Advances in Numerical Analysis
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© 2016 , Hindawi Publishing Corporation . All rights reserved.

Remarks on Numerical Experiments of the AllenCahn 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 onedimensional AllenCahn 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.

EighthOrder 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 highorder compact finite difference method for solving onedimensional (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 eighthorder compact finite difference method for the entire domain and fourthorder 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 CrankNicholson 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 ConvectionDiffusion Problems Discretized on ShishkinType Meshes
Sun, 28 Feb 2016 10:40:31 +0000
http://www.hindawi.com/journals/ana/2016/2161279/
A onedimensional linear convectiondiffusion problem with a perturbation parameter multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layeradapted 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 convectiondiffusion 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 Shishkintype meshes.
Thái Anh Nhan and Relja Vulanović
Copyright © 2016 Thái Anh Nhan and Relja Vulanović. All rights reserved.

The Exponential Cubic BSpline Algorithm for Kortewegde Vries Equation
Tue, 17 Feb 2015 13:07:04 +0000
http://www.hindawi.com/journals/ana/2015/367056/
The exponential cubic Bspline algorithm is presented to find the numerical solutions of the Kortewegde 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 twodimensional case.
Bishnu P. Lamichhane and Adam McNeilly
Copyright © 2015 Bishnu P. Lamichhane and Adam McNeilly. All rights reserved.

A FourthOrder Collocation Scheme for TwoPoint Interface Boundary Value Problems
Mon, 24 Nov 2014 12:55:44 +0000
http://www.hindawi.com/journals/ana/2014/875013/
A fourthorder accurate orthogonal spline collocation scheme is formulated to approximate linear twopoint 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 rootsearching algorithm for a general function . We exploit the fact that the combination of two Secant steps leads to an improved, socalled firstorder approximant of the root. The original Secant algorithm can be modified to a firstorder accelerated algorithm which generates a sequence of firstorder 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 QuasiNonexpansive 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 quasinonexpansive 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 ReactionDiffusion 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 initialboundary 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 ThirdOrder NewtonType 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 Newtontype 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 mthorder ndimensional 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 FunctionVorticity 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 functionvorticity 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.
FengGong Lang and XiaoPing Xu
Copyright © 2014 FengGong Lang and XiaoPing Xu. All rights reserved.

A Meshless Method for the Numerical Solution of a TwoDimension 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 twodimension 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 TraubSteffensenType 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 TraubSteffensen method, we present a derivative free threestep family of sixthorder methods for solving systems of nonlinear equations. The local convergence order of the family is determined using firstorder 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 wellposedness nonlinear model of dehydration of thermic problems. We summarize the general formulations for the boundary control for initialboundary 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 onepoint 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 DoubleLayer 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 twolayered 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 SharpEdged Boundaries
Tue, 24 Dec 2013 09:51:56 +0000
http://www.hindawi.com/journals/ana/2013/967342/
We present a new secondorder accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with
sharpedged boundaries. Nontraditional finite element method with nonbodyfitting 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 nonsmooth 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 secondorder 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 ProperOrthogonal 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 reducedorder modeling of incompressible flows. It is well known that the proper orthogonal decomposition (POD) technique in reducedorder 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 PODGalerkin 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. AhmadiAsl, and A. Aminataei
Copyright © 2013 Z. Kalateh Bojdi et al. All rights reserved.

Some Results on Preconditioned MixedType Splitting Iterative Method
Tue, 03 Dec 2013 13:54:44 +0000
http://www.hindawi.com/journals/ana/2013/512084/
We present a preconditioned mixedtype splitting iterative method for solving the linear system , where is a Zmatrix. And we give some comparison theorems to show that the rate of convergence of the preconditioned mixedtype splitting iterative method is faster than that of the mixedtype 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.

On Some Efficient Techniques for Solving Systems of Nonlinear Equations
Thu, 31 Oct 2013 11:47:05 +0000
http://www.hindawi.com/journals/ana/2013/252798/
We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Thirdorder method is composed of two steps, namely, Newton iteration as the first step and weightedNewton iteration as the second step. Fifth and sixthorder methods are composed of three steps of which the first two steps are same as that of the thirdorder method whereas the third is again a weightedNewton step. Computational efficiency in its general form is discussed and a comparison between the efficiencies of proposed techniques with existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present methods have an edge over similar existing methods, particularly when applied to large systems of equations.
Janak Raj Sharma and Puneet Gupta
Copyright © 2013 Janak Raj Sharma and Puneet Gupta. All rights reserved.

Geometric Mesh ThreePoint Discretization for FourthOrder Nonlinear Singular Differential Equations in Polar System
Thu, 24 Oct 2013 11:09:33 +0000
http://www.hindawi.com/journals/ana/2013/614508/
Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourthorder ordinary differential equations. The method can be easily extended to the sixthorder differential equations. Convergence analysis proves the thirdorder convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block GaussSeidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.
Navnit Jha, R. K. Mohanty, and Vinod Chauhan
Copyright © 2013 Navnit Jha et al. All rights reserved.

A Global Convergence of LSCD Hybrid Conjugate Gradient Method
Tue, 22 Oct 2013 08:26:11 +0000
http://www.hindawi.com/journals/ana/2013/517452/
Conjugate gradient method is one of the most effective algorithms for solving unconstrained optimization problem. In this paper, a modified conjugate gradient method is presented and analyzed which is a hybridization of known LS and CD conjugate gradient algorithms. Under some mild conditions, the Wolfetype line search can guarantee the global convergence of the LSCD method. The numerical results show that the algorithm is efficient.
Xiangfei Yang, Zhijun Luo, and Xiaoyu Dai
Copyright © 2013 Xiangfei Yang et al. All rights reserved.

Convergent Homotopy Analysis Method for Solving Linear Systems
Tue, 08 Oct 2013 08:34:16 +0000
http://www.hindawi.com/journals/ana/2013/732032/
By using homotopy analysis method (HAM), we introduce an iterative method for solving linear systems. This method (HAM) can be used to accelerate the convergence of the basic iterative methods. We also show that by applying HAM to a divergent iterative scheme, it is possible to construct a convergent homotopyseries solution when the iteration matrix G of the iterative scheme has particular properties such as being symmetric, having real eigenvalues. Numerical experiments are given to show the efficiency of the new method.
H. Nasabzadeh and F. Toutounian
Copyright © 2013 H. Nasabzadeh and F. Toutounian. All rights reserved.

New Nonpolynomial Spline in Compression Method of for the Solution of 1D Wave Equation in Polar Coordinates
Mon, 30 Sep 2013 12:50:43 +0000
http://www.hindawi.com/journals/ana/2013/470480/
We propose a threelevel implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in direction and finite difference approximation in direction for the numerical solution of onedimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method.
Venu Gopal, R. K. Mohanty, and Navnit Jha
Copyright © 2013 Venu Gopal et al. All rights reserved.

Iterative Method for Solving a Beam Equation with Nonlinear Boundary Conditions
Wed, 28 Aug 2013 11:27:21 +0000
http://www.hindawi.com/journals/ana/2013/470258/
In this paper, we propose an iterative method for solving a beam problem which is described by a nonlinear fourthorder equation with nonlinear boundary conditions. The method reduces this nonlinear fourthorder problem to a sequence of linear secondorder problems with linear boundary conditions. The convergence of the method is proved, and some numerical examples demonstrate the efficiency of the method.
Quang A. Dang and Nguyen Thanh Huong
Copyright © 2013 Quang A. Dang and Nguyen Thanh Huong. All rights reserved.