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Abstract In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size. [ABSTRACT FROM AUTHOR]

Abstract: In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the -stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation. [Copyright &y& Elsevier]

Abstract: In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating limits with approximation order 2. [Copyright &y& Elsevier]

Abstract: The problems of calculating a dominant eigenvector or a dominant pair of singular vectors, arise in several large scale matrix computations. In this paper we propose a minimum norm approach for solving these problems. Given a matrix, , the new method computes a rank-one matrix that is nearest to , regarding the Frobenius matrix norm. This formulation paves the way for effective minimization techniques. The methods proposed in this paper illustrate the usefulness of this idea. The basic iteration is similar to that of the power method, but the rate of convergence is considerably faster. Numerical experiments are included. [Copyright &y& Elsevier]

In this paper, a series of new high-order numerical approximations to α th ( 0 < α < 1 ) order Caputo derivative is constructed by using r th degree interpolation approximation for the integral function, where r ≥ 4 is a positive integer. As a result, the new formulas can be viewed as the extensions of the existing jobs (Cao et al., 2015; Li et al., 2014), the convergence orders are O ( τ r + 1 − α ) , where τ is the time stepsize. Two test examples are given to demonstrate the efficiency of these schemes. Then we adopt the derived schemes to solve the Caputo type advection–diffusion equation with Dirichlet boundary conditions. The local truncation error of the derived difference scheme is O ( τ r + 1 − α + h 2 ) , where τ is the time stepsize, and h the space one. The stability and convergence of the proposed schemes for r = 4 are also considered. Without loss of generality, we only display the numerical examples for r = 4 , 5 , which support the numerical algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a semilinear parabolic problem with an unknown time-dependent source function p ( t ) is studied. This missing parameter is reconstructed from a given measurement of the total energy/mass in the domain. The existence and uniqueness of a solution in suitable function spaces is established under minimal regularity assumptions on the data. A numerical time-discrete scheme to approximate the unique weak solution and the unknown source parameter is designed and convergence of the approximations is proved. Finally, the theoretically obtained results are supported by a numerical experiment. [ABSTRACT FROM AUTHOR]

Abstract: This paper treats definite integrations numerically using Gegenbauer quadratures. The novel numerical scheme introduces the idea of exploiting the strengths of the Chebyshev, Legendre, and Gegenbauer polynomials through a unified approach, and using a unique numerical quadrature. In particular, the developed numerical scheme employs the Gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms. For a large-scale number of expansion terms, the numerical quadrature has the advantage of converging to the optimal Chebyshev and Legendre quadratures in the -norm and -norm, respectively. The key idea is to construct the Gegenbauer quadrature through discretizations at some optimal sets of points of the Gegenbauer–Gauss (GG) type in a certain optimality sense. We show that the Gegenbauer polynomial expansions can produce higher-order approximations to the definite integrals of a smooth function for the small range by minimizing the quadrature error at each integration point through a pointwise approach. The developed Gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes. Exact integrations are obtained for polynomials of any arbitrary degree if the number of columns in the developed Gegenbauer integration matrix (GIM) is greater than or equal to . The error formula for the Gegenbauer quadrature is derived. Moreover, a study on the error bounds and the convergence rate shows that the optimal Gegenbauer quadrature exhibits very rapid convergence rates, faster than any finite power of the number of Gegenbauer expansion terms. Two efficient computational algorithms are presented for optimally constructing the Gegenbauer quadrature. We illustrate the high-order approximations of the optimal Gegenbauer quadrature through extensive numerical experiments, including comparisons with conventional Chebyshev, Legendre, and Gegenbauer polynomial expansion methods. The present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods. [Copyright &y& Elsevier]

Abstract: Hybrid systems, and Piecewise Deterministic Markov Processes in particular, are widely used to model and numerically study systems exhibiting multiple time scales in biochemical reaction kinetics and related areas. In this paper an almost sure convergence analysis for numerical simulation algorithms for Piecewise Deterministic Markov Processes is presented. The discussed numerical methods arise through discretising a constructive method defining these processes. The stochastic problem of simulating the random, path-dependent jump times of such processes is reformulated as a hitting time problem for a system of ordinary differential equations with random threshold. Then deterministic continuous methods (methods with dense output) are serially employed to solve these problems numerically. We show that the almost sure convergence rate of the stochastic algorithm is identical to the order of the embedded deterministic method. We illustrate our theoretical findings by numerical examples from mathematical neuroscience, Piecewise Deterministic Markov Processes are used as biophysically accurate stochastic models of neuronal membranes. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique. [Copyright &y& Elsevier]

Abstract: A deflated restarting Krylov subspace method for approximating a function of a matrix times a vector is proposed. In contrast to other Krylov subspace methods, the performance of the method in this paper is better. We further show that the deflating algorithm inherits the superlinear convergence property of its unrestarted counterpart for the entire function and present the results of numerical experiments. [Copyright &y& Elsevier]

Abstract: In this paper, a least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an inverse problem for a fractional diffusion equation which is highly ill-posed. Such a problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order . We show that the problem is severely ill-posed and further apply an optimal regularization method to solve it based on the solution in the frequency domain. We can prove the optimal convergence estimate, which shows that the regularized solution depends continuously on the data and is a good approximation to the exact solution. Numerical examples show that the proposed method works well. [Copyright &y& Elsevier]

Abstract: This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed. [Copyright &y& Elsevier]

Abstract: In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski’s method and in a sixth-order improved Ostrowski’s method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the numerical approximation of stochastic differential equations with discontinuous/nondifferentiable drifts. We show that under one-sided Lipschitz and general growth conditions on the drift and global Lipschitz condition on the diffusion, a variant of the implicit Euler method known as the split-step backward Euler (SSBE) method converges with strong order of one half to the true solution. Our analysis relies on the framework developed in [D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002) 1041–1063] and exploits the relationship which exists between explicit and implicit Euler methods to establish the convergence rate results. [Copyright &y& Elsevier]

Abstract: The subject of this paper is the analytic approximation method for solving stochastic differential equations with time-dependent delay. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. It will be shown, without making any restrictive assumption for the delay function, that the approximate solutions converge in -norm and with probability 1 to the solution of the initial equation. Also, the rate of the convergence increases when the degrees in the Taylor approximations increase, analogously to what is found in real analysis. At the end, a procedure will be presented which allows the application of this method, with the assumption of continuity of the delay function. [Copyright &y& Elsevier]

Abstract: This paper considers a stochastic mathematical program with hybrid equilibrium constraints (SMPHEC), which includes either “here-and-now” or “wait-and-see” type complementarity constraints. An example is given to describe the necessity to study SMPHEC. In order to solve the problem, the sampling average approximation techniques are employed to approximate the expectations and smoothing and penalty techniques are used to deal with the complementarity constraints. Limiting behaviors of the proposed approach are discussed. Preliminary numerical experiments show that the proposed approach is applicable. [Copyright &y& Elsevier]

Abstract: Recently, Hager and Zhang (2005) proposed a new conjugate gradient method which generates sufficient descent direction , this property is independent of the line search used. In this paper, we take a modification of this method, such that the sufficient descent direction satisfies , this property is also independent of the line search used. Under appropriate conditions, we prove that the proposed method is globally convergent. Moreover, we give a sufficient condition for the global convergence of the proposed general method. The numerical results show that the proposed method is efficient. [Copyright &y& Elsevier]

Abstract: Many researches attempt to improve the efficiency of the usual quasi-Newton (QN) methods by accelerating the performance of the algorithm without causing more storage demand. They aim to employ more available information from the function values and gradient to approximate the curvature of the objective function. In this paper we derive a new QN method of this type using a fourth order tensor model and show that it is superior with respect to the prior modification of Wei et al. (2006) . Convergence analysis gives the local convergence property of this method and numerical results show the advantage of the modified QN method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the authors study “truncated” quadrature rules based on the zeros of Generalized Laguerre polynomials. Then, they prove the stability and the convergence of the introduced integration rules. Some numerical tests confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations. [Copyright &y& Elsevier]

Abstract: In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra–Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture Kung and Traub (1974) , that establishes for an iterative method based on evaluations an optimal order is fulfilled, getting the highest efficiency indices for orders and , which are 1.587 and 1.682. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution , defined for , (), that satisfies this equation almost everywhere on and assumes specified values on the intervals and . We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence. [Copyright &y& Elsevier]

Abstract: This paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction–diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given. [Copyright &y& Elsevier]

Abstract: In this work, we solve the elliptic partial differential equation by coupling the meshless mixed Galerkin approximation using radial basis function with the three-field domain decomposition method. The formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function. Convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the superconvergence properties of the - version of the finite element method (FEM) for two-point boundary value problems. A postprocessing technique for the - finite element approximation is analyzed. The analysis shows that the postprocess improves the order of convergence. Furthermore, we obtain asymptotically exact a posteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis. [Copyright &y& Elsevier]

Abstract: Recently, numerical solutions of stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet. In the paper, the Euler–Maruyama method for neutral stochastic delay differential equations is developed. The key aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition. [Copyright &y& Elsevier]

Abstract: During numerical time integration, the accuracy of the numerical solution obtained with a given step size often proves unsatisfactory. In this case one usually reduces the step size and repeats the computation, while the results obtained for the coarser grid are not used. However, we can also combine the two solutions and obtain a better result. This idea is based on the Richardson extrapolation, a general technique for increasing the order of an approximation method. This technique also allows us to estimate the absolute error of the underlying method. In this paper we apply Richardson extrapolation to the sequential splitting, and investigate the performance of the resulting scheme on several test examples. [Copyright &y& Elsevier]

Abstract: In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations. They come from two modified conjugate gradient methods [W.Y. Cheng, A two term PRP based descent Method, Numer. Funct. Anal. Optim. 28 (2007) 1217–1230; L. Zhang, W.J. Zhou, D.H. Li, A descent modified Polak–Ribiére–Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal. 26 (2006) 629–640] recently proposed for unconstrained optimization problems. Under appropriate conditions, the global convergence of the proposed method is established. Preliminary numerical results show that the proposed method is promising. [Copyright &y& Elsevier]

Abstract: We consider an inverse problem of reconstructing the coefficient q in the parabolic equation from the final measurement , where q is in some subset of . The optimization method, combined with the finite element method, is applied to get the numerical solution under some assumption on q. The existence of minimizer, as well as the convergence of approximate solution in finite-dimensional space, is proven. The new ingredient in this paper is that we do not need uniformly a priori bounds of -norm on q. Numerical implementations are also presented. [Copyright &y& Elsevier]