Solutions to Nonlinear Partial Differential Equations from Symmetry-Enhancing and Symmetry-Preserving Constraints

AbstractWe show how solutions to practical partial differential equations can be found by classical symmetry reductions of a larger system of equations including constraints that result in the enlarged system having a larger symmetry group or an identical symmetry group compared to the original target equation. Symmetry-enhancing constraints provide additional similarity solutions beyond those of the standard Lie algorithm for scalar equations. Symmetry-preserving constraints enable solutions to be found more easily, after reduction of variables. Examples are given for the cylindrical boundary layer equations and for a Navier–Stokes formulation of Schrödinger wave mechanics

Spherically restricted random hyperbolic diffusion

This paper investigates solutions of hyperbolic diffusion equations in R 3 R3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 S2 are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short- or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings

Similitude applied to centrifugal scaling of unsaturated flow

Centrifuge experiments modeling single-phase flow in prototype porous media typically use the same porous medium and permeant. Then, well-known scaling laws are used to transfer the results to the prototype. More general scaling laws that relax these restrictions are presented. For permeants that are immiscible with an accompanying gas phase, model-prototype (i.e., centrifuge model experiment-target system) scaling is demonstrated. Scaling is shown to be feasible for Miller-similar (or geometrically similar) media. Scalings are presented for a more, general class, Lisle-similar media, based on the equivalence mapping of Richards' equation onto itself. Whereas model-prototype scaling of Miller-similar media can be realized easily for arbitrary boundary conditions, Lisle-similarity in a finite length medium generally, but not always, involves a mapping to a moving boundary problem. An exception occurs for redistribution in Lisle-similar porous media, which is shown to map to spatially fixed boundary conditions. Complete model-prototype scalings for this example are derived

Conditionally Integrable Nonlinear Diffusion with Diffusivity 1/u

An explicit mapping is given from the space of general complex meromorphic functions to a space of special time-dependent solutions of the 1 + 2-dimensional nonlinear diffusion equation with diffusivity depending on concentration as D = 1/u. These solutions have constant-flux boundary conditions. Some simple examples are constructed, including that of a line source enclosed by a cylindrical barrier. This has direct application to electron diffusion in a laser-heated plasma