Integral Representation of the Solutions for Neutral Linear Fractional System with Distributed Delays

In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) and Caputo type derivatives, in the case of initial functions with first kind discontinuities. This result allows to prove that the corresponding homogeneous system possesses a fundamental matrix C(t,s) continuous in t,t∈[a,∞),a∈R. As an application, integral representations of the solutions of the Cauchy problem for the considered inhomogeneous systems are obtained.

Geometry and complexity of path integrals in inhomogeneous CFTs

In this work we develop the path integral optimization in a class of inhomogeneous 2d CFTs constructed by putting an ordinary CFT on a space with a position dependent metric. After setting up and solving the general optimization problem, we study specific examples, including the Möbius, SSD and Rainbow deformed CFTs, and analyze path integral geometries and complexity for universal classes of states in these models. We find that metrics for optimal path integrals coincide with particular slices of AdS3 geometries, on which Einstein’s equations are equivalent to the condition for minimal path integral complexity. We also find that while leading divergences of path integral complexity remain unchanged, constant contributions are modified in a universal, position dependent manner. Moreover, we analyze entanglement entropies in inhomogeneous CFTs and show that they satisfy Hill’s equations, which can be used to extract the energy density consistent with the first law of entanglement. Our findings not only support comparisons between slices of bulk spacetimes and circuits of path integrations, but also demonstrate that path integral geometries and complexity serve as a powerful tool for understanding the interesting physics of inhomogeneous systems.

On the Wave Transmission in a Gently Perturbed Weakly Inhomogeneous Non-Linear Force Chain

We have obtained rigorous analytic and numerical solutions of the equations which govern the transport of mechanical perturbations in a gently precompressed 1D Hertz chain. Both finite-length and infinite-length systems have been studied. We examine both discrete and continuousformulations of the mentioned problem. A few families of analytic solutions of the problem given in the form of quasinormal waves and specific resonance modes have been obtained in the linear approximation for weakly perturbed inhomogeneous systems. Resonance modes are proposed to be interpreted as the Ramsauer–Townsend effect which happens due to the inhomogeneity. The obtained analytic results have been compared with numerical solutions of the discrete equations. We observe a multiscaled scenario of the impulse transport in an inhomogeneous force chain which could happens either asymptotically or at the intermittency between discrete- and continuous limits of the formulated problem. The role of a disorder has been also analyzed with the help of the Dyson concept.

A Model to Improve Granular Temperature in CFD-DEM Simulations

CFD-DEM (computational fluid dynamic-discrete element method) is a promising approach for simulating fluid–solid flows in fluidized beds. This approach generally under-predicts the granular temperature due to the use of drag models for the average drag force. This work develops a simple model to improve the granular temperature through increasing the drag force fluctuations on the particles. The increased drag force fluctuations are designed to match those obtained from PR-DNSs (particle-resolved direct numerical simulations). The impacts of the present model on the granular temperatures are demonstrated by posteriori tests. The posteriori tests of tri-periodic gas–solid flows show that simulations with the present model can obtain transient as well as steady-state granular temperature correctly. Moreover, the posteriori tests of fluidized beds indicated that the present model could significantly improve the granular temperature for the homogenous or slightly inhomogeneous systems, while it showed negligible improvement on the granular temperature for the significantly inhomogeneous systems.