AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS

Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new fractional operators,namely the CaputoFabrizio operator,the Atangana-Baleanu operator,the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator.Usually,the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator.Here,we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral.Then,a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown,respectively.In terms of the above analysis,we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators.

Numerical Simulation and Parallel Computing of Acoustic Wave Equation in Isotropic-Heterogeneous Media

In this paper,we consider the numerical implementation of the 2D wave equation in isotropic-heterogeneous media.The stability analysis of the scheme using the von Neumann stability method has been studied.We conducted a study on modeling the propagation of acoustic waves in a heterogeneous medium and performed numerical simulations in various heterogeneous media at different time steps.Developed parallel code using Compute Unified Device Architecture(CUDA)technology and tested on domains of various sizes.Performance analysis showed that our parallel approach showed significant speedup compared to sequential code on the Central Processing Unit(CPU).The proposed parallel visualization simulator can be an important tool for numerous wave control systems in engineering practice.

QUASIPERIODICITY OF TRANSCENDENTAL MEROMORPHIC FUNCTIONS

This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.

Numerical Simulation of Thermocapillary Convection with Evaporation Induced by Boundary Heating

The dynamics of a bilayer system filling a rectangular cuvette subjected to external heating is studied.The influence of two types of thermal exposure on the flow pattern and on the dynamic contact angle is analyzed.In particular,the cases of local heating from below and distributed thermal load from the lateral walls are considered.The simulation is carried out within the frame of a two-sided evaporative convection model based on the Boussinesq approximation.A benzine–air system is considered as reference system.The variation in time of the contact angle is described for both heating modes.Under lateral heating,near-wall boundary layers emerge together with strong convection,whereas the local thermal load from the lower wall results in the formation of multicellular motion in the entire volume of the fluids and the appearance of transition regimes followed by a steady-state mode.The results of the present study can aid the design of equipment for thermal coating or drying and the development of methods for the formation of patterns with required structure and morphology.

Stability Study of Low Voltage Electrical Distribution Network: Audit and Improvement of DJEGBE Mini Solar Photovoltaic Power Plant in the Commune of OUESSE (Benin)

The supply of quality energy is a major concern for distribution network managers. This is the case for the company ASEMI, whose subscribers on the DJEGBE mini-power station network are faced with problems of current instability, voltage drops, and repetitive outages. This work is part of the search for the stability of the electrical distribution network by focusing on the audit of the DJEGBE mini photovoltaic solar power plant electrical network in the commune of OUESSE (Benin). This aims to highlight malfunctions on the low-voltage network to propose solutions for improving current stability among subscribers. Irregularities were noted, notably the overloading of certain lines of the PV network, implying poor distribution of loads by phase, which is the main cause of voltage drops;repetitive outages linked to overvoltage caused by lightning and overcurrent due to overload;faulty meters, absence of earth connection at subscribers. Peaks in consumption were obtained at night, which shows that consumption is greater in the evening. We examined the existing situation and processed the data collected, then simulated the energy consumption profiles with the network analyzer “LANGLOIS 6830” and “Excel”. The power factor value recorded is an average of 1, and the minimum value is 0.85. The daily output is 131.08 kWh, for a daily demand of 120 kWh and the average daily consumption is 109.92 kWh, or 83.86% of the energy produced per day. These results showed that the dysfunctions are linked to the distribution and the use of produced energy. Finally, we proposed possible solutions for improving the electrical distribution network. Thus, measures without investment and those requiring investment have been proposed.

Fluctuating Asymmetry of Elopes lacerta (Valenciennes, 1847) Otoliths in the Western African Waters

The importance of this study lies in the in-depth exploration of the ecological diversity of otoliths in Elopes lacerta, based on the analysis of data from 260 individuals collected from various sites, including the Porto Novo lagoon, the Cotonou lagoon, Lake Nokou, and the Atlantic coast in southern Benin. The results highlight significant variations in otolith morphology, establishing relevant links with the biological parameters of the fish at each site. Exploration of the asymmetry between the right and left sides reveals notable distinctions between these two aspects. Analysis of otolith shape thus emerges as a valuable tool for discriminating between stocks and providing a better understanding of ecological variations. The local diversity observed highlights the crucial importance of implementing adaptive management strategies to ensure effective conservation of the species and its habitat.

Analysis of Otolith Shape as a Tool for Discriminating Stocks of Cassava Croaker (Pseudotolithus senegalensis) in Beninese Waters

This study examined the right and left otoliths of 174 individuals of Pseudotolithus senegalensis from the Porto-Novo lagoon, Lake Nokoué and the Atlantic coast. The results show a significant variation in the population according to geographic location. Mixed-effects linear analysis showed no significant variation by site, side or sex (p > 0.05). Analysis of otolith shape using ANOVA showed significant differences for length, width and area (p 0.05). Canonical discriminant analysis revealed significant differences between sites (p < 0.05). Finally, shape analyses showed significant differences between sites (p = 0.0001) and between right and left sides (p = 0.007), but no difference by sex (p = 0.395). The study of otolith morphology is therefore proving to be a valuable tool for differentiating stocks and understanding ecological variations.

Analysis of the COVID-19, Outbreak in Brazil Using Topological Weighted Centroid: An Intelligent Geographic Information System Approach

This study used Topological Weighted Centroid (TWC) to analyze the Coronavirus outbreak in Brazil. This analysis only uses latitude and longitude in formation of the capitals with the confirmed cases on May 24, 2020 to illustrate the usefulness of TWC though any date could have been used. There are three types of TWC analyses, each type having five associated algorithms that produce fifteen maps, TWC-Original, TWC-Frequency and TWC-Windowing. We focus on TWC-Original to illustrate our approach. The TWC method without using the transportation information predicts the network for COVID-19 outbreak that matches very well with the main radial transportation routes network in Brazil.

Contribution of Satellite Observations in the Optical and Microphysical Characterization of Aerosols in Burkina Faso, West Africa

In this work, we proceed to an optical and microphysical analysis of the observations reversed by the MODIS, SeaWiFS, MISR and OMI sensors with the aim of proposing the best-adapted airborne sensor for better monitoring of aerosols in Burkina Faso. To this end, a comparison of AOD between satellite observations and in situ measurements at the Ouagadougou site reveals an underestimation of AERONET AOD except for OMI which overestimates them. Also, an inter-comparison done based on the linear regression line representation shows the correlation between the aerosol models incorporated in the airborne sensor inversion algorithms and the aerosol population probed. This can be seen through the correlation coefficients R which are 0.84, 0.64, 0.55 and 0.054 for MODIS, SeaWiFS, MISR and OMI respectively. Furthermore, an optical analysis of aerosols in Burkina Faso by the MODIS sensor from 2001 to 2016 indicates a large spatial and temporal variability of particles strongly dominated by desert dust. This is corroborated by the annual and seasonal cycles of the AOD at 550 nm and the Angström coefficient measured in the spectral range between 412 nm and 470 nm. A zoom on a few sites chosen according to the three climatic zones confirms the majority presence of mineral aerosols in Burkina Faso, whose maxima are observed in spring and summer.

THE EXISTENCE OF GLOBAL SOLUTIONS FOR THE FULL NAVIER-STOKES-KORTEWEG SYSTEM OF VAN DER WAALS GAS

The aim of this work is to prove the existence for the global solution of a nonisothermal or non-isentropic model of capillary compressible fluids derived by J.E.Dunn and J.Serrin(1985),in the case of van der Waals gas.Under the small initial perturbation,the proof of the global existence is based on an elementary energy method using the continuation argument of local solution.Moreover,the uniqueness of global solutions and large time behavior of the density are given.It is one of the main difficulties that the pressure p is not the increasing function of the densityρ.

THE ASYMPTOTIC STABILITY OF PHASE SEPARATION STATES FOR COMPRESSIBLE IMMISCIBLE TWO-PHASE FLOW IN 3D

This paper is concerned with a diffuse interface model called Navier-Stokes/CahnHilliard system.This model is usually used to describe the motion of immiscible two-phase flows with a diffusion interface.For the periodic boundary value problem of this system in torus T3,we prove that there exists a global unique strong solution near the phase separation state,which means that no vacuum,shock wave,mass concentration,interface collision or rupture will be developed in finite time.Furthermore,we establish the large time behavior of the global strong solution of this system.In particular,we find that the phase field decays algebraically to the phase separation state.

A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem:Application to the 1D Euler Equations

We show how to combine in a natural way(i.e.,without any test nor switch)the conservative and non-conservative formulations of an hyperbolic system that has a conservative form.This is inspired from two different classes of schemes:the residual distribution one(Abgrall in Commun Appl Math Comput 2(3):341–368,2020),and the active flux formulations(Eyman and Roe in 49th AIAA Aerospace Science Meeting,2011;Eyman in active flux.PhD thesis,University of Michigan,2013;Helzel et al.in J Sci Comput 80(3):35–61,2019;Barsukow in J Sci Comput 86(1):paper No.3,34,2021;Roe in J Sci Comput 73:1094–1114,2017).The solution is globally continuous,and as in the active flux method,described by a combination of point values and average values.Unlike the“classical”active flux methods,the meaning of the point-wise and cell average degrees of freedom is different,and hence follow different forms of PDEs;it is a conservative version of the cell average,and a possibly non-conservative one for the points.This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem.We also develop a method to perform nonlinear stability.We illustrate the behaviour on several benchmarks,some quite challenging.

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws.The basic idea is that the“meaningful objects”are the fluxes,evaluated across domain boundaries over time intervals.The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting.It implies that a weak solution indeed satisfies the balance law.In fact,it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.It should be emphasized that the weak solutions considered here need not be entropy solutions.Furthermore,the assumption imposed on the flux f(u)is quite minimal-just that it is locally bounded.

Vibration of a Two-Layer“Metal+PZT”Plate Contacting with Viscous Fluid

The present work investigates the mechanically forced vibration of the hydro-elasto-piezoelectric system consisting of a two-layer plate“elastic+PZT”,a compressible viscous fluid,and a rigid wall.It is assumed that the PZT(piezoelectric)layer of the plate is in contact with the fluid and time-harmonic linear forces act on the free surface of the elastic-metallic layer.This study is valuable because it considers for the first time the mechanical vibration of the metal+piezoelectric bilayer plate in contact with a fluid.It is also the first time that the influence of the volumetric concentration of the constituents on the vibration of the hydro-elasto-piezoelectric system is studied.Another value of the present work is the use of the exact equations and relations of elasto-electrodynamics for elastic and piezoelectric materials to describe the motion of the plate layers within the framework of the piecewise homogeneous body model and the use of the linearized Navier-Stokes equations to describe the flow of the compressible viscous fluid.The plane-strain state in the plate and the plane flow in the fluid take place.For the solution of the corresponding boundary-value problem,the Fourier transform is used with respect to the spatial coordinate on the axis along the laying direction of the plate.The analytical expressions of the Fourier transform of all the sought values of each component of the system are determined.The origins of the searched values are determined numerically,after which numerical results on the stress on the fluid and plate interface planes are presented and discussed.These results are obtained for the case where PZT-2 is chosen as the piezoelectric material,steel and aluminum as the elastic metal materials,and Glycerin as the fluid.Analysis of these results allows conclusions to be drawn about the character of the problem parameters on the frequency response of the interfacial stress.In particular,it was found that after a certain value of the vibration frequency,the presence of the metal layer in the two-layer plate led to an increase in the absolute values of the above interfacial stress.

Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids

Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case,such as involutions and non-trivial stationary states.These features need to be captured by numerical methods without excessive grid refine-ment.The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary.For the equations of linear acoustics,an exact evolution operator can be used for the update of these point values.It incorporates all multi-dimensional information.The active flux method is stationarity preserving,i.e.,it discretizes all the stationary states of the PDE.This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state.

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part Ⅱ:Entropy Stability

In the hyperbolic research community,there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability.In the first part of the series[6],the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly.By applying this technique,the authors demonstrate that a pure continu-ous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way.In this work,we extend this investigation to the nonlinear case and focus on entropy conservation.By switching to entropy variables,we provide an estimation of the boundary operators also for nonlinear problems,that guarantee conservation.In numerical simulations,we verify our theoretical analysis.

Digital Simulation of Projective Non-Abelian Anyons with 68 Superconducting Qubits

Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter.They break the fermion-boson dichotomy and obey non-Abelian braiding statistics:their interchanges yield unitary operations,rather than merely a phase factor,in a space spanned by topologically degenerate wavefunctions.They are the building blocks of topological quantum computing.However,experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto,in spite of various theoretical proposals.Here,we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice.By implementing the ground states of the toric-code model with twists through quantum circuits,we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons,i.e.,the Ising anyons.In particular,we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type,and can be explored to encode topological logical qubits.Furthermore,we demonstrate how to implement both single-and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits.Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons,offering a new lens into the study of such peculiar quasiparticles.

Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday

December 9,2022 is the 90th birthday of Tong Zhang,a mathematician in Institute of Mathematics,Chinese Academy of Sciences where he was always working on the Riemann problem for gas dynamics in his mathematical life.To celebrate his 90th birthday and great contributions to this specifc feld,we organize this focused issue in the journal Communications on Applied Mathematics and Computation,since the Riemann problem has been proven to be a building block in all felds of theory,numerics and applications of hyperbolic conservation laws.

Two Families of Multipoint Root-Solvers Using Inverse Interpolation with Memory

In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case.These kinetic models depend on a small parameter that can be seen as a"Knudsen"number.The method is asymptotic preserving in this Knudsen number.Also,the computational costs of the method are of the same order of a fully explicit scheme.This work is the extension of Abgrall et al.(2022)[3]to multidimensional systems.We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.