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The aim of this article is to present a penalty finite element method (PFEM) in fully discrete form for the unsteady incompressible magnetohydrodynamic (MHD) equations. The proposed method is applied to address the incompressible constraint "div v =0". The backward Euler scheme is used for temporal discretization, and the (P 1 b , P 1 , P 1) finite element pair is used for spatial discretization, which satisfies the discrete LBB condition. Moreover, rigorous analysis of the optimal error estimate for the fully discrete PFEM is provided, which depends on penalty parameter ϵ , the mesh size h and the time step size Δ t. Finally, some benchmark numerical experiments which include the hydromagnetic Kelvin-Helmholtz instability, flow around a cylinder and lid driven cavity flow, are carried out to illustrate the theoretical results and the effectiveness of the our proposed method. • We propose a first order penalty decoupled method. • The saddle point problem is transformed into two small problems to solve. • The optimal error estimate for the fully discrete penalty finite element method is derived. • The results of theoretical analysis are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

• Spectral approximation method for variable coefficient fractional diffusion equation. • A priori error estimates for the approximation scheme. • Numerical experiments support the derived theoretical results. In this article we consider the approximation of a variable coefficient (two-sided) fractional diffusion equation (FDE), having unknown u. By introducing an intermediate unknown, q , the variable coefficient FDE is rewritten as a lower order, constant coefficient FDE. A spectral approximation scheme, using Jacobi polynomials, is presented for the approximation of q, q N. The approximate solution to u, u N , is obtained by post processing q N. An a priori error analysis is given for (q − q N) and (u − u N). Two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates. [ABSTRACT FROM AUTHOR]

In this study, the parametric cubic spline scheme is implemented to find the approximate solution of the coupled nonlinear Schrödinger equations. This scheme is based on the Crank–Nicolson method in time and parametric cubic spline functions in space. The error analysis and stability of the scheme are investigated and the numerical results show that we can get different precision schemes by choosing suitably parameter values and this scheme is unconditionally stable. Two problems are solved to illustrate the efficiency of the methods as well as to compare with other methods. [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, the consistency of the smoothed particle hydrodynamics (SPH) interpolation formula is investigated by analytical means. A novel error analysis is developed in n -dimensional space using the Poisson summation formula, which enables the simultaneous treatment of both the kernel and particle approximation errors for arbitrary particle distributions. New consistency integral relations are derived for the particle approximation, which correspond to the cosine Fourier transform of the kernel consistency conditions. The functional dependence of the error bounds on the SPH interpolation parameters, namely the smoothing length, h , and the number of particles within the kernel support, N , is demonstrated explicitly from which consistency conditions arise. As N → ∞ , the particle approximation converges to the kernel approximation independently of h provided that the particle mass scales with h as m ∝ hβ with β > n , where n is the spatial dimension. This implies that as h → 0, the joint limit m → 0, N → ∞ , and N → ∞ is necessary for complete convergence to the continuum, where N is the total number of particles. The analysis also reveals a dominant error term of the form (ln N) n / N for finite N , as it has long been conjectured based on the similarity between the SPH and the quasi-Monte Carlo estimates. When N ≫ 1 , the error of the SPH interpolant decays as N − 1 independently of the dimension. This ensures approximate partition of unity of the kernel volume. [ABSTRACT FROM AUTHOR]

Abstract In the recent paper, "Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis [Applied Mathematics and Computation 265 (2015) 304–312]", The authors have approximated the solutions of nonlinear Fredholm integral equations (NFIEs) of the second type by using the successive approximations method. In any stage, the rationalized Haar wavelets (RHWs) and the corresponding operational matrices were applied to approximate the integral operator. In Theorem 4.2, page 307 of the reference [4], the authors introduced an upper bound for the error and explicitly stated that the rate of convergence of u i to u is O (qi), in which i is the number of iterations, q is the contraction constant, u is the exact solution, and u i is the approximate solution in i th iteration. This statement is not true and we prove carefully that the rate of convergence will be O (iqi). [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 Based on the finite difference scheme in temporal and the direct discontinuous Galerkin (DDG) method in spatial, a fully discrete DDG scheme is first proposed to solve the two-dimensional fractional diffusion-wave equation with Caputo derivative of order 1 < α < 2. The proposed scheme is unconditional stable, and the spatial global convergence and the temporal convergence order of O (Δ t + h k + 1) is derived in L 2 norm with Pk polynomial. Numerical experiments are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a new numerical approach for a quad-curl model problem which arises in the inverse electromagnetic scattering problems and magnetohydrodynamics (MHD). We first split the quad-curl problem with homogeneous boundary conditions into a system of second order equations, and then apply a mixed finite element method to solve the resulting system. The perturbed mixed finite element method is constructed by using edge elements. The well posedness of the numerical scheme is derived. The optimal error estimates in H (curl) and L 2 norms for the primal and auxiliary variables are obtained, respectively. The theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract In this work, the solution of the Navier–Stokes equations (NSE) is addressed by a Finite Element (FE) Local Projection Stabilization (LPS) method combined with an efficient time-splitting approximation strategy. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an alternative interpolation-stabilized structure. The main contribution is on numerically analyzing for the cited LPS method the proposed time approximation via stable velocity–pressure segregation, using semi-implicit Backward Differentiation Formulas (BDF). An overview about theoretical results from the numerical analysis of the proposed method is given, by highlighting the used mathematical tools. Numerical studies support the analytical results and illustrate the potential of the method for the efficient and accurate simulation of turbulent flows on relatively coarse grids. Smooth unsteady flows are simulated with optimal order of accuracy. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the problem of robust fault estimation observer design for continuous-time switched systems subject to unknown input is investigated. Firstly, based on the average dwell-time method and the switched Lyapunov function technique, an unknown input observer is designed for the augmented switched system by eliminating the unknown input, which can guarantee that the error system is globally uniformly asymptotically stable with a prescribed H ∞ performance index. Next, the method is extended to the measurement disturbances case. Finally, an example is presented to illustrate the design and efficiency of the proposed fault estimation observer technique. [ABSTRACT FROM AUTHOR]

Abstract In the present paper, two different modifications are proposed for (p, q)-Szász-Mirakjan-Kantorovich operators which preserve some test functions. Some approximation results with the help of better-known Korovkin's theorem and weighted Korovkin's theorem for these operators are presented. Furthermore, convergence properties in terms of modulus of continuity and class of Lipschitz functions are studied. It has been shown that for a given absolute error bound, King type modified (p, q)-Szász-Mirakjan-Kantorovich operators require lesser value of m and elapsed time within some subintervals. Further for comparisons, some graphics and error estimation tables are presented using MATLAB(R2018a). [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the master-slave synchronization in a class of neutral neural networks with both leakage and distributed delays. Firstly, based on state feedback controller, the closed-loop error system can be obtained. Then by fully utilizing the information and internal relationship of time-delays, three augmented Lyapunov–Krasovskii functionals (LKFs) are constructed, in which infinite distributed delay and finite one are respectively studied. Especially, some Wirtinger-based inequalities are exploited and an extended reciprocal combination technique (ERCT) is proposed, which can reconsider the previously ignored information. Sufficient conditions are formulated in terms of linear matrix inequalities (LMIs) and can be easily tested. Finally, three numerical examples are given to illustrate the designed schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the error and stability analysis of the method of fundamental solution (MFS) is explored for biharmonic equations. The bounds of errors are derived for the fundamental solutions r 2ln r in bounded simply-connected domains, and the polynomial convergence rates are obtained for certain smooth solutions. The bounds of condition number are also derived to show the exponential growth rates for disk domains. Numerical experiments are carried out to support the above analysis, which is the first time to provide the rigorous analysis of the MFS using r 2ln r for biharmonic equations. [ABSTRACT FROM AUTHOR]

In this article, we obtain explicit approximations of the modified error function introduced in Cho and Sunderland (1974), as part of a Stefan problem with a temperature-dependent thermal conductivity. This function depends on a parameter δ , which is related to the thermal conductivity in the original phase-change process. We propose a method to obtain approximations, which is based on the assumption that the modified error function admits a power series representation in δ . Accurate approximations are obtained through functions involving error and exponential functions only. For the special case in which δ assumes small positive values, we show that the modified error function presents some characteristic features of the classical error function, such as monotony, concavity, and boundedness. Moreover, we prove that the modified error function converges to the classical one when δ goes to zero. [ABSTRACT FROM AUTHOR]

In this paper, numerical techniques are presented for solving initial value problems of nonlinear fractional differential equations. The method is implemented by applying alternative Legendre polynomials. The operational matrix of fractional integration and the product for the alternative Legendre polynomials are derived in order to transform the nonlinear equations into a system of algebraic equations. The study of the error analysis of the obtained method is also considered. Furthermore, numerical examples demonstrate that this method is applicable and accurate. [ABSTRACT FROM AUTHOR]

We present a new finite-difference predictor-corrector method (L1 - PCM) to solve nonlinear fractional differential equations (FDEs) along with its error and stability analysis. The method is further extended for systems of FDEs. The proposed method is applied to fractional version of chaotic system introduced by Bhalekar and Daftardar-Gejji to explore its rich dynamics. The proposed method is accurate, time-efficient and performs well even for very small values of the order of the derivatives. [ABSTRACT FROM AUTHOR]

A local projection stabilization (LPS) method in space is considered to approximate the evolutionary Oseen equations. Optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous-in-time case for both the velocity and pressure approximation. In addition, the fully discrete case in combination with higher order continuous Galerkin–Petrov (cGP) methods is studied. Error estimates of order k + 1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms. [ABSTRACT FROM AUTHOR]

Two efficient splitting schemes are developed for 3D Maxwell’s equations. The schemes are energy-preserving and unconditionally stable, while being implemented explicitly. Rigorous optimal error estimates are established for the proposed schemes, and especially the constant in the error estimates is only O ( T ) . Numerical results confirm the theoretical analysis, and numerical comparison with some existing methods shows the good performance of the present schemes. [ABSTRACT FROM AUTHOR]

In this paper, the problem of finding numerical solution for a system of Cauchy type singular integral equations of first kind with index zero is considered. The analytic solution of such system is known. But it is of limited use as it is a nontrivial task to use it practically due to the presence of singularity in the known solution itself. Therefore, a residual based Galerkin method is proposed with Legendre polynomials as basis functions to find its numerical solution. The proposed method converts the system of Cauchy type singular integral equations into a system of linear algebraic equations which can be solved easily. Further, Hadamard conditions of well-posedness are established for system of Cauchy singular integral equations as well as for system of linear algebraic equations which is obtained as a result of approximation of system of singular integral equations with Cauchy kernel. The theoretical error bound is derived which can be used to obtain any desired accuracy in the approximate solution of system of Cauchy singular integral equations. The derived theoretical error bound is also validated with the help of numerical examples. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to use the Legendre collocation spectral method for solving the high-order linear Volterra–Fredholm integro-differential equations under the mixed conditions. Avoiding integration of both sides of the equation, we expressed mixed conditions as equivalent integral equations, by adding the neutral term to the equation. Error analysis for approximate solution and approximate derivatives up to order k of the solution is obtained in both L 2 norm and L ∞ norm. To illustrate the accuracy of the spectral method, some numerical examples are presented. [ABSTRACT FROM AUTHOR]

An element-free Galerkin (EFG) method for solving two-dimensional hyperbolic problems is derived. The proposed method is based on the weak form of the EFG method and the moving least square (MLS) approximation. Error analysis shows that the error between the numerical and exact solutions are directly related to the radius of the weight functions which influence the space and time steps. Three numerical examples are given to validate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The Sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis. Even for other functions, the Sinc approximation works accurately when combined with a proper variable transformation. The convergence rate has been analyzed for typical cases including finite, semi-infinite, and infinite intervals. Recently, for verified numerical computations, a more explicit, “computable” error bound has been given in the case of a finite interval. In this paper, such explicit error bounds are derived for other cases. [ABSTRACT FROM AUTHOR]

Based on Jacobi polynomials, an operational method is proposed to solve the generalized Abel’s integral equations (a class of singular integral equations). These equations appear in various fields of science such as physics, astrophysics, solid mechanics, scattering theory, spectroscopy, stereology, elasticity theory, and plasma physics. To solve the Abel’s singular integral equations, a fast algorithm is used for simplifying the problem under study. The Laplace transform and Jacobi collocation methods are merged, and thus, a novel approach is presented. Some theorems are given and established to theoretically support the computational simplifications which reduce costs. Also, a new procedure for estimating the absolute error of the proposed method is introduced. In order to show the efficiency and accuracy of the proposed method some numerical results are provided. It is found that the proposed method has lesser computational size compared to other common methods, such as Adomian decomposition, Homotopy perturbation, Block-Pulse function, mid-point, trapezoidal quadrature, and product-integration. It is further found that the absolute errors are almost constant in the studied interval. [ABSTRACT FROM AUTHOR]

Reaction–diffusion models with time delay have been widely applied in population biology as well as epidemiology. This type of models can possibly exhibit complex dynamical behaviors such as traveling wave, self-organized spatial pattern, or chaos. Numerical methods play an essential role in the study of these dynamical behaviors. This paper concerns the finite element approximation for reaction–diffusion models with time delay. Two fully discrete schemes and corresponding a priori error estimates are derived. Generally, the research on evolution of population and epidemic needs to survey long-time dynamical behaviors of these models, so that it is important to improve the speed of numerical simulation. To this end, interpolation technique is used in our schemes to avoid numerical integration of reaction term. An outstanding advantage of using interpolation of reaction term is that it improves the operation speed greatly, meanwhile does not reduce convergence order. Applications are given to some model problems arising from population biology and epidemiology. From these simulations some interesting phenomena can be found and we try to explain them in biological significance. [ABSTRACT FROM AUTHOR]

We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either relative or absolute errors. We show that the value of the magic constant can depend on the number of Newton–Raphson iterations. We present results for one and two iterations. [ABSTRACT FROM AUTHOR]

Combining the simplified interpolating moving least-squares (IMLS) method with the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method is presented for the numerical study of the regularized long wave (RLW) equation. The trial function in the IMLS method has one less unknown coefficient than in the traditional MLS approximation, and the shape functions satisfy the interpolating property at nodes. Then the IEFG method can apply the essential boundary conditions naturally and directly, and has high computational accuracy. And then this paper studies the error estimates of the IEFG method for RLW equation. The error estimate of the semi-discrete IEFG method for the RLW equation on the influence domain radius is first studied, and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied. The theoretical results show that the IEFG method for RLW equation has good convergence rate. For the purpose of demonstration, three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of quantized feedback stabilization for linear systems. In the controlled systems, the measurement output and control input signals are transmitted via the digital communication link, and the quantization errors are treated as sector bound uncertainties. Two different approaches to designing output feedback are developed and the corresponding design conditions in terms of solutions to linear matrix inequalities (LMIs) are presented. The resulting design is such that the homologous closed-loop system is asymptotically stable with respect to the quantization effects. Finally, we illustrate the efficiency of our main results by a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we consider the posteriori error estimates of Galerkin finite element method for the unsteady Navier–Stokes equations. By constructing the approximate Navier–Stokes reconstructions, the errors of velocity and pressure are split into two parts. For the estimates of time part, the energy method and other skills are used, for the estimates of spatial part, the well-developed theoretical analysis of posteriori error estimates for the elliptic problem can be adopted. More important, the error estimates of time part can be controlled by the estimates of spatial part. As a consequence, the posteriori error estimates in L ∞ (0, T ; L 2 (Ω)),  L ∞ (0, T ; H 1 (Ω)) and L 2 (0, T ; L 2 (Ω)) norms for velocity and pressure are derived in both spatial discrete and time-space fully discrete schemes. [ABSTRACT FROM AUTHOR]

We design and analyze the interior penalty discontinuous Galerkin discretization of the magnetic induction equation with resistivity. The resulting semi-discrete scheme is shown to be energy stable and consistent. Numerical experiments are performed in order to demonstrate the accuracy and convergence of the DG scheme through the L 2 -error and divergence error analysis by incorporating several time discretization schemes. [ABSTRACT FROM AUTHOR]

The compensated quotient-difference ( Compqd ) algorithm is proposed along with some applications. The main motivation is based on the fact that the standard quotient-difference ( qd ) algorithm can be numerically unstable. The Compqd algorithm is obtained by applying error-free transformations to improve the traditional qd algorithm. We study in detail the error analysis of the qd and Compqd algorithms and we introduce new condition numbers so that the relative forward rounding error bounds can be derived directly. Our numerical experiments illustrate that the Compqd algorithm is much more accurate than the qd algorithm, relegating the influence of the condition numbers up to second order in the rounding unit of the computer. Three applications of the new algorithm in the obtention of continued fractions and in pole and zero detection are shown. [ABSTRACT FROM AUTHOR]

This paper investigates two classes of synchronization problems of multiple chaotic systems with unknown uncertainties and disturbances by employing sliding mode control. Modified projective synchronization and transmission synchronization are discussed here. For the modified projective synchronization problem, sliding mode controllers are designed to ensure that multiple response systems synchronize with one drive system under the effects of external disturbances. For the transmission synchronization problem, based on adaptive sliding mode control, an integral sliding surface is selected and the adaptive laws are derived to tackle unknown uncertainties and disturbances for such systems. A class of nonlinear adaptive sliding mode controllers is developed to guarantee asymptotical stability of the error systems so that all chaotic systems can synchronize with each other. Simulation results are given to illustrate the effectiveness of the proposed schemes by comparing with the existing methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exponential synchronization of Lur’e type complex dynamical networks with uncertain coupling strength. A novel pinning impulsive controller is designed to achieve synchronization, in which, only selection of nodes are chosen to control instead of the whole network. In addition, the proposed control scheme is digital and it can lead to low installation cost and less time. In order to realize synchronization of Lur’e type complex dynamical networks, crucial delay-dependent stability conditions are derived by utilizing Lyapunov functional together with information of the time-varying delay and convex combination technique. The resulting stability conditions are formulated in the form of linear matrix inequalities (LMIs) and by solving the obtained LMIs, desired impulsive control gain matrices are calculated, which are capable to guarantee the exponential stability of error system. Finally, the effectiveness of the proposed method is demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]

Block-centered finite difference methods are constructed to solve the general Darcy–Forchheimer problems with Neumann boundary conditions, in which the velocity and pressure can be approximated simultaneously. We demonstrate that with sufficiently smooth analytical solution, the errors for both pressure and velocity in discrete L 2 -norms are second-order accurate on a nonuniform rectangular grid. Numerical experiments carried out using the scheme show the consistency of the convergence rates of our method with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we derive several error estimates that are pertinent to the study of Chebyshev-type spectral approximations on the real line. The results are applied to construct a stable and accurate pseudo-spectral Chebyshev scheme for the nonlinear Schrödinger equation. The new technique has several computational advantages as compared to Fourier and Hermite-type spectral schemes, described in the literature (see e.g., [1]–[3]. Similar to Hermite-type methods, we do not require domain truncation and/or use of artificial boundary conditions. At the same time, the computational complexity is comparable to the best Fourier-type spectral methods described in the literature. [ABSTRACT FROM AUTHOR]

The method of fundamental solutions (MFS) was first proposed by Kupradze in 1963. Since then, the MFS has been extensively applied for solving various kinds of problems in science and engineering. However, few theoretical works have been reported in the literature. In this paper, we devote our work to the error analysis and stability of the MFS for the case of the modified Helmholtz equation. For disk domains, a convergence analysis of the MFS was provided by Li (J. Comput. Appl. Math., 159:113–125, 2004) for solving the modified Helmholtz equation. In this paper, the error bounds of the MFS for bounded and simply connected domains are derived for smooth solutions of the modified Helmholtz equation. The exponential convergence rates can be achieved for analytic solutions. The bounds of condition numbers of the MFS are derived for both disk domains and the bounded and simply connected domains, to give the exponential growth via the number of fundamental solutions used. Numerical experiments are carried out to support the theoretical analysis. Moreover, the analysis of this paper is applied to parabolic equations, and some reviews of proof techniques and the analytical characteristics of the MFS are addressed. [ABSTRACT FROM AUTHOR]

In this paper, a porous thermoelastic problem is numerically considered. The variational formulation is written as a coupled system of two hyperbolic equations for the displacement and the porosity fields and a parabolic equation for the temperature field. An existence and uniqueness result as well as an energy decay property are recalled. Then, fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the first-order time derivatives. A priori error estimates are proved, from which the linear convergence is deduced under some additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution. [ABSTRACT FROM AUTHOR]

An analysis is reported here for three-time level integration methods following the global spectral analysis (GSA) described in High Accuracy Computing Methods , T.K. Sengupta, Cambridge Univ. Press, USA. The focus is on the second order Adams–Bashforth (AB2) and the extrapolation in time (EXT2) methods. Careful distinction is made for the first time step at t = 0 by either Euler forward or four-stage, fourth order Runge–Kutta (RK4) time schemes. The latter is used to solve a benchmark aeroacoustic problem. Several one-dimensional wave propagation models are analyzed: pure advection and advection-diffusion equations. Various spatial discretizations are discussed, including Fourier spectral method. Attention is paid to the presence of physical and numerical modes as noted in the quadratic equation obtained from the difference equation for the model 1D convection equation. It is shown that AB2 method is less stable and accurate than EXT2 method, with respect to numerical dissipation and dispersion. This is true for the methods, in which the physical mode dominates over the numerical mode. Presented analysis provides useful guide to analyze any three-time level methods. [ABSTRACT FROM AUTHOR]

In this paper, we perform a second-order two-scale analysis and introduce a numerical algorithm for the damped wave equations of composite materials with a quasi-periodic structure. Firstly, second-order two-scale asymptotic expansion solutions for these problems are constructed by a multiscale asymptotic analysis. In addition, we explain the importance of the second-order two-scale solutions by the error analysis in the pointwise sense. Moreover, explicit convergence rates of these second-order two-scale solutions are obtained in the integral sense. Then a second-order two-scale numerical method based on a Newmark scheme is presented to solve these multiscale problems. Finally, some numerical examples show the effectiveness and efficiency of the multiscale numerical method we proposed. [ABSTRACT FROM AUTHOR]

• Using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. • We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. • Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. • Our proposed method is simple to construct, the algorithm is easy to implement and has less running times. Moreover, only a small number of interpolation nodes is needed to achieve high accuracy. Our proposed method is applicable to many kinds of highly oscillatory Fourier-type integrals. This fully shows the wide application of this method. In this article, we propose and analyze the Clenshaw–Curtis–Filon-type method for computing many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities at the endpoints. First, by using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. Numerical experiments verify the efficacy and correctness of the presented method. [ABSTRACT FROM AUTHOR]

• In this paper, the weak Galerkin finite element method is studied for a singularly perturbed problem with two parameters. On a piecewise uniform Shishkin mesh, we prove uniform convergence of the weak Galerkin finite element method with piecewise higher order discontinuous polynomials. • One important feature of the WG-FEM is that it is possible to eliminate the element basis functions on each local interval by static condensation with the help of the weak derivatives and using the basis functions in the element and element boundary separately. Then, the resulting global coupled linear system will only involve the degrees of freedom on the skeleton of the mesh. Therefore, the degrees of freedom in the WG-FEM is comparable to the standard finite element method and the WG-FEM has much less unknown than the discontinuous Galerkin methods. • Using a special interpolation operator and its good features on anisotropic mesh, we established a robust uniform optimal convergence in the corresponding energy norm. Moreover, we presented error analysis in a stronger balanced norm with a locally defined L2-projection for the first time. • Extensive numerical experiments were carried out to support the theoretical findings in the paper. A weak Galerkin finite element method is proposed for solving singularly perturbed problems with two parameters. A robust uniform optimal convergence has been proved in the corresponding energy and a stronger balanced norms using piecewise higher order discontinuous polynomials on a piecewise uniform Shishkin mesh. Finally, we give some numerical experiments to support theoretical results. [ABSTRACT FROM AUTHOR]

• We propose and analyze two different efficient and accurate quadrature methods for a class of singularly oscillatory integral. • We perform the rigorous error analysis of the proposed methods and obtain asymptotic error estimates in inverse powers of the frequency parameter ω. • The proposed methods are compared with the asymptotic Filon-type method given in this work (J. Math. Anal. Appl. 494 (2021), Article number: 124448) and the modified Filon-type method. At the same computational cost, the two-point Taylor interpolation method and the asymptotic Filon-type method have a very close accuracy level, and their accuracy is higher than that of the modified Filon-type method, and the precision of the contour integration method is much higher than that of the asymptotic Filon-type method, the modified Filon-type method, and the two-point Taylor interpolation method. • The advantage of the proposed methods is that the quadrature precision greatly improves when the number of nodes increases and ω is fixed and when ω increases and the number of nodes is fixed. For medium or large ω , high precision can be achieved with only a few nodes. The recent article (J. Math. Anal. Appl. 494 (2021), Article number: 124448) presented an asymptotic Filon-type method for computing the oscillatory integral with a special oscillator and weak singularities, ∫ 0 b x α (b − x) β f (x) e i ω x r d x , − 1 < α , β ≤ 0 , 0 < b < + ∞ , ω ∈ R , r ∈ N +. In this article, we propose and analyze two different efficient and accurate quadrature methods for this singularly oscillatory integral. First, we give a two-point Taylor interpolation method by using a two-point Taylor polynomial instead of f (x). In addition, we propose a more efficient contour integration method. By exploiting the Taylor polynomial of the function f at x = 0 , and then based on the additivity of the integration interval, we change the considered integral into two integrals. One integral can be efficiently computed by the contour integration method based on Cauchy Residue Theorem and generalized Gaussian-Laguerre quadrature rule. The other integral can be explicitly calculated by special functions. Specifically, we perform the rigorous error analysis of the proposed methods and obtain asymptotic error estimates in inverse powers of the frequency parameter ω. Ultimately, the proposed methods are compared with the asymptotic Filon-type method given in this work (J. Math. Anal. Appl. 494 (2021), Article number: 124448) and the modified Filon-type method. At the same computational cost, the two-point Taylor interpolation method and the asymptotic Filon-type method have a very close accuracy level. Their accuracy is higher than that of the modified Filon-type method, and the precision of the contour integration method is much higher than that of the asymptotic Filon-type method, the modified Filon-type method, and the two-point Taylor interpolation method. We verify error analyses of the proposed methods by experimental results. Numerical experiments can also verify the efficiency and precision of the proposed methods. [ABSTRACT FROM AUTHOR]

Obtaining the transient probability distribution vector of a continuous-time Markov chain (CTMC) using an implicit ordinary differential equation (ODE) solver tends to be advantageous in terms of run-time computational cost when the product of the maximum output rate of the CTMC and the largest time of interest is large. In this paper, we show that when applied to the transient analysis of CTMCs, many implicit ODE solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1-norm of the error. This allows the development of implementations of those ODE solvers for the transient analysis of CTMCs that can be more efficient and more accurate than more standard implementations. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of multi-degree reduction of a composite Bézier curve with the parametric continuity constraints at the endpoints of the segments. We present a novel method which is based on the idea of using constrained dual Bernstein polynomials to compute the control points of the reduced composite curve. In contrast to other methods, ours minimizes the L 2 -error for the whole composite curve instead of minimizing the L 2 -errors for each segment separately. As a result, an additional optimization is possible. Examples show that the new method gives much better results than multiple application of the degree reduction of a single Bézier curve. [ABSTRACT FROM AUTHOR]

We present a local as well a semilocal convergence analysis for Newton’s method in a Banach space setting. Using the same Lipschitz constants as in earlier studies, we extend the applicability of Newton’s method as follows: local case: a larger radius is given as well as more precise error estimates on the distances involved. Semilocal case: the convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. Numerical examples further justify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we modify the idea of Haar wavelets method to obtain semi-analytical solutions for the systems of three-dimensional nonlinear partial differential equations. Theoretical considerations are discussed. To demonstrate the efficiency of the method, a test problem is presented. The approximate solutions of the system of three-dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature. The numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the spatially semi-discrete piecewise linear finite volume element method for the time fractional sub-diffusion problem in two dimensions, and give an approximate solution of this problem. At first, we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods. Furthermore, we use the standard finite element Ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Finally, some numerical examples are included to illustrate the effectiveness of the new technique. [ABSTRACT FROM AUTHOR]

We enrich the known results about coupled fixed and best proximity points of cyclic contraction ordered pair of maps. The uniqueness of the coupled best proximity points for cyclic contraction ordered pair of maps in a uniformly convex Banach space is proven. We find a priori and a posteriori error estimates for the coupled best proximity points, obtained by sequences of successive iterations, when the underlying Banach space has modulus of convexity of power type. A looser conditions are presented for the existence and uniqueness of coupled fixed points of a cyclic contraction ordered pair of maps in a complete metric space and a priori, a posteriori error estimates and the rate of convergence for the coupled fixed points are obtained for the sequences of successive iterations. We apply these results for solving systems of integral equations, systems of linear and nonlinear equations. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of the asynchronous H ∞ filtering for a class of two-dimensional(2D) Markovian jump systems with sensor failure. The asynchronous phenomenon is considered to be random mismatch between system modes and candidate filters. The mismatch behavior is assumed to be stochastic with a certain probability and a new Markov chain is proposed to model the jumping process of the filtering error system. Besides, a switching method is proposed to dealing with the sensor failure. Sufficient conditions for the existence of a desired filter such that the filtering error system is stochastically stable with a guaranteed H ∞ performance index are obtained in the form of linear matrix inequalities (LMIs). Two examples are provided to illustrate the effectiveness and advantage of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, solitary-wave solutions of the generalized regularized long wave (GRLW) equation are obtained by using septic B-spline collocation method with two different linearization techniques. To demonstrate the accuracy and efficiency of the numerical scheme, three test problems are studied by calculating the error norms L 2 and L ∞ and the invariants I 1 , I 2 and I 3 . A linear stability analysis based on the von Neumann method of the numerical scheme is also investigated. Consequently, our findings indicate that our numerical scheme is preferable to some recent numerical schemes. [ABSTRACT FROM AUTHOR]