Exact Traveling Wave Solutions of Nonlinear PDEs in Mathematical Physics Using the Modified Simple Equation Method

In this article, we apply the modified simple equation method to find the exact solutions with parameters of the (1+1)-dimensional nonlinear Burgers-Huxley equation, the (2+1) dimensional cubic nonlinear Klein-Gordon equation and the (2+1)-dimensional nonlinear Kadomtsev- Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The new exact solutions of these three equations are obtained. When these parameters are given special values, the solitary solutions are obtained

An exponential expansion method and its application to the strain wave equation in microstructured solids

AbstractThe modeling of wave propagation in microstructured materials should be able to account for various scales of microstructure. Based on the proposed new exponential expansion method, we obtained the multiple explicit and exact traveling wave solutions of the strain wave equation for describing different types of wave propagation in microstructured solids. The solutions obtained in this paper include the solitary wave solutions of topological kink, singular kink, non-topological bell type solutions, solitons, compacton, cuspon, periodic solutions, and solitary wave solutions of rational functions. It is shown that the new exponential method, with the help of symbolic computation, provides an effective and straightforward mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering

Bouncing and walking droplets : towards a hydrodynamic pilot-wave theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 197-205).Coalescence of a liquid drop with a liquid bath can be prevented by vibration of the bath. In a certain parameter regime, a purely vertical bouncing motion may ensue. In another, this bouncing state is destabilized by the droplet's wavefield, leading to drop motion with a horizontal component called walking. The walking drops are of particular scientific interest because Couder and coworkers have demonstrated that they exhibit many phenomena reminiscent of microscopic quantum particles. Nevertheless, prior to this work, no quantitative theoretical model had been developed to rationalize and inform the experiments before our work. In this thesis, we develop a hierarchy of theoretical models of increasing complexity in order to describe the drop's vertical and horizontal motion in the relevant parameter range. Modeling the drop-bath interaction via a linear spring is found lacking; therefore, a logarithmic spring model is developed. We first introduce this model in the context of a drop impacting a rigid substrate, and demonstrate its accuracy by comparison with existing numerical and experimental data. We then extend the model to the case of impact on a liquid substrate, and apply it to rationalize the dependence of the bouncing droplet's behaviour on the system parameters. The theoretical developments have motivated further experiments, which have in turn lead to refinements of the theory. We proceed by modeling the evolution of the standing waves created by impact on the bath, which enables us to predict the onset of walking and the dependence of the walking speed on the system parameters. New complex walking states are predicted, and subsequently validated by our detailed experimental study. A trajectory equation for the horizontal motion is obtained by averaging over the vertical bouncing.by Jan Molác̆ek.Ph.D

Goddard Laboratory for Atmospheric Sciences, collected reprints 1978 - 1979, volume 2

Information about the Earth hydrosphere, obtained in the field and from aircraft and satellite imagery is reported. Particular emphasis is given to the use of microwave sensors in the study of soil moisture, sea ice, snow cover and atmospheric parameters associated with watersheds

Proceedings of the First International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics

1st International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Kruger Park, 8-10 April 2002.This lecture is a principle-based review of a growing body of fundamental work stimulated by multiple opportunities to optimize geometric form (shape, structure, configuration, rhythm, topology, architecture, geography) in systems for heat and fluid flow. Currents flow against resistances, and by generating entropy (irreversibility) they force the system global performance to levels lower than the theoretical limit. The system design is destined to remain imperfect because of constraints (finite sizes, costs, times). Improvements can be achieved by properly balancing the resistances, i.e., by spreading the imperfections through the system. Optimal spreading means to endow the system with geometric form. The system construction springs out of the constrained maximization of global performance. This 'constructal' design principle is reviewed by highlighting applications from heat transfer engineering. Several examples illustrate the optimized internal structure of convection cooled packages of electronics. The origin of optimal geometric features lies in the global effort to use every volume element to the maximum, i.e., to pack the element not only with the most heat generating components, but also with the most flow, in such a way that every fluid packet is effectively engaged in cooling. In flows that connect a point to a volume or an area, the resulting structure is a tree with high conductivity branches and low-conductivity interstices.tm201