Dynamic evolution of the averaged distribution function

In this article we derive an alternative to the classical quasi-linear method to describe the evolution of the averaged distribution function avoiding the difficulties associated to the resolution of a three dimensional diffusion equation. We consider explicitly the case of the electron species in presence of a current-driven ion-acoustic turbulence. Our formulation aims to confirm the arguments developed by Balescu in a recent work, where it has been pointed out that a self-similar state which behaves as exp — (w5/ν 50) is not consistent with the marginal stability hypothesis imposed in the model of Vekstein, Ryutov and Sagdeev.Dans cet article nous décrivons une alternative à la méthode quasi linéaire classique pour décrire l'évolution de la fonction de distribution moyenne, évitant les difficultés associées à la résolution d'une équation de diffusion à trois dimensions. Nous considérons explicitement le cas des électrons en présence de la turbulence acoustique-ionique générée par un courant de dérive. Notre formulation apporte une confirmation aux arguments développés par Balescu dans un travail récent, selon lesquels un état auto-similaire qui se comporte comme exp — (w5/ν50) n'est pas compatible avec l'hypothèse de stabilité marginale imposée dans le modèle de Vekstein, Ryutov et Sagdeev

An approach to solutions of systems of linear partial differential equations with applications

AbstractInitial and boundary value problems governed by a system of linear partial differential equations can be solved by using the classical methods. This holds in solving problems which are governed by a unique system of equations over the whole region of interest. But if a problem is governed by a given system of equations over a region and by another system over the complementary one, classical methods may fail in treating this problem. A typical problem is that of evaluating the time-dependent electric field in the conductive half-space (the substratum) as a model in geophysical prospecting. The electric field in the air above the substratum is time independent. This problem has been solved numerically. Here, we solve it analytically. We proceed by presenting an approach for finding the solutions of systems of linear partial differential equations. Eigen operators and fractional powers of matrices of operators have been introduced. The formal solutions obtained are adequate for studying initial and boundary value problems whose solutions are anharmonic ones. They are used to solve the above-mentioned problem

A new technique for solving Burgers-Kadomtsev-Petviashvili equation with an external source. Suppression of wave breaking and shock wave

It was found in the literature that traveling wave solutions of the Burgers-Kadomtsev-Petviashivili equation exhibit wave breaking or shock wave formation in shallow water. Shock waves can be formed due to steepening of ordinary waves.This can be seen in ocean waves that forming breakers near the shore. The shock wave can be prevented by injecting water source or by hot- air shield. This problem was not considered in the literature, especially in a fluid. Here, we study this problem, which is completely novel, by considering the aforementioned equation with an external source. To this issue, we propose a new technique to solve this equation. It is based on representing the external source term in rational form in an auxiliary function, together with implementing the unified method and the extended unified method.Here, the external source may be source (injection) or sink (suction). This stands for the process of water beam injection (or suction). The results found show the presence of wave breaking and shock wave in the absence of the source. While, in the presence of the source, it is shown that wave breaking and shock wave can suppressed. This is visualized via the formation of waves with smooth boundary. In the present work, it is found that shock wave and wave breaking that arise in traveling and semi-self-similar solutions can be suppressed by source injection or by suction via sink. While, in self-similar waves fronts they are robust against the suppression procedure. These results are consolidated graphically

Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion

Herein, the extended coupled Kadomtsev–Petviashvili equation (CKPE) with lateral dispersion is investigated for studying the atmospheric blocking in two layers. A variety of new types of polynomial solutions for the CKPE is obtained using the unified method. Furthermore, we use the Hamiltonian systems with two degrees of freedom to discuss the stability of the obtained solutions through the bifurcation diagrams. MSC 2010: 76D05, 76M60, 76M99, 76S05, 76W05, Keywords: Two-atmospheric-layer, Extended coupled Kadomtsev–Petviashvili equation, Atmospheric blocking, Unified metho

Approximate solutions of fractional dynamical systems based on the invariant exponential functions with an application. A novel double-kernel fractional derivative

The systems with fractional time derivative (FTD) and fractional space derivative (FSD), with different versions, have occupied a vast area of research in the literature. It is worth mentioning that in the literature when establishing the solutions of fractional systems, the only function dealt with is the Mittage-Leffler function. In the present work, our objectives are; First is to the reversibility identity (RI) of FTD. Second, is to construct the fractional exponential function invariant under a fractional derivative. In consequence, the trigonometric and hyperbolic fractional functions are constructed. So, exact solutions of linear fractional systems can be obtained. When RI holds then a FD establishes a calculus analog the ordinary ones. Although, these concepts are simple, they were not considered in the literature. Here, various FDs are considered. Furthermore, an approach for approximate analytic solutions for the fractional-prey-predator dynamical system with harvesting is presented via an iteration scheme and the convergence theorem is proved. Finally, a novel double kernel FD is introduced and as a particular case the Hilfer FD is established