simple (but less general) methods.

ϕ n ( x)ψ n ( y), with 2n unknown functions, is equivalent to prescribing a differential constraint of order 2 n; in general, the number of unknown functions ϕ i ( x), ψ i ( y) corresponds to the order of the differential equation representing the differential constraint.

For the types of solutions listed in Tables 20 and 21, it is preferable to use the methods of generalized and functional separation of variables, since these methods require less steps where it is necessary to solve intermediate differential equations. Furthermore, the method of differential constraints is ill-suited for the construction of exact solutions of higher (arbitrary) order equations.

S.8.4-2. Generalized similarity reductions and differential constraints. Consider a generalized similarity reduction based on a prescribed form of the desired solution,

(60) where

w(x, t) = F x, t, u(z) ,

z = z(x, t),

F (x, t, u) and z(x, t) should be selected so as to obtain ultimately a single ordinary differential equation for u(z); see Subsection S.6.2. Let us show that employing the solution structure (60) is equivalent to searching for a solution with the help of a first-order quasilinear differential constraint

∂w

∂w

ξ(x, t)

+ η(x, t)

= ζ(x, t, w).

Indeed, first integrals of the characteristic system of ordinary differential equations

ξ(x, t)

η(x, t)

ζ(x, t, w)

have the form

(62) where C 1 and C 2 are arbitrary constants. Therefore, the general solution of equation (61) can be

z(x, t) = C 1 ,

ϕ(x, t, w) = C 2 ,

written as follows:

(63) where u(z) is an arbitrary function. On solving (63) for w, we obtain a representation of the solution

ϕ(x, t, w) = u z(x, t) ,

in the form (60). ❊✂❋

Reference : P. J. Olver (1994).

S.8.4-3. Group analysis and differential constraints. The group analysis method for differential equations can be restated in terms of the differential

constraints method. This can be demonstrated by the following example with a general second- order equation

∂w ∂w ∂ 2 w ∂ 2 w ∂ 2 w

(64) Let us supplement equation (64) with two differential constraints

F x, y, w,

∂x ∂y ∂x 2 ∂x∂y ∂y 2

∂w yy where ξ = ξ(x, y, w), η = η(x, y, w), and ζ = ζ(x, y, w) are unknown functions, and the coordinates

of the first and the second prolongations ζ i and ζ ij are defined by formulas (13) and (14) of

(64); see (11) in Subsection S.7.1. The method for the construction of exact solutions to equation (64) based on using the first-order partial differential equation (65) and the invariance condition (66) corresponds to the nonclassical method of group analysis (see Subsection S.7.2).

Remark. When the classical schemes of group analysis are employed, one first considers two equations, (64) and (66). From these, one eliminates one of the highest-order derivatives, say w yy , while the remaining derivatives ( w x , w y , w xx , and w xy ) are assumed “independent.” The resulting expression splits into powers of independent derivatives (see Subsection S.7.1). As a result, one arrives at an overdetermined system of equations, from which the functions ξ, η, and ζ are found. Then, these functions are inserted into the quasilinear first-order equation (65), whose solution allows

us to determine the general form of a solution (this solution contains some arbitrary functions). Next, using (64), one can refine the structure of the solution obtained on the preceding step.

The classical scheme may result in the loss of some solutions, since at the first step of splitting it is assumed that the first derivatives w x and w y are independent, whereas these derivatives are in fact linearly dependent due to equation (65). ●✂❍

References for Subsection S.8.4: S. V. Meleshko (1983), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Ro- dionov (1999).