Advances in Numerical Analysis The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . 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 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 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 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 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 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 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 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 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 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 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 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 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 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. On Some Efficient Techniques for Solving Systems of Nonlinear Equations Thu, 31 Oct 2013 11:47:05 +0000 We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Third-order method is composed of two steps, namely, Newton iteration as the first step and weighted-Newton iteration as the second step. Fifth and sixth-order methods are composed of three steps of which the first two steps are same as that of the third-order method whereas the third is again a weighted-Newton 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 Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System Thu, 24 Oct 2013 11:09:33 +0000 Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel 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 LS-CD Hybrid Conjugate Gradient Method Tue, 22 Oct 2013 08:26:11 +0000 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 Wolfe-type line search can guarantee the global convergence of the LS-CD 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 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 homotopy-series 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 We propose a three-level 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 one-dimensional 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 In this paper, we propose an iterative method for solving a beam problem which is described by a nonlinear fourth-order equation with nonlinear boundary conditions. The method reduces this nonlinear fourth-order problem to a sequence of linear second-order 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. Codimension-m Bifurcation Theorems Applicable to the Numerical Verification Methods Thu, 27 Jun 2013 11:59:43 +0000 We establish codimension-m bifurcation theorems applicable to the numerical verification methods. They are generalization of codimension-1 bifurcation theorems established by (Kawanago, 2004). As a numerical example, we treat Hopf bifurcation, which is codimension-2 bifurcation. Tadashi Kawanago Copyright © 2013 Tadashi Kawanago. All rights reserved. Mixed Finite Element Methods for the Poisson Equation Using Biorthogonal and Quasi-Biorthogonal Systems Thu, 11 Apr 2013 17:54:10 +0000 We introduce two three-field mixed formulations for the Poisson equation and propose finite element methods for their approximation. Both mixed formulations are obtained by introducing a weak equation for the gradient of the solution by means of a Lagrange multiplier space. Two efficient numerical schemes are proposed based on using a pair of bases for the gradient of the solution and the Lagrange multiplier space forming biorthogonal and quasi-biorthogonal systems, respectively. We also establish an optimal a priori error estimate for both finite element approximations. Bishnu P. Lamichhane Copyright © 2013 Bishnu P. Lamichhane. All rights reserved. Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations Sun, 24 Mar 2013 09:02:14 +0000 We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published. Gustavo Fernández-Torres and Juan Vásquez-Aquino Copyright © 2013 Gustavo Fernández-Torres and Juan Vásquez-Aquino. All rights reserved. On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation Thu, 21 Mar 2013 18:51:45 +0000 We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice. Arnak Poghosyan Copyright © 2013 Arnak Poghosyan. All rights reserved. A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence Tue, 19 Mar 2013 13:53:34 +0000 An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method. Ababu Teklemariam Tiruneh, W. N. Ndlela, and S. J. Nkambule Copyright © 2013 Ababu Teklemariam Tiruneh et al. All rights reserved. Modified Bézier Curves with Shape-Preserving Characteristics Using Differential Evolution Optimization Algorithm Wed, 13 Mar 2013 17:54:31 +0000 A parametric equation for a modified Bézier curve is proposed for curve fitting applications. The proposed equation contains shaping parameters to adjust the shape of the fitted curve. This flexibility of shape control is expected to produce a curve which is capable of following any sets of discrete data points. A Differential Evolution (DE) optimization based technique is proposed to find the optimum value of these shaping parameters. The optimality of the fitted curve is defined in terms of some proposed cost parameters. These parameters are defined based on sum of squares errors. Numerical results are presented highlighting the effectiveness of the proposed curves compared with conventional Bézier curves. From the obtained results, it is observed that the proposed method produces a curve that fits the data points more accurately. Mohammad Asif Zaman and Shuvro Chowdhury Copyright © 2013 Mohammad Asif Zaman and Shuvro Chowdhury. All rights reserved. Nonlinear Finite Element Analysis of Sloshing Wed, 27 Feb 2013 13:31:19 +0000 The disturbance on the free surface of the liquid when the liquid-filled tanks are excited is called sloshing. This paper examines the nonlinear sloshing response of the liquid free surface in partially filled two-dimensional rectangular tanks using finite element method. The liquid is assumed to be inviscid, irrotational, and incompressible; fully nonlinear potential wave theory is considered and mixed Eulerian-Lagrangian scheme is adopted. The velocities are obtained from potential using least square method for accurate evaluation. The fourth-order Runge-Kutta method is employed to advance the solution in time. A regridding technique based on cubic spline is employed to avoid numerical instabilities. Regular harmonic excitations and random excitations are used as the external disturbance to the container. The results obtained are compared with published results to validate the numerical method developed. Siva Srinivas Kolukula and P. Chellapandi Copyright © 2013 Siva Srinivas Kolukula and P. Chellapandi. All rights reserved. Modular Analysis of Sequential Solution Methods for Almost Block Diagonal Systems of Equations Sun, 24 Feb 2013 07:56:20 +0000 Almost block diagonal linear systems of equations can be exemplified by two modules. This makes it possible to construct all sequential forms of band and/or block elimination methods. It also allows easy assessment of the methods on the basis of their operation counts, storage needs, and admissibility of partial pivoting. The outcome of the analysis and implementation is to discover new methods that outperform a well-known method, a modification of which is, therefore, advocated. Tarek M. A. El-Mistikawy Copyright © 2013 Tarek M. A. El-Mistikawy. All rights reserved. Parallel Nonoverlapping DDM Combined with the Characteristic Method for Incompressible Miscible Displacements in Porous Media Tue, 19 Feb 2013 09:51:26 +0000 Two types of approximation schemes are established for incompressible miscible displacements in porous media. First, standard mixed finite element method is used to approximate the velocity and pressure. And then parallel non-overlapping domain decomposition methods combined with the characteristics method are presented for the concentration. These methods use the characteristic method to handle the material derivative term of the concentration equation in the subdomains and explicit flux calculations on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the velocity and pressure can be approximated simultaneously, and the parallelism can be achieved for the concentration equation. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability. These schemes hold the advantages of nonoverlapping domain decomposition methods and the characteristic method. Optimal error estimates in -norm are derived for these two schemes, respectively. Keying Ma and Tongjun Sun Copyright © 2013 Keying Ma and Tongjun Sun. All rights reserved. New Approach for Solving a Class of Doubly Singular Two-Point Boundary Value Problems Using Adomian Decomposition Method Wed, 26 Dec 2012 14:35:44 +0000 We propose two new modified recursive schemes for solving a class of doubly singular two-point boundary value problems. These schemes are based on Adomian decomposition method (ADM) and new proposed integral operators. We use all the boundary conditions to derive an integral equation before establishing the recursive schemes for the solution components. Thus we develop recursive schemes without any undetermined coefficients while computing successive solution components, whereas several previous recursive schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete calculation of the solution by several earlier modified recursion schemes using the ADM. The approximate solution is computed in the form of series with easily calculable components. The effectiveness of the proposed approach is tested by considering four examples and results are compared with previous known results. Randhir Singh, Jitendra Kumar, and Gnaneshwar Nelakanti Copyright © 2012 Randhir Singh et al. All rights reserved. Solution of Nonlinear Volterra-Fredholm Integrodifferential Equations via Hybrid of Block-Pulse Functions and Lagrange Interpolating Polynomials Mon, 10 Dec 2012 12:42:49 +0000 An efficient hybrid method is developed to approximate the solution of the high-order nonlinear Volterra-Fredholm integro-differential equations. The properties of hybrid functions consisting of block-pulse functions and Lagrange interpolating polynomials are first presented. These properties are then used to reduce the solution of the nonlinear Volterra-Fredholm integro-differential equations to the solution of algebraic equations whose solution is much more easier than the original one. The validity and applicability of the proposed method are demonstrated through illustrative examples. The method is simple, easy to implement and yields very accurate results. Hamid Reza Marzban and Sayyed Mohammad Hoseini Copyright © 2012 Hamid Reza Marzban and Sayyed Mohammad Hoseini. All rights reserved.