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Of concern is the dynamics of anti-periodic solutions on Clifford-valued high-order Hopfield neural networks (CVHOHNNs) with multi-proportional delays. By utilizing differential inequality techniques, Lyapunov method and a rigorous mathematical proof, we establish some sufficient criteria on the existence and global stability of anti-periodic solutions. Finally, a numerical example is offered to verify and extend the analytical results and visualize the interesting phenomenon.

This paper develops a high order semi-implicit weighted compact nonlinear scheme (WCNS) for one- and two-dimensional viscous Burgers’ equations. This semi-implicit WCNS combines a fifth-order WCNS with a third-order implicit-explicit Runge–Kutta (IMEX-RK) time-stepping method. The advection terms of viscous Burgers’ equations are treated explicitly, while the viscosity terms are treated implicitly. Stability analysis shows that the CFL condition of the semi-implicit WCNS is controlled only by the advection terms. Compared to the explicit time-stepping method, the semi-implicit method has the advantage in terms of computational efficiency. Numerical results validate the accuracy and efficiency of the semi-implicit WCNS. This method can also solve viscous Burgers’ equations with large Reynolds numbers and has high-resolution shock-capturing ability.

Using a general computational framework, we derive an optimal error estimate in the L 2 norm for a semi discrete method based on high order B-splines Galerkin spatial discretizations, applied to a coupled nonlinear Schrodinger system with cubic nonlinearity. A fully discrete method based on a conservative nonlinear splitting Crank-Nicolson time step is then proposed; and conservation of the mass and the energy is theoretically proven. To validate its accuracy in space and time, and its conservation properties, several numerical experiments are carried out with B-splines up to order 7.

This paper devotes to solving time-varying linear matrix equations (TVLMEs) from the viewpoint of high-order neural networks . For this purpose, high-order zeroing neural network (ZNN) models are designed and studied to solve TVLMEs. Compared with the first-order ZNN model for TVLMEs, the proposed high-order ZNN models are based on the design of the high-order error functions, and different order choices will generate different high-order ZNN models. Two nonlinear activation functions [i.e., tunable activation function (TunAF) and sign-bi-power activation function (SBPAF)] are used to speedup the high-order ZNN models for achieving the finite-time convergence. Furthermore, the strict theoretical analyses are provided to show that high-order ZNN models have better properties (especially in terms of convergence), when the nonlinear activation functions are used. Two numerical simulations are given to reveal the superior convergence property of the proposed high-order ZNN models, as compared to the first-order ZNN model for solving TVLMEs.

In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grunwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank–Nicolson scheme in time. A third-order scheme for the tempered Black–Scholes equation is also proposed and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.

In this paper, we investigate the cooperative control problem of high-order integrators under heterogeneous couplings. A new class of distributed control algorithms are developed for the designated convergence rate (DCR) problem of high-order integrators, which could explicitly show the convergence margin of the closed-loop system, and has better robustness than conventional consensus algorithms. We first propose state consensus control algorithms for high-order integrators, where necessary and sufficient convergence conditions are proposed by theoretical analysis. Then we extend the results to the case of output leaderless consensus of heterogeneous high-order integrators with heterogeneous couplings. Finally, simulation examples are given to validate the effectiveness of the proposed algorithms.

International audience; In this paper, we consider the numerical approximation of hyperbolic systems of conservation laws with sti source terms and parabolic degeneracy in the asymptotic limit. We are more precisely interested in the design of high-order asymptotic-preserving schemes on unstructured meshes. Our approach is based on a very simple modication of the numerical ux associated with the usual HLL scheme and boils down to a sharp control of the underlying numerical diusion. The strategy allows to capture the correct asymptotic parabolic behavior and to preserve the high-order accuracy also in the asymptotic limit. Numerical experiments are proposed to illustrate these properties.

This study presents a parametric approach to design output feedback for descriptor quasi-linear high-order systems. Utilizing the solution to the high-order generalized Sylvester equations (HGSEs), the more unified parametric form of output feedback is developed. With the proposed approach, the closed-loop is transformed into a linear constant system with an expected eigenstructure. Meanwhile, the regularity of the closed-loop system is maintained easily and effectively. Finally, a synchronization problem of Genesio-Tesi and Coullet systems is proposed to illustrate the effectiveness of the parametric approach.

Recently, a 4th-order asymptotic preserving multiderivative implicit-explicit (IMEX) scheme was developed (Sch\"utz and Seal 2020, arXiv:2001.08268). This scheme is based on a 4th-order Hermite interpolation in time, and uses an approach based on operator splitting that converges to the underlying quadrature if iterated sufficiently. Hermite schemes have been used in astrophysics for decades, particularly for N-body calculations, but not in a form suitable for solving stiff equations. In this work, we extend the scheme presented in Sch\"utz and Seal 2020 to higher orders. Such high-order schemes offer advantages when one aims to find high-precision solutions to systems of differential equations containing stiff terms, which occur throughout the physical sciences. We begin by deriving Hermite schemes of arbitrary order and discussing the stability of these formulas. Afterwards, we demonstrate how the method of Sch\"utz and Seal 2020 generalises in a straightforward manner to any of these schemes, and prove convergence properties of the resulting IMEX schemes. We then present results for methods ranging from 6th to 12th order and explore a selection of test problems, including both linear and nonlinear ordinary differential equations and Burgers' equation. To our knowledge this is also the first time that Hermite time-stepping methods have been applied to partial differential equations. We then discuss some benefits of these schemes, such as their potential for parallelism and low memory usage, as well as limitations and potential drawbacks., Comment: 23 pages, 4 figures, submitted to the Journal of Computational Physics. Comments welcome

In this paper, an improved sub-step time-marching procedure is proposed for structural dynamics. The proposed procedure has two free parameters which can be adopted to control basic algorithmic properties including computation accuracy, algorithmic damping, period elongation, and overshooting behavior. The influences of the algorithmic parameters on these basic properties of the method are studied, and the optimal parameter value is thus obtained. Theoretical analysis and numerical simulations demonstrate that new procedure is desirable for linear and nonlinear dynamics due to its higher-order accuracy, desirable numerical dissipation, and high-efficient energy conservation.

We study positive solutions of the Yamabe equation with isolated singularity and prove the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.

We use a double mollification method to study a family of high order nonlinear partial differential equations with peakon weak solutions. This method gives the approximated N-peakon weak solutions which satisfy weak consistent property. By a limiting process, we prove the global existence of N-peakon weak solutions. Moreover, the traveling speeds of single peakon weak solutions (soliton waves) are obtained directly by the limit of the mollification.

In this work, dynamic controllers are designed for reactor power, pressurizer water level, and pressure control in the primary circuit of a pressurized water reactor. These nonlinear controllers use super-twisting sliding-mode estimators to enhance their robustness versus parameter variations and external disturbances. Hence, the perturbative terms can be canceled by the control, thus improving the dynamic behavior of the controlled system. The designed controllers ensure good performances and better transient behavior, also in the presence of uncertainties and disturbances. A performance study of the proposed controllers is carried out in the presence also of unmodeled dynamics.

When undertaking a numerical solution of Helmholtz problems using the Boundary Element Method (BEM) it is common to employ low-order Lagrange polynomials, or more recently Non-Uniform Rational B-Splines (NURBS), as basis functions. A popular alternative for high frequency problems is to use an enriched basis, such as the plane wave basis used in the Partition of Unity Boundary Element Method (PUBEM). To the authors’ knowledge there is yet to be a thorough quantification of the numerical error incurred as a result of employing high-order NURBS and Lagrange polynomials for wave-based problems in a BEM setting. This is the focus of the current work, along with comparison of the results against PUBEM. The results show expected improvements in the convergence rates of a Lagrange or NURBS scheme as the order of the basis functions is increased, with the NURBS basis slightly outperforming the Lagrange basis. High-order Lagrange and NURBS formulations can compare favourably against PUBEM for certain cases. In addition, the recently observed pollution effect in BEM is studied for a travelling wave in a duct and the numerical dispersion presented for all three sets of basis functions.

Numerical results for polynomial (p-version) refinement in the three-dimensional boundary element method are presented. Results are based on a weakly singular form of the gradient boundary element formulation in potential theory. The weakly singular formulation facilitates direct application of numerical quadrature, completely removing the need for closed-form integration. The new results, which include interpolations as high as tenth order, show the efficiency of the approach while confirming a previous assertion that whole-body regularization in conjunction with only linear or quadratic elements will lead to extremely poor results.

A powerful automated high-order unstructured curved mesh generator is proposed in this work with a high-level dynamic programming language Julia. This generator uses higher-order one-sided curved triangular finite elements for the domains having curved borders with parabolic arcs. The proposed approach of the mesh generator can be successfully applied for solving several industrial problems inclusive of photonics with the finite element method. The use of the parabolic arcs method to obtain node relations for the curved geometry enhances the performance of the technique with the subparametric mappings. The mesh generator suggested is based on the prominent Gmsh mesh generator. For all geometry, the presented technique can be implemented. The methodology is applied in this paper to illustrate a few photonic crystal domains.

We present a fractional series solution (FSS) for a class of higher-order linear fractional PDEs. The fractional derivative in this class is considered in the conformable fractional derivative (Co-FD) sense. An appropriate expansion was introduced to reach an FSS that is consistent with the target equations in this research. The residual power series technique is used to determine the coefficients of the FSS. Five applications are tested to verify the effectiveness of the used method, as well as to compare the current results with the previous results for the same applications in which the fractional derivative was considered in the Caputo sense. Numerical and graphical comparisons are made to determine the compatibility of the behavior of the solution in the case of the use of the concept of Co-FD as a suitable alternative to the use of the concept of Caputo fractional derivative (Ca-FD) in the modeling of natural phenomena.

Purpose To describe the visual, refractive, topographic and aberrometric outcomes of six different accelerated cross-linking (ACXL) protocols in progressive keratoconus (KC) at 12 months. # Material and methods This observational retrospective study included 62 eyes of 49 patients with progressive KC that received one of the following accelerated cross-linking protocols: Group I (8 patients, 5.4 J/cm2, 15mWx12min, pulsed), Group II (11 patients, 5.4 J/cm2, 6mWx15min, continuous), Group III (16 patients, 5.4 J/cm2, 9mWx10min, continuous), Group IV (13 patients, 5.4 J/cm2, 30mWx6min, pulsed), Group V (8 patients, 7.2 J/cm2, 15mWx16min, pulsed) and Group VI (6 patients, 7.2 J/cm2, 30mWx8min, pulsed). Best corrected visual acuity (BCVA), sphere, refractive cylinder, spherical equivalent (SE), maximum keratometry (Kmax), mean keratometry (Km), flat keratometry (K1), steep keratometry (K2), thinnest pachymetry, total aberrations, high order aberrations (RMS-HOA), spherical aberration, coma and trefoil were studied previously and at 12 months. Intragroup and intergroup statistical analysis was performed. Postoperative complications were noted. # Results 11 patients were females(22,45%) and 38 males(77,55%). Improvement in the BCVA was observed in all groups (P > 0.05). Kmax reduced in all patients and was statistically significant in groups II (p = 0.041), III (p = 0.017), IV (p = 0.018) and V (p = 0.018). Flattening of the Km was significant in groups II (p = 0.028), IV (p = 0.008) and V (p = 0.028), as well as the K1 in groups IV(p = 0.01) and V(p = 0.028) and K2 in group IV(p = 0.036). There was no significant difference in the thinnest pachymetry in any of the groups. Total aberrations, RMS-HOA, spherical aberration and coma reduced in all groups with statistical significance in Group V (P = 0.016). Two patients in Group I had anterior stromal scarring and one patient in Group V presented an unexpected overflattening. Progression was noticed in one patient in Group II. # Conclusions Accelerated CXL protocols included in this study can be considered effective and safe procedures in stopping the progression of keratoconus at 12 months. Improvement in BCVA and wavefront analysis is observed, but results are limited by the number of patients. Caution should be taken when applying high radiance and long duration protocols to prevent undesirable events.

In this paper we propose a new distribution of Gaussian points to compute the weak form integrals of the Generalized Finite Element Method (GFEM). The relevance of this new distribution is the possibility of evaluating the integrals of oscillatory functions inside the reference triangle in an alternate way. A simple scheme of relocation of the quadrature points allows to improve the efficacy of the method. A wave propagation problem is solved with the proposed technique and its performance is compared to conventional and other existing proposals. In addition, we propose a new distribution of Gaussian points to the tetrahedral reference element

The plasmonics of two–dimensional materials, such as graphene, has become an important field over the past decade. The active tunability of graphene via electrical gating or chemical doping has generated a great deal of excitement among engineers seeking sensing devices. Consequently there is significant demand for robust and highly accurate computational capabilities which can simulate such materials. The class of High–Order Perturbation of Surfaces methods have proven to be particularly appropriate for this purpose. In this contribution we describe our recent efforts to utilize both Dirichlet–Neumann Operators and Impedance–Impedance Operators in these schemes. In addition, we present detailed numerical results which not only validate our simulations using the Method of Manufactured Solutions, but we also describe Localized Surface Plasmon Resonances in graphene nanotubes enclosing rod–shaped dielectric materials.

In this paper, we consider a class of Clifford-valued higher order Hopfield neural networks (HNNs) with multiple discrete delays whose coefficients of the leakage terms are also Clifford numbers. By a direct approach, based on the fixed point theorem and differential inequality techniques, we gain the existence and global exponential stability of ( μ , ν ) -pseudo almost periodic solutions of the proposed networks. Finally, we use an example to illustrate the feasibility of our results. Our results are new even when the considered HNNs degenerate into real-valued, complex-valued and quaternion-valued ones.

The classical dual mixed finite element method for flow simulations is based on H ( div , Ω ) conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in L 2 ( Ω ) for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance. The main purpose of this paper is to present a high-order finite element methodology to solve the Darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of H ( div , Ω ) and H 1 ( Ω ) . The chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables, without requiring any compatibility constraint, as required in mixed methods for these problems. Convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with Raviart–Thomas elements.

Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.

In this paper, we go beyond what was proposed in theory by Melnikov ( [15] ) to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time-periodic equations. Not only we derive integral formula for splitting distances of all orders, but also we develop an analytic theory to evaluate the acquired multiple integrals. We reveal that the dominance of the exponentially small Poincare/Melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equations.

In this work, a new limiting method for bicompact schemes is proposed that preserves them conservative. The method is based upon a finite-element treatment of the bicompact approximation. An analogy between collocation finite-element schemes and bicompact schemes is established. The proposed method is tested on one-dimensional gas dynamics problems that include the Sedov problem, the “peak test” Riemann problem, the Shu–Osher problem, and the “blast wave” problem. Additionally, the method is tested as applied to a two-dimensional problem for the quasilinear Hopf equation. It is shown on these examples that bicompact schemes with conservative limiting are significantly more accurate than hybrid bicompact schemes.

We consider quasilinear elliptic 2m-order ( m ⩾ 2 ) partial differential equations which prototype is ∑ | α | = m ( − 1 ) | α | D α ( | D m u | p − 2 D α u ) = f ( x ) , x ∈ Ω , where Ω is a bounded open set in R n , n = m p and f ∈ L 1 ( Ω ) . Using an analogue of the Kilpelainen-Malý method, we obtain the local boundedness and continuity of arbitrary weak solution u ∈ W m , p ( Ω ) via the Wolff potential W m , p f .

Assume that u ( x ) satisfies the problem L u ( x ) ≡ − ∂ ∂ x i ( a i j ∂ u ∂ x j ) = f ( x ) , ∀ x ∈ Ω , u ( x ) = 0 , ∀ x ∈ ∂ Ω . In this article, using interpolation postprocessing technique, we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u ( x ) using piecewise polynomials of degrees bi- k ( k ≥ 3 ) over a rectangular partition. Assume that k ≥ 3 is odd and x 0 is an interior vertex satisfying ρ ( x 0 , ∂ Ω ) ≥ c . Using the new interpolation postprocessing formula presented in this study, we show that the primal variable and the derivative of the post-processed finite element solution using piecewise of degrees bi- k ( k ≥ 3 ) at x 0 converge to the primal variable and the derivative of the exact solution with order O ( h k + 3 | ln h | ) under suitable regularity and mesh conditions, respectively. Finally, we use numerical experiments to illustrate our theoretical findings.

This paper presents the solution for synchronizing of perturbed Genesio–Tesi systems in finite time. A new homogeneous high order sliding mode control approach is introduced ensuring the finite-time synchronization despite perturbations in the considered systems. By combining a proposed Lyapunov function and the homogeneity concept, the proposed controller possesses superior features such as requiring only one control input signal to obtain finite-time synchronization, resulting in a continuous control signal that significantly reduces the chattering effect without an additional differentiator, and compensating a wide class of perturbations. The computer simulation results are provided to demonstrate the effectiveness of the proposed control approach, compared with two existing control schemes.

Building individual brain networks form the single volume of anatomical MRI is a challenging task. Furthermore, the high-order connectivity of morphological networks remains unexplored. This paper aimed to investigate the individual high-order morphological connectivity from anatomical MRI. Towards this goal, a unified framework based on six feature distances (euclidean, seuclidean, mahalanobis, cityblock, minkowski, and chebychev) was proposed to derive high-order interregional morphological features. The test-retest datasets and the healthy aging datasets were applied to analyze the reliability and the inter-subject variability of the novel features. In addition, the predictive models based on these novel features were established for age estimation. The proposed six neuroanatomical features exhibited significant high-to-excellent reliability. Certain connections were significantly correlated to biological age based on the six novel metrics (p .05, FDR corrected). Moreover, the predicted age were significantly correlated to the original age in each regression task (r 0.5, p 10

In this paper, meshless RBFs method is proposed to solve two-dimensional time-fractional Sobolev equations. The proposed method uses RBFs for approximation of spatial operator. Finite difference formula of O ( δ t 2 − α ) ( 0 α ≤ 1 ) is used for time-fractional derivative approximation while θ -rule ( 0 ≤ θ ≤ 1 ) as time stepping scheme for the advancement of solution in time. Validation of the proposed method is made by considering various test examples from literature. Simulated results are found in very good agreement with available exact solutions. A rigorous comparative analysis made with other methods testifies proposed method’s superiority in higher dimensions. Efficiency and accuracy of RBFs method are examined by varying number of nodes N in the domain of influence, time-step size δ t , as well as L 2 , L ∞ and L rms error norms. Linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis.

Electron paramagnetic resonance (EPR) technology had been adopted to analyze the structures and properties of Cu2+ and VO2+ in Biochar (BG1 and BG2), but the local structural properties have not been clarified up to now. In present work, three possible structures (orthorhombically, rhombically and tetragonally elongated octahedra) are proposed for Cu2+ in BG1 and BG2 samples, and the former seems most reasonable in view of the best agreement between the theoretical EPR parameters based on the perturbation formulae and the experimental data. As for the VO2+ center, unlike the previous assignment of the tetragonally elongated [VO5]6− cluster based on the simple formulae of EPR parameters, presently proposed tetragonally compressed [VO6]8− cluster and the high order perturbation formulae of the EPR parameters yields good agreement with the observed values. This point is also supported by the d-d optical absorption 2B1g →2B2g for tetragonally compressed octahedral 3 d1 systems.

Cyclodextrins (CDs) had been regarded as destructors in molecular assembly systems for a long time until CD/surfactants were found to assemble into high order structure driven by hydrogen bonding between CDs. Thereafter, intensive researches have been conducted on construction and regulation of CD–amphiphile systems. Here, we summarized the recent progress on construction and regulation of CDs and amphiphiles assembly. The scope of amphiphiles have been extended from surfactants (ionic surfactants, zwitterion surfactants, nonionic surfactants, gemini surfactant, and so on), to nontypical amphiphiles (amines, aromatic molecules, alkanes, and so on). Owing to the abundant choices of guest amphiphiles and dynamic nature of host–guest inclusive interaction, numerous regulation methods (such as enzyme, light, pH, concentration, temperature, and so on) have been used in CD–amphiphile systems. Moreover, remarks and future perspectives are also discussed at the end of this review, which is expected to stimulate progress on both mechanism and application level.

Density Functional Theory calculations using Quantum ESPRESSO under systematically increasing high pressure have been calculated and performed with difference of 10 GPa upto 200 GPa for Ruthenium Diboride (RuB2). The structural deformation and phase transformation at high pressure have also been achieved. This calculation is well optimized with high order of convergence threshold and well agrees with the previous calculated values upto 100 GPa.

We consider sliding-mode control systems subject to unmatched disturbances. Classical first-order sliding-mode techniques are capable to compensate unmatched disturbances if differentiations of the output of sufficiently high order are included in the sliding variable. For such disturbances it is commonly assumed that they do not affect the relative degree of the system. In this contribution we consider disturbances that alter the relative degree of the process and study their impact on the closed-loop control system with the classical first-order sliding-mode design. We analyse the reaching and sliding phase of the resulting closed-loop system. We show that uniqueness of the solution may be lost and derive conditions for such behaviour. We present conditions for the stability of the sliding-mode dynamics and analyse the disturbance rejection properties. A simulation case study of a two-mass spring-damper system illustrates the various closed-loop behaviours.

In this paper, a new limiter using weighted essentially non-oscillatory (WENO) methodology is investigated for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving hyperbolic conservation laws. The idea is to use the high-order DG solution polynomial itself in the target cell and the linear polynomials which are reconstructed by the cell averages of solution in the target cell and its neighboring cells to reconstruct a new high-order polynomial in a manner of WENO methodology. Since only the linear polynomials need to be prepared for reconstruction, this limiter is very simple and compact with a stencil including only the target cell and its immediate neighboring cells. Numerical examples of various problems show that the new limiting procedure can simultaneously achieve uniform high-order accuracy and sharp, non-oscillatory shock transitions.

This article addresses the problem of average consensus by a multi-agent system when the desired consensus quantity is a time varying signal, in particular the average of individual time varying signals localized at the agents. Although this problem has been addresses in existing literature by linear schemes, only bounded steady-state errors has been achieved. In this work, we propose a new exact dynamic consensus algorithm which leverages high order sliding modes to achieve zero steady-state error of the average of time varying reference signals in a group of agents. Moreover, our proposal is also able to achieve consensus to high order derivatives of the average signal, if desired. Finally, the effectiveness and advantages of our proposal are shown with concrete simulation scenarios.

It turns out that higher-order RC is useful for improving the robustness of the entire control system, and its importance is only increasing. In this paper, we propose a new high-order RC design method. The proposed high-order RC design method includes more design parameters than the same order conventional high-order RC, which increases design flexibility. Further improvement of robustness can be expected. It will also prove useful for improving time response. Further, it is possible to suppress a disturbance having a frequency different from the frequency of the target periodic signal. This is not possible with conventional high-order RC.

In this paper, we present a stability criterion for a high-order neural network with one proportional delay. Unlike most existing results about time-delayed high-order neural networks, the proportional delay mentioned here is unbounded and time-varying. The stability criterion obtained in this paper has some improvements and advantages compared with ones in some existing results, which is further explained by numerical examples. The approach proposed in this paper is also available for (high-order) neural networks with multiple proportional delays.

In this paper, numerical methods for the modified Zakharov system with high-order fractional Laplacian and a quantum correction (FMZS) are considered. A conservative linearly-implicit difference scheme for the FMZS is proposed. This scheme is shown to conserve the mass and energy in the discrete level. On the basis of some priori estimates and Sobolev norm inequalities, it is proven that the difference scheme is stable and convergent of order O ( τ 2 + h 2 ) in the maximum norm. Numerical examples are given to demonstrate the theoretical results. In particular, some complex dynamical behaviors including pattern dynamics are observed and analyzed in the numerical results.

Numerical solution of time-dependent differential equations with quasi-interpolation usually uses derivatives of a quasi-interpolant directly to approximate corresponding spatial derivatives (called the direct technique) in equations. This in turn requires that the quasi-interpolant should possess high-order derivatives for solving high-order differential equations. In addition, the resulting numerical solution usually gives a lower approximation accuracy. To circumvent these limitations, the paper proposes a new scheme for getting high-order numerical solutions of time-dependent differential equations based on quasi-interpolation. The scheme uses an iterated technique instead of the direct quasi-interpolation technique for approximating spatial derivatives. Moreover, it requires only computing the first-order derivative of the involved quasi-interpolant. Numerical examples of solving several benchmark equations using the proposed scheme are provided at the end of the paper to demonstrate these features vividly.

Copyright © 2022 The Author(s). Artificial compressibility methods intend to offer divergence-free free velocity fields at the incompressible limit for compressible solvers. Three major approaches for this are compared within a high-order flux reconstruction framework: the established method (ACM) of Chorin (1967) and a new entropically damped method (EDAC) of Clausen (2013) which can keep velocity divergence sufficiently low to be run explicitly without the non-linear solver required by ACM. Furthermore, the ACM approach with hyperbolised diffusion is investigated. The accuracy and computational efficiency of these methods is investigated for a series of turbulent test cases over a range of Reynolds numbers. It is found for EDAC that velocity divergence scales linearly with the square root of compressibility, whereas for ACM a clear relation is not observed. EDAC is found to accurately resolve the low Reynolds number Taylor–Green vortex case; however, for the circular cylinder at Reynolds number 3900, earlier transition of the free shear-layer is observed due to an over-production of the turbulence kinetic energy. This over production of turbulent kinetic energy is attributed to the increased spatial pressure gradients of the EDAC method, and similar behaviour is observed for an aerofoil at Reynolds number 60 000 with an attached transitional boundary layer. These issues were not observed for the other ACM approaches. It is concluded that hyperbolic diffusion of ACM can be beneficial in terms of convergence but at the cost of case setup time, and EDAC can be a time efficient method for unsteady incompressible flows. However, care must be taken when reducing the stiffness of EDAC as the resulting pressure fluctuations can have a significant impact on transition.

In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N, the Nth method of this family has the order of convergence 2 N + 1 . In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family.

On the basis of the previous work by Tang and Zhang [37], in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations. Instead of analyzing order conditions step by step as shown in the previous work, the new technique of this paper is using Legendre expansions to deal with the simplifying assumptions for order conditions. With the new technique, high-order symplectic integrators can be conveniently devised by truncating an orthogonal series.

International audience; High order Whitney finite element spaces generally lack natural choices of bases but they do have spanning families. In these pages, we recall such a family on simplicial meshes and we prove theoretically its effectiveness. We also comment on some aspects of a new set of degrees of freedom, the so-called weights on the small simplices, to represent discrete functions in these spaces.

This paper studies the high-order moment control problem for discrete-time Markov jump linear systems (MJLSs) with certain dynamic response performance and disturbance rejection specifications. An appropriate cumulant generating function is employed to express the original stochastic system in high-order component form. This facilities the high-order moment stabilization of MJLSs. Moreover, a pole region assignment approach is utilized to ensure desired dynamic response specifications with a certain attenuation rate. An arithmetic and geometric inequality approach is utilized to extract sufficient conditions ensuring the designed controller existence. These conditions ensure the high-order moment steady-state property and certain dynamic specifications for the MJLSs. The effectiveness of the proposed method is demonstrated through numerical and practical examples.

This paper concerns the synchronization problem of memristive high-order competitive neural networks with time-varying delay. First, a novel control scheme with a linear term and a discontinuous term is proposed. Then, based on the Lyapunov stability theory, several criteria with algebraic form or matrix form are derived to ensure global exponential synchronization of the networks by adopting some inequality techniques. Finally, two numerical examples are presented to substantiate the effectiveness of the results.

This paper investigates the time-varying formation problem for high-order multi-agent systems subject to external disturbances. The interaction topology among agents is assumed to be directed and has a directed spanning tree. A novel event-triggered integral sliding mode control strategy is proposed. It can be proved that, with the designed control law and the event-triggered condition, the desired time-varying formation will be reached. Moreover, the Zeno behavior of triggering time sequences can be avoided. Finally, a simulation example is presented to illustrate the effectiveness of the theoretical results.