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In the current study, an efficient iterative scheme which is based on the two powerful tools called the quasilinearization method and the Jacobi-Galerkin method is proposed to solve a type of functional Volterra integral equations. According to the iterative structure of the proposed method, an error analysis of the method is provided. This method can overcome the difficulty of solving nonlinear system of algebraic equations associated with the Galerkin method in treating nonlinear functional Volterra integral equations directly. Moreover, under sufficient conditions, the suggested method has spectral accuracy. In the end, several numerical experiments are reported to confirm the efficiency and accuracy of the proposed scheme. • A family of strong nonlinear functional Volterra integral equations investigated. • The method is a combination of the quasilinearization and the Galerkin methods. • The convergence analysis of the numerical method is studied in details. • Applicability and accuracy of the method are checked on some examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating the stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established, which shows the second-order accuracy. Finally, the numerical experiments are presented confirming the accuracy of the proposed scheme. • In this paper, a new approach to solve numerically neutral Volterra integro-differential equation (NVIDE) has been considered. • The last works related to numerical analysis of NVIDE were concerned with collocation method, Legendre spectral method and etc. But the most important feature of our study is a finite difference scheme has been constructed on a uniform mesh. • Using composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points, we construct the finite difference scheme which have a second-order accuracy. • Therefore we obtain numerical methods which are as high accuracy as possible. [ABSTRACT FROM AUTHOR]

In this paper, the cost function of a single-server queueing system with Poisson input stream and Erlang- k service time will be analyzed. Treated as a function of the traffic intensity ρ , with respect to some known constants, we will show that its stationary points are solutions of a fourth-degree polynomial equation with real coefficients. Moreover, an explicit form of these solutions is given and it is shown the function reaches a minimum value at some of these points. For illustration, a numerical analysis of the cost function is carried out by changing the values of the costs, which are changed according to the principle of arithmetic progression. Also, a statistical analysis of the relationship between optimal solutions ρ ⁎ and Φ (ρ ⁎) is done. • A cost function is formed for single-server queueing system with Poisson input stream and Erlang-k service time. • Stationary points of the proposed cost function are solutions of fourth-degree polynomial equation with real coefficients. • The sensitivity of the parameters is tested according to the principle of arithmetic progression. • A statistical analysis of the optimal solutions is made and an economic interpretation of the results is given. [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]

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]

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]

• 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 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]

Abstract A multiscale coupled model is developed to simulate the compressible fluid flow in fractured media aquifer system, where the flow is governed by Forchheimer's law in the fracture and continuum porous medium. Due to the fact that the thickness of fracture is much smaller than characteristic diameters of surrounding porous medium, the fracture is reduced to a lower dimensional interface and a more complicated transmission condition is derived on the fracture-interface. The coupled model is numerically solved by the finite difference method with an implicit iteration procedure. The fewest nodal points are used to construct the optimal scheme for approximating the multiscale Forchheimer–Forchheimer coupled model. Different degrees of freedom are located on both sides of fracture-interface in order to capture the jump of velocity. Second-order error estimates in discrete norms are derived on nonuniform staggered grids for both pressure and velocity. The proposed scheme can also be extended to high dimensional model without loss of accuracy. Numerical experiments are performed to demonstrate the efficiency and accuracy of the numerical method. [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 For n × n complex matrices A, C and H , where H is non-singular Hermitian, the Krein space C -numerical range of A induced by H is the subset of the complex plane given by { Tr (C U [ * ] A U) : U − 1 = U [ * ] } with U [ * ] = H − 1 U * H the H -adjoint matrix of U. We revisit several results on the geometry of Krein space C -numerical range of A and in particular we obtain a condition for the Krein space C -numerical range to be a subset of the real line. [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 An analytical method to obtain the exact solution of q -fractional order differential equations was presented in article Koca (2015) which shows that: "Solving a q -fractional order differential equation is equivalent to solve a q -integer order differential equation". In this article, we first show that the analytical method given in [3] is wrong. Then, we propose analytical and numerical methods for the Caputo type q -fractional differential equations by using the q -beta function. Some examples are given to disconfirm the solving method in [3] and valid the effectiveness of our methods. [ABSTRACT FROM AUTHOR]

Abstract This article describes a numerical scheme to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method estimates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin method results from the numerical integration of all integrals associated with the scheme. In the current work, we employ the composite Gauss-Legendre integration rule to approximate the integrals appearing in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The algorithm of the described scheme is computationally attractive and easy to implement on computers. The error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates. [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 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 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 A numerical approximation is developed, analyzed, and investigated for quenching solutions to a degenerate Kawarada problem with a left and right Riemann-Liouville fractional Laplacian over a finite one dimensional domain. The numerical analysis provides criterion for the numerical approximations to be monotonic, nonnegative, and linearly stable throughout the computation. The numerical algorithm is used to develop an experimental scaling law relating the critical quenching domain size to the order of fractional derivative. Additional experiments indicate that imbalanced left and right derivative transport coefficients can attenuate or prevent quenching from occurring. [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 work, we present a new implicit numerical scheme for fractional subdiffusion equations. In this approach we use the Keller Box method [1] to spatially discretise the fractional subdiffusion equation and we use a modified L1 scheme (ML1), similar to the L1 scheme originally developed by Oldham and Spanier [2], to approximate the fractional derivative. The stability of the proposed method was investigated by using Von-Neumann stability analysis. We have proved the method is unconditionally stable when 0 < λ q < min (1 μ 0 , 2 γ) and 0 < γ ≤ 1, and demonstrated the method is also stable numerically in the case 1 μ 0 < λ q ≤ 2 γ and log 3 2 ≤ γ ≤ 1. The accuracy and convergence of the scheme was also investigated and found to be of order O (Δ t 1 + γ) in time and O (Δx 2) in space. To confirm the accuracy and stability of the proposed method we provide three examples with one including a linear reaction term. [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]

Abstract We present a rigorous numerical method for location of simple zeros of a system of two analytic functions in a rectangular cuboid domain based on the logarithmic integral. We compare this to a simpler, also rigorous, method based on bisection. The latter is determined to be more efficient in the examples considered. This is mainly due to inefficient methods for computing the logarithmic integral occurring in the former method. [ABSTRACT FROM AUTHOR]

Abstract Superconvergence analysis of nonconforming finite element methods (FEMs) are discussed for solving the second order variational inequality problem with displacement obstacle. The elements employed have a common typical character, i.e., the consistency error can reach order O (h 3 / 2 − ɛ ) , nearly 1/2 order higher than their interpolation error when the exact solution of the considered problem belongs to H 5 / 2 − ɛ (Ω) for any ε > 0. By making full use of special properties of the element's interpolations and Bramble–Hilbert lemma, the superconvergence error estimates of order O (h 3 / 2 − ɛ ) in the broken H 1-norm are derived. Finally, some numerical results are provided to confirm the theoretical results. [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 Structure based partitioning of a system of ordinary differential equations is considered. A general form of the explicit multischeme Runge–Kutta type method for such systems is presented. Order conditions and simplifying conditions are written down. An algorithm of derivation of the sixth order method with seven stages and reuse with two free parameters is given. It embeds a fourth order error estimator. Numerical comparison to the Dormand–Prince method with the same computation cost but of lower order is performed. [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 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]

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 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 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]

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 The impact of multiple time delays on the dynamics of two disc dynamos with viscous friction is studied in this paper. We consider the stability of equilibrium states for different delay values, and determine the location of relevant Hopf bifurcations using the normal form method and the center manifold theory. By performing numerical calculations and analysis, we verify the validity of our analytically obtained results. Our research results reveal a classical period-doubling route towards deterministic chaos in the studied system, and play an important role for the better understanding of the complex dynamics of two disc dynamos with viscous friction subject to multiple time delays. [ABSTRACT FROM AUTHOR]

Abstract In this work, we adapt the implicity linear collocation method for solving nonlinear two dimensional mixed Volterra–Fredholm integral equations of Hammerstein type by using IMQ-RBFs on a non-rectangular domain. The proposed method is mesh-free, and it is independent of the geometry of domain. Convergence analysis of the proposed method together with some benchmark examples is provided which support its reliability and numerical stability. [ABSTRACT FROM AUTHOR]

Abstract For transient analysis of structural dynamics with reduced order model (ROM), data recovery procedures that use modal methods such as the classical mode-acceleration (MA) and modal-displacement (MD) methods are an important step in order to increase the convergence and accuracy of the solution. In this work, we propose novel MA and MD methods for highly accurate transient analysis with a reduced order model based on enhanced Craig–Bampton (ECB) formulation, which is an extension of the classical Craig–Bampton (CB) method that is a way to reduce the size of a finite element (FE) model. The performance of the proposed data recovery approach is demonstrated with two numerical examples. We also investigate the relation between the proposed and classical MA and MD methods. [ABSTRACT FROM AUTHOR]

Abstract This contribution (part two) focuses on numerical implementation and efficiency aspects of an extended family of nonconforming quasi-Wilson elements type. Such a class of nonconforming elements has been introduced recently in Achchab et al. (2015). Here, based on a rectangular mesh, it is used for the approximate solution of a planar elasticity problem. It is shown that this family passes the patch-test, and that by suitably adding some orthogonality conditions, on a general class of enrichment functions, we can derive higher order consistency error estimates. Our general theoretical results, see Theorems 4.1 and 4.2, unify, simplify and extend a number of existing works on the improvement of the order of consistency error. Numerical experiments are carried out to demonstrate that our method is optimal for various Lamé parameter μ , shear modulus λ and locking free when the Poisson parameter ν approaches close to 0.5. [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]

Abstract Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. The paper presents a meshless method based on spatial trial spaces spanned by the radial basis functions (RBFs) for the numerical solution of a class of initial-boundary value fractional diffusion equations with variable coefficients on a finite domain. The space fractional derivatives are defined by using Riemann–Liouville fractional derivative. We first provide Riemann–Liouville fractional derivatives for the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matérn and Thin-plate splines, in one dimension. The time-dependent fractional diffusion equation is discretized in space with the RBF collocation method and the remaining system of ordinary differential equations (ODEs) is advanced in time with an ODE method using a method of lines approach. Some numerical results are given in order to demonstrate the efficiency and accuracy of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method. The stability of the linear systems arising from discretizing Riemann–Liouville fractional differential operator with RBFs is also analysed. [ABSTRACT FROM AUTHOR]

Abstract We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class of partial differential equations of fractional order. We demonstrate the applicability of the method by implementing it to solve a model fractional problem. [ABSTRACT FROM AUTHOR]

A double Roman dominating function of a graph G is a labeling f : V ( G ) → {0, 1, 2, 3} such that if f ( v ) = 0 , then the vertex v must have at least two neighbors labeled 2 under f or one neighbor with f ( w ) = 3 , and if f ( v ) = 1 , then v must have at least one neighbor with f ( w ) ≥ 2. The double Roman domination number γ dR ( G ) of G is the minimum value of Σ v  ∈  V ( G ) f ( v ) over such functions. In this paper, we firstly give some bounds of the double Roman domination numbers of graphs with given minimum degree and graphs of diameter 2, and further we get that the double Roman domination numbers of almost all graphs are at most n . Then we obtain sharp upper and lower bounds for γ d R ( G ) + γ d R ( G ¯ ) . Moreover, a linear time algorithm for the double Roman domination number of a cograph is given and a characterization of the double Roman cographs is provided. Those results partially answer two open problems posed by Beeler et al. (2016). [ABSTRACT FROM AUTHOR]

Numerical performance of CFD codes in convection dominated flows where in diffusion is feeble, rely majorly on the numerical scheme. In such cases, numerical schemes play a very important role in the prediction of flow property, depending upon their order. To overcome this inadequacy, a new numerical scheme is presented which is in context of finite volume method. The proposed scheme possesses high accuracy as it depends on the contribution from far-off neighboring grid points and thus advects better flow property than lower-order schemes. Furthermore, the scheme is formulated based on the symmetric limiter property and minimized downwind effect, which ensures a better convergence than the existing higher-order schemes. The presented scheme is henceforth named as High Resolution Equi-Gradient (HREG) scheme. Pure advection tests are conducted to assess the performance of this scheme with the existing schemes. Results clearly suggest that the HREG scheme advects accurate and symmetric profiles. Further , to assess the convergence behaviour of the proposed scheme, incompressible Newtonian and non-Newtonian fluids are being studied. Results generated strongly depict that the HREG scheme can be used as a substitute over existing schemes for studying highly convective flows. [ABSTRACT FROM AUTHOR]

In this paper, the existence of periodic solutions are obtained for a class of non-autonomous second-order p -Laplacian systems by the least action principle. [ABSTRACT FROM AUTHOR]

In this article, Ritz approximation have been employed to obtain numerical solutions of fractional partial differential equations (FPDEs) based on the Caputo fractional derivative. Transforming fractional partial differential equations into optimization problem and using polynomial basis functions, we obtain the system of algebraic equation. Then, we solve the system of nonlinear algebraic equation using Mathematica7 and we have the coefficients of polynomial expansion. We extensively discuss the convergence of the method. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. [ABSTRACT FROM AUTHOR]

This paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one-sided Lipschitz drift. Strong convergence rate is revealed under a global one-sided Lipschitz condition, while for a local one-sided Lipschitz condition, the tamed theta scheme is modified to ensure the well-posedness of implicit numerical schemes, then we show the convergence of the numerical solutions. [ABSTRACT FROM AUTHOR]