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In this paper, the impulsive synchronization of a class of nonlinear multiple neural networks (MNNs) with multi-delays was considered under a dynamic event-based mechanism. To achieve a more comprehensive synchronization outcome and mitigate the conservativeness of impulsive control due to predetermined time sequences, we integrated a dynamic event-triggered strategy. This approach formed a novel control framework for generalized MNNs, where impulsive inputs were applied only under specific conditions governed by event-triggering rules. Towards the above objectives, the impulsive jumping system, resulting from dynamic component, and matrix measure method were invoked to directly increase the computational simplicity and extensibility of the study. As the outcome, the synchronization criteria for the MNNs could be achieved, and the exponential convergence rate is resolved by considering both the generalized comparison principle regarding impulsive systems and the variable parameter formula. Moreover, Zeno-freeness of the achieved triggering regulation is ensured. Finally, two numerical examples confirmed the validity of the designed approach. [ABSTRACT FROM AUTHOR]

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]

In this paper we consider the distributed optimal control of the Navier—Stokes equations in the presence of pointwise mixed control-state constraints. After deriving a first order necessary condition, the regularity of the mixed constraint multiplier is investigated. Second order sufficient optimality conditions are studied as well. In the last part of the paper, a semismooth Newton method is applied for the numerical solution of the control problem. The convergence of the method is proved and numerical experiments are carried out. [ABSTRACT FROM AUTHOR]

In this paper, we apply the Adomian decomposition method (ADM) for solving Fractional Differential Equations (FDEs) with some modifications to the traditional method. The aim of this paper is to make ADM more efficient, rapid in convergence, and easy to use, so we will discuss two modifications. We use the reliable modification to simplify calculations. For difficulties in symbolic integration, we use a numerical implementation method. All these modifications were applied to the integer-order case, so we would apply it to FDEs. Some numerical results are given from solving these cases and comparing the solution with the ADM method. [ABSTRACT FROM AUTHOR]

In this paper, three nonlinear finite difference schemes are proposed for solving a generalized nonlinear derivative Schrödinger equation which exposits the propagation of ultrashort pulse through optical fiber and has been illustrated to admit exact soliton-solutions. Two of the three schemes are two-level ones and the third scheme is a three-level one. It is proved that the two-level schemes only preserve the total mass or the total energy in the discrete sense and the three-level scheme preserves both the total mass and total energy. Furthermore, many numerical results are presented to test the conservative properties and convergence rates of the proposed schemes. Several dynamical behaviors including solitary-wave collisions and the first-order rogue wave solution are also simulated, which further illustrates the effectiveness of the proposed method for the generalized nonlinear derivative Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper we propose an alternative for the study of local convergence radius and the uniqueness radius for some third-order methods for multiple roots whose multiplicity is known. The main goal is to provide an alternative that avoids the use of sophisticated properties of divided differences that are used in already published papers about local convergence for multiple roots. We defined the local study by using a technique taking into consideration a bounding condition for the (m + i) th derivative of the function f (x) with i=1,2. In the case that the method uses first and second derivative in its iterative expression and i=1 in case the method only uses first derivative. Furthermore we implement a numerical analysis in the following sense. Since the radius of local convergence for high-order methods decreases with the order, we must take into account the analysis of ITS behaviour when we introduce a new iterative method. Finally, we have used these iterative methods for multiple roots for the case where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behaviour of the corresponding estimations and how this fact affect to the original method. [ABSTRACT FROM AUTHOR]

In this paper, a generalized full orthogonalization method (GFOM) based on weighted inner products is discussed for computing PageRank. In order to improve convergence performance, the GFOM algorithm is accelerated by two cheap methods respectively, one is the power method and the other is the extrapolation method based on Ritz values. Such that two new algorithms called GFOM-Power and GFOM-Extrapolation are proposed for computing PageRank. Their implementations and convergence analyses are studied in detail. Numerical experiments are used to show the efficiency of our proposed algorithms. [ABSTRACT FROM AUTHOR]

Consider the problem of finding the maximal nonpositive solvent Φ of the quadratic matrix equation (QME) X 2 + B X + C = 0 with B being a nonsingular M -matrix and C an M -matrix such that B − 1 C ≥ 0. Such QME arises from an overdamped vibrating system. Recently, under the condition that B − C − I is a nonsingular M -matrix, Yu et al. (Appl. Math. Comput. , 218 (2011): 3303–3310) proved that ρ (Φ) ≤ 1 for this QME. In this paper, under the same condition, we slightly improve their result and prove that ρ (Φ) < 1 , which is important for the quadratic convergence of the structure-preserving doubling algorithm. Then, a new globally monotonically and quadratically convergent structure-preserving doubling algorithm for solving the QME is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, the temporal fractional Black-Scholes model (TFBSM) is discussed in the limited specific domain which the time derivative of this template is the Caputo fractional function. The value variance of the associated fractal transmission method was applied to forecast TFBSM. For solving, at first the semi-discrete scheme is obtained by using linear interpolation with a temporally τ2-α order accuracy. Then, the full scheme is collected by approximating the spatial derivative terms with the help of the Chebyshev collocation system focused on the fourth form. Finally, the unconditional stability and convergence order are evaluated by performing the energy process. As an implementation of this method, two examples of the TFBSM were reported to demonstrate the accuracy of the developed scheme. Simulation and comparison show that the suggested strategy is very accurate and effective. [ABSTRACT FROM AUTHOR]

Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals. [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]

In this paper, we trace back the genesis of Aitken's Δ2 process and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture. [ABSTRACT FROM AUTHOR]

In this paper, we establish the truncated Euler-Maruyama (EM) method for stochastic functional differential equation (SFDE) and consider the strong convergence theory for the numerical solutions of SFDEs under the local Lipschitz condition plus Khasminskii-type condition instead of the linear growth condition. The type of convergence specifically addressed in this paper is strong- convergence for , and p is a parameter in Khasminskii-type condition. We also discussed the rates of -convergence for the truncated EM method. [ABSTRACT FROM AUTHOR]

From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization , and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

Correlation operators based on the solution of an implicitly formulated diffusion equation can be implemented numerically using the Chebyshev iteration method. The attractive properties of the algorithm for modelling correlation functions on high‐performance computers have been discussed in a recent paper. The current paper describes a straightforward variant of that algorithm that allows the matrix–vector products involved in the sequential pseudo‐time diffusion process to be performed in parallel. Contrary to the original algorithm, which requires solving, in sequence, linear systems involving a symmetric positive‐definite (SPD) matrix, the "time"‐parallel algorithm requires solving a single linear system involving a non‐symmetric positive‐definite (NSPD) matrix. The key information required by the Chebyshev iteration for solving the NSPD problem is an estimate of the extreme eigenvalues of the NSPD matrix. For the problem under consideration, the extreme eigenvalues of the NSPD matrix are the same as those of the original SPD matrix, and can be pre‐computed using a Lanczos algorithm applied to the latter. The convergence properties of the algorithm are studied from a theoretical perspective and using numerical experiments with a diffusion‐based covariance model in a variational data assimilation system for the global ocean. Results suggest that time‐parallelization can reduce the run‐time of an implicit diffusion‐based correlation operator by greater than a factor of two. It can be implemented practically using a hybrid parallelization approach that combines Message Passing Interface tasks in the spatial domain with Open Multi‐Processing threads spanning the pseudo‐time dimension. The sensitivity of the results to preconditioning, to the choice of first guess and to the stopping criterion is discussed. This paper describes a diffusion‐based correlation model in which the M matrix–vector products involved in the sequential M pseudo‐time‐step implicit diffusion process can be performed in parallel. The "time"‐parallel algorithm requires solving a single linear system involving a non‐symmetric positive definite matrix. An approximate solution can be obtained using the Chebyshev iteration. Results suggest that the "time"‐parallel algorithm can reduce run‐time of the correlation operator by a factor between two and three. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the τ-leaping framework to past information. Using the Θ-trapezoidal τ-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k ≥ 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach. [ABSTRACT FROM AUTHOR]

This paper deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton's method, using majorizing sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to verify the given theory. [ABSTRACT FROM AUTHOR]

This paper presents a robust finite-time guidance (RFTG) law to a short-range interception problem. The main challenge is that the evasive strategy of the target is unpredictable because it is determined not only by the states of both the interceptor and the target, but also by external un-modeled factors. By robustly stabilizing a line-of-sight rate, this paper proposes an integrated continuous finite-time disturbance observer/bounded continuous finite-time stabilizer strategy. The design of this integrated strategy has two points: 1) effect of a target maneuver is modeled as disturbance and then is estimated by the second-order homogeneous observer; 2) the finite-time stabilizer is actively coupled with the observer. Based on homogeneity technique, the local finite-time input-to-state stability is established for the closed-loop guidance system, thus implying the proposed RFTG law can quickly render the LOS rate within a bounded error throughout intercept. Moreover, convergence properties of the LOS rate in the presence of control saturation are discussed. Numerical comparison studies demonstrate the guidance performance. [ABSTRACT FROM AUTHOR]

The paper revisits the classical problem of evaluating f ( A ) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix A . We present several numerical examples to illustrate our analysis. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for initial values satisfying a slightly stronger condition we obtain an optimal estimate on the rate of convergence. By establishing a connection with a related problem in continuous time, we are able to use this optimal estimate to improve the rate of convergence in the continuous setting obtained by the authors in a previous paper. We illustrate the power of the general approach by using it to study several concrete examples, both in continuous and in discrete time. [ABSTRACT FROM AUTHOR]

The paper proposes a new extragradient algorithm for solving strongly pseudomonotone equilibrium problems which satisfy a Lipschitz-type condition recently introduced by Mastroeni in auxiliary problem principle. The main novelty of the paper is that the algorithm generates the strongly convergent sequences in Hilbert spaces without the prior knowledge of Lipschitz-type constants and any hybrid method. Several numerical experiments on a test problem are also presented to illustrate the convergence of the algorithm. [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]

In this paper, a modified weak Galerkin method is proposed for the Stokes problem. The numerical scheme is based on a novel variational form of the Stokes problem. The degree of freedoms in the modified weak Galerkin method is less than that in the original weak Galerkin method, while the accuracy stays the same. In this paper, the optimal convergence orders are given and some numerical experiments are presented to verify the theory. [ABSTRACT FROM AUTHOR]

In this paper, the complete convergence and the complete moment convergence for extended negatively dependent (END, in short) random variables without identical distribution are investigated. Under some suitable conditions, the equivalence between the moment of random variables and the complete convergence is established. In addition, the equivalence between the moment of random variables and the complete moment convergence is also proved. As applications, the Marcinkiewicz-Zygmund-type strong law of large numbers and the Baum-Katz-type result for END random variables are established. The results obtained in this paper extend the corresponding ones for independent random variables and some dependent random variables. [ABSTRACT FROM AUTHOR]

A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forward-backward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom. [ABSTRACT FROM AUTHOR]

Calculations of three-electron atomic systems in Hylleraas coordinates require integrals involving all the interparticle distances rij, which have usually been evaluated by introducing series expansions. For integrals with the smallest powers of rij these expansions do not converge at a satisfactory rate, leading some investigators to introduce convergence-acceleration procedures. This paper recommends the alternative of evaluating these integrals in closed form and presents stable explicit formulas for so doing. Some of the formulas are more compact versions of those in the literature; others have not been previously reported. It is also shown that finite-difference methods can be used with advantage to obtain additional low-order integrals. Sample integral values have been provided for test purposes. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A new fast direct linear equation solver for the boundary element method (BEM) is presented in this paper. The idea of the new fast direct solver stems from the concept of the hierarchical off-diagonal low-rank matrix. The hierarchical off-diagonal low-rank matrix can be decomposed into the multiplication of several diagonal block matrices. The inverse of the hierarchical off-diagonal low-rank matrix can be calculated efficiently with the Sherman-Morrison-Woodbury formula. In this paper, a more general and efficient approach to approximate the coefficient matrix of the BEM with the hierarchical off-diagonal low-rank matrix is proposed. Compared to the current fast direct solver based on the hierarchical off-diagonal low-rank matrix, the proposed method is suitable for solving general 3-D boundary element models. Several numerical examples of 3-D potential problems with the total number of unknowns up to above 200,000 are presented. The results show that the new fast direct solver can be applied to solve large 3-D BEM models accurately and with better efficiency compared with the conventional BEM. [ABSTRACT FROM AUTHOR]

In this paper, we present a new two step iterative method to solve the nonlinear equation f (x) - 0 and discuss about its convergence. Few numerical examples are considered to show the efficiency of the new method in comparison with the other methods considered in this paper. [ABSTRACT FROM AUTHOR]

The objective of this paper is to discuss finite-time bounded synchronization for a class of the growing complex network with nondelayed and delayed coupling. In order to realize finite-time synchronization of complex networks, a new finite-time stable theory is proposed; effective criteria are developed to realize synchronization of the growing complex dynamical network in finite time. Moreover, the error of two growing networks is bounded simultaneously in the process of finite-time synchronization. Finally, some numerical examples are provided to verify the theoretical results established in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with the nonmonotone projection algorithm for constrained nonlinear equations. For some starting points, the previous projection algorithms for the problem may encounter slow convergence which is related to the monotone behavior of the iterative sequence as well as the iterative direction. To circumvent this situation, we adopt the nonmonotone technique introduced by Dang to develop a nonmonotone projection algorithm. After constructing the nonmonotone projection algorithm, we show its convergence under some suitable condition. Preliminary numerical experiment is reported at the end of this paper, from which we can see that the algorithm we propose converges more quickly than that of the usual projection algorithm for some starting points. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of a first order fractional-step time-scheme (via decomposition of the viscosity) and “inf-sup” stable finite-element spatial approximations applied to the Primitive Equations of the Ocean. The aim of the paper is twofold. First, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Second, optimal error estimates for velocity and pressure are provided of order O ( k + h l ) for l = 1 or l = 2 considering either first or second order finite-element approximations ( k and h being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint k ≤ C h 2 . [ABSTRACT FROM AUTHOR]

This paper discusses the continuous-time consensus problems for directed multi-agent networks under certain coupling or control. Since agents are driven not only by their neighbors' states but also by their derivatives in many realistic situations, dynamical networks described by singular systems are appropriate to study. Consensus problems for such multi-agent networks are considered when the agents communicate in the presence or absence of time delays. The maximum tolerated time-delay is obtained when the multi-agent network asymptotically reaches consensus. The results of this paper indicate that such multi-agent networks can achieve consensus with a demanding convergent speed through agents' interactions. Numerical examples are given to demonstrate the effectiveness of the 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]

In this paper, the author considers the one dimensional initial-boundary problem for a pseudo-parabolic equation with time delay in second spatial derivative. To solve this problem numerically, the author constructs higher order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Some numerical examples illustrate the convergence and effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a coupled system of m (≥ 2) singularly perturbed semilinear reaction–diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain. The diffusion term of each equation is multiplied by small singular perturbation parameter, but these parameters are assumed to be different in magnitude. A numerical method is constructed on a variant of Shishkin mesh. The approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters. Numerical results are in support of the theoretical results. Highlights • A coupled system of m (≥ 2) singularly perturbed semilinear reaction-diffusion equations is considered. • The source term is having a discontinuity at a point in the interior of the domain. • The diffusion term of each equation is multiplied by a small perturbation parameter. • All the perturbation parameters are different in magnitude. • The constructed numerical method has almost second parameter uniform convergence. [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]

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them. [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]

Highlights • A new variational model for image retinex is proposed. • The existence of minimizer for the variational model is proved. • An efficient algorithm is derived to solve the model by adding auxiliary variables. • The convergence of the numerical algorithm is proved under some assumptions. • The proposed method can be extended to other regularization and fidelity terms. Abstract In this paper, we propose a detail preserving variational model for Retinex to simultaneously estimate the illumination and the reflectance from an observed image. Most previous models use the log-transform as pretreatment which results in loss of details in reflectance. From this observation, a detail preserving variational method is proposed for better decomposition. Different from the log-transform based models, the proposed model performs the decomposition directly in the image domain. Mathematically, we prove the existence of a solution for the proposed model. Numerically, we derive an efficient iterative algorithm by utilizing alternating direction method of multipliers (ADMM) method. Experimental results demonstrate the effectiveness of the proposed method. Compared with other closely related Retinex methods, the proposed method achieves competitive results on both subjective and objective assessments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the two MAC schemes are introduced and analyzed to solve the time fractional Stokes equation on non-uniform grids. One is the standard MAC scheme and another is the efficient MAC scheme, where the fast evaluation of the Caputo fractional derivative is used. The stability results are derived. We obtain the second order superconvergence in discrete L 2 norm for both velocity and pressure. We also obtain the second order superconvergence for some terms of the H 1 norm of the velocity on non-uniform grids. Besides, the efficient algorithm for the evaluation of the Caputo fractional derivative is used to save the storage and computation cost greatly. Finally, some numerical experiments are presented to show the efficiency and accuracy of MAC schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger–Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O (τ 2 + h 4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]