<abstract><p>In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.</p></abstract>

A mathematical model of the nutrient-phytoplankton-zooplankton associated with viral infection in phytoplankton under the Atangana-Baleanu derivative in Caputo sense is investigated in this study. We prove the theoretical results for the existence and uniqueness of the solutions by using Banach’s and Sadovskii’s fixed point theorems. The notion of various Ulam’s stability is used to guarantee the context of the stability analysis. Furthermore, the equilibrium points and the basic reproduction numbers for the proposed model are provided. The Adams type predictor-corrector algorithm has been applied for the theoretical confirmation to establish the approximate solutions. A variety of numerical plots corresponding to various fractional orders between zero and one are presented to describe the dynamical behavior of the fractional model under consideration.

The transport of energetic charged particles in turbulent magnetic fields is a topic of interest in various astrophysical contexts. In order to estimate the mean free path of a particle in the direction parallel to the mean magnetic field, one can use theoretical expressions that employ pitch-angle diffusion coefficients. In this work we review some of the methods used in estimating pitch-angle diffusion coefficients from test particle computer simulations. We examine if these methods and theoretical approaches are able to provide consistent estimates of the parallel mean free path, that can also be obtained directly from computer simulations. We perform test particle simulations for synthetic turbulence models over a range of turbulence parameters and particle energies. From the trajectories of test particles, pitch-angle distribution functions and statistics of pitch-angle displacements are obtained, which are then used to estimate the pitch-angle diffusion coefficients. We find that a method using the pitch-angle flux and derivative of the pitch-angle distribution is able to provide accurate values for the parallel mean free path over the range of parameters considered. Other methods considered are accurate only for a limited range of the turbulent fluctuation strength, or must be evaluated at a specific time to provide a reasonable estimate.

The transport of energetic charged particles in turbulent magnetic fields is a topic of interest in various astrophysical and laboratory plasma contexts. In order to estimate the mean free path λ∥ of a particle in the direction parallel to the mean magnetic field, one can use theoretical expressions that include the pitch-angle diffusion coefficient Dμμ. In this work we evaluate theories for Dμμ in the context of the noisy reduced magnetohydrodynamic (NRMHD) model where turbulent fluctuations are absent at large parallel wavenumbers. For most turbulence models, the standard quasilinear theory predicts zero pitch-angle diffusion only for particles with a 90◦ pitch angle, but for NRMHD a range of pitch angles is affected, leading to infinite λ∥. We examine two theories that include resonance broadening which yield finite λ∥ and compare them with test-particle computer simulations in which the parallel mean free path can be readily obtained. We find that both theories are quite accurate in some regions of the parameter space considered, but neither is particularly good when the particle Larmor radius RL becomes much smaller than the quasilinear resonance limit.