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Provides information on the discussion of the importance of similarity transformations and their applications to partial differential equations. Information on scaling transformation; Detailed information on translation transformation.

This work is devoted to the investigation of self-similar solutions for steady wall jets. The problem is considered in the context of two- and three-dimensional Prandtl boundary layer equations, and three-dimensional parabolized Navier-Stokes equations for Newtonian and nonNewtonian fluids. In contrast to dimensional analysis, which does not allow the determination of self-similar solutions in this case, a generating functions approach elaborated by Vinogradov [Soviet Math. Dokl., 19 (1978), pp. 144-148] enables one to derive conservation laws for the above-mentioned problems and, as a consequence, to find new self-similarities of the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

We wish to predict ionic currents that flow through narrow protein channels of biological membranes in response to applied potential and concentration differences across the channel when some features of channel structure are known. We propose to apply singular perturbation analysis to the coupled Poisson-Nernst-Planck equations, which are the basic continuum model of ionic permeation and semiconductor physics. In semiconductor physics the problem is a singular perturbation, because the ratio of the Debye length to the width of the channel is a very small parameter that multiplies the Laplacian term in the Poisson equation. In contrast to semiconductors, the atomic scale geometry of narrow ion channels sometimes makes this ratio a large parameter, which, surprisingly, renders the problem a singular perturbation in a different sense. We construct boundary layers and match them asymptotically across the different regions of the channel to derive good approximations for Fick's and Ohm's laws. Our aim is to extend the asymptotic analysis to a class of nonlinear problems hitherto intractable. Analytical and numerical results for the mass flux and the electric current serve as a tool for molecular biophysicists and physiologists to understand, study, and control protein channels, thereby aiding clinical and technological applications. [ABSTRACT FROM AUTHOR]

Given α ϵ [0, 2) and f ϵ L²((0, T) × (0, 1)), we derive new Carleman estimates for the degenerate parabolic problem wt + (xαwx)x = f, where (t, x) ϵ (0, T) × (0, 1), associated to the boundary conditions w(t, 1) = 0 and w(t, 0) = 0 if 0 = ⩽ 1 or (xαwx)(t, 0) = 0 if 1 = ⩽ α < 2. The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all 0 ⩽ α < 2 and w ⊂ ⊂ (0, 1), we deduce null controllability results for the degenerate one-dimensional heat equation ut - (xαux)x = hXw with the same boundary conditions as above. [ABSTRACT FROM AUTHOR]

The radiation (reactive or Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary; however, the relation between the reaction constant in the Robin boundary condition and the reflection probability is not well defined. In this paper we define the partially reflected process as a limit of the Markovian jump process generated by the Euler scheme for the underlying Itô dynamics with partial boundary reflection. Trajectories that cross the boundary are terminated with probability P√Δt and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the Fokker- Planck equation in a half-space, with the Robin constantκ. The boundary layer equation is of the Wiener-Hopf type. We show that the Robin boundary condition is recovered if and only if trajectories are reflected in the conormal direction σn, where σ is the (possibly anisotropic) constant diffusion matrix and n is the unit normal to the boundary. Otherwise, the density satisfies an oblique derivative boundary condition. The constant ? is related to P by κ = ϒP√σn, where ϒ = 1/√π and σn = nTσn. The reflection law and the relation are new for diffusion in higher dimensions. [ABSTRACT FROM AUTHOR]

An asymptotic reduction of the Gierer-Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity leads to a singularly perturbed nonlocal reaction diffusion equation for the activator concentration. In the limit of small activator diffusivity, a one-spike solution to this nonlocal model is constructed. The spectrum of the eigenvalue problem associated with the linearization of the nonlocal model around such an isolated spike solution is studied in both a one-dimensional and a multidimensional context. It is shown that the principal eigenvalues in the spectrum are exponentially small in the limit of small activator diffusivity. The nonlocal term in the eigenvalue problem is essential for ensuring the existence of such exponentially small principal eigenvalues. These eigenvalues are responsible for the occurrence of an exponentially slow, or metastable, spike-layer motion for the time-dependent problem. Explicit metastable spike dynamics are derived by using a projection method, which enforces a limiting solvability condition on the solution to the linearized problem. [ABSTRACT FROM AUTHOR]

In highly diffusive regimes, the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path ∈ tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis. A numerical scheme for solving this problem has an ∈-1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small ∈, thus degrad- ing its performance in diffusive regimes. In this paper we show that whenever the discrete-ordinate method has the correct diffusion limit, both in the interior and at the boundaries, its solutions converge to the solution of the transport equation uniformly in ∈. Our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete-ordinate method. [ABSTRACT FROM AUTHOR]

Examines the findings of a survey on the developments of a similarity approach to the numerical solution of free boundary value problems (BVPs). Treatment of free BVPs; Definition of the most general class of BVPs; Proposal of an approach to the numerical solution of free BVPs; Findings of the survey.