New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

Acoustic and Large Eddy Simulation studies of azimuthal modes in annular combustion chambers

The objectives of this paper are the description of azimuthal instability modes found in annular combus- tion chambers using two numerical tools: (1) Large Eddy Simulation (LES) methods and (2) acoustic solv- ers. These strong combustion instabilities are difficult to study experimentally and the present study is based on a LES of a full aeronautical combustion chamber. The LES exhibits a self-excited oscillation at the frequency of the first azimuthal eigenmode. The mesh independence of the LES is verified before ana- lysing the nature of this mode using various indicators over more than 100 cycles: the mode is mostly a pure standing mode but it transitions from time to time to a turning mode because of turbulent fluctu- ations, confirming experimental observations and theoretical results. The correlation between pressure and heat release fluctuations (Rayleigh criterion) is not verified locally but it is satisfied when pressure and heat release are averaged over sectors. LES is also used to check modes predicted by an acoustic Helmholtz solver where the flow is frozen and flames are modelled using a Flame Transfer Function (FTF) as done in most present tools. The results in terms of mode structure compare well confirming that the mode appearing in the LES is the first azimuthal mode of the chamber. Moreover, the acoustic solver provides stability maps suggesting that a reduction of the time delay of the FTF would be enough to sta- bilise the mode. This is confirmed with LES by increasing the flame speed and verifying that this modi- fication leads to a damped mode in a few cycles

The Spherically Symmetric Gravitational Collapse of a Clump of Solids in a Gas

Several mechanisms have been identified that create dense particle clumps in the solar nebula. The present work is concerned with the gravitational collapse of such clumps, idealized as being spherically symmetric. Calculations using the two-fluid model are performed (almost) up to the time when a central density singularity forms. The end result of the study is a parametrization for this time, in order that it may be compared with timescales for various disruptive effects to which clumps may be subject. An important effect is that as the clump compresses, it also compresses the gas due to drag. This increases gas pressure which retards particle collapse and leads to oscillation in the size and density of the clump. The ratio of gravitational force to gas pressure gives a two-phase Jeans parameter, $J_t$, which is the classical Jeans parameter with the sound speed replaced by an the wave speed in a coupled two-fluid medium. Its use makes the results insensitive to the initial density ratio of particles to gas as a separate parameter. An ordinary differential equation model is developed which takes the form of two coupled non-linear oscillators and reproduces key features of the simulations. Finally, a parametric study of the time to collapse is performed and a formula (fit to the simulations) is developed. In the incompressible limit $J_t \to 0$, collapse time equals sedimentation time. As $J_t$ increases, the collapse time decreases roughly linearly with $J_t$ until $J_t \gtrsim 0.4$ when it becomes approximately equal to the dynamical time

Steady state analysis of Chua's circuit with RLCG transmission line

In this paper we present a new technique to compute the steady state response of nonlinear autonomous circuits with RLCG transmission lines. Using multipoint Pade approximants, instead of the commonly used expansions around s=0 or s/spl rarr//spl infin/ accurate, low-order lumped equivalent circuits of the characteristic impedance and the exponential propagation function are obtained in an explicit way. Then, with the temporal discretization of the equations that describe the transformed circuit, we obtain a nonlinear algebraic formulation where the unknowns to be determined are the samples of the variables directly in the steady state, along with the oscillation period, the main unknown in autonomous circuits. An efficient scheme to build the Jacobian matrix with exact partial derivatives with respect to the oscillation period and with respect to the samples of the unknowns is obtained. Steady state solutions of the Chua's circuit with RLCG transmission line are computed for selected circuit parameters.Peer ReviewedPostprint (published version

An analytical approach for predicting pilot induced oscillations

The optimal control model (OCM) of the human pilot is applied to the study of aircraft handling qualities. Attention is focused primarily on longitudinal tasks. The modeling technique differs from previous applications of the OCM in that considerable effort is expended in simplifying the pilot/vehicle analysis. After briefly reviewing the OCM, a technique for modeling the pilot controlling higher order systems is introduced. Following this, a simple criterion or determining the susceptability of an aircraft to pilot induced oscillations (PIO) is formulated. Finally, a model-based metric for pilot rating prediction is discussed. The resulting modeling procedure provides a relatively simple, yet unified approach to the study of a variety of handling qualities problems