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• The convergence analysis and the techniques used in Lemma 3.3 by X. Wang et al. in [1] are based essentially on Sobolev's inequality in one dimension and they cannot be extended to two dimensions. • All results based on Lemma 3.3 are false, in particular Theorem 3.3, Theorem 4.1, Theorem 5.1 and Theorem 5.2. • Regarding the conservation of mass and energy, the initial terms Q 0 and E 0 depends on U 0 and on U 1 and the authors consider only Eq. (4) without taking into account Eq. (7). Therfore, there is an error in Theorem 3.1 and Theorem 3.2. • In Theorem 4.1, the authors did not show that u1 is uniquely determined by Eq. (7). In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments. [ABSTRACT FROM AUTHOR]

Abstract: In this short note an addition to the paper [A.G. Bratsos, M. Ehrhardt, I.Th. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations, Appl. Math. Comput. 197(1) (2008) 190–205] using the modulus of the terms evaluated from the Adomian decomposition method on p. 194 and their relation to the convergence of the resulting series is presented. Conclusions for the accuracy of the approximated solution are derived. [Copyright &y& Elsevier]

Abstract: In this note, we will modify several gaps in Gurudwan and Sharma [N. Gurudwan, B.K. Sharma, Convergence theorem for the common solution for a finite family of -strongly accretive operator equations, Appl. Math. Comput. 217 (2011) 6748–6754]. [Copyright &y& Elsevier]

Abstract: In abstract of the paper [A. Rafiq, A note on “A family of methods for solving nonlinear equations”, Appl. Math. Comput. 195 (2008) 819–821] we can find the following sentences. We cite: Ujević et al. introduced a family of methods for solving nonlinear equations. However the main Algorithm 1 put forward by Ujević et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujević et al. – the end of the citation. Here it is shown that all of the mentioned assertions are not true. In other words, the Algorithm 1 is correct (up to an obvious misprint, which is not mentioned in the above paper) and there are no major bugs in the paper by Ujević et al. In fact, these observations, which will be given in this note, show that the main aim of the paper by Rafiq is wrong. [Copyright &y& Elsevier]

Abstract In this paper, we propose the split-step theta method for stochastic delay integro-differential equations by the Lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme. It is shown that the split-step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper. A numerical example is given at the end of the paper to illustrate the method and conclusion of the paper. In addition, the convergence of the split-step theta method is proved in the Appendix. [ABSTRACT FROM AUTHOR]

Highlights • The heat equation with nonlocal initial condition is considered. • Several finite difference schemes are constructed and analyzed. • The stability of the schemes is the main objective of investigation. • Time step size bound obtained in previous paper is revised. • Numerical experiments with linear and nonlinear problems confirm the theoretical results. Abstract In this paper, the two-level finite difference schemes for the one-dimensional heat equation with a nonlocal initial condition are analyzed. As the main result, we obtain conditions for the numerical stability of the schemes. In addition, we revise the stability conditions obtained in [21] for the Crank–Nicolson scheme. We present several numerical examples that confirm the theoretical results within linear, as well as nonlinear problems. In some particular cases, it is shown that for small regions of the time step size values, the explicit FTCS scheme is stable while certain implicit methods, such as Crank–Nicolson scheme, are not. [ABSTRACT FROM AUTHOR]

Abstract In this paper we focus on two sources of enhancement in accuracy and computational demanding in approximating a function and its derivatives by means of the Smoothed Particle Hydrodynamics method. The approximating power of the standard method is perceived to be poor and improvements can be gained making use of the Taylor series expansion of the kernel approximation of the function and its derivatives. The modified formulation is appealing providing more accurate results of the function and its derivatives simultaneously without changing the kernel function adopted in the computation. The request for greater accuracy needs kernel function derivatives with order up to the desidered accuracy order in approximating the function or higher for the derivatives. In this paper we discuss on the scheme dealing with the infinitely differentiable Gaussian kernel function. Studies on the accuracy, convergency and computational efforts with various sets of data sites are provided. Moreover, to make large scale problems tractable the improved fast Gaussian transform is considered picking up the computational cost at an acceptable level preserving the accuracy of the computation. [ABSTRACT FROM AUTHOR]

This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson’s Rule and two-step BDF method are of order p = 0 when applied to impulsive delay differential equations. An improved linear multistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion. [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 class of non-negativity-preserving and maximum-principle-satisfying finite difference methods have been derived by Vieta theorem for one-dimensional and two-dimensional Fisher's equation. By using the positivity and boundedness of numerical and exact solutions, it is shown that numerical solutions obtained by current methods converge to exact solutions with orders of O (Δ t + (Δ t / h x) 2 + h x 2) for one-dimensional case and O (Δ t + (Δ t / h x) 2 + (Δ t / h y) 2 + h x 2 + h y 2) for two-dimensional case in the maximum norm, respectively. Here, Δ t , h x and h y are meshsizes in t -, x - and y -directions, respectively. Finally, numerical results verify that the proposed method can inherit the monotonicity, boundedness and non-negativity of the continuous problems. • New Du Fort-Frankel methods are devised for Fisher's equation. • They are non-negativity-preserving and maximum-principle-satisfying schemes. • Errors in maximum norm are given for them. • They are very good at long-term simulations. • They are very easy to be implemented. [ABSTRACT FROM AUTHOR]

This paper focuses on the convergence of a class of collocation methods for Volterra integral equations of the second kind with highly oscillatory Bessel functions. Compared to existing theoretical results, sharper frequency-related convergence rates of these methods are established by exploring the asymptotic expansions of solutions and solving error equations. Theoretical results in this paper show the direct Filon method and continuous linear collocation method share the same convergence rate. Both of them admit a better convergence rate compared to the piecewise constant collocation method in solving Volterra integral equations with highly oscillatory Bessel kernels. These results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Recently, Bai, Parlett and Wang presented a class of parameterized inexact Uzawa (PIU) methods for solving saddle point problems (Bai et al., 2005). In this paper, we develop a new generalized PIU method for solving both nonsingular and singular saddle point problems. The necessary and sufficient conditions of the convergence (semi-convergence) for solving nonsingular (singular) saddle point problems are derived. Meanwhile, the characteristic of eigenvalues of the iteration matrix corresponding to the above iteration method is discussed. We further show that the generalized PIU-type method proposed in this paper has a wider convergence (semi-convergence) region than some classical Uzawa methods, such as the inexact Uzawa method, the SOR-like method, the GSOR method and so on. Finally, numerical examples are given to illustrate the feasibility and efficiency of this method. [ABSTRACT FROM AUTHOR]

In this paper, a new split-step θ (SS θ ) method for stochastic age-dependent population equations with Poisson jumps is constructed. The main aim of this paper is to investigate the convergence of the SS θ method for stochastic age-dependent population equations with Poisson jumps. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory, the results show that the SS θ method has better accuracy compared to the Euler method. [ABSTRACT FROM AUTHOR]

• A real option is a contract which gives its holder the flexibility to expand the scale of an investment project or production. Real options are often used to hedge risks or capture opportunities in investments. In this paper, we establish a mathematical model for pricing a real option of expansion whose underlying asset price and its volatility/variance satisfy two separate stochastic equations. Based on Ito's lemma and a hedging technique, we show that the option price satisfies a 2 nd -order parabolic partial differential equation (PDE) in two spatial dimensions. We also derive the boundary and terminal conditions for the PDE and some of these conditions are also determined by PDEs. • We propose a novel 9-point finite difference scheme with a upwind technique is designed for solving the PDE system, as well that for determining the terminal (payoff) condition, established. We show that the coefficient matrix of the system from this discretization is an M-matrix and the numerical solution generated by the finite difference scheme converge to the exact one by proving that the scheme is consistent, monotone and stable. • Extensive numerical experiments on the model and numerical methods using a model investment problem in an iron-ore industry have been performed. The numerical results show that our model and numerical methods for solving the model are able to produce numerical results which are financially meaningful. In this paper we develop a PDE-based mathematical model for valuing real options on the expansion of an investment project whose underlying commodity price and its volatility follow their respective geometric Brownian motions. This mathematical model is of the form of a 2-dimensional Black-Scholes equation whose payoff condition is determined also by a PDE system. A novel 9-point finite difference scheme is proposed for the discretization of the spatial derivatives and the fully implicit time-stepping scheme is used for the time discretization of the PDE systems. We show that the coefficient matrix of the fully discretized system is an M -matrix and prove that the solution generated by this finite difference scheme converges to the exact one when the mesh sizes approach zero. To demonstrate the usefulness and effectiveness of the mathematical model and numerical method, we present a case study on a real option pricing problem in the iron-ore mining industry. Numerical experiments show that our model and methods are able to produce numerical results which are financially meaningful. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that the system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation, we propose and analyze some parallel schemes for solving a system of extended general quasi variational inequalities. The convergence of these new schemes is discussed under some mild conditions. Several special cases are discussed. Results obtained in this paper continue to hold for these problems. The ideas and techniques of this paper may stimulate further research in this dynamic field. [ABSTRACT FROM AUTHOR]

Abstract: For solving Volterra functional differential equations (VFDEs), a class of discrete Runge–Kutta methods based on canonical interpolation is studied, which was first presented by the same author in a previous published paper, classical stability and convergence theories for this class of methods are established. The methods studied and the theories established in this paper can be directly applied to non-stiff non-linear initial value problems in delay differential equations (DDEs), integro-differential equations (IDEs), delay integro-differential equations (DIDEs), and VFDEs of other type which appear in practice, and can be used as a necessary basis for the study of splitting methods for complex nonlinear stiff VFDEs. [Copyright &y& Elsevier]

Abstract: This paper deals with a family of balanced implicit methods with linear interpolation for the stochastic pantograph equations with jumps. In this paper, the strong mean-square convergence theory is established for the numerical solutions of the system. It is shown that the balanced implicit methods, which are fully implicit methods, give strong convergence rate of at least 1/2. For a linear scalar test equation, the balanced implicit methods are shown to capture the mean-square stability for all sufficiently small time-steps under appropriate conditions. Furthermore, weak variants are also considered and their mean-square stability analyzed. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities. [Copyright &y& Elsevier]

Abstract: The problem to describe the Bernstein polynomials of unbounded functions goes back to Lorentz. The aim of this paper is to investigate the convergence properties of the q-Bernstein polynomials of the Cauchy kernel with a pole for . The previously obtained results allow one to describe these properties when a pole is different from for some . In this context, the focus of the paper is on the behavior of polynomials for the functions of the form and . Here, the problem is examined both theoretically and numerically in detail. [Copyright &y& Elsevier]

Abstract: In stochastic differential equations (SDEs), there is a class of stochastic functional differential equations with random jump magnitudes, which aries in many financial models. In general most equations of stochastic age-dependent capital system do not have explicit solutions. Thus numerical approximation schemes are invaluable tools for exploring their properties. In this paper, the numerical approximation is established for a class of stochastic age-dependent capital system with random jump magnitudes. The main aim of this paper is to investigate the convergence of the numerical approximation for a class of stochastic age-dependent capital system with random jump magnitudes. It is proved that the numerical approximate solutions converge to the analytical solutions of the equations under given conditions. The numerical approximate results in Zhang et al. (2011) [2] are improved. An example is given for illustration. [Copyright &y& Elsevier]

A stability theory of analytic and numerical solutions to linear impulsive delay differential equations(IDDEs) is established. The stability results in existing literature are extended to IDDEs with variable impulses. A convergent numerical process is proposed to calculate numerical solutions to IDDEs with variable impulses. Convergence and stability of the numerical solutions are studied in the paper. Numerical experiments are given in the end to confirm the conclusion. [ABSTRACT FROM AUTHOR]

Abstract In this paper, combined the regular splittings of the coefficient matrix I − α P with the inner-outer iteration framework [9], a general inner-outer (GIO) iteration method is presented for solving the PageRank problem. Firstly, the AOR and modified AOR (MAOR) methods for solving the PageRank problem are constructed, and several comparison results are also given. Next, the GIO iteration scheme is developed, and its overall convergence is analyzed in detail. Furthermore, the preconditioner derived from the GIO iteration can be used to accelerate the Krylov subspace methods, such as GMRES method. Finally, some numerical experiments on several PageRank problems are provided to illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the max-product version of the generalized sampling operators based upon a general kernel function. In particular, we prove pointwise and uniform convergence for the above operators, together with a certain quantitative Jackson-type estimate based on the first order modulus of continuity of the function being approximated. The proof of the proposed results are based on the definition of the so-called generalized absolute moments. By the proposed approach, the achieved approximation results can be applied for several type of kernels, not necessarily duration-limited, such as the sinc-function, the Fejér kernel and many others. Examples of kernels with compact support for which the above theory holds can be given, for example, by the well-known central B-splines. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a fast and efficient numerical method to solve the two-dimensional integro-differential equation with a weakly singular kernel. The numerical method are considered by finite difference approach for spatial discretization and alternating direction implicit (ADI) method in time, combined with the second-order fractional quadrature convolution rule introduced by Lubich and the classical L 1 approximation for Caputo fractional derivative. The detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order O (τ min { 1 + α , 2 − α } + h 1 2 + h 2 2). Some numerical results are also given to confirm our theoretical prediction. [ABSTRACT FROM AUTHOR]

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

Abstract In this paper, we present a numerical solution for a finite time complete tracking problem based on the iterative learning control technique for dynamical systems governed by partial differential inclusions of parabolic type with noninstantaneous impulses. By imposing a standard Lipschitz condition on a set-valued mapping and applying conventional P-type updating laws with an initial iterative learning mechanism, we successfully establish an iterative learning process for the tracking problem and conduct a novel convergence analysis with the help of Steiner-type selectors. Sufficient conditions are presented for ensuring asymptotical convergence of the tracking error to zero. Numerical examples are provided to verify the effectiveness of the proposed method with a suitable selection of set-valued mappings. [ABSTRACT FROM AUTHOR]

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

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, we propose an efficient linearly implicit conservative difference solver for the fractional Klein–Gordon–Zakharov system. First of all, we present a detailed derivation of the energy conservation property of the system in the discrete setting. Then, by using the mathematical induction, it is proved that the proposed scheme is uniquely solvable. Subsequently, by virtue of the discrete energy method and a 'cut-off' function technique, it is shown that the proposed solver possesses the convergence rates of O (Δ t 2 + h 2) in the sense of L ∞- and L 2- norms, respectively, and is unconditionally stable. Finally, numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results. [ABSTRACT FROM AUTHOR]

Abstract The classical continuous Runge–Kutta methods are widely applied to compute the numerical solutions of delay differential equations without impulsive perturbations. However, the classical continuous Runge–Kutta methods cannot be applied directly to impulsive delay differential equations, because the exact solutions of the impulsive delay differential equations are not continuous. In this paper, impulsive continuous Runge–Kutta methods are constructed for impulsive delay differential equations with the variable delay based on the theory of continuous Runge–Kutta methods, convergence of the constructed numerical methods is studied and some numerical examples are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract Recently, Njeru and Guo presented an accelerated SOR-like (ASOR) method for solving the large and sparse augmented systems. In this paper, we establish an accelerated symmetric SOR-like (ASSOR) method, which is an extension of the ASOR method. Furthermore, the convergence properties of the ASSOR method for augmented systems are studied under suitable restrictions, and the functional equation between the iteration parameters and the eigenvalues of the relevant iteration matrix is established in detail. Finally, numerical examples show that the ASSOR is an efficient iteration method. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical solutions of nonlinear time fractional reaction–diffusion equations with time delay. A linearized compact finite difference scheme is proposed to solve the equations. In terms of a new developed fractional Gronwall type inequality, convergence and stability of the proposed scheme are obtained. Numerical experiments are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations is considered with nonsmooth initial data. Our numerical scheme is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms for the temporal discretization, standard Galerkin finite element method is used to the spatial discretization. In order to improve the computational efficiency, the decoupled method is introduced, as a consequence the original problem is split into two linear subproblems, and these subproblems can be solved in parallel. We verify that our numerical scheme is almost unconditionally stable for the nonsmooth initial data ( u 0 , θ 0 ) with the divergence-free condition. Furthermore, under some stability conditions, we show that the error estimates for velocity and temperature in L 2 norm is of the order O ( h 2 + Δ t 3 2 ) , in H 1 norm is of the order O ( h 2 + Δ t ) , and the error estimate for pressure in a certain norm is of the order O ( h 2 + Δ t ) . Finally, some numerical examples are provided to verify the established theoretical findings and test the performances of the developed numerical scheme. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a kind of Ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings is introduced and studied in convex metric spaces. Under some suitable conditions, the convergence theorems concerned with the Ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings were proved in convex metric spaces. The results presented in the paper generalize and improve some recent results of Wang and Liu (C. Wang, L.W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal., TMA 70 (2009), 2067–2071). [Copyright &y& Elsevier]

Abstract: In this paper, it is considered for a class of stochastic age-structured population equations with diffusions and Markovian switching. Most kind of equations are nonlinear and cannot be solved explicitly, so the construction of efficient computational methods is of great importance. The main aim of this paper is to develop a numerical scheme and investigate the convergence of numerical approximation. An example is given for illustration. [Copyright &y& Elsevier]

Abstract: For the augmented system of linear equations, Golub et al. [G.H. Golub, X. Wu, J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] studied an SOR-like method, by further accelerating it with another parameter, Bai et al. [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems Numer. Math. 102 (2005) 1–38] gave out a generalized SOR method. By considering a new splitting of the coefficient matrix, this paper presents another generalization of the SOR-like method (GSOR-like) which is different from the method in the Bai et al.’s paper (2005), and mainly discusses the selection of the optimal parameters. Theoretical analyses show that the convergence region for the relaxation parameter in our method properly contains that of the Golub et al.’s paper (2001) and our method has the same optimal asymptotic convergence rate with the method in the Bai et al.’s paper (2005). Further, the numerical example given shows the superiority of the GSOR-like method. [Copyright &y& Elsevier]

Abstract: In this paper, we point out that the generalized iteration method presented in a recent paper by Ling and Hu [X.-F. Ling, X.-Z. Hu, On the iterative algorithm for large sparse saddle point problems, Appl. Math. Comput. 178 (2006) 372–379] belongs to the inexact Uzawa algorithm, and an improper statement in a theorem in that paper is corrected. [Copyright &y& Elsevier]

Abstract: In a recent paper [M.A. Noor, K.I. Noor, Three-step iterative methods for nonlinear equations, Applied Mathematics and Computation 183 (2006) 322–327], a three-step iterative method with third-order convergence for solving nonlinear equations has been presented. In this paper, we show that the iterative method given by the authors has quadratic convergence, not the third-order one suggested in their work. [Copyright &y& Elsevier]

Abstract: In this paper, a feedback neural network model is presented by two steps. Firstly, a convex sub-optimization problem with bound constraints is established by introducing an energy function and the neural subnetwork for solving the sub-optimization problem is constructed based on the projection method. Secondly, a feedback neural network is proposed by using the subnetwork and can converge to an exact optimal solution of primal optimization problem. The distinguishing features of the proposed feedback network are no Lagrange multipliers, no dual variables, and no penalty parameters. It has the least number of state variables, simple structure, and is suitable for hardware implementation. Two simulation examples are provided to show the feasibility and efficiency of proposed method in the paper. [Copyright &y& Elsevier]

Abstract: In a recent paper, Noor [M. Aslam Noor, New classes of iterative methods for nonlinear equations, Appl. Math. Comput., 2007, doi:10.1016/j.amc:2007], suggested and analyzed a generalized one parameter Halley method for solving nonlinear equations using. In this paper, we modified this method which has fourth order convergence. As special cases, we obtain a family of third-order iterative methods for appropriate and suitable choice of the parameter. We have compared this modified Noor method with some other iterative methods which shows that this new iterative method is robust and efficient one. Several examples are given to illustrate the efficiency and the performance of this new method. [Copyright &y& Elsevier]

Abstract: In a recent paper, Noor and Noor [K. Inayat Noor, M. Aslam Noor, Predictor–corrector Halley method for nonlinear equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.11.023] have suggested and analyzed a predictor–corrector method Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method is a robust one. Several examples are given to illustrate the efficiency and the performance of this new method. [Copyright &y& Elsevier]

Abstract: In this paper, we present a simple and elegant procedure for the construction of iterative methods for the simultaneous determination of (simple or multiple) zeros of an algebraic polynomial. This procedure is based on the application of a special type of functions, called Weierstrass’ functions, to suitable zero-finding methods for a single zero. For demonstration, using this approach we derive many known iterative methods in a simpler way compared with original derivations, as well as some new methods. Aside from the presented methodology in developing zero-finding methods, the paper offers a short review of simultaneous methods including some historical notes. [Copyright &y& Elsevier]

Abstract: This paper deals with the convergence of discrete approximations to the optimization problem (P) for a neutral functional–differential inclusion subject to general endpoint constraints. In the first part of the paper, discrete approximations to the neutral functional–differential inclusion are constructed using Euler finite difference methods and the convergence of discrete approximations is proved. In the second part of the paper, a family of discrete optimization problems (P N ) to (P) is provided and the strong convergence of optimal solutions for (P N ) to the optimal solution of (P) is discussed. [Copyright &y& Elsevier]

The linear complementarity problem can be solved by modified AOR method given in [Appl. Math. Comput. 140 (2003) 53]. In the same paper the convergence was proved for the H-matrix case, using the estimation of spectral radius of corresponding matrix. In this paper we present the other possibility for obtaining convergence result. We use the estimation of maximum norm, and surprisingly, obtain convergence area which can be better. First, we consider SDD (strictly diagonally dominant) matrix case, and after that H-matrix case. [Copyright &y& Elsevier]

• We have added two words "A" and "the" in the title, added two new references and provided a new rigorous proof of convergence of the new iteration method in Section 2. Furthermore, we have further improved the English writing of the paper. • As suggested by Referee 1, we have revised some citation problems and given some symbol introductions. • As suggested by Referee 2, we have revised some equations and further improved the English writing of the paper. • As suggested by Referee 3, we have proposed a more rigorous proof to discuss the convergence of the new iteration method; see Lemma 2.2 and Theorem 2.2. For the non-Hermitian generalized saddle point problems, we propose a new constraint preconditioner. The new constraint preconditioner is constructed based on the preconditioned generalized shift-splitting (PGSS) iteration method. We also analyze the invertibility condition of the new preconditioner in detail. Moreover, the convergence properties of the new constraint preconditioning iteration method are derived. Finally, the effectiveness of the proposed preconditioner is illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a discrete form is proposed for solving time fractional convection–diffusion equation. Firstly, we obtain a time discrete scheme based on finite difference method. Secondly, we prove that the time discrete scheme is unconditionally stable, and the numerical solution converges to the exact one with order O ( τ 2 − α ) , where τ is the time step size. Finally, two numerical examples are proposed respectively, to verify the order of convergence. [ABSTRACT FROM AUTHOR]

In this paper, by transforming the original problem equivalently, we propose a new preconditioned iterative method for solving saddle point problem. We call the new method as PTU (preconditioned transformative Uzawa) method. And we study the convergence of the PTU method under suitable restrictions on the iteration parameters. Moreover, we show the choices of the optimal parameters and the spectrum of the preconditioned matrix deriving from the PTU method. Based on the PTU iterative method, we propose another iterative method – nonlinear inexact PTU method – for solving saddle point problem. We also prove its convergence and study the choices of the optimal parameters. In addition, we present some numerical results to illustrate the behavior of the considered algorithms. [ABSTRACT FROM AUTHOR]

The main goal of this paper is the numerical computation of solutions of the so-called Yang–Baxter-like matrix equation A X A = X A X , where A is a given complex square matrix. Two novel matrix iterations are proposed, both having second-order convergence. A sign modification in one of the iterations gives rise to a third matrix iteration. Strategies for finding starting approximations are discussed as well as a technique for estimating the relative error. One of the methods involves a very small cost per iteration and is shown to be stable. Numerical experiments are carried out to illustrate the effectiveness of the new methods. [ABSTRACT FROM AUTHOR]

Nonlinear wave equation is extensively applied in a wide variety of scientific fields, such as nonlinear optics, solid state physics and quantum field theory. In this paper, two high-performance compact alternating direction implicit (ADI) methods are developed for the nonlinear wave equations. The first scheme is developed a three-level nonlinear difference scheme for nonlinear wave equations, where in x -direction, series of linear tridiagonal systems are solved by Thomas algorithm, while in y -direction, nonlinear algebraic system are computed by Newton’s iterative method. In contrast, the second scheme is linear, and permits the multiple uses of the Thomas algorithm in both x - and y -directions, thus it saves much time cost. By using the discrete energy analysis method, it is shown that both the developed schemes can attain numerical accuracy of order O ( τ 4 + h x 4 + h y 4 ) in H 1 -norm. Meanwhile, by the fixed point theorem and symmetric positive-definite properties of coefficient matrix, it is proved that they are both uniquely solvable. Besides, the proposed schemes are extended to the numerical solutions of the coupled sine-Gordon wave equations and damped wave equations. Finally, numerical results confirm the convergence orders and exhibit efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the convergence and semi-convergence of a class of SSOR-like methods with four real functions for augmented systems. The class takes most existed SSOR-like methods as its special cases. For nonsingular systems, we obtain the minimum of convergence factors of all the SSOR-like methods in the class, and study when it can be reached by the convergence factors of methods in the class. By considering the equivalence of methods, we show that most of the existed SSOR-like methods have the same minimum of convergence factors. [ABSTRACT FROM AUTHOR]

Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with the Robin condition are approximated in the paper by solutions of associated boundedness-preserving implicit difference functional equations. It is proved that the discrete solutions uniquely exist, they are uniformly bounded with respect to meshes and the numerical method is convergent and stable. We also find the error estimate and its asymptotic behavior. The properties of some auxiliary nonlinear discrete recurrent equations are showed. The proofs are based on the comparison technique and the Banach fixed-point theorem. [ABSTRACT FROM AUTHOR]