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Abstract: Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form where f(x) and g(x) are continuous on [a, b], and A i and B i (i =1,2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments. [Copyright &y& Elsevier]

Abstract: In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the formThe results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations where and . On the other hand, some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations where . The results obtained here complement and correct some known results in . Some examples are given to illustrate the main results. [Copyright &y& Elsevier]

• We build a pairwise (PW) nonlinear fermi dynamics to study the rock-paper-scissors game's under evolutionary dynamics. • The model also investigates the effect of environmental noise and demographic noise. • We perform analytical analysis and numerical simulation to show the impact of linear and nonlinear to explore the noise effect. • Both linear and non-linear dynamics show comparable stability equilibria. The difference between conventional replicator dynamics and pairwise (PW) nonlinear Fermi dynamics can be discerned by studying the evolutionary dynamics of the interactions between the symmetric cyclic structure in the rock–paper–scissors game and inter- and intraspecific competitions. Often, conventional replicator models presume that the payoff difference among species is a linear function (a linear benefit). This study introduces a PW contrast under the properties of the well-known Fermi rule, where species play against one another in pairs. To model a PW nonlinear evolutionary environment (a nonlinear benefit) within this framework, both analytical and numerical approaches are applied. It is determined that the dynamics of the linear and nonlinear benefits can present the same stability conditions at equilibrium. Moreover, it is also demonstrated that, even in an identical equilibrium condition for both dynamics, the numerical result run by a deterministic approach presents a faster stability state for nonlinear benefit dynamics. This study also suggests that introducing mutation as demographic noise can effectively disrupt the phase regions and show the different relationships between linear and nonlinear dynamics. The symmetric bidirectional mutation among all the species reduced to the stable limit cycle by an arbitrary small mutation rate is also explored. Due to the environmental noise, however, linear and nonlinear exhibit the same steady state. Nevertheless, non-linearity illustrates more stable and faster stability situations. Our result suggests that environmental and demographic noise on the evolutionary dynamic framework can serve as a mechanism for supporting PW nonlinear dynamics in multi-species games. [ABSTRACT FROM AUTHOR]

In this paper we present and analyze efficient numerical schemes for computing the Bessel transforms with an irregular oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points for the general oscillator g (t) , we derive a series of new quadrature formulae for such transforms and carry out rigorous analysis for the proposed numerical methods. The error analysis demonstrates that this methods exhibit high asymptotic order, and the accuracy improves drastically with either increasing the frequency ω or adding more nodes. Compared with the existing modified Filon-type method, the established approaches show higher precision and order of the error estimate at the same computational cost. The extensive numerical examples are provided to verify the theoretical results and illustrate the efficiency and accuracy of the proposed method. • New quadrature formulae for a class of Bessel transforms with an irregular oscillator. • The accuracy improves drastically by increasing the frequency ω or adding nodes. • High precision and high order of the error estimate. [ABSTRACT FROM AUTHOR]

Among the numerous applications of the Ginzburg-Landau theory to the analysis of significant reaction-diffusion systems, modeling of the behavior of promising polar dielectric materials should be especially highlighted. The paper is devoted to the theoretical and numerical analysis of the Landau-Khalatnikov model describing the dynamics of 2D domain pattern formation in ferroelectrics. The unique solvability of the initial-boundary value problem for the system of 2D cubic-quintic Landau-Khalatnikov equations is proved. The proof is based on the derivation of new a priori estimates for the solution of the system of nonlinear parabolic equations. A series of computational experiments are conducted to examine both spontaneous and polar-induced domain pattern formation in biaxial ferroelectrics. Finite-element simulations allow us to visualize different types of ferroelectric domain structures depending on the varying boundary conditions. • The Landau-Khalatnikov model describing the dynamics of 2D domain pattern formation in ferroelectrics is studied. • A priori estimates for the solution of the initial-boundary value problem for the 2D Landau-Khalatnikov model are derived. • The unique solvability of the initial-boundary value problem for the 2D Landau-Khalatnikov model is proved. • Computational experiments to examine both spontaneous and polar-induced ferroelectric domain patterns are conducted. [ABSTRACT FROM AUTHOR]

Abstract Twenty years after the classical book of Ortega and Rheinboldt was published, five definitions for the Q -convergence orders of sequences were independently and rigorously studied (i.e., some orders characterized in terms of others), by Potra (1989), resp. Beyer, Ebanks and Qualls (1990). The relationship between all the five definitions (only partially analyzed in each of the two papers) was not subsequently followed and, moreover, the second paper slept from the readers attention. The main aim of this paper is to provide a rigorous, selfcontained, and, as much as possible, a comprehensive picture of the theoretical aspects of this topic, as the current literature has taken away the credit from authors who obtained important results long ago. Moreover, this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers, where the authors check the convergence orders of different iterative methods for solving nonlinear (systems of) equations. Tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold. [ABSTRACT FROM AUTHOR]

The paper presents a developed numerical tool in the form of a CFD code solving Reynolds-averaged Navier–Stokes equations for transonic flows of a compressible gas which is used to model the process of atmospheric air expansion in nozzles. The numerical model takes account of condensation of water vapour contained in atmospheric air. The paper presents results of numerical modelling of both homo- and heterogeneous condensation taking place as air expands in the nozzle and demonstrates the impact of the air relative humidity and pollutants on the condensation process. [ABSTRACT FROM AUTHOR]

It is shown that many systems of nonlinear differential equations of interest in various fields are naturally embedded in a new family of differential equations. In this paper, we improve new and effective methods for nonautonomous systems and they produce new exact solutions to some important biological systems. The exact solution of predator and prey population for different particular cases has been derived. The numerical examples show that new exact solutions can be obtained for many biological systems such as SIR model, Lotka–Volterra model. The methods perform extremely well in terms of efficiency and simplicity to solve this historical biological models. The Lotka–Volterra nonlinear differential equations for two competing species, namely X and Y, contain six independent parameters. Their general analytic solutions, valid for arbitrary values of the parameters, are at present unknown. However, when two or more of these parameters are interrelated, it is possible to obtain the exact solutions in the X, Y phase plane, and six cases of solvability are given in this paper. The dependence of the solutions on the parameters and the initial conditions can thus be readily investigated. [ABSTRACT FROM AUTHOR]

In this paper we consider the two-dimensional nonlocal phase-field and hyperbolic nonlocal phase-field models to obtain their numerical solutions. For this purpose, we propose a localized method based on radial basis functions (RBFs), namely localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for spatial discretization. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. This approach does not require meshing in spatial domain and hence inherits the meshless and spectral convergence properties of the global radial basis functions collocation method (GRBFCM). Some numerical results indicate that the obtained simulations via the LRBF-PSM is effective and stable for approximating the solution of nonlocal models investigated in the current paper. [ABSTRACT FROM AUTHOR]

As a type of uncertain differential equations, uncertain spring vibration equation is driven by Liu process. This paper proposes a concept of α -path, and shows that the solution of an uncertain spring vibration equation can be expressed by a family of solutions of second-order ordinary differential equations. This paper also proves that the inverse uncertainty distribution of solution of uncertain spring vibration equation is just the α -path of uncertain spring vibration equation, and a numerical algorithm is designed. Moreover, a formula to calculate the expected value of solution of uncertain spring vibration equation is derived. Finally, several numerical examples are provided to illustrate the efficiency of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper we propose a method to solve the Kadomtsev–Petviashvili equation based on splitting the linear part of the equation from the nonlinear part. The linear part is treated using FFTs, while the nonlinear part is approximated using a semi-Lagrangian discontinuous Galerkin approach of arbitrary order. We demonstrate the efficiency and accuracy of the numerical method by providing a range of numerical simulations. In particular, we find that our approach can outperform the numerical methods considered in the literature by up to a factor of five. Although we focus on the Kadomtsev–Petviashvili equation in this paper, the proposed numerical scheme can be extended to a range of related models as well. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mean square error of numerical methods for SPDEs driven by Gaussian and non-Gaussian noises. The Gaussian noise considered here is a Hilbert space valued Q -Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Lévy process. As the models consider the influences of Gaussian and non-Gaussian noises simultaneously, this makes the models more realistic when the models are also influenced by some randomly abrupt factors, but more complicated. As a consequence, the numerical analysis of the problems becomes more involved. We first study the regularity for the mild solution. Next, we propose a semidiscrete finite element scheme in space and a fully discrete linear implicit Euler scheme for the SPDEs, and rigorously obtain their error estimates. Both the regularity results of the mild solution and error estimates obtained in the paper are novel. [ABSTRACT FROM AUTHOR]

In this manuscript, we present an innovative discretization algorithm designed to address the challenges posed by the three-dimensional (3D) Bratu problem, a well-known problem characterized by non-unique solutions. Our algorithm aims to achieve exceptional precision and accuracy in determining all potential solutions, as previous studies in the literature have only managed to produce limited accurate results. Additionally, our computational scheme approximates the critical values of the transition parameter. Moreover, we establish a rigorous proof demonstrating the uniform convergence of the approximation sequence to the exact solution of the original problem, under the condition that the initial guess is sufficiently close to the true solution. This theoretical result further establishes the reliability of our algorithm. To evaluate the effectiveness of our proposed approach, we conduct extensive numerical simulations, which convincingly demonstrate its capability to accurately solve the 3D Bratu problem while effectively determining the critical values of the transition parameter. Furthermore, we investigate the bifurcated behavior of the solution by analyzing the infinity norm for various values of the transition parameter. The outcomes of our study offer a robust and efficient method for tackling the 3D Bratu problem, making a significant contribution to the field of numerical analysis of partial differential equations (PDEs) where non-unique solutions are commonly encountered. Our algorithm's ability to produce accurate results for this challenging problem showcases its potential for broader applications in diverse scientific and engineering domains. • Continuation of a series of publications on the Bratu problem. • Three-dimensional nonlinear problem. • Only two papers studied the 3D Bratu problem. • Existence of three critical values of the transition parameter. • Highly accurate solution to be considered as benchmark for future work. [ABSTRACT FROM AUTHOR]

• A detailed analysis and proof of optimality. • The construction of the optimal approximant. • Remarks on approximation of spherical rectangles. • Numerical examples. • The correction of the previously published result. In [1], the author considered the problem of the optimal approximation of symmetric surfaces by biquadratic Bézier patches. Unfortunately, the results therein are incorrect, which is shown in this paper by considering the optimal approximation of spherical squares. A detailed analysis and a numerical algorithm are given, providing the best approximant according to the (simplified) radial error, which differs from the one obtained in [1]. The sphere is then approximated by the continuous spline of two and six tensor product quadratic Bézier patches. It is further shown that the G 1 smooth spline of six patches approximating the sphere exists, but it is not a good approximation. The problem of an approximation of spherical rectangles is also addressed and numerical examples indicate that several optimal approximants might exist in some cases, making the problem extremely difficult to handle. Finally, numerical examples are provided that confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In the recent paper entitled “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli, they proposed the following new four-dimensional (4-D) quadratic autonomous hyper-chaotic system: x 1 ˙ = a ( x 2 − x 1 ) , x 2 ˙ = b x 1 − x 2 + e x 4 − x 1 x 3 , x 3 ˙ = − c x 3 + x 1 x 2 + x 1 2 , x 4 ˙ = − d x 2 , which generates double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Combining theoretical analysis and numerical simulations, they investigated some dynamical properties of that system like Lyapunov exponent spectrum, bifurcation diagram, phase portrait, Hopf bifurcation, etc. In particular, they formulated a conclusion that the system has the ellipsoidal ultimate bound by employing the method presented in the paper entitled “Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems” [Int. J. Bifurc. Chaos, 21(09) (2011), 2679–2694] by P. Wang et al. However, by means of detailed theoretical analysis, we show that both the conclusion itself and the derivation of its proof in [Appl. Math. Comput. 291 (2016) 323–339] are erroneous. Furthermore, we point out that the method adopted for studying the ultimate bound of that system is not applicable at all. Therefore, the ultimate bound estimation of that system needs further studying in future work. [ABSTRACT FROM AUTHOR]

In this paper, several methods for approximating the solution of Hammerstein equations with weakly singular kernels are considered. The paper is motivated by the results reported in papers [7, 12]. The orders of convergence of the proposed methods and those of superconvergence of the iterated methods are analyzed. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper discusses the multiscale analysis and numerical algorithms for coupled conduction, convection and radiation heat transfer problem in periodic porous materials. First, the multiscale asymptotic expansion of the solution for the coupled problem is presented, and high-order correctors are constructed. Then, error estimates and their proofs will be given on some regularity hypothesis. Finally, the corresponding finite element algorithms based on multiscale method are introduced and some numerical results are given in detail. The numerical tests demonstrate that the developed method is feasible and valid for predicting the heat transfer performance of periodic porous materials, and support the approximate convergence results proposed in this paper. [ABSTRACT FROM AUTHOR]

Solar radiation numerical models need the implementation of an accurate method for determining cast shadows on the terrain or on solar collectors. The aim of this work is the development of a new methodology to detect the shadows on a particular terrain. The paper addresses the detection of self and cast shadows produced by the orography as well as those caused by clouds. The paper presents important enhancements on the methodology proposed by the authors in previous works, to detect the shadows caused by the orography. The domain is the terrain surface discretised using an adaptive mesh of triangles. A triangle of terrain will be under cast shadows when, looking at the mesh from the Sun, you can find another triangle that covers all or partially the first one. For each time step, all the triangles should be checked to see if there are cast or self shadows on it. The computational cost of this procedure eventually resulted unaffordable when dealing with complex topography such as that in Canary Islands thus, a new methodology was developed. This one includes a filtering system to identify which triangles are those likely to be shadowed. If there are no self shadowed triangles, the entire mesh will be illuminated and there will not be any shadows. Only triangles that have their backs towards the Sun will be able to cast shadows on other triangles. Detection of shadows generated by clouds is achieved by a shadow algorithm using satellite images. In this paper, Landsat 8 images have been used. The code was done in python programming language. Finally, the outputs of both approaches, shadows generated by the topography and generated by clouds, can be combined in one map. The whole problem has been tested in Gran Canaria and Tenerife Island (Canary Islands – Spain), and in the Tatra Mountains (Poland and Slovakia). [ABSTRACT FROM AUTHOR]

This paper is devoted to numerical global optimization algorithms applying several ideas to reduce the problem dimension. Two approaches to the dimensionality reduction are considered. The first one is based on the nested optimization scheme that reduces the multidimensional problem to a family of one-dimensional subproblems connected in a recursive way. The second approach as a reduction scheme uses Peano-type space-filling curves mapping multidimensional domains onto one-dimensional intervals. In the frameworks of both the approaches, several univariate algorithms belonging to the characteristical class of optimization techniques are used for carrying out the one-dimensional optimization. Theoretical part of the paper contains a substantiation of global convergence for the considered methods. The efficiency of the compared global search methods is evaluated experimentally on the well-known GKLS test class generator used broadly for testing global optimization algorithms. Results for representative problem sets of different dimensions demonstrate a convincing advantage of the adaptive nested optimization scheme with respect to other tested methods. [ABSTRACT FROM AUTHOR]

Under the assumption of known root multiplicity m ∈ N , a triparametric family of two-point optimal quartic-order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function-to-function ratio. Special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper. In addition, we explore the relevant dynamics of proposed methods via Möbius conjugacy map when applied to a prototype polynomial ( z − a ) m ( z − b ) m . The results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes. [ABSTRACT FROM AUTHOR]

Abstract: A integro-differential equation (IDE) model of a cyclic competitive system is analyzed. Linear stability analysis are performed about the spatial constant solutions, and conditions are derived under which they are stable or unstable. Spatial patterns along with the instability of the nontrivial equilibrium are also observed. [Copyright &y& Elsevier]

In this paper, we theoretically and numerically deal with nonlinear Volterra integro-differential equations with Itô integral under a one-sided Lipschitz condition and polynomially growth conditions. It is proved that both the exact solutions and vector fields are bounded and satisfy a Hölder condition in the p th moment sense. Analogously, the boundedness and Hölder condition in the p th moment sense are preserved by the semi-implicit Euler method for sufficiently small step-size. Moreover, by the local truncated errors, we prove the strong convergence order 1. Finally, numerical simulations on stochastic control models and stochastic Ginzburg–Landau equation illustrate our results. [ABSTRACT FROM AUTHOR]

In computer aided geometric design a polynomial is usually represented in Bernstein form. This paper presents a family of compensated algorithms to accurately evaluate a polynomial in Bernstein form with floating point coefficients. The principle is to apply error-free transformations to improve the traditional de Casteljau algorithm. At each stage of computation, round-off error is passed on to first order errors, then to second order errors, and so on. After the computation has been "filtered" (K − 1) times via this process, the resulting output is as accurate as the de Casteljau algorithm performed in K times the working precision. Forward error analysis and numerical experiments illustrate the accuracy of this family of algorithms. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an alternating direction implicit (ADI) scheme of a fractional-order diffusion tensor image (DTI) registration model. Furthermore, compatibility, stability and convergence of ADI scheme are proved. Moreover, five numerical tests are also performed to show the efficiency of this scheme. [ABSTRACT FROM AUTHOR]

Abstract We consider the initial boundary value problems for a homogeneous Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain Ω. The memory kernel of the equation is made up in a complicated way from the (distinct) moduli of stress relaxation for compression and shear, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will study the second order continuous time Galerkin approximation for its space discretization and propose a fully discrete scheme employing the Crank–Nicolson method for the time discretization. Then we derive the uniform long time error estimates in the norm L t 1 (0 , ∞ ; L x 2) for the finite element solutions. Some numerical results are presented to illustrate our theoretical error bounds. [ABSTRACT FROM AUTHOR]

In this paper, the Jacobi and Gauss–Seidel-type iteration methods are proposed for solving the matrix equation A X B = C , which are based on the splitting schemes of the matrices A and B . The convergence and computational cost of these iteration methods are discussed. Furthermore, we give the preconditioned Jacobi and Gauss–Seidel-type iteration methods. Numerical examples are given to demonstrate the efficiency of these methods proposed in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose an impulsive predator–prey model with communicable disease in the prey species only and investigate its interesting biological dynamics. By the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we have deduced the sufficient conditions for the globally asymptotical stability of the semi-trivial periodic solution and the permanence of the proposed model. We also give the existences of the “infection-free” periodic solution and the “predator-free” solution. Finally, numerical results validate the effectiveness of theoretical analysis for the proposed model in this paper. [ABSTRACT FROM AUTHOR]

• Probability theory and stochastic processes: Discrete-time risk processes with surplus-dependent insurance premium corrections, which allow adjustments in premium transition rules according to the insurance company's current surplus. • Numerical analyses: Recursive calculation of finite-time ruin probabilities and finite-time Parisian ruin probabilities. This paper studies discrete-time risk models with insurance premiums adjusted according to claims experience. The premium correction mechanism follows the well-known principle in the non-life insurance industry, the so-called bonus-malus system. The bonus-malus framework that we study here extends the current literature by allowing the premium correction rules to vary according to the current surplus level of the insurance company. The main goal of this paper is to evaluate the risk of ruin for the insurer who implements the proposed bonus-malus system. Two premiums correction principles are examined: by aggregate claims or by claim frequency. Further, the Parisian type of ruin is also considered, where the premium adjustment rules are different in positive- and negative-surplus environment. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new viscosity approximation method based on the conjugate gradient method and an averaged mapping approach for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem. Under suitable conditions, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of the split variational inclusion problem and the set of solutions of the constrained convex minimization problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area. Finally, preliminary numerical results indicate the feasibility and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to prove that the implicit numerical approximation can converge to the true solution to highly nonlinear hybrid stochastic pantograph differential equation. After providing the boundedness of the exact solution, the paper proves that the backward Euler–Maruyama numerical method can preserve boundedness of moments, and the numerical approximation converges strongly to the true solution. Finally, the exponential stability criterion on the backward Euler–Maruyama scheme is given, and a high order example is provided to illustrate the main result. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to construct a numerical method to preserve positivity and mean-square dissipativity of stochastic age-dependent capital system with Poisson jumps. We use the balanced implicit numerical techniques to maintain the nonnegative path of the exact solution. It is proved that the balanced implicit method(BIM) preserves positivity and converges with order 1 2 under given conditions. In addition, some sufficient conditions are obtained for ensuring the system and the balanced implicit method(BIM) are mean-square dissipative. Finally, a numerical example is simulated to illustrate the efficiency of theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present an incremental pressure correction finite element method for the time-dependent Oldroyd flows. This method is a fully discrete projection method. As we all know, most projection methods have been studied without space discretization. Then the ensuing analysis may not extend to this case. We also give the stability analysis and the optimal error analysis. The analysis is based on a time discrete error and a spatial discrete error. In order to show the effectiveness of the method, we also present some numerical results. The numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time-dependent Oldroyd flows. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a block format of left-looking AINV preconditioner for a nonsymmetric matrix. This preconditioner has block 1 × 1 or 2 × 2 pivot entries. It is introduced based on a block format of Gaussian Elimination process which has been studied in [14]. We have applied the multilevel nested dissection reordering as the preprocessing and have compared this block preconditioner by the plain left-looking AINV preconditioner. If we mix the multilevel nested dissection by the maximum weighted matching process, then the numerical experiments indicate that the number of 2 × 2 pivot entries in the block preconditioner will grow up. In this case, the block preconditioner makes GMRES method convergent in a smaller number of iterations. In the numerical section, we have also compared the ILUT and block left-looking AINV preconditioners. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose Fourier spectral method to solve space fractional Klein–Gordon–Schrödinger equations with periodic boundary condition. First, the semi-discrete scheme is given by using Fourier spectral method in spatial direction, and conservativeness and convergence of the semi-discrete scheme are discussed. Second, the fully discrete scheme is obtained based on Crank–Nicolson/leap-frog methods in time direction. It is shown that the scheme can be decoupled, and preserves mass and energy conservation laws. It is proven that the scheme is of the accuracy O (τ 2 + N − r). Last, based on the numerical experiments, the correctness of theoretical results is verified, and the effects of the fractional orders α , β on the solitary solution behaviors are investigated. In particular, some interesting phenomena including the quantum subdiffusion are observed, and complex dynamical behaviors are shown clearly by many intuitionistic images. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the flocking problem in the nonlinear multi-agent systems with time-varying delay. By adopting the event-triggered control strategy, some sufficient conditions for the flocking problem are given. The results show that the stable flocking motion is achieved when a leader could be tracked. Finally, several numerical examples are presented to illustrate the effectiveness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Abstract The paper proposes some estimators for the population mean using robust regression methods presented in Zaman and Bulut (2018) [6] and shows that all proposed estimators more efficient than the robust ratio estimators under all conditions. This result is also supported by a numerical example. [ABSTRACT FROM AUTHOR]

Highlights • A type of extended one-leg methods are constructed for a class of nonlinear stiff fractional differential equations. • Under some suitable conditions, the extended one-leg methods are proved to be stable and convergent of order min { p , 2 − γ }. • Several interesting numerical examples are presented to illustrate the computational efficiency and accuracy of the extended one-leg methods. Abstract This paper is concerned with numerical solutions for a class of nonlinear stiff fractional differential equations (SFDEs). By combining the underlying one-leg methods with piecewise linear interpolation, a type of extended one-leg methods for nonlinear SFDEs with γ -order (0 < γ < 1) Caputo derivatives are constructed. It is proved under some suitable conditions that the extended one-leg methods are stable and convergent of order min { p , 2 − γ } , where p is the consistency order of the underlying one-leg methods. Several numerical examples are given to illustrate the computational efficiency and accuracy of the methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an M -matrix is proposed. The convergence analysis of the presented method is given. In particular, we provide a comparison theorem between preconditioned two-step modulus-based Gauss–Seidel (PTMGS) iteration method and two-step modulus-based Gauss–Seidel (TMGS) iteration method, which shows that PTMGS method improves the convergence rate of original TMGS method for linear complementarity problem. Numerical tested examples are used to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose the restarted global full orthogonalization method (Gl-FOM) and global generalized minimum residual (Gl-GMRES) method to solve the Stein-like matrix equation X + M (X) = C with M (X) = A X B , M (X) = A X ⊤ B , M (X) = A X ¯ B or M (X) = A X H B , respectively, where X is an unknown matrix to be solved. First, by using a real inner product in complex matrix spaces, a generalized global Arnoldi process is given. Then we demonstrate how to employ the restarted Gl-FOM and Gl-GMRES algorithms for solving the Stein-like matrix equation. The proposed algorithms take advantage of the shifted structure of the matrix equation and are implemented by the original coefficient matrices. Finally, some numerical examples are given to illustrate the effectiveness with comparison to some existing methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce a split-step theta Milstein (SSTM) method for n -dimensional stochastic delay differential equations (SDDEs). The exponential mean-square stability of the numerical solutions is analyzed, and in accordance with previous findings, we prove that the method is exponentially mean-square stable if the employed time-step is smaller than a given and easily computable upper bound. In particular, according to our investigation, larger time-steps can be used in the case θ ∈ (1 2 , 1 ] than in the case θ ∈ [ 0 , 1 2 ]. Numerical results are presented which reveal that the SSTM method is conditionally mean-square stable and that in the case θ ∈ (1 2 , 1 ] the interval of time-steps for which the SSTM method is theoretically shown to be mean-square stable is significantly larger than in the case θ ∈ [ 0 , 1 2 ]. It is worth mentioning that the SSTM method has never been employed or analyzed for the numerical approximation of SDDEs, at least to the very best of our knowledge. [ABSTRACT FROM AUTHOR]

Highlights • Employing extended feedback policy to effectually control bifurcation of the fractional delayed predator–prey model. • Effects of extended feedback delay on bifurcation control are numerically explored. • Impact of the single order on the devised control strategy is thoroughly researched. • The bifurcation diagrams are nicely plotted. Abstract This paper aims at controlling bifurcation of a fractional predator–prey system by an original extended delayed feedback controller. Firstly, some sufficient conditions of delay-induced bifurcations for such uncontrolled system are captured regarding time delay as a bifurcation parameter. Secondly, a generalised delayed feedback controller is subtly designed to control Hopf bifurcation for the proposed system. It suggests that bifurcation dynamics can be controlled efficaciously for such system by carefully adjusting extended feedback delay or fractional order so long as the other parameters are established. Thirdly, the bifurcation diagrams are meticulously plotted. The obtained results consumedly popularize the previous studies concerning bifurcation control of delayed fractional-order systems. To underline the effectiveness of the proposed control scheme, some numerical simulations are ultimately addressed. [ABSTRACT FROM AUTHOR]

Abstract Research on the theories and efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years. In this paper, we consider numerical solution of M-matrix algebraic Riccati equation and propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. Convergence of the MALI method is proved by choosing proper parameters for the nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the MALI method is effective and efficient in some cases. [ABSTRACT FROM AUTHOR]

Abstract Let S ⊆ V (G) and κ G (S) denote the maximum number k of edge-disjoint trees T 1 , T 2 , ... , T k in G such that V (T i) ⋂ V (T j) = S for any i , j ∈ { 1 , 2 , ... , k } and i ≠ j. For an integer r with 2 ≤ r ≤ n , the generalized r-connectivity of a graph G is defined as κ r (G) = min { κ G (S) | S ⊆ V (G) and | S | = r }. The parameter is a generalization of traditional connectivity. So far, almost all known results of κ r (G) are about regular graphs and r = 3. In this paper, we focus on κ r (EH (s, t)) of the exchanged hypercube for r = 4 , where the exchanged hypercube EH (s, t) is not regular if s ≠ t. We show that κ 4 (E H (s , t)) = m i n { s , t } for min { s, t } ≥ 3. As a corollary, we obtain that κ 3 (E H (s , t)) = m i n { s , t } for min { s, t } ≥ 3. [ABSTRACT FROM AUTHOR]

Abstract In this paper, spectral methods based on conformal mappings are proposed to solve the Steklov eigenvalue problem and its related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use a gradient ascent approach to find the optimal domain which maximizes k th Steklov eigenvalue with a fixed area for a given k. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different k. [ABSTRACT FROM AUTHOR]

Abstract This paper contains results on the backward Euler method for a class of neutral stochastic differential equations with both unbounded and bounded delays and Markovian switching. The convergence in probability of the backward Euler method is proved under nonlinear growth conditions including the one-sided Lipschitz condition in order for the backward Euler method to be well defined. The presence of the neutral term, which is hybrid, that is, it depends on the Markov chain, is essential for consideration of these equations. It is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable without the linear growth condition on the drift coefficient. [ABSTRACT FROM AUTHOR]

Abstract Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. They have been used in a wide range of applications such as biology, medicine and engineering and play an important role in the theory of random dynamical systems. RODEs can be investigated pathwise as deterministic ODEs, however, the classical numerical methods for ODEs do not attain original order of convergence because the stochastic process has at most Hölder continuous sample paths and the resulting vector is also at most Hölder continuous in time. Recently, Jenzen & Kloeden derived new class of numerical methods for RODEs using integral versions of implicit Taylor-like expansions and developed arbitrary higher order schemes for RODEs. Their idea can be applied to random ordinary delay differential equations (RODDEs) by implementing Taylor-like expansions in the corresponding delay term. In this paper, numerical methods for RODDEs are systematically constructed based on Taylor-like expansions and they are applied to virus dynamics model with random fluctuations and time delay. [ABSTRACT FROM AUTHOR]

Abstract The current flow betweenness centrality is a useful tool to estimate traffic status in spatial networks and, in general, to measure the intermediation of nodes in networks where the transition between them takes place in a random way. The main drawback of this centrality is its high computational cost, especially for very large networks, as it is the case of urban networks. In this paper, a new approach to the current flow betweenness centrality for its practical application in urban networks with data is presented and discussed. The new centrality measure allows the estimation of pedestrian flow developed in urban networks, taking into account both the network topology and its associated data. In addition, its computational cost makes it suitable for application in networks with a large number of nodes. Some examples are studied in order to better understand the characteristics and behaviour of the proposed centrality in the context of the city. [ABSTRACT FROM AUTHOR]

Abstract This paper presents an optimal control problem to analyze the efficacy of counter-terrorism tactics. We present an algorithm that efficiently combines the Minimum Principle of Pontryagin, the shooting method and the cyclic descent of coordinates. We also present a result that allows us to know a priori the steady state solutions. Using this technique we are able to choose parameters that reach a specific solution, of which there are two. Numerical examples are presented to illustrate the possibilities of the method. Finally, we study the sufficient conditions for optimality and suggest an improvement on the functional which also guarantees local optimality. [ABSTRACT FROM AUTHOR]