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Abstract: This paper deals with the rate of approximation for the Bézier variant of certain operators. Here we estimate the rate of convergence in simultaneous approximation. [Copyright &y& Elsevier]

Abstract: An extension of some smooth parametrized Newton-like method to the semismooth case is presented in this paper. Thus, the problems due to the fact the B-differential is singular are solved. These methods are shown to be superlinear convergence. Some numerical experiments are given. [Copyright &y& Elsevier]

In this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations (ODEs), y ′ ′ ′ = f ( x , y ) ; namely, hybrid methods for solving third order ODEs directly (HMTD). Using the theory of B-series, order of convergence of the HMTD methods is investigated. The main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two-step hybrid method. A three-stage explicit HMTD is constructed. Results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we suggest and analyze a class of unified iteration schemes with errors for the variational inequalities coupled with nonexpansive mappings in Hilbert spaces. The main results of this paper extend the main results of Iiduka and Takahashi [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350], and Takahasi and Toyoda [W. Takahasi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2) (2003) 417–428] to the case of strong convergence for relaxed (γ, r)-cocoercive mappings involved in variational inequalities. Results obtained in this paper may be viewed as an refinement of the previously known results in this field. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new three-step iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality using the technique of updating the solution. We show that the sequence converges strongly to a common element of two sets under some control conditions. Results proved in this paper may be viewed as an improvement and refinement of the recent results of Noor and Huang [M. Aslam Noor, Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput., in press] and Yao and Yao [Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., in press]. [Copyright &y& Elsevier]

Abstract: This paper demonstrates the use of higher order methods to solve some time-dependent stiff PDEs. In the past, the most popular numerical methods for solving system of reaction–diffusion equations was based on the combination of low order finite difference method with low order time-stepping method. We extend in this report the compatibility of fourth-order finite difference scheme (in space) coupled with fourth-order time-stepping methods such as IMEXLM4, IMEXPC4, IMEXRK4 and ETDRK4-B (in time), for direct integration of reaction–diffusion equations in one space dimension. Some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries. Our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio-temporal dynamical structures. [Copyright &y& Elsevier]

Abstract: In this paper, we present a variant of Jarratt method for solving non-linear equations. Per iteration the new method adds the evaluation of the function at another point in the procedure iterated by Jarratt method. As a consequence, the local order of convergence is improved from four for Jarratt method to six for the new method. A new multistep iteration scheme, based on the new method, is developed and numerical tests verifying the theory are also given. [Copyright &y& Elsevier]

Abstract: This paper studies the multi-splitting algorithm and the two-stage iterative method for waveform relaxation (WR) method, solving the initial value problems for linear systems of ordinary differential equations (ODEs) in the form . Convergence of both cases are analyzed when the coefficient matrix is Hermitian positive definite and the splittings are P-regular. Finally, numerical experiment is presented. [Copyright &y& Elsevier]

Abstract: In this paper, we study the oscillatory and asymptotic properties of solutions of higher order nonlinear neutral difference equations. Using Krasnoselskii’s fixed point theorem, Schauder’s fixed point theorem and Lebesgue’s dominated convergence theorem, we obtain some necessary and sufficient conditions for all bounded solutions to be oscillatory or tend to zero. Our results extend and improve some known results. [Copyright &y& Elsevier]

Abstract: The unsymmetric block overrelaxation-type (UBOR-type) method is proposed to attack saddle point problems in this paper. The convergence and the optimal parameters for the method are studied when the iteration parameters satisfy some relationship. Theoretical analyses show that the UBOR-type method has faster asymptotic convergence rate than the SOR-like method and its convergence rate can reach the same as that of the GSOR method at least. Numerical experiments support our theoretical results. Moveover, the numerical results further reveal that the new method can be much more effective than the GSOR method in terms of iteration steps. [Copyright &y& Elsevier]

Abstract: In this paper, we show one way of deriving a fourth order scheme from any third order Newton type method without any additional evaluation of the function or its derivatives. For this purpose, the rectangular rule may be employed in Newton’s theorem about a special point. This special point may be selected from a variant of any third order scheme. Two such cases are illustrated here. The first case leads to a new fourth order scheme while the second case leads to the well known Jarratt’s method. The efficiency index of the proposed scheme is 1.587. In order to compare its performance with some of the existing third and fourth order schemes several numerical examples are furnished here. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to develop an improved method for finding the multiple roots of nonlinear equations in along with an approximation to it’s multiplicity when it is not known explicitly. This is done by first describing a derivative free transformation of which reduces the multiple roots of to simple roots of . Then, a second order Newton-type method is used to compute the simple roots of and the approximation to the multiplicity of the roots of . The method is tested on two numerical examples considered by King [R.F. King, A secant method for multiple roots, BIT 17 (1977) 321–328] and results obtained are compared. It is found that our method is more efficient than that given by King and obtain multiple roots as well as it’s multiplicity much faster. [Copyright &y& Elsevier]

Abstract: In this paper, control of Hopf bifurcation for a plankton population model with a non-integer exponent of closure is studied. For the plankton model with the closure term varies continuously between the commonly used linear and quadratic forms, washout filter aided feedback controllers are employed to control the stability of the Hopf bifurcation, and thus the chaos could be eliminated. By employing the stability theorems derived, the stability of a control system is investigated. Finally numerical simulations are provided to show the effectiveness and feasibility of the developed methods. [Copyright &y& Elsevier]

Abstract: In the present paper, we introduce the q-analog of Chlodowsky operators. We study some approximation properties of these new operators, which include the rate of convergence and investigation of the monotonocity property of q-Chlodowsky operators. [Copyright &y& Elsevier]

Abstract: For current sequential programming (SQP) type algorithms, there exist two problems. One is that there are many quadratic programming subproblems are needed to be solved per iteration, the other is that the search direction may be unbounded and caused the sequence to be divergent. In this paper, we presented a modified SQP-filter method based on the modified quadratic subproblem proposed in Zhou [G.L. Zhou, A modified SQP method and its global convergence, J. Global Optim. 11 (1997) 193–205]. This method has no demand on the initial point, and need not using a penalty parameter, which could be problematic to obtain. What is more, the subproblem is feasible at each iterate point. Under some conditions, the global convergence property is obtained. [Copyright &y& Elsevier]

Abstract: It is well known that a possibly nonsmooth convex minimization problem can be transformed into a differentiable convex optimization problem by using the Moreau-Yosida regularization. This paper presents a new trust region method to solve the latter problem. Under some reasonable assumptions, the proposed algorithm is shown to be globally and Q-superlinearly convergent. [Copyright &y& Elsevier]

The study of stochastic partial different equations is motivated by the fact that when one wants to model some evolution phenomena arising in physics, biology and engineering etc. However, in many applications the resulting equations have so far proved intractable to direct analytical solution. Numerical approximations, such as the Euler scheme, are therefore a vital tool in exploring model behaviour. Unfortunately, current results concerning the convergence of such schemes impose conditions on the drift and diffusion coefficients of the stochastic partial differential equation, namely the linear growth and global Lipschitz conditions, which are often not met by systems of interest. In this paper, these conditions are relaxed and prove that numerical solutions based on the Euler scheme will converge to the true solution of a broad class of stochastic partial differential equations. Several examples are studied to illustrate the results. [Copyright &y& Elsevier]