Showing papers in "International Journal of Computer Mathematics in 1999"
TL;DR: A matrix method called the Taylor collocation method is presented for numerically solving the linear integro-differential equations by a truncated Taylor series and it can be used for linear differential and integral equations.
Abstract: In this study, a matrix method called the Taylor collocation method is presented for numerically solving the linear integro-differential equations by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Also the method can be used for linear differential and integral equations. To illustrate the method, it is applied to certain linear differential, integral, and integro-differential equations and the results are compared.
130 citations
TL;DR: Third-order linear and non-linear boundary-value problems are solved using fourth-degree B-splines and the method is tested on two problems from the literature.
Abstract: Third-order linear and non-linear boundary-value problems are solved using fourth-degree B-splines. The convergence of the method is discussed. The method is tested on two problems from the literature
73 citations
TL;DR: In this paper, a new approach to a linear Korteweg-de Vries like equation is implemented by the Adomian decomposition method, which allows to obtain a decomposition series analytic solution of the equation which is calculated in the form of a convergent power series with easily computable components.
Abstract: A new approach to a linear Korteweg-de Vries like equation is implemented by the Adomian decomposition method. The approach is based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear, deterministic or stochastic [1–4]. It allows to obtain a decomposition series analytic solution of the equation which is calculated in the form of a convergent power series with easily computable components. The inhomogeneous problem is quickly solved by observing the self-canceling “noise” terms where the sum of components vanishes in the limit. Many test modeling problems from mathematical physics, linear and nonlinear, are discussed to illustrate the effectiveness and the performance of the decomposition method. This paper is particularly concerned with the accuracy for the modeling of various linear Korteweg-de Vries like equations by the Adomian decomposition method. Its remarkable accuracy is finally demonstrated in the study of several test problems.
72 citations
TL;DR: This research compares two normalization methods for Backpropagation in terms of prediction accuracy on the test sets through experiments with real world image data.
Abstract: Neural networks have been increasingly applied to many problems in many areas, and Backpropagation has been the most popular neural network model. Despite its wide application, there are some major issues to be considered before using the model, such as the network topology, learning parameter, and normalization methods for the input and output vectors. Input and output vectors for Backpropagation need to be normalized properly in order to achieve the best performance of the network. In this research, several normalization methods have been studied theoretically and two methods have been compared for performance in terms of prediction accuracy on the test sets through experiments with real world image data
57 citations
TL;DR: A class of continuous methods for the solution of initial value problems of general second order ordinary differential equations based on collocation of the differential systems at the grid points is proposed.
Abstract: In this paper we propose a class of continuous methods for the solution of initial value problems of general second order ordinary differential equations . The procedure which yields a solution matrix equation for different stepnumber k≥2 is based on collocation of the differential systems at the grid points. For n=2, three discrete schemes of order four for k=2 and k=3, and of order five for k=4 are recovered from the methods. Numerical examples are given to demonstrate and compare the efficiency of the methods for the stepnumbers k=2 and k=3 respectively.
56 citations
TL;DR: This paper shows that some continuous FD formulae can be used to provide a uniform treatment of both the ivp and the boundary value problem (bvp) without using the shooting method for the latter.
Abstract: In recent papers continuous finite difference (FD) approximations have been developed for the solution of the initial value problem (ivp) for first order ordinary differential equations (odes) They provide dense output of accurate solutions and global error estimates for the ivp economically In this paper we show that some continuous FD formulae can be used to provide a uniform treatment of both the ivp and the boundary value problem (bvp) without using the shooting method for the latter Higher order accurate solutions can be obtained on the same meshes with constant spacing used by one-step method without using the iterated deferred correction technique No additional conditions are required to ensure low order continuity and this leads to fewer necessary equations than those required by most of the popular methods for bvps No quadratures are involved in this non-overlapping piecewise continuous polynomial technique Some computed results are given to show the effectiveness of the proposed method and
54 citations
TL;DR: A remote password authentication scheme based on the digital signature methods is proposed that does not require the system to maintain a password file, and it can withstand attacks based on message replaying.
Abstract: Conventional password authentication schemes require password files or verification tables to validate the legitimacy of the login user. In addition, for remote access, these schemes cannot withstand an attack by replaying a previously intercepted login request. In this paper, we propose a remote password authentication scheme based on the digital signature methods. This scheme does not require the system to maintain a password file, and it can withstand attacks based on message replaying.
51 citations
TL;DR: A new Runge–Kutta RK(4,4) method with error control is introduced and the theory and analysis of its properties are investigated and compared with the more popular RK (4,5) method.
Abstract: In this paper a new Runge–Kutta RK(4,4) method with error control is introduced. The theory and analysis of its properties are investigated and compared with the more popular RK(4,5) method for the numerical solution of ordinary differential equations
39 citations
TL;DR: The method is based on combining the well-known Newton iterative method and the Alternating Group Explicit (A.G.E.) method and is suitable for use on parallel computers.
Abstract: In this paper a new iterative method is proposed for solving non-linear two point boundary value problems The method is based on combining the well-known Newton iterative method and the Alternating Group Explicit (AGE) method and is suitable for use on parallel computers
37 citations
TL;DR: Numerical experiments demonstrate the superiority of the present GTF-FDS, especially for the case of problems with discontinuities in the boundary and the initial conditions, compared with the popular Crank -Nicolson scheme.
Abstract: We investigate the application of the one-parameter family of generalized trapezoidal formulas (GTFs) introduced in Chawla et al. [2] for the time-integration of parabolic equations. The resulting GTF finite-difference schemes (GTF-FDS) are, in general, second order in both time and space and unconditionally stable. Interestingly, there exists a method of the family which is third order in time. Unlike the popular Crank -Nicolson scheme, our present GTF-FDS can cope with discontinuities in the boundary conditions and the initial conditions. We consider extensions of the GTF-FDS for equations with derivative boundary conditions and to a nonlinear problem. Numerical experiments demonstrate the superiority of the present GTF-FDS, especially for the case of problems with discontinuities in the boundary and the initial conditions.
33 citations
TL;DR: This paper uses cubic polynomial splines to derive some consistency relations which are used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle, unilateral and contact problems.
Abstract: In this paper, we use cubic polynomial splines to derive some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method.
TL;DR: A new second-order finite difference technique based upon the Peaceman and Rachford (P - R) alternating direction implicit (ADI) scheme, and also a fourth- order finite difference scheme based on the Mitchell and Fairweather (M - F) ADI method, are used as the basis to solve the two-dimensional time dependent diffusion equation with non-local boundary conditions.
Abstract: A new second-order finite difference technique based upon the Peaceman and Rachford (P - R) alternating direction implicit (ADI) scheme, and also a fourth-order finite difference scheme based on the Mitchell and Fairweather (M - F) ADI method, are used as the basis to solve the two-dimensional time dependent diffusion equation with non-local boundary conditions. The results of numerical experiments for these new methods are presented. These schemes use less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. They also have a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method.
TL;DR: An efficient technique known as a quarter sweeps multigrid method for solving two dimensional Poisson equation with the Dirichlet boundary condition with the red black Gauss-Seidel smoothing scheme is introduced.
Abstract: In this paper, we introduce an efficient technique known as a quarter sweeps multigrid method for solving two dimensional Poisson equation with the Dirichlet boundary condition. The method with the red black Gauss-Seidel smoothing scheme is shown to be the most superior than the half- and full-sweeps multigrid methods due to Othman et at. [8] and Gupta et al. [5], respectively. Some numerical experiments are included to confirm our recommendation
TL;DR: It is found that the rational cubic spline with linear denominator gives the best approximation to the function being interpolated.
Abstract: This paper deals with the approximation properties of some typical rational cubic splines, including the case with cubic, quadratic or linear denominator. From the point of view of the magnitude of the optimal error constant, it is found that the rational cubic spline with linear denominator gives the best approximation to the function being interpolated.
TL;DR: Using a particular geometric property of trapezoids graph, an efficient algorithm is designed which enables to generate all maximal independent sets for a trapezoid graph in time, where [mbar] and a are respectively the number of edges of the complement graph of the given graph and the numberof generated maximalIndependent sets.
Abstract: Generation of all maximal independent sets (MIS) is a NP-complete problem for general graphs. In this paper, using a particular geometric property of trapezoid graph, an efficient algorithm is designed which enables to generate all maximal independent sets for a trapezoid graph in time, where [mbar] and a are respectively the number of edges of the complement graph of the given graph and the number of generated maximal independent sets. There is no prior algorithm available to solve this problem of finding all MIS for trapezoid graphs.
TL;DR: Two new methods for reducing the complexity of size graphs are described and some theoretical results are given together with various examples.
Abstract: In this paper two new methods for reducing the complexity of size graphs are described. Some theoretical results are given together with various examples.
TL;DR: The discrete solutions of the second order multivariable linear system are obtained using the Single Term Walsh Series (STWS) technique and fourth order Runge-Kutta method to prove the effectiveness and simplicity of the methods.
Abstract: The discrete solutions of the second order multivariable linear system are obtained using the Single Term Walsh Series (STWS) technique and fourth order Runge-Kutta (RK) method. The results have been compared with the exact solutions and analysed to prove the effectiveness and simplicity of the methods.
TL;DR: By using collocation techniques, a stable pth-order explicit pseudo two-step Runge-Kutta method requiring only one effective sequential right-hand side evaluation per step on multiprocessor computers is obtained.
Abstract: The aim of this paper is to investigate a class of explicit pseudo two-step Runge-Kutta methods of arbitrarily high order for nonstiff problems for systems of first-order differential equations. By using collocation techniques we can obtain for any given order of accuracy p, a stable pth-order explicit pseudo two-step Runge-Kutta method requiring only one effective sequential right-hand side evaluation per step on multiprocessor computers. By a few widely-used test problems, we show the superiority of the methods considered in this paper over both sequential and parallel methods available in the literature.
TL;DR: It has been proven that fine-grained parallel genetic algorithm is an ergodic Markov chain and that it converges to the stationary distribution and the theoretical result has been applied to the context of optimisation of a deceptive function of 4-th order.
Abstract: This paper presents a fine-grained parallel genetic algorithm with mutation rate as a control parameter. The function of the mutation rate is similar to the temperature parameter in the simulated annealing [3,8,10]. The motivation behind this research is to develop a global convergence theory for the fine-grained parallel genetic algorithms based on the simulated annealing model There is a mathematical difficulty associated with the genetic algorithms as they do not strictly come under die definition of an algorithm. Algorithms normally have a starting point and a defined point of termination which genetic algorithms lack. The parallel genetic algorithm presented here is a stochastic process based on Markov chain [2] model It has been proven that fine-grained parallel genetic algorithm is an ergodic Markov chain and that it converges to the stationary distribution. The theoretical result has been applied to in the context of optimisation of a deceptive function of 4-th order.
TL;DR: The effectiveness of the Runge-Kutta method is demonstrated through different types of second order systems and it is observed that for singular second order system, the method is more suitable than any other systems.
Abstract: The effectiveness of the Runge-Kutta method is demonstrated through different types of second order systems. It is observed that for singular second order system, the method is more suitable than any other systems.
TL;DR: New time-integration schemes for the linear convection-diffusion equation and for the (viscous) Burgers' equation are described, employing the generalized trapezoidal formulas of Chawla et al.
Abstract: We describe new time-integration schemes for the linear convection-diffusion equation and for the (viscous) Burgers' equation, employing the generalized trapezoidal formulas of Chawla et al [2]. The obtained generalized trapezoidal formula finite-difference schemes (GTF(α)-FDS) are second order in both time and space and unconditionally stable. The better known existing schemes employ the Euler, the backward Euler or the classical arithmetic-mean trapezoidal formula (AM-TF) for integration in time. The performance of our present GTF(α)-FDS is compared with the AM-TF scheme by considering three test problems wherein the significance of the role played by the parameter α becomes evident in providing both stability and accuracy of the computed solution in the presence of diffusivity.
TL;DR: A new approach to implicit method for solving the convection-diffusion equation is developed, which will exhibit several advantageous features: highly accurate, fast and with good results whatever the exact solution is too large i.e., the absolute error still very small.
Abstract: In this paper, we shall develop a new approach to implicit method for solving the convection-diffusion equation, which will exhibit several advantageous features: highly accurate, fast and with good results whatever the exact solution is too large i.e., the absolute error still very small. The stability region is discussed and the obtained results for a test problem is compared with the exact solution and with Crank-Nicolson approximation, proves the mentioned advantages.
TL;DR: A variant of the step by step construction which associates standard and trim automata to regular languages and shows that the automaton constructed by the variant and the Glushkov automaton (computed by Berry-Sethi algorithm) are isomorphic.
Abstract: There are three classical algorithms to compute a finite automaton from a regular expression. The Brzozowski algorithm yields a deterministic automaton, the Glushkov algorithm a nondeterministic one, and the general step by step method generally yields a NFA with e-transitions. Berry and Sethi have adapted Brzozowski's algorithm to compute the Glushkov automaton of an expression. We describe a variant of the step by step construction which associates standard and trim automata to regular languages. We show that the automaton constructed by the variant and the Glushkov automaton (computed by Berry-Sethi algorithm) are isomorphic.
TL;DR: New improved fourth and fifth order linear and nonlinear Runge-Kutta methods are used in starting procedures for the well known k step multistep methods to give greater accuracy in the application to problems involving discontinuities and severe gradients where the stepsize is frequently changed.
Abstract: In this paper the use of new improved fourth and fifth order linear and nonlinear Runge-Kutta methods in starting procedures for the well known k step multistep methods is shown to give greater accuracy in the application to problems involving discontinuities and severe gradients where the stepsize is frequently changed
TL;DR: A new method for approximating the solution of nonlinear first order Fredholm delay integro-differential equation and results as well as numerical examples are given.
Abstract: In this paper, a new method for approximating the solution of nonlinear first order Fredholm delay integro-differential equation is presented. Boundness of the approximate solution, convergence results as well as numerical examples are given.
TL;DR: A matrix factorisation method based on the product of quadrant interlocking factors (Q.I.F.) which can be denoted mnemonically as W and Z and are of ‘butterfly’ or bow-tie form when the given matrix is symmetric then the factorisation resolves into a more simpler form i.e., A = WW T.
Abstract: Recently Evans [1] introduced a matrix factorisation method based on the product of quadrant interlocking factors (Q.I.F.) which can be denoted mnemonically as W and Z and are of ‘butterfly’ or bow-tie form. However when the given matrix is symmetric then the factorisation resolves into a more simpler formi.e., A = WW T and is presented in this short note.
TL;DR: A modified algebraic multigrid (AMG) method for queueing networks is presented and it is demonstrated that this method is more efficient and robust than conventional AMG method.
Abstract: A modified algebraic multigrid (AMG) method for queueing networks is presented. The method keeps the singularity of queueing networks in the coarse grid by modifying the restriction operators. Numerical results demonstrate that this method is more efficient and robust than conventional AMG method.
TL;DR: The iterative method showed in [5, 6] with case ∊ = 0,.
Abstract: Recently, some iterative methods have been proposed for distinguishing generalized diagonally dominant matrices. But these methods need to introduce a parameter ∊, it seems hard to decide the best value of ∊. In this paper, we discuss the iterative method showed in [5, 6] with case ∊ = 0,. prove the convergence of this method, and confirm this method is effective for distinguishing generalized diagonally dominant matrices by some examples
TL;DR: This paper, which is an extension of Fatunla (1984, 1985) presents P-stable one- leg LMM and its corresponding one-leg hybrid methods.
Abstract: Dahlquist (1978) barrier theorem which have been generalised by Hairer (1979) has greatly influenced the development of Linear Multistep methods (LMM) for second order initial value problems. Currently, emphasis is now on hybrid methods because of their high order and P-Stability characteristics. This paper which is an extension of Fatunla (1984, 1985) presents P-stable one-leg LMM and its corresponding one-leg hybrid methods.
TL;DR: It is shown the best iterative method is the preconditioned conjugate gradient method for solving rank deficient least squares problems.
Abstract: Rank deficient least squares problems appear in obtaining numerical solution of differential equations, computational genetics and other applications. The usual methods to solve the problem are QR decomposition. It is well-known that for large sparse problems, iterative methods are preferable. Miller and Neumann (1987) proposed the 4-block SOR method, and Santos, Silva and Yuan (1997) proposed the 2-block SOR method and the 3-block SOR method for solving the problem. Here some preconditioned conjugate gradient methods are proposed for solving the problem. The error bound and comparison with block SOR methods are studied. We show the best iterative method is the preconditioned conjugate gradient method for solving rank deficient least squares problems.