Showing papers in "International Journal of Computer Mathematics in 2004"
TL;DR: An improved method of fuzzy time series to forecast university enrollments of the University of Alabama is presented, which is as simple as Chen's method but more accurate.
Abstract: This article presents an improved method of fuzzy time series to forecast university enrollments. The historical enrollment data of the University of Alabama were first adopted by Song and Chissom (Song, Q. and Chissom, B. S. (1993). Forecasting enrollment with fuzzy time series-part I, Fuzzy Sets and Systems, 54, 1–9; Song, Q. and Chissom, B. S. (1994). Forecasting enrollment with fuzzy time series-part II, Fuzzy Sets and Systems, 54, 267–277) to illustrate the forecasting process of the fuzzy time series. Later, Chen proposed a simpler method. In this article, we show that our method is as simple as Chen's method but more accurate. In forecasting the enrollment of the University of Alabama, the root mean square percentage error (RMSPE) of our method is 3.1113% while the RMSPE of Chen's method is 4.0516%, which shows that our method is doing much better. E-mail: asdwx@axp1.stm.ntou.edu.tw
134 citations
TL;DR: In this article, an approximate method for solving higher-order ODEs is proposed based on a rational Chebyshev (RC) tau method, where the operational matrices of the derivative functions of the ODE are derived from the same matrix.
Abstract: An approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev (RC) tau method. The operational matrices of the derivative...
96 citations
TL;DR: It is concluded that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.
Abstract: Hypercube is one of the most popular topologies for connecting processors in multicomputer systems. In this paper we address the maximum order of a connected component in a faulty cube. The results established include several known conclusions as special cases. We conclude that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.
86 citations
TL;DR: A finite difference scheme to solve the coupled nonlinear Schrödinger equation is written and is unconditionally stable and extrapolation is used in the temporal direction, which makes the method fourth-order in the two directions, space and time.
Abstract: The coupled nonlinear Schrodinger equation models several interesting physical phenomena. It presents a model equation for optical fiber with linear birefringence. In this article, we write a finite difference scheme to solve this equation. The method is fourth-order in space and second-order in time. It is unconditionally stable and extrapolation is used in the temporal direction and this makes the method fourth-order in the two directions, space and time. Many numerical tests have been conducted to display the robustness of the scheme.
79 citations
TL;DR: A method to construct the exact solutions of some nonlinear evolution equations is presented by the hyperbolic function method, demonstrating the efficiency and the properties of these equations shown with figures.
Abstract: A method to construct the exact solutions of some nonlinear evolution equations is presented by the hyperbolic function method. The efficiency of the method can be demonstrated by some nonlinear PDEs such as the Benjamin-Bona-Mahony equation, the coupled KdV and MKdV equations. New exact travelling wave solutions are presented. In addition, the properties of these equations are shown with figures. †E-mail: dj.evans@ntu.ac.uk
68 citations
TL;DR: It is shown that parameter estimation of linear system can be done easily using the idea proposed and the results of the article, which include the time information, are illustrated in two examples.
Abstract: In this article, a computational method based on Haar wavelet in time-domain for solving the problem of optimal control of the linear time invariant systems for any finite time interval is proposed. Haar wavelet integral operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the linear systems with respect to a quadratic cost function by solving only the linear algebraic equations. It is shown that parameter estimation of linear system can be done easily using the idea proposed. On the basis of Haar function properties, the results of the article, which include the time information, are illustrated in two examples.
65 citations
TL;DR: A numerical solution of the Regularised Long Wave (RLW) Equation is obtained using space-splitting technique and quadratic B-spline Galerkin finite element method.
Abstract: A numerical solution of the Regularised Long Wave (RLW) Equation is obtained using space-splitting technique and quadratic B-spline Galerkin finite element method. Solitary wave motion, interaction of two solitary waves and wave generation are studied using the proposed method. Comparisons are made with analytical solutions and with some spline finite element method calculations at selected times. Accuracy and efficiency are discussed by computing the numerical conserved laws and L 2, L ∞ norms.
59 citations
TL;DR: The approximate solutions to the eighth-order boundary-value differential equations are solved by using the Adomian decomposition method (ADM), which is more reliable, efficient and accurate than the traditional schemes.
Abstract: In this paper, the approximate solutions to the eighth-order boundary-value differential equations are solved by using the Adomian decomposition method (ADM). The numerical solutions of the problem are calculated in the form of a series with easily computable components. The numerical illustrations show that this technique is more reliable, efficient and accurate than the traditional schemes. E-mail: dj.evans@ntu.ac.uk
48 citations
TL;DR: The QSIADE method has been shown to be very fast as compared with the standard IADE method, and some numerical tests were included to support this statement.
Abstract: The aim of this article is to describe the formulation of the quarter-sweep iterative alternating decomposition explicit (QSIADE) method using the finite difference approach for solving one-dimensional diffusion equations. The concept of the QSIADE method is inspired via combination between the quarter-sweep iterative and the iterative alternating decomposition explicit (IADE) methods known as one of the technique in two-step iterative methods. The QSIADE method has been shown to be very fast as compared with the standard IADE method. Some numerical tests were included to support our statement.
46 citations
TL;DR: Numerical solutions for the Korteweg–de Vries equation based on the collocation method using septic splines as element shape functions are set up and are shown to be accurate and efficient.
Abstract: Numerical solutions for the Korteweg–de Vries equation based on the collocation method using septic splines as element shape functions are set up. A linear stability analysis of the scheme shows the method to be unconditionally stable. A test problem concerning the development and motion of solitons is used to validate the method. The numerical scheme is compared with other published methods and shown to be accurate and efficient.
44 citations
TL;DR: The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the second-order linear parabolic partial differential equation with nonlocal boundary specifications replacing the standard boundary conditions.
Abstract: Over the last 20 years, the Adomian decomposition approach has been applied to obtain formal solutions to a wide class of stochastic and deterministic problems involving algebraic, differential, integro-differential, differential delay, integral and partial differential equations. This method leads to computable, efficient, solutions to linear and nonlinear operator equations. Furthermore in the past, only classical boundary value problems have been considered. The parabolic partial differential equations with non-classical conditions model various physical problems. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the second-order linear parabolic partial differential equation with nonlocal boundary specifications replacing the standard boundary conditions. This scheme is employed for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval. The Adomian decomposition metho...
TL;DR: This article presents a stochastic simulation-based genetic algorithm for solving chance constraint programming problems, where the random variables involved in the parameters follow any continuous distribution.
Abstract: In this article, we present a stochastic simulation-based genetic algorithm for solving chance constraint programming problems, where the random variables involved in the parameters follow any continuous distribution. Generally, deriving the deterministic equivalent of a chance constraint is very difficult due to complicated multivariate integration and is only possible if the random variables involved in the chance constraint follow some specific distribution such as normal, uniform, exponential and lognormal distribution. In the proposed method, the stochastic model is directly used. The feasibility of the chance constraints are checked using stochastic simulation, and the genetic algorithm is used to obtain the optimal solution. A numerical example is presented to prove the efficiency of the proposed method. E-mail: rabin@maths.iitkgp.ernet.in
TL;DR: A new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation with series-based technique of the Adomian decomposition technique that provides the solution in terms of rapid convergent series.
Abstract: In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional...
TL;DR: An ϵ-uniform fitted mesh method is presented to solve boundary-value problems for singularly perturbed differential-difference equations containing negative as well as positive shifts with layer behavior.
Abstract: An ϵ-uniform fitted mesh method is presented to solve boundary-value problems for singularly perturbed differential-difference equations containing negative as well as positive shifts with layer behavior. Such types of BVPs arise at various places in the literature such as the variational problem in control theory and in the determination of the expected time for the generation of action potentials in nerve cells. The method consists of the standard upwind finite difference operator on a special type of mesh. Here, we consider a piecewise uniform fitted mesh, which turns out to be sufficient for the construction of ϵ-uniform method. One may use some more complicated meshes, but the simplicity of the piecewise uniform mesh is supposed to be one of their major attractions. The error estimate is established which shows that the method is ϵ-uniform. Several numerical examples are solved to show the effect of small shifts on the boundary layer solution. Numerical results in terms of maximum errors are tabulate...
TL;DR: The aim of the present paper is to investigate the starlikeness and convexity of these partial sums fn (z) and gn (z).
Abstract: In Geometric Function Theory, it is well known that the familiar Koebe function f(z) = z/(1 − z)2 is the extremal function for the class 𝒮* of starlike functions in the open unit disk 𝕌 and also that the function g(z) = z/(1 − z) is the extremal function for the class 𝒦 of convex functions in the open unit disk 𝕌. However, the partial sum fn (z) of f(z) is not starlike in 𝕌 and the partial sum gn (z) of g(z) is not convex in 𝕌. The aim of the present paper is to investigate the starlikeness and convexity of these partial sums fn (z) and gn (z). Computational and graphical usages of Mathematica (Version 4.0) as well as geometrical descriptions of the image domains in several illustrative examples are also presented. E-mail: owa@math.kindai.ac.jp
TL;DR: In this article, a local convergence study for Secant-type methods is carried out to enlarge the radius of convergence, without increasing the necessary hypothesis.
Abstract: In this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed. E-mail: sonia.busquier@upct.es E-mail: sergio.amat@upct.es
TL;DR: The continuous Legendre wavelets on the interval [0, 1) constructed by M. Razzaghi and S. Yousefi are used to solve the linear second kind integro-differential equations and construct the quadrature formulae for the calculation of inner products of any functions, which are required in the approximation for the Integro- differential equations.
Abstract: In this article, we use the continuous Legendre wavelets on the interval [0, 1) constructed by [M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32(4) (2001) 495–502.] to solve the linear second kind integro-differential equations and construct the quadrature formulae for the calculation of inner products of any functions, which are required in the approximation for the integro-differential equations. Then we reduce the integro-differential equation to the solution of linear algebraic equations. E-mail: ahadi@khuisf.ac.ir
TL;DR: The GIPF-CG method, an approximate version of the MSD- CG method, only requires communication between neighboring subdomains and eliminate global inner product entirely, which is well suited for massively parallel computation.
Abstract: In this article, we proposed a new CG-type method based on domain decomposition method, which is called multiple search direction conjugate gradient (MSD-CG) method. In each iteration, it uses a search direction in each subdomain. Instead of making all search directions conjugate to each other, as in the block CG method [O'Leary, D. P. (1980). The block conjugate gradient algorithm and related methods. Lin. Alg. Appl., 29, 293–322.], we require that they are nonzero in corresponding subdomains only. The GIPF-CG method, an approximate version of the MSD-CG method, only requires communication between neighboring subdomains and eliminate global inner product entirely. This method is therefore well suited for massively parallel computation. We give some propositions and a preconditioned version of the MSD-CG method.
TL;DR: The main goal of this article is to demonstrate the use of the decomposition method that was developed by George Adomian to obtain solitary wave solutions for a new Hirota–Satsuma coupled KdV equations and a coupled MKdV equation.
Abstract: The main goal of this article is to demonstrate the use of the decomposition method that was developed by George Adomian [Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston, MA.] to obtain solitary wave solutions for a new Hirota–Satsuma coupled KdV equation and a coupled MKdV equation [Fan, E. G. (2001). Soliton solution for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation. Phys. Lett. A, 282, 18–22 and Wu, Y. T., Geng, X. G., Hu, X. B. and Zhu, S. M. (1999). Phys. Lett. A, 255, 259.]. The algorithm is illustrated by studying an initial value problem. The obtained results are presented, and only few terms are required to obtain an approximation solution that is found to be accurate and efficient.
TL;DR: A linear stability analysis shows the proposed scheme to be unconditionally stable and compared with published numerical and exact solutions performs well.
Abstract: A numerical solution of the one-dimensional Burgers equation is obtained using a lumped Galerkin method with quadratic B-spline finite elements. The scheme is implemented to solve a set of test problems with known exact solutions. Results are compared with published numerical and exact solutions. The proposed scheme performs well. A linear stability analysis shows the scheme to be unconditionally stable. E-mail: aesen@inonu.edu.tr
TL;DR: Using spline in compression, three difference schemes for the numerical solution of singularly perturbed two-point singular boundary-value problems are discussed and numerical results are given to illustrate the utility of the proposed methods.
Abstract: In this article, using spline in compression, we discuss three difference schemes for the numerical solution of singularly perturbed two-point singular boundary-value problems. The proposed schemes are second-order accurate and applicable to problems both in singular and non-singular cases. Convergence analysis of a difference scheme is discussed and numerical results are given to illustrate the utility of the proposed methods.
TL;DR: Some properties of EP matrix and its generalized inverse are given, the solutions of EP systems are analyzed and some applications in Navier-Stokes equations are made.
Abstract: In this article, we deal with EP matrix and EP singular linear systems. We give some properties of EP matrix and its generalized inverse, analyze the solutions of EP systems and make some applications in Navier-Stokes equations.
TL;DR: Compactons, solitary patterns, solitons, and periodic solutions are formally derived in nonlinear dispersive special types of the Zakharov–Kuznetsov equation with positive and negative exponents.
Abstract: In this article, we study nonlinear dispersive special types of the Zakharov–Kuznetsov equation with positive and negative exponents. The approach depends mainly on the sine–cosine algorithm. Compactons, solitary patterns, solitons, and periodic solutions are formally derived.
TL;DR: A B-spline finite element method is used to solve the equal width equation numerically and the resulting system of ordinary differential equations is integrated with respect to time using the fourth order Runge–Kutta method.
Abstract: A B-spline finite element method is used to solve the equal width equation numerically. This approach involves a collocation method using quintic B-splines at the knot points as element shape. Time integration of the resulting system of ordinary differential equations is effected using the fourth order Runge–Kutta method, instead of the finite difference method, the resulting system of ordinary differential equations is integrated with respect to time. Standard problems are used to validate the algorithm which is then used to model the smooth development of an undular bore.
TL;DR: An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed, based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method.
Abstract: An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed. The method is based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the diffusion equation with nonlocal boundary conditions are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.
TL;DR: A new finite difference method for solving a system of fourth-order boundary value problems associated with obstacle, unilateral and contact problems is introduced and it gives approximations, which are better than those produced by other collocation, finite difference and spline methods.
Abstract: In this article, we introduce and develop a new finite difference method for solving a system of fourth-order boundary value problems associated with obstacle, unilateral and contact problems. The convergence analysis of the new method has been discussed and it was shown that the order is four and it gives approximations, which are better than those produced by other collocation, finite difference and spline methods. Numerical examples are presented to illustrate the applications of this method. E-mail: dkaya@firat.edu.tr E-mail: eisasaid@ksu.edu.sa E-mail: kamel.alkhaled@uaeu.ac.ae
TL;DR: A new digital signature scheme based on the difficulty of simultaneously factoring a composite number and computing discrete logarithms is proposed, which each user uses common arithmetic moduli and only owns one private key and one public key.
Abstract: This article proposes a new digital signature scheme based on the difficulty of simultaneously factoring a composite number and computing discrete logarithms. In the proposed scheme, each user uses...
TL;DR: In this article, the accelerated over-relaxation (AOR) method is generalized for solving the saddle point problem or the augmented system and the connection between the parameters and the eigenvalues of the iteration matrix of the GAOR method is given.
Abstract: In this article, the accelerated over-relaxation (AOR) method is generalized for solving the saddle point problem or the augmented system. The successive over-relaxation (SOR)-like method is the special case of the generalized AOR (GAOR) method. The connection between the parameters and the eigenvalues of the iteration matrix of the GAOR method is given. Therefore, a necessary and sufficient condition for the convergence of the GAOR method is derived. Numerical examples are also given to show that the GAOR method is better than the SOR-like method in certain cases.
TL;DR: The existing implicit Adam–Bashforth method was extended and modified to gain a new explicit Taylor-liked method in solving stiff differential equations and the stability property for this method was considered.
Abstract: In this article, we extended the existing explicit Taylor method and modified it to gain a new explicit Taylor-liked method in solving stiff differential equations. We also considered the stability property for this method since the stability property of the classical explicit fourth order Runge–Kutta (RK4) method is not adequate for the solution of stiff problems. Implicit methods could work well for stiff problems but have certain drawbacks especially when discussing about the cost. A comparison of the existing implicit Adam–Bashforth, the classical explicit (RK4) and the new explicit Taylor-liked method is presented.
TL;DR: The obtained discrete solutions using the RK–Butcher algorithms are compared with the exact solutions of the optimal control problem and are found to be very accurate.
Abstract: In this article, the Runge–Kutta (RK)–Butcher algorithm is used to study the optimal control of linear singular systems with quadratic performance cost. The obtained discrete solutions using the RK–Butcher algorithms are compared with the exact solutions of the optimal control problem and are found to be very accurate. Stability analysis for the RK–Butcher algorithm is presented. Error graphs for discrete and exact solutions are presented in a graphical form to show the efficiency of this method. This RK–Butcher algorithm can be easily implemented in a digital computer and the solution can be obtained for any length of time. †On leave from National Institute of Technology, Thiruchirappalli, India