International Journal of Computational Mathematics The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. On the Dynamics of Laguerre’s Iteration Method for Finding the th Roots of Unity Wed, 26 Nov 2014 11:44:16 +0000 Previous analyses of Laguerre’s iteration method have provided results on the behavior of this popular method when applied to the polynomials , . In this paper, we summarize known analytical results and provide new results. In particular, we study symmetry properties of the Laguerre iteration function and clarify the dynamics of the method. We show analytically and demonstrate computationally that for each the basin of attraction to the roots is a subset of an annulus that contains the unit circle and whose Lebesgue measure shrinks to zero as . We obtain a good estimate of the size of the bounding annulus. We show that the boundary of the basin of convergence exhibits fractal nature and quasi self-similarity. We also discuss the connectedness of the basin for large values of . We also numerically find some short finite cycles on the boundary of the basin of convergence for . Finally, we demonstrate that when using the floating point arithmetic and the general formulation of the method, convergence occurs even from starting values outside of the basin of convergence due to the loss of significance during the evaluation of the iteration function. Pavel Bělík, HeeChan Kang, Andrew Walsh, and Emma Winegar Copyright © 2014 Pavel Bělík et al. All rights reserved. Plane Elastostatic Solution in an Infinite Functionally Graded Layer Weakened by a Crack Lying in the Middle of the Layer Tue, 25 Nov 2014 16:57:14 +0000 This paper is concerned with an internal crack problem in an infinite functionally graded elastic layer. The crack is opened by an internal uniform pressure along its surface. The layer surfaces are supposed to be acted on by symmetrically applied concentrated forces of magnitude with respect to the centre of the crack. The applied concentrated force may be compressive or tensile in nature. Elastic parameters λ and μ are assumed to vary along the normal to the plane of crack. The problem is solved by using integral transform technique. The solution of the problem has been reduced to the solution of a Cauchy-type singular integral equation, which requires numerical treatment. The stress-intensity factors and the crack opening displacements are determined and the effects of graded parameters on them are shown graphically. R. Patra, S. P. Barik, M. Kundu, and P. K. Chaudhuri Copyright © 2014 R. Patra et al. All rights reserved. A Numerical Method for 1-D Parabolic Equation with Nonlocal Boundary Conditions Thu, 20 Nov 2014 09:44:34 +0000 This paper is concerned with a local method for the solution of one-dimensional parabolic equation with nonlocal boundary conditions. The method uses a coordinate transformation. After the coordinate transformation, it is then possible to obtain exact solutions for the resulting equations in terms of the local variables. These exact solutions are in terms of constants of integration that are unknown. By imposing the given boundary conditions and smoothness requirements for the solution, it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration. A number of examples are used to study the applicability of the method. In particular, three nonlinear problems are used to show the novelty of the method. M. Tadi and Miloje Radenkovic Copyright © 2014 M. Tadi and Miloje Radenkovic. All rights reserved. A Numerical Test of Padé Approximation for Some Functions with Singularity Thu, 20 Nov 2014 00:00:00 +0000 The aim of this study is to examine some numerical tests of Padé approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut, and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Padé approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series. Hiroaki S. Yamada and Kensuke S. Ikeda Copyright © 2014 Hiroaki S. Yamada and Kensuke S. Ikeda. All rights reserved. Block Hybrid -Step Backward Differentiation Formulas for Large Stiff Systems Mon, 20 Oct 2014 07:21:02 +0000 This paper presents a generalized high order block hybrid -step backward differentiation formula (HBDF) for solving stiff systems, including large systems resulting from the semidiscretization parabolic partial differential equations (PDEs). A block scheme in which two off-grid points are specified by the zeros of the second degree Chebyshev polynomial of the first kind is examined for convergence, and stabilities. Numerical simulations that illustrate the accuracy of a Chebyshev based method are given for selected stiff systems and partial differential equations. S. N. Jator and E. Agyingi Copyright © 2014 S. N. Jator and E. Agyingi. All rights reserved. Modelling Hepatotoxicity of Antiretroviral Therapy in the Liver during HIV Monoinfection Sun, 19 Oct 2014 00:00:00 +0000 Liver related complications are currently the leading cause of morbidity and mortality among human immunodeficiency virus (HIV) infected individuals. In HIV monoinfected individuals on therapy, liver injury has been associated with the use of antiretroviral agents as most of them exhibit some degree of toxicity. In this study we proposed a mathematical model with the aim of investigating hepatotoxicity of combinational therapy of antiretroviral drugs. Therapy efficacy and toxicity were incorporated in the model as dose-response functions. With the parameter values used in the study, protease inhibitors-based regimens were found to be more toxic than nonnucleoside reverse transcriptase inhibitors-based regimens. In both regimens, the combination of stavudine and zidovudine was the most toxic baseline nucleoside reverse transcriptase inhibitors followed by didanosine with stavudine. However, the least toxic combinations were zidovudine and lamivudine followed by didanosine and lamivudine. The study proposed that, under the same second line regimens, the most toxic first line combination gives the highest viral load and vice versa. Hasifa Nampala, Livingstone S. Luboobi, Joseph Y. T. Mugisha, and Celestino Obua Copyright © 2014 Hasifa Nampala et al. All rights reserved. Mathematical Modeling of Multienzyme Biosensor System Wed, 01 Oct 2014 12:16:02 +0000 A mathematical model of hybrid inhibitor biosensor system is discussed. This model consists of five nonlinear partial differential equations for bisubstrate sensitive amperometric system. Simple and closed form of analytical expressions for concentration of glucose-6-phosphate (substrate), potassium dihydrogen phosphate (inhibitor), oxygen (co-substrate), glucose (product 1), and hydrogen peroxide (product 3) is obtained in terms of rate constant using modified Adomian decomposition method (MADM). In this study, behavior of biokinetic parameters is analyzed using this theoretical result. The obtained analytical results (concentrations) are compared with the numerical results and are found to be in satisfactory agreement. SP. Ganesan, K. Saravanakumar, and L. Rajendran Copyright © 2014 SP. Ganesan et al. All rights reserved. Approximate Periodic Solution for the Nonlinear Helmholtz-Duffing Oscillator via Analytical Approaches Mon, 29 Sep 2014 11:52:45 +0000 The conservative Helmholtz-Duffing oscillator is analyzed by means of three analytical techniques. The max-min, second-order of the Hamiltonian, and the global error minimization approaches are applied to achieve natural frequencies. The obtained results are compared with the homotopy perturbation method and numerical solutions. The results show that second-order of the global error minimization method is very accurate, so it can be widely applicable in engineering problems. A. Mirzabeigy, M. K. Yazdi, and M. H. Nasehi Copyright © 2014 A. Mirzabeigy et al. All rights reserved. A Collocation Method for Numerical Solution of Hyperbolic Telegraph Equation with Neumann Boundary Conditions Sun, 28 Sep 2014 11:46:26 +0000 We present a technique based on collocation of cubic B-spline basis functions to solve second order one-dimensional hyperbolic telegraph equation with Neumann boundary conditions. The use of cubic B-spline basis functions for spatial variable and its derivatives reduces the problem into system of first order ordinary differential equations. The resulting system subsequently has been solved by SSP-RK54 scheme. The accuracy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and in good agreement with the exact solution. R. C. Mittal and Rachna Bhatia Copyright © 2014 R. C. Mittal and Rachna Bhatia. All rights reserved. Hermitian Positive Definite Solution of the Matrix Equation Wed, 24 Sep 2014 08:25:11 +0000 We consider the Hermitian positive definite solution of the nonlinear matrix equation . Some new sufficient conditions and necessary conditions for the existence of Hermitian positive definite solutions are derived. An iterative method is proposed to compute the Hermitian positive definite solution. In the end, an example is used to illustrate the correctness and application of our results. Chun-Mei Li and Jing-Jing Peng Copyright © 2014 Chun-Mei Li and Jing-Jing Peng. All rights reserved. Solving Operator Equation Based on Expansion Approach Sun, 07 Sep 2014 07:08:56 +0000 To date, researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partial differential equations and also based on polynomial basis. But the principal importance of this paper is to develop the expansion approach based on general basis functions (in particular case polynomial basis) for solving general operator equations, wherein the particular cases of our development are integral equations, ordinary differential equations, difference equations, partial differential equations, and fractional differential equations. In other words, this paper presents the expansion approach for solving general operator equations in the form , with respect to boundary condition , where , and are linear, nonlinear, and boundary operators, respectively, related to a suitable Hilbert space, is the domain of approximation, is an arbitrary constant, and is an arbitrary function. Also the other importance of this paper is to introduce the general version of pseudospectral method based on general interpolation problem. Finally some experiments show the accuracy of our development and the error analysis is presented in norm. A. Aminataei, S. Ahmadi-Asl, and M. Pakbaz Copyright © 2014 A. Aminataei et al. All rights reserved. On the Spectrum and Spectral Norms of -Circulant Matrices with Generalized -Horadam Numbers Entries Sun, 31 Aug 2014 10:56:35 +0000 This work is concerned with the spectrum and spectral norms of -circulant matrices with generalized -Horadam numbers entries. By using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them. In addition, a sufficient condition for an -circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral norms of normal -circulant matrix with generalized -Horadam numbers, which generalize and improve the known results. Lele Liu Copyright © 2014 Lele Liu. All rights reserved. Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method Tue, 12 Aug 2014 12:53:08 +0000 The generalized regularized long wave (GRLW) equation is solved numerically by using a distributed approximating functional (DAF) method realized by the regularized Hermite local spectral kernel. Test problems including propagation of single solitons, interaction of two and three solitons, and conservation properties of mass, energy, and momentum of the GRLW equation are discussed to test the efficiency and accuracy of the method. Furthermore, using the Maxwellian initial condition, we show that the number of solitons which are generated can be approximately determined. Comparisons are made between the results of the proposed method, analytical solutions, and numerical methods. It is found that the method under consideration is a viable alternative to existing numerical methods. Edson Pindza and Eben Maré Copyright © 2014 Edson Pindza and Eben Maré. All rights reserved. The Convergence of Geometric Mesh Cubic Spline Finite Difference Scheme for Nonlinear Higher Order Two-Point Boundary Value Problems Wed, 23 Jul 2014 09:30:37 +0000 An efficient algorithm for the numerical solution of higher (even) orders two-point nonlinear boundary value problems has been developed. The method is third order accurate and applicable to both singular and nonsingular cases. We have used cubic spline polynomial basis and geometric mesh finite difference technique for the generation of this new scheme. The irreducibility and monotone property of the iteration matrix have been established and the convergence analysis of the proposed method has been discussed. Some numerical experiments have been carried out to demonstrate the computational efficiency in terms of convergence order, maximum absolute errors, and root mean square errors. The numerical results justify the reliability and efficiency of the method in terms of both order and accuracy. Navnit Jha, R. K. Mohanty, and Vinod Chauhan Copyright © 2014 Navnit Jha et al. All rights reserved.