Advances in Numerical Analysis http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Analysis of Subgrid Stabilization Method for Stokes-Darcy Problems Thu, 22 Sep 2016 14:19:10 +0000 http://www.hindawi.com/journals/ana/2016/7389102/ A number of techniques, used as remedy to the instability of the Galerkin finite element formulation for Stokes like problems, are found in the literature. In this work we consider a coupled Stokes-Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous mixed finite elements in the two parts. A better method, from a computational point of view, consists in using a unified approach on both subdomains. Here, the coupled Stokes-Darcy problem is analyzed using equal-order velocity and pressure approximation combined with subgrid stabilization. We prove that the obtained finite element solution is stable and converges to the classical solution with optimal rates for both velocity and pressure. Kamel Nafa Copyright © 2016 Kamel Nafa. All rights reserved. Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows Mon, 19 Sep 2016 13:01:27 +0000 http://www.hindawi.com/journals/ana/2016/7010645/ Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction. S. S. Ravindran Copyright © 2016 S. S. Ravindran. All rights reserved. Lebesgue Constant Using Sinc Points Mon, 19 Sep 2016 09:21:25 +0000 http://www.hindawi.com/journals/ana/2016/6758283/ Lebesgue constant for Lagrange approximation at Sinc points will be examined. We introduce a new barycentric form for Lagrange approximation at Sinc points. Using Thiele’s algorithm we show that the Lebesgue constant grows logarithmically as the number of interpolation Sinc points increases. A comparison between the obtained upper bound of Lebesgue constant using Sinc points and other upper bounds for different set of points, like equidistant and Chebyshev points, is introduced. Maha Youssef, Hany A. El-Sharkawy, and Gerd Baumann Copyright © 2016 Maha Youssef et al. All rights reserved. A Finite Volume-Complete Flux Scheme for a Polluted Groundwater Site Wed, 24 Aug 2016 13:44:03 +0000 http://www.hindawi.com/journals/ana/2016/1945958/ We present a model of a polluted groundwater site. The model consists of a coupled system of advection-diffusion-reaction equations for the groundwater level and the concentration of the pollutant. We use the complete flux scheme for the space discretization in combination with the -method for time integration and we prove a new stability result for the scheme. Numerical results are computed for the Guarani Aquifer in South America and they show good agreement with results in literature. M. F. P. ten Eikelder, J. H. M. ten Thije Boonkkamp, M. P. T. Moonen, and B. V. Rathish Kumar Copyright © 2016 M. F. P. ten Eikelder et al. All rights reserved. Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions Thu, 18 Aug 2016 11:55:30 +0000 http://www.hindawi.com/journals/ana/2016/1397849/ In radial basis function approximation, the shape parameter can be variable. The values of the variable shape parameter strategies are selected from an interval which is usually determined by trial and error. As yet there is not any algorithm for determining an appropriate interval, although there are some recipes for optimal values. In this paper, a novel algorithm for determining an interval is proposed. Different variable shape parameter strategies are examined. The results show that the determined interval significantly improved the accuracy and is suitable enough to count on in variable shape parameter strategies. Jafar Biazar and Mohammad Hosami Copyright © 2016 Jafar Biazar and Mohammad Hosami. All rights reserved. Analysis of Linear Piecewise Constant Delay Systems Using a Hybrid Numerical Scheme Sun, 07 Aug 2016 13:29:54 +0000 http://www.hindawi.com/journals/ana/2016/8903184/ An efficient computational technique for solving linear delay differential equations with a piecewise constant delay function is presented. The new approach is based on a hybrid of block-pulse functions and Legendre polynomials. A key feature of the proposed framework is the excellent representation of smooth and especially piecewise smooth functions. The operational matrices of delay, derivative, and product corresponding to the mentioned hybrid functions are implemented to transform the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the proposed numerical scheme. H. R. Marzban and S. M. Hoseini Copyright © 2016 H. R. Marzban and S. M. Hoseini. All rights reserved. Remarks on Numerical Experiments of the Allen-Cahn Equations with Constraint via Yosida Approximation Tue, 07 Jun 2016 12:12:45 +0000 http://www.hindawi.com/journals/ana/2016/1492812/ We consider a one-dimensional Allen-Cahn equation with constraint from the viewpoint of numerical analysis. The constraint is provided by the subdifferential of the indicator function on the closed interval, which is the multivalued function. Therefore, it is very difficult to perform a numerical experiment of our equation. In this paper we approximate the constraint by the Yosida approximation. Then, we study the approximating system of the original model numerically. In particular, we give the criteria for the standard forward Euler method to give the stable numerical experiments of the approximating equation. Moreover, we provide the numerical experiments of the approximating equation. Tomoyuki Suzuki, Keisuke Takasao, and Noriaki Yamazaki Copyright © 2016 Tomoyuki Suzuki et al. All rights reserved. Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation Mon, 16 May 2016 10:05:30 +0000 http://www.hindawi.com/journals/ana/2016/8376061/ The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods. Asma Yosaf, Shafiq Ur Rehman, Fayyaz Ahmad, Malik Zaka Ullah, and Ali Saleh Alshomrani Copyright © 2016 Asma Yosaf et al. All rights reserved. Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes Sun, 28 Feb 2016 10:40:31 +0000 http://www.hindawi.com/journals/ana/2016/2161279/ A one-dimensional linear convection-diffusion problem with a perturbation parameter multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is -uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “-uniform stability plus -uniform consistency implies -uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in . At the same time, the condition number of the discrete system becomes independent of due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes. Thái Anh Nhan and Relja Vulanović Copyright © 2016 Thái Anh Nhan and Relja Vulanović. All rights reserved. The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation Tue, 17 Feb 2015 13:07:04 +0000 http://www.hindawi.com/journals/ana/2015/367056/ The exponential cubic B-spline algorithm is presented to find the numerical solutions of the Korteweg-de Vries (KdV) equation. The problem is reduced to a system of algebraic equations, which is solved by using a variant of Thomas algorithm. Numerical experiments are carried out to demonstrate the efficiency of the suggested algorithm. Ozlem Ersoy and Idris Dag Copyright © 2015 Ozlem Ersoy and Idris Dag. All rights reserved. Approximation Properties of a Gradient Recovery Operator Using a Biorthogonal System Tue, 03 Feb 2015 06:34:06 +0000 http://www.hindawi.com/journals/ana/2015/187604/ A gradient recovery operator based on projecting the discrete gradient onto the standard finite element space is considered. We use an oblique projection, where the test and trial spaces are different, and the bases of these two spaces form a biorthogonal system. Biorthogonality allows efficient computation of the recovery operator. We analyze the approximation properties of the gradient recovery operator. Numerical results are presented in the two-dimensional case. Bishnu P. Lamichhane and Adam McNeilly Copyright © 2015 Bishnu P. Lamichhane and Adam McNeilly. All rights reserved. A Fourth-Order Collocation Scheme for Two-Point Interface Boundary Value Problems Mon, 24 Nov 2014 12:55:44 +0000 http://www.hindawi.com/journals/ana/2014/875013/ A fourth-order accurate orthogonal spline collocation scheme is formulated to approximate linear two-point boundary value problems with interface conditions. The coefficients of the differential operator may have jump discontinuities at the interface point, a nodal point of the scheme. Existence and uniqueness of the numerical solution are proved. Optimal order error estimates in the maximum norm are obtained, and a superconvergence property of the numerical solution in the maximal nodal norm is proved. Numerical results are presented confirming the theoretical estimates. Rakhim Aitbayev and Nazgul Yergaliyeva Copyright © 2014 Rakhim Aitbayev and Nazgul Yergaliyeva. All rights reserved. A Method to Accelerate the Convergence of the Secant Algorithm Wed, 19 Nov 2014 09:08:14 +0000 http://www.hindawi.com/journals/ana/2014/321592/ We present an acceleration technique for the Secant method. The Secant method is a root-searching algorithm for a general function . We exploit the fact that the combination of two Secant steps leads to an improved, so-called first-order approximant of the root. The original Secant algorithm can be modified to a first-order accelerated algorithm which generates a sequence of first-order approximants. This process can be repeated: two th order approximants can be combined in a th order approximant and the algorithm can be modified to an th order accelerated algorithm which generates a sequence of such approximants. We show that the sequence of th order approximants converges to the root with the same order as methods using polynomial fits of of degree . M. J. P. Nijmeijer Copyright © 2014 M. J. P. Nijmeijer. All rights reserved. Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings Thu, 13 Nov 2014 07:18:11 +0000 http://www.hindawi.com/journals/ana/2014/181049/ Let be a nonempty closed and convex subset of a uniformly convex real Banach space and let be multivalued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence is proved to be an approximating fixed point sequence of each ; that is, . Then, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)). C. Diop, M. Sene, and N. Djitté Copyright © 2014 C. Diop et al. All rights reserved. Identifying an Unknown Coefficient in the Reaction-Diffusion Equation Using He’s VIM Sun, 19 Oct 2014 11:20:05 +0000 http://www.hindawi.com/journals/ana/2014/547973/ An inverse heat problem of finding an unknown parameter p(t) in the parabolic initial-boundary value problem is solved with variational iteration method (VIM). For solving the discussed inverse problem, at first we transform it into a nonlinear direct problem and then use the proposed method. Also an error analysis is presented for the method and prior and posterior error bounds of the approximate solution are estimated. The main property of the method is in its flexibility and ability to solve nonlinear equation accurately and conveniently. Some examples are given to illustrate the effectiveness and convenience of the method. F. Parzlivand and A. M. Shahrezaee Copyright © 2014 F. Parzlivand and A. M. Shahrezaee. All rights reserved. A Third-Order Newton-Type Method for Finding Polar Decomposition Tue, 30 Sep 2014 12:08:45 +0000 http://www.hindawi.com/journals/ana/2014/576325/ It is attempted to present an iteration method for finding polar decomposition. The approach is categorized in the scope of Newton-type methods. Error analysis and rate of convergence are studied. Some illustrations are also given to disclose the numerical behavior of the proposed method. F. Khaksar Haghani Copyright © 2014 F. Khaksar Haghani. All rights reserved. The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor Sun, 28 Sep 2014 08:06:00 +0000 http://www.hindawi.com/journals/ana/2014/109525/ We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor . The main contribution of this paper is to show that when is perturbed to a nonnegative tensor by , the absolute difference between the spectral radii of and is bounded by the largest magnitude of the ratio of the ith component of and the ith component , where is an eigenvector associated with the largest eigenvalue of in magnitude and its entries are positive. We further derive the bound in terms of the entries of only when is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis. Wen Li and Michael K. Ng Copyright © 2014 Wen Li and Michael K. Ng. All rights reserved. On the Stream Function-Vorticity Finite Element Formulation for Incompressible Flow in Porous Media Thu, 18 Sep 2014 06:14:19 +0000 http://www.hindawi.com/journals/ana/2014/459761/ Stream function-vorticity finite element formulation for incompressible flow in porous media is presented. The model consists of the heat equation, the equation for the concentration, and the equations of motion under the Darcy law. The existence of solution for the discrete problem is established. Optimal a priori error estimates are given. Abdellatif Agouzal, Karam Allali, and Siham Binna Copyright © 2014 Abdellatif Agouzal et al. All rights reserved. Error Analysis for a Noisy Lacunary Cubic Spline Interpolation and a Simple Noisy Cubic Spline Quasi Interpolation Wed, 10 Sep 2014 09:10:39 +0000 http://www.hindawi.com/journals/ana/2014/353194/ We mainly present the error analysis for two new cubic spline based methods; one is a lacunary interpolation method and the other is a very simple quasi interpolation method. The new methods are able to reconstruct a function and its first two derivatives from noisy function data. The explicit error bounds for the methods are given and proved. Numerical tests and comparisons are performed. Numerical results verify the efficiency of our methods. Feng-Gong Lang and Xiao-Ping Xu Copyright © 2014 Feng-Gong Lang and Xiao-Ping Xu. All rights reserved. A Meshless Method for the Numerical Solution of a Two-Dimension IHCP Sun, 03 Aug 2014 06:45:04 +0000 http://www.hindawi.com/journals/ana/2014/456142/ This paper uses the collocation method and radial basis functions (RBFs) to analyze the solution of a two-dimension inverse heat conduction problem (IHCP). The accuracy of the method is tested in terms of Error and RMS errors. Also, the stability of the technique is investigated by perturbing the additional specification data by increasing the amounts of random noise. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme. F. Parzlivand and A. M. Shahrezaee Copyright © 2014 F. Parzlivand and A. M. Shahrezaee. All rights reserved. An Efficient Family of Traub-Steffensen-Type Methods for Solving Systems of Nonlinear Equations Wed, 02 Jul 2014 09:14:36 +0000 http://www.hindawi.com/journals/ana/2014/152187/ Based on Traub-Steffensen method, we present a derivative free three-step family of sixth-order methods for solving systems of nonlinear equations. The local convergence order of the family is determined using first-order divided difference operator for functions of several variables and direct computation by Taylor's expansion. Computational efficiency is discussed, and a comparison between the efficiencies of the proposed techniques with the existing ones is made. Numerical tests are performed to compare the methods of the proposed family with the existing methods and to confirm the theoretical results. It is shown that the new family is especially efficient in solving large systems. Janak Raj Sharma and Puneet Gupta Copyright © 2014 Janak Raj Sharma and Puneet Gupta. All rights reserved. On Nonlinear Inverse Problems of Heat Transfer with Radiation Boundary Conditions: Application to Dehydration of Gypsum Plasterboards Exposed to Fire Sun, 13 Apr 2014 13:39:37 +0000 http://www.hindawi.com/journals/ana/2014/634712/ The paper investigates boundary optimal controls and parameter estimates to the well-posedness nonlinear model of dehydration of thermic problems. We summarize the general formulations for the boundary control for initial-boundary value problem for nonlinear partial differential equations modeling the heat transfer and derive necessary optimality conditions, including the adjoint equation, for the optimal set of parameters minimizing objective functions . Numerical simulations illustrate several numerical optimization methods, examples, and realistic cases, in which several interesting phenomena are observed. A large amount of computational effort is required to solve the coupled state equation and the adjoint equation (which is backwards in time), and the algebraic gradient equation (which implements the coupling between the adjoint and control variables). The state and adjoint equations are solved using the finite element method. A. Belmiloudi and F. Mahé Copyright © 2014 A. Belmiloudi and F. Mahé. All rights reserved. General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods Mon, 31 Mar 2014 16:39:23 +0000 http://www.hindawi.com/journals/ana/2014/963878/ A general scheme of third order convergence is described for finding multiple roots of nonlinear equations. The proposed scheme requires one evaluation of , and each per iteration and contains several known one-point third order methods for finding multiple roots, as particular cases. Numerical examples are included to confirm the theoretical results and demonstrate convergence behavior of the proposed methods. In the end, we provide the basins of attraction for some methods to observe their dynamics in the complex plane. Rajni Sharma and Ashu Bahl Copyright © 2014 Rajni Sharma and Ashu Bahl. All rights reserved. Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions Thu, 30 Jan 2014 07:25:34 +0000 http://www.hindawi.com/journals/ana/2014/504825/ Fractal interpolation is an advanced technique for analysis and synthesis of scientific and engineering data. We introduce the -rational quadratic fractal interpolation functions (FIFs) through a suitable rational quadratic iterated function system (IFS). The novel notion of shape preserving fractal interpolation without any shape parameter is introduced through the rational fractal interpolation model in the literature for the first time. For a prescribed set of monotonic data, we derive the sufficient conditions by restricting the scaling factors for shape preserving -rational quadratic FIFs. A local modification pertaining to any subinterval is possible in this model if the scaling factors are chosen appropriately. We establish the convergence results of a monotonic rational quadratic FIF to the original function in . For given data with derivatives at grids, our approach generates several monotonicity preserving rational quadratic FIFs, whereas this flexibility is not available in the classical approach. Finally, numerical experiments support the importance of the developed rational quadratic IFS scheme through construction of visually pleasing monotonic rational fractal curves including the classical one. A. K. B. Chand and N. Vijender Copyright © 2014 A. K. B. Chand and N. Vijender. All rights reserved. Nonlinear Double-Layer Bragg Waveguide: Analytical and Numerical Approaches to Investigate Waveguiding Problem Wed, 22 Jan 2014 08:08:52 +0000 http://www.hindawi.com/journals/ana/2014/231498/ The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function. The existence of propagating TE waves for chosen nonlinearity (the Kerr law) is proved using the contraction mapping method. Conditions under which k waves can propagate are obtained, and intervals of localization of the corresponding propagation constants are found. For numerical solution of the problem, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. In numerical experiment, two types of nonlinearities are considered and compared: the Kerr nonlinearity and nonlinearity with saturation. New propagation regime is found. Yury G. Smirnov, Eugenii Yu. Smol’kin, and Dmitry V. Valovik Copyright © 2014 Yury G. Smirnov et al. All rights reserved. A New Upper Bound for of a Strictly -Diagonally Dominant -Matrix Thu, 26 Dec 2013 15:23:54 +0000 http://www.hindawi.com/journals/ana/2013/980615/ A new upper bound for of a real strictly diagonally dominant -matrix is present, and a new lower bound of the smallest eigenvalue of is given, which improved the results in the literature. Furthermore, an upper bound for of a real strictly -diagonally dominant -matrix is shown. Zhanshan Yang, Bing Zheng, and Xilan Liu Copyright © 2013 Zhanshan Yang et al. All rights reserved. Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries Tue, 24 Dec 2013 09:51:56 +0000 http://www.hindawi.com/journals/ana/2013/967342/ We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries. Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded. First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface. The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface. This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient. The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised. Extensive numerical experiments indicate that this method is second-order accurate in the norm in both two and three dimensions and numerically very stable. Liqun Wang and Liwei Shi Copyright © 2013 Liqun Wang and Liwei Shi. All rights reserved. A Proper-Orthogonal Decomposition Variational Multiscale Approximation Method for a Generalized Oseen Problem Wed, 18 Dec 2013 13:13:31 +0000 http://www.hindawi.com/journals/ana/2013/974284/ We introduce the variational multiscale (VMS) stabilization for the reduced-order modeling of incompressible flows. It is well known that the proper orthogonal decomposition (POD) technique in reduced-order modeling experiences numerical instability when applied to complex flow problems. In this case a POD discretization naturally separates out structures which corresponding to the energy cascade on large and small scales, in order, a VMS approach is natural. In this paper, we provide the mathematical background necessary for implementing VMS to a POD-Galerkin model of a generalized Oseen problem. We provide theoretical evidence which indicates the consistency of utilizing a VMS approach in the stabilization of reduced order flows. In addition we provide numerical experiments indicating that VMS improves fidelity in reproducing the qualitative properties of the flow. John Paul Roop Copyright © 2013 John Paul Roop. All rights reserved. A New Extended Padé Approximation and Its Application Tue, 10 Dec 2013 12:02:37 +0000 http://www.hindawi.com/journals/ana/2013/263467/ We extend ordinary Padé approximation, which is based on a set of standard polynomials as , to a new extended Padé approximation (Müntz Padé approximation), based on the general basic function set    (the particular case of Müntz polynomials) using general Taylor series (based on fractional calculus) with error convergency. The importance of this extension is that the ordinary Padé approximation is a particular case of our extended Padé approximation. Also the parameterization ( is the corresponding parameter) of new extended Padé approximation is an important subject which, obtaining the optimal value of this parameter, can be a good question for a new research. Z. Kalateh Bojdi, S. Ahmadi-Asl, and A. Aminataei Copyright © 2013 Z. Kalateh Bojdi et al. All rights reserved. Some Results on Preconditioned Mixed-Type Splitting Iterative Method Tue, 03 Dec 2013 13:54:44 +0000 http://www.hindawi.com/journals/ana/2013/512084/ We present a preconditioned mixed-type splitting iterative method for solving the linear system , where is a Z-matrix. And we give some comparison theorems to show that the rate of convergence of the preconditioned mixed-type splitting iterative method is faster than that of the mixed-type splitting iterative method. Finally, we give one numerical example to illustrate our results. Guangbin Wang and Fuping Tan Copyright © 2013 Guangbin Wang and Fuping Tan. All rights reserved.