A fast numerical algorithm for solving nearly penta-diagonal linear systems

In this paper, we present a fast numerical algorithm for solving nearly penta-diagonal linear systems and show that the computational cost is less than those of three algorithms in El-Mikkawy and Rahmo, [Symbolic algorithm for inverting cyclic penta-diagonal matrices recursively–Derivation and implementation, Comput. Math. Appl. 59 (2010), pp. 1386–1396], Lv and Le [A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707–712] and Neossi Nguetchue and Abelman [A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629–634.]. In addition, an efficient way of evaluating the determinant of a nearly penta-diagonal matrix is also discussed. The algorithm is suited for implementation using computer algebra systems (CAS) such as MATLAB, MACSYMA and MAPLE. Some numerical examples are given in order to illustrate the efficiency of our algorithm.     

Frequency domain formulation of active parametric deformable models

Active deformable models are simple tools, very popular in computer vision and computer graphics, for solving ill-posed problems or mimic real physical systems. The classical formulation is given in the spatial domain, the motor of the procedure is a second-order linear system, and rigidity and elasticity are the basic parameters for its characterization. This paper proposes a novel formulation based on a frequency-domain analysis: the internal energy functional and the Lagrange minimization are performed entirely in the frequency domain, which leads to a simple formulation and design. The frequency-based implementation offers important computational savings in comparison to the original one, a feature that is improved by the efficient hardware and software computation of the FFT algorithm. This new formulation focuses on the stiffness spectrum, allowing the possibility of constructing deformable models apart from the elasticity and rigidity-based original formulation. Simulation examples validate the theoretical results.     

An approach to multivariable combustion control design

In this paper, an approach to multivariable combustion control design within the Individual Channel Design (ICD) framework for analysis and control design is presented. ICD is a framework which involves an interplay between customer specification, uncertain plant characteristics, and the multivariable feed-back design itself. Established multivariable methods and process engineering knowledge can be incorporated or evaluated within the ICD framework. The combustion control has been designed and evaluated with a computer simulation of both a linearized model and a nonlinear model of the closed-loop system. The ICD multivariable framework shows in a highly transparent manner, by way of simple graphical frequency response indicators, what the main possibilities and difficulties posed by a combustion process for multivariable control are, and how much trade-off between control specifications is possible. Solutions are also presented for problems such as: integrity of closed-loop control, balance of input-output channels, simple and transparent controller structure, and robustness.     

An analysis and comparison of conservative upwind difference schemes for ideal and non-ideal gases

In a recent paper [P. Glaister, Conservative upwind difference schemes for the Euler equations, Comput. Math. Appl. 45 (2003) 1673–1682] a number of numerical schemes were presented for the Euler equations governing compressible flows of an ideal gas, the principal one of which is based on a conservative linearisation approach. This scheme was subsequently extended to encompass compressible flows of real gases where the equation of state allows for non-ideal gases [P. Glaister, Conservative upwind difference schemes for compressible flows of a real gas, Comput. Math. Appl. 48 (2004) 469–480]. These schemes use different parameter vectors in their construction and, consequently, the scheme in [P. Glaister, Conservative upwind difference schemes for compressible flows of a real gas, Comput. Math. Appl. 48 (2004) 469–480] when applied to the special case of an ideal gas is not identical to the principal ideal gas scheme in [P. Glaister, Conservative upwind difference schemes for the Euler equations, Comput. Math. Appl. 45 (2003) 1673–1682]. In this paper it is shown how these schemes are related, followed by a numerical comparison when each is applied to two standard test problems.     

Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients

The purpose of this study is to give a Taylor polynomial approximation for the solution of hyperbolic type partial differential equations with constant coefficients. The technique used is an improved Taylor matrix method, which has been given for solving ordinary differential, integral and integro-differential equations [M. Gülsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82(5) (2005), pp. 629–642; M. Gülsu and M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput. 188 (2007), pp. 446–449; A. Karamete and M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79(9) (2002), pp. 987–1000; N. Kurt and M. Çevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mech. Res. Commun. 35 (2008), pp. 530–536; N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (2008), pp. 839–850; ?. Nas, S. Yalçinba?, and M. Sezer, A method for approximate solution of the high-order linear Fredholm integro-differential equations, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Edu. Sci. Technol. 25(5) (1994), pp. 625–633; M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, M. Gülsu, and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst. 343 (2006), pp. 647–659; S. Yalçinba?, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinba? and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), pp. 291–308]. Some numerical examples, which consist of initial and boundary conditions, are given to illustrate the reliability and efficiency of the method. Also, the results obtained are compared by the known results; the error analysis is performed and the accuracy of the solution is shown.     

Sturm–Liouville problems with discontinuities at two points

In this paper we extend some spectral properties of regular Sturm–Liouville problems to those which consist of a Sturm–Liouville equation with piecewise continuous potentials together with eigenparameter-dependent boundary conditions and four supplementary transmission conditions. By modifying some techniques of [C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293–308; E. Tunç, O.Sh. Muhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput. 157 (2004) 347–355; O.Sh. Mukhtarov, E. Tunç, Eigenvalue problems for Sturm–Liouville equations with transmission conditions, Israel J. Math. 144 (2004) 367–380] and [O.Sh. Mukhtarov, M. Kadakal, F.Ş. Muhtarov, Eigenvalues and normalized eigenfunctions of discontinuous Sturm–Liouville problem with transmission conditions, Rep. Math. Phys. 54 (2004) 41–56], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulae for the eigenvalues and eigenfunctions.     

A computer program for the statistical analysis of cancer inhibition/promotion experiments

The computer program discussed and presented in this paper is designed for the statistical analysis of data from cancer inhibition/promotion experiments that involve Type I censoring. The theoretical derivation of the method is described in Biometrics 43 (1987) 525-534 and Appl. Math. Lett. 1 (1988) 197-201. Turbo Pascal is used to compute maximum likelihood estimates of the parameters of interest: the mean number of induced tumors per animal, and the mean time to tumor detection. The program conducts a likelihood ratio test to determine if there is an overall experiment effect, and allows the user to select options to isolate the pair(s) of groups contributing to the experiments effect, to isolate group differences in terms of tumor number and growth rate, and to graph 95% confidence regions for the vector parameter of interest.     

Notes on Implicit finite difference approximation for time fractional diffusion equations [Comput. Math. Appl. 56 (2008) 1138–1145]

In a recent paper [Diego A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl. 56 (2008) 1138–1145], the author has proposed an unconditional stable implicit difference scheme for solving time fractional diffusion equations. In this paper, we show that the method of stability analysis by the author is not accurate, so we analyze the stability by using a correct method.     

Technology Trends and Disk Array Performance

The original design of disk arrays was motivated primarily by the desire for greater fault tolerance and higher data transfer rates than those achievable with a single disk. However, disk technology has continued to evolve, driven largely by the needs of the commodity personal computer market, rather than those of the “niche markets” of parallel and fault-tolerant computing. Using a combination of analytic modeling and simulation, we consider four disk array designs and extrapolate the behavior of these designs as a function of predicted changes in disk technology. The results of this analysis suggest that changes in disk technology may lessen, though not eliminate, the current performance advantages of disk arrays and may shift current design points.     

Influence of centrifugal forces on the development of wave packets in boundary layers with a uniformly valid model

The behavior of three-dimensional wave packets in the boundary layer on curved surfaces is analyzed in this study based on a modification of the triple-deck theory referred to as the “criss-cross” interaction model. The equations of the criss-cross interaction describe a particular type of boundary layer instability mode that possesses underlying properties of both the Tollmien-Schlichting waves and Taylor-Görtler vortices. Previous analysis of the criss-cross interaction regime suggests a possibility for upstream propagation of perturbations in the boundary layer and possible absolute instability of the flow. However, these results cannot be considered as conclusive because the initial-value problem for the criss-cross interaction equations is ill-posed. In a recent work [Turkyilmazoglu M, Ruban AI. A uniformly valid well-posed asymptotic approach to the inviscid-viscous interaction theory. Stud Appl Math 2002;108:161-85] a regularized non-asymptotic model to describe criss-cross interaction has been proposed. Whereas in the original version of the theory, perturbations have an unbounded growth rate as the longitudinal wave number ∣k∣ → ∞, in the new model of [Turkyilmazoglu M, Ruban AI. A uniformly valid well-posed asymptotic approach to the inviscid-viscous interaction theory. Stud Appl Math 2002;108:161-85], as physically expected the amplification rate remains bounded for both spatially growing and temporally growing waves. A Fourier transform method is used in the present study to solve the linearized equations for the flow over concave roughness and humps and it is found that disturbances develop and are convected downstream as wave packets. The behavior of the wave packets is consistent with convective instability, and the upstream influence is no longer present at large times.     

Non-reflecting artificial boundaries for modelling scalar wave propagation problems in two-dimensional half space

Using the operator splitting method, the non-reflecting artificial boundary condition, which has the same form as those originally presented by Hagstrom and Hariharan [Appl. Numer. Math. 27 (1998) 403] as well as Thompson et al. [Comput. Meth. Appl. Mech. Engrg. 191 (2001) 311], is re-derived in this paper to deal with transient scalar wave propagation problems in a two-dimensional homogeneous half space. In particular, the original non-reflecting artificial boundary condition is extended to approximately solve transient scalar wave propagation problems in a two-dimensional layered half space, which has a far field consisting of non-homogeneous materials. The proposed non-reflecting artificial boundary condition can be decomposed into three types of elements: physically meaningful springs and dashpots as well as the generalized energy absorbers, which represent the effects of incident waves at previous time instants on the truncated artificial boundary. Due to the meaningfulness in physics, the proposed non-reflecting artificial boundary is straightforwardly applied to the finite element analysis of wave propagation problems in infinite media. The related numerical results from three application examples have demonstrated the effectiveness, efficiency and usefulness of the proposed non-reflecting artificial boundary in dealing with scalar wave propagation problems in two-dimensional homogeneous and layered half spaces.     

基于 SPOC 的“计算机辅助产品表现”课程教学模式重构

针对“计算机辅助产品表现”课程传统教学模式中存在的局限与问题,结合对其形 成原因的具体剖析,提出融合互联网技术的混合式教学是切实可行的改革方向,并引入小规模 限制性在线课程(SPOC)模式对原有的课程内容与授课形式进行重构与创新。通过教学实践对 SPOC 课程的内容建设、教学实施与效果评估等重要环节进行了分析与研究,并通过对测验成 绩的数据分析,验证了 SPOC 模式对于该课程教学效果提升所具备的有效性,为今后计算机辅 助设计类课程的教学改革做出了有益的探索与尝试。     

A note on block-diagonally preconditioned PIU methods for singular saddle point problems

Recently, Bai and Wang [On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008), pp. 2900–2932] and Gao and Kong [Block diagonally preconditioned PIU methods of saddle point problems, Appl. Math. Comput. 216 (2010), pp. 1880–1887] discussed the parameterized inexact Uzawa (PIU) method and the preconditioned parameterized inexact Uzawa (PPIU) method. In this paper, we further study the block-diagonally preconditioned PIU methods for solving singular saddle point problems, and give the corresponding convergence analysis.     

Design techniques of a computer program for the identification of processes and the simulation of water quality in aquatic systems

In order to support environmental scientists in finding an “adequate” model for the system they are investigating, a computer program is necessary that allows its users to perform simulations for different models, to assess the identifiability and to estimate the values of model parameters (using measured data), and to estimate prediction uncertainty. These requirements, especially that of providing much freedom in model formulation, are difficult to realize in such a program. In this paper, it is shown how object-oriented program design techniques were employed to facilitate the realization of an identification and simulation program for aquatic systems (AQUASIM) that is very flexible with regard to model formulation and that provides methods of sensitivity analysis, parameter estimation and uncertainty analysis in addition to simulation. It is the goal of this paper to encourage developers of environmental software to revise previously used program structures and to employ modern program design techniques.     

On local convergence of a Newton-type method in Banach space

In this study we are concerned with the local convergence of a Newton-type method introduced by us [I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7–9.] for approximating a solution of a nonlinear equation in a Banach space setting. This method has also been studied by Homeier [H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227–230.] and Özban [A.Y. Özban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677–682.] in real or complex space. The benefits of using this method over other methods using the same information have been explained in [I.K. Argyros, Computational theory of iterative methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Science Inc., New York, USA, 2007.; I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7–9.; H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227–230.; A.Y. Özban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677–682.]. Here, we give the convergence radii for this method under a type of weak Lipschitz conditions proven to be fruitful by Wang in the case of Newton's method [X. Wang, Convergence of Newton's method and inverse function in Banach space, Math. Comput. 68 (1999), pp. 169–186 and X. Wang, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000), pp. 123–134.]. Numerical examples are also provided.     

Numerical methods for a class of nonlinear integro-differential equations

In a previous article (Glowinski, J. Math. Anal. Appl. 41, 67–96, 1973) the first author discussed several methods for the numerical solution of nonlinear equations of the integro-differential type with periodic boundary conditions. In this article we discuss an alternative methodology largely based on the Strang’s symmetrized operator-splitting scheme. Several numerical experiments suggest that the new method is robust and accurate. It is also easier to implement than the various methods discussed by Glowinski in J. Math. Anal. Appl. 41, 67–96 (1973).