Dynamic Partial Differential Equations Research Articles

Mathematical model of building behavior of Apis mellifera

Following experimental observations on the building techniques of Apis mellifera , a model describing the construction of parallel combs in beehive is proposed. The model assumes as the essential elements in building activities the bee-wax and bee-bee interactions coupled by means of a feedback mechanism. The construction is represented by a set of dynamical non-linear partial differential equations for the density of bees and the quantity of wax distributed in the hive. Using linear stability analysis the conditions for growth of the coherent spatially ordered inhomogeneous structure are derived. Numerical simulations depicting growth process in the nonlinear regime have been performed, illuminating the role of the competition between bees and the bee-wax interaction. The model reproduces the beginning of the growth of parallel and equidistant combs satisfactorily.

Extrapolation methods for dynamic partial differential equations

Several extrapolation procedures are presented for increasing the order of accuracy in time for evolutionary partial differential equations. These formulas are based on finite difference schemes in both the spatial and temporal directions. One of these schemes reduces to a Runge-Kutta type formula when the equations are linear. On practical grounds the methods are restricted to schemes that are fourth order in time and either second, fourth or sixth order in space. For hyperbolic problems the second order in space methods are not useful while the fourth order methods offer no advantage over the Kreiss-Oliger method unless very fine meshes are used. Advantages are first achieved using sixth order methods in space coupled with fourth order accuracy in time. The averaging procedure advocated by Gragg does not increase the efficiency of the scheme. For parabolic problems severe stability restrictions are encountered that limit the applicability to problems with large cell Reynolds number. Computational results are presented confirming the analytic discussions.

On the identification of inhomogeneous parameters in dynamic linear partial differential equations

The task of identifying inhomogeneous (position-dependent) coefficients of linear dynamic partial differential equations on the basis of a finite collection of points of the solution has practical importance and is the subject of many published analyses, some of which are described herein. The purpose of the present paper is to present new developments on a simple yet appealing method due to the hydrologist B. Sagar. The technique exploits the viewpoint that the coefficient values of the partial differential equation at any point x are uniquely determined by the solution values in a small neighborhood of x. The identification algorithm which results from these considerations is extremely simple, and yet, in view of technical considerations and experimental evidence set forth here, it seems effective. In particular, we have been able to derive error bounds, which the authors believe is a new feature in the literature of identification of partial differential systems.

Propagation of laser radiation in randomly inhomogeneous media

A review is given of recent investigations of the propagation of laser radiation in randomly inhomogeneous media with large-scale fluctuations of the refractive index. The review is mainly concerned with aspects of the correlation theory of fluctuations of the intensity of radiation traveling over long distances when the effects of multiple scattering become important. Considerable attention is paid to the results relating to the broadening of laser beams and to phase fluctuations. The relatively new, but important in practice, subject of spatial intensity peaks is discussed. Whenever possible, the theoretical results are compared systematically with the experimental data. The analytic methods discussed in the review are applicable to a wide class of stochastic and dynamic partial differential equations and they may be of interest in other branches of physics.

Laser irradiance propagation in turbulent media

The results obtained in recent years on laser irradiance propagation in random weakly inhomogeneous media with large scale index of refraction fluctuations are reviewed. Of particular concern are the problems of the cozrelation theory of fluctuations of irradiance propagating over large distances, where the effects of multiple scattering are greatly pronounced. Much attention is paid to the results on laser beams spread and phase fluctuations. Consideration is given to problem of the study of spatial spikes of irradiance that had passed through a turbulent media; which is comparatively new but important for practical applications. Systematic comparison of the theory with experiment is given where appropriate. The methods of analyses reviewed in this paper are applicable to a class of stochastic and dynamic partial differential equations and thus may be of interest in other areas of engineering reseach.

Dynamic programming and Partial Differential Equations (Edward Angel and Richard Bellman)

Dynamic programming and Partial Differential Equations (Edward Angel and Richard Bellman)

Dynamic programming and linear partial differential equations

Dynamic programming and linear partial differential equations