Discrete Integration Research Articles

Theory and applications of new fractional-order chaotic system under Caputo operator

This paper introduces the properties of a fractional-order chaotic system described by the Caputo derivative. The impact of the fractional-order derivative has been focused on. The phase portraits in different orders are obtained with the aids of the proposed numerical discretization, including the discretization of the Riemann-Liouville fractional integral. The stability analysis has been used to help us to delimit the chaotic region. In other words, the region where the order of the Caputo derivative involves and where the presented system in this paper is chaotic. The nature of the chaos has been established using the Lyapunov exponents in the fractional context. The schematic circuit of the proposed fractional-order chaotic system has been presented and simulated in via Mutltisim. The results obtained via Multisim simulation of the chaotic circuit are in good agreement with the results with Matlab simulations. That provided the fractional operators can be applied in real- worlds applications as modeling electrical circuits. The presence of coexisting attractors for particular values of the parameters of the presented fractional-order chaotic model has been studied.

High-order explicit conservative exponential integrator schemes for fractional Hamiltonian PDEs

The paper aims to develop a class of high-order explicit exponential time differencing energy-preserving schemes for some conservative fractional PDEs based on the general Hamiltonian form of these equations. The equation is first reformulated into an equivalent system that possesses a new quadratic energy by introducing an auxiliary variable. Then, explicit schemes are derived via applying the exponential time differencing Runge-Kutta method approximations for time integration and Fourier pseudo-spectral discretization in space, which can maintain the accuracy and improve the stability to the system with stiffness. Subsequently, highly efficient energy-preserving schemes are given by combining the proposed explicit schemes and the projection technique. Finally, the advantages of the proposed schemes are illustrated by the numerical results of the fractional nonlinear Schrödinger equation.

The Ponzano–Regge cylinder and propagator for 3d quantum gravity

We investigate the propagator of 3d quantum gravity, formulated as a discrete topological path integral. We define it as the Ponzano–Regge amplitude of the solid cylinder swept by a 2d disk evolving in time. Quantum states for a 2d disk live in the tensor products of N spins, where N is the number of holonomy insertions connecting to the disk boundary. We formulate the cylindric amplitude in terms of a transfer matrix and identify its eigen-modes in terms of spin recoupling. We show that the propagator distinguishes subspaces with different total recoupled spin. This may select the vanishing overall spin sector at late time depending on the chosen cylinder boundary data, leading to an emergent symmetry scenario in the continuum limit. We discuss applications to quantum circuits and the possibility of experimental simulations of this 3d quantum gravity propagator.

Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized m , N − m -fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.

Discrete IV dG-Choquet integrals with respect to admissible orders

In this work, we introduce the notion of dG-Choquet integral, which generalizes the discrete Choquet integral replacing, in the first place, the difference between inputs represented by closed subintervals of the unit interval [0,1] by a dissimilarity function; and we also replace the sum by more general appropriate functions. We show that particular cases of dG-Choquet integral are both the discrete Choquet integral and the d-Choquet integral. We define interval-valued fuzzy measures and we show how they can be used with dG-Choquet integrals to define an interval-valued discrete Choquet integral which is monotone with respect to admissible orders. We finally study the validity of this interval-valued Choquet integral by means of an illustrative example in a classification problem.

Full-scale tests of unsteady aerodynamic loads and pressure distribution on fast trains in crosswinds

Unsteady aerodynamic loads have a significant influence on the running safety of a train in crosswinds. Full-scale tests were carried out to measure the pressure on a fast streamlined train in crosswinds, and the unsteady aerodynamic loads were obtained by the discrete integration of the pressure to assess the influence of different crosswinds on the loads. The maximum wind speed was 34.0 m/s in the full-scale tests, and the maximum turbulence intensity was 0.202. The means and fluctuations of the side force coefficient, lift coefficient, and rolling moment coefficient around the lee rail at different yaw angles were studied. The means first increased and then decreased with the increase in the yaw angle, and the maximum values occurred in the range of 54.5°–59.6°. The coefficients of variation of the aerodynamic load coefficients were approximately twice the turbulence intensity, which indicated that the fluctuations of the aerodynamic load coefficients were due to the oscillations of the wind speed. For the frequency characteristics of the aerodynamic load coefficients, the most dominant frequencies were at Strouhal numbers of 0–0.4.

Anisotropic Landau-Lifshitz model in discrete space-time

We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.

A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation

A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schrödinger equation. The method can be implemented by using fast Fourier transform with \(O(N\ln N)\) operations at every time level, and is proved to have an \(L^2\)-norm error bound of \(O(\tau \sqrt{\ln (1/\tau )}+N^{-1})\) for \(H^1\) initial data, without requiring any CFL condition, where \(\tau \) and N denote the temporal stepsize and the degree of freedoms in the spatial discretisation, respectively.

Bending analysis of FG-GPLRC axisymmetric circular/annular sector plates by considering elastic foundation and horizontal friction force using 3D-poroelasticity theory

Composite structures encounter different types of imperfections and defects during the installation, working environment, and fabrication process. In the present research, a 3D-poroflexibility theory has been presented to study the bending responses of the functionally graded graphene nanoplatelets reinforced composite (FG-GPLRC) axisymmetric circular/annular sector plates on the elastic substrate. For modeling elastic substrate, horizontal friction force and the elastic foundation with five elastic parameters have been formulated. For obtaining the exact displacement and stress/strain responses of the current structure, a discrete singular convolution integration method (DSC-IM) has been utilized. Also, the outputs of the current research have been verified with the results of COMSOL Multi-physics software. Consequently, the outcomes demonstrate that GPL’s weight fraction, in-plane boundary condition, radial and circumferential initially stresses, and Biot's coefficient have a remarkable impact on the bending responses of the circular/annular sector plates. Also, when the normal compressive stress increases, the negative effect from normal tensile stress on the bending characteristics of the structures is more remarkable than the positive effect from normal compressive stress. Also, the effect of GPL’s weight fraction on the stress and displacement fields is more considerable at the higher value of the opening angle of the current composite structure.

Choquet-Sugeno-like operator based on relation and conditional aggregation operators

We introduce a Choquet-Sugeno-like operator generalizing many operators for bounded nonnegative functions and monotone measures from the literature, e.g., the Sugeno-like operator, the Lovász and Owen measure extensions, the F-decomposition integral with respect to a partition decomposition system, and others. The new operator is based on concepts of dependence relation and conditional aggregation operators, but it does not depend on α-level sets. We also provide conditions under which the Choquet-Sugeno-like operator coincides with some Choquet-like integrals defined on finite spaces and appeared recently in the literature, e.g., the reverse Choquet integral, the d-Choquet integral, the F-based discrete Choquet-like integral, some version of the CF1F2-integral, the CC-integrals (or Choquet-like Copula-based integral) and the discrete inclusion-exclusion integral. Some basic properties of the Choquet-Sugeno-like operator are studied.

Integrable discretizations and soliton solutions of an Eckhaus–Kundu equation

We consider the integrable discretizations of an Eckhaus–Kundu equation. The x-discrete, t-discrete and one fully discrete version of the equation are presented on the basic of the bilinear form. Furthermore, soliton solutions of the derived discrete equations are constructed successfully via Hirota’s bilinear method.

Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere hat {mathbb {C}}. Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

Accurate Open Channel Flowrate Estimation Using 2D RANS Modelization and ADCP Measurements

Boat-mounted Acoustic Doppler Current Profilers (ADCP) are commonly used to measure the streamwise velocity distribution and discharge in rivers and open channels. Generally, the method used to integrate the measurements is the velocity-area method, which consists of a discrete integration of flow velocity over the whole cross-section. The discrete integration is accomplished independently in the vertical and transversal direction without assessing the hydraulic coherence between both dimensions. To address these limitations, a new alternative method for estimating the discharge and its associated uncertainty is here proposed. The new approach uses a validated 2D RANS hydraulic model to numerically compute the streamwise velocity distribution. The hydraulic model is fitted using state estimation (SE) techniques to accurately reproduce the measurement field and hydraulic behaviour of the free-surface stream. The performance of the hydraulic model has been validated with measurements on two different trapezoidal cross-sections in a real channel, even with asymmetric velocity distribution. The proposed method allows extrapolation of measurement information to other points where there are no measurements with a solid and consistent hydraulic basis. The 2D-hydraulic velocity model (2D-HVM) approach discharge values have been proven more accurate than the ones obtained using velocity-area method, thank to the enhanced use of the measurements in addition to the hydraulic behaviour represented by the 2D RANS model.

Analytical design of discrete PI–PR controllers via dominant pole assignment

In this study, a new discrete proportional integral–proportional retarded (D-PI–PR) controller design method is proposed. The proposed controller involves replacing the PD controller in the classical PI–PD control structure with a PR controller in discrete time and is implemented using dominant pole assignment, which is carried out by help of the modified Nyquist plot approach. The method starts with the determination of the controller parameters Ki and Kr in terms of the parameter Kp for a chosen delay parameter h after the computation of the desired dominant poles depending on the required closed-loop system performance criteria. The next step is to find the feasible Kp interval, in which the dominant pole assignment is guaranteed. To eliminate the unwanted impact of the controller zeros on the transient response of the system, the D-PIR structure is then transformed into the D-PI–PR structure. For the demonstration of the success of the proposed algorithm, first, second and third-order systems with time delays are used and the simulation results are compared with some other PID-type controllers design methods from the literature.

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

By combining a certain approximation property in the spatial domain, and weighted 𝓁2-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L1 (ℝ∞, V, γ) norm of the generating methods where γ is the Gaussian probability measure on ℝ∞ and V is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

Lattice elasticity, waves and temperature from an interaction potential

Kinetic theory is taught to first-year students of physics as a plausible account of thermal equilibrium on the microscopic scale. However, it does not adequately clarify important properties like elasticity, lattice waves, phonons, optical oscillations, bandgap and so on, that are postulated through atomic vibrations. In this article, we apply simple computational tools in four computational experiments to study the effects of the vibrations of 373 atoms of a face-centered cubic crystal such as copper. We also discuss the perturbation modelling approach, the Sutton–Chen potential, and discrete integration. The experiments have different aims e.g. determining the steady-state atom arrangement starting from an arbitrarily specified cluster, the elasticity, the ensuing wave motion, lattice resonance, acoustic properties, and temperature rise. The calculated results agree well with their literature values.

The Primitive Derivation and Discrete Integrals

he modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type A root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements.

A Novel Discretization Method for Multiple Second-Order Generalized Integrators

Multiple second-order generalized integrators (MSOGI) has been widely used to extract the signal that contains multiple harmonics. The MSOGI needs to be implemented in the discrete-time domain for its practical applications. More critically, the discretization method can greatly affect its performance. The existing discretization methods for MSOGI can be divided into three categories: entire transfer-function discretization, SOGI-transfer-function discretization, and integrator discretization. The theoretical analysis indicates that the first method can avoid the errors caused by unit delay, but it is hard to be digitally realized for high-order transfer functions. The other two methods have nonnegligible errors in both the phase and the quadrature characteristic due to the unit delay. In this letter, we propose a novel discretization method by modifying the structure of integrator discretization. This method can enhance the performance in both the phase and the quadrature characteristic with an acceptable computation burden, which has been theoretically analyzed and experimentally verified.

Numerical simulation of tethered–wing power systems based on variational integration

This paper describes dynamic formulations for a multi-physics system consisting of an electrical generator and a finite element model of a variable-length cable attached at one end to a reeling mechanism and the other to a rigid body that represents a flying wing. Lie group methods are utilized to obtain singularity-free dynamics of the wing. Two different methods are employed for the derivation and integration of the Euler–Lagrange equations. In the first method, the equations of motion are derived in continuous-time and integrated by a general-purpose implicit ODE solver. The second is the application of the discrete analogue of Lagrange–d’Alembert principle that yields a variational integrator. Extensive simulations are carried out to calculate the computational complexity of the discrete integrator and compare the results for CPU time and accuracy to those of the ODE solver. It is shown that the variational integrator has a significant advantage in terms of preservation of the orthogonality of the wing’s attitude matrix and reduction of the CPU time. The descriptions of the implemented aerodynamic models and controllers, along with the results for the simulation of a tethered-wing system operating in a turbulent wind environment are also presented.

Thermodynamically consistent nonlinear viscoplastic formulation with exact solution for the linear case and well-conditioned recovery of the inviscid one

In this work, a consistent viscoplasticity formulation is derived from thermodynamical principles and employing the concept of continuum elastic corrector rate. The proposed model is developed based on the principle of maximum viscoplastic dissipation for determining the flow direction. The model uses both the equivalent viscoplastic strain and its rate as state variables. Power balance and energy balance give, respectively, separate evolution equations for the equivalent viscoplastic strain rate and the viscoplastic strain, the former written in terms of inviscid rates. Several key points distinguish our formulation from other proposals. First, the viscoplastic strain rate (instead of a yield function) consistently distinguishes conservative from dissipative behaviours during reverse loading; and the discrete implicit integration algorithm is an immediate discretization of the continuum theory based on the mentioned principles considered as separate equations. Second, the inviscid solution is recovered in a well-conditioned manner by simply setting the viscosity to zero. Indeed, inviscid plasticity, viscoelasticity and viscoplasticity are particular cases of our formulation and integration algorithm, and are recovered just by setting the corresponding parameters to zero (viscosity or yield stress). Third, the linear viscoplasticity solution is obtained in an exact manner for proportional loading cases, independently of the time step employed. Four, general nonlinear models (Perzyna, Norton, etc) may be immediately incorporated as particular cases both in the theory and the computational implementation.