We study the existence of spatial periodic solutions for nonlinear elliptic equations where is a continuous function, nondecreasing w.r.t. . We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations....
We study the existence of spatial periodic solutions for nonlinear
elliptic equations
where is a continuous function, nondecreasing w.r.t. . We
give necessary and sufficient conditions for the existence of
periodic solutions. Some cases with nonincreasing functions
are investigated as well. As an application we analyze the
mathematical model of electron beam focusing system and we prove
the existence of positive periodic solutions for the envelope
equation. We present also numerical simulations.
...
This work is devoted to the numerical simulation of the Vlasov equation using a phase space grid. In contrast to Particle-In-Cell (PIC) methods, which are known to be noisy, we propose a semi-Lagrangian-type method to discretize the Vlasov equation in the two-dimensional phase space. As this kind of method requires a huge computational effort, one has to carry out the simulations on parallel machines. For this purpose, we present a method using patches decomposing the phase domain, each patch being...
We present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus "light waves" are somewhat supressed, which in turn allows thenumerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian...
The equations of physics are mathematical models consisting of geometric objects and relationships between then. There are many methods to discretize equations, but few maintain the physical nature of objects that constitute them. To respect the geometrical nature elements of physics, it is necessary to change the point of view and using differential geometry, including the numerical study. We propose to construct discrete differential forms using B-splines and a formulation discrete for different...
A new numerical scheme called particle-in-wavelets is proposed for the Vlasov-Poisson
equations, and tested in the simplest case of one spatial dimension. The plasma
distribution function is discretized using tracer particles, and the charge distribution
is reconstructed using wavelet-based density estimation. The latter consists in projecting
the Delta distributions corresponding to the particles onto a finite dimensional linear
space spanned by...
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