Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Abstract

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

Separation of variables in PDEs using nonlinear transformations: Applications to reaction–diffusion type equations

“…Formulas (26) and equation (27) define the functional coefficients of equation (8) and its solution (2). It is noteworthy that for a(x) = a 0 x k , equation (27) admits the exact solution…”

“…Hence, in certain cases, it may be more effective than the nonclassical method of symmetry reductions based on an invariant surface condition. For details, see [26].…”

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

“…The efficiency of all compared methods in terms of the maximum absolute global norm error versus the function evaluations is presented in Figure 5 for t ∈ [0, 30π]. For each algorithm, the most efficient step size control is used, if applicable, which for all embedded pairs is the modified algorithm presented in Equation (10). Otherwise, a constant step is used.…”

“…The analytical solution of the NLS with varying coefficients has attracted great interest in the recent past (e.g., see [5][6][7] or more recent works in [8][9][10][11] and references therein). Additionally, the computation of the NLS is a critical part of the verification process of the analytical theories.…”

Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

“…This method allowed to find more than 40 exact solutions of nonlinear reaction-diffusion equations and wave type equations with variable coefficients involving one or more arbitrary functions. In [36], it was shown that some of the solutions given in [34,35] cannot be obtained using the nonclassical method of symmetry reductions [38][39][40][41][42][43][44][45] (see also [18,23]) based on the use of the invariant surface condition (a first-order differential constraint equivalent to the relation (4)). Note that constructing solutions in implicit form with the integral term (4) often allows us to reduce the order of the resulting functional differential equations [33,34].…”