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Showing papers in "International Journal of Computer Mathematics in 2005"


Journal ArticleDOI
TL;DR: A numerical method based on the Adomian decomposition method is introduced for the approximate solution of delay differential equations and it is shown that only a few terms are required to obtain an approximate solution which is found to be accurate and efficient.
Abstract: A numerical method based on the Adomian decomposition method which has been developed by Adomian [Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston, MA] is introduced in this paper for the approximate solution of delay differential equations. The algorithm is illustrated by studying an initial value problem. The results obtained are presented and show that only a few terms are required to obtain an approximate solution which is found to be accurate and efficient.

223 citations


Journal ArticleDOI
TL;DR: A simple routing algorithm is developed for LTQ n, which creates a shortest path from the source to the destination in O(n) time, and it is shown thatLTQ n consists of two disjoint copies of Q n −1 by adding a matching between their nodes.
Abstract: This paper introduces a new variant of the popular n-dimensional hypercube network Q n , known as the n-dimensional locally twisted cube LTQ n , which has the same number of nodes and the same number of connections per node as Q n . Furthermore, LTQ n is similar to Q n in the sense that the nodes can be one-to-one labeled with 0–1 binary sequences of length n, so that the labels of any two adjacent nodes differ in at most two successive bits. One advantage of LTQ n is that the diameter is only about half of the diameter of Q n . We develop a simple routing algorithm for LTQ n , which creates a shortest path from the source to the destination in O(n) time. We find that LTQ n consists of two disjoint copies of Q n −1 by adding a matching between their nodes. On this basis, we show that LTQ n has a connectivity of n.

193 citations


Journal ArticleDOI
TL;DR: The tanh function method is used to solve non-linear partial differential equations in one and two dimensions by applying it to a variety of selected equations.
Abstract: Exact solutions of some important non-linear partial differential equations in one and two dimensions are obtained using the tanh function method. The efficiency of the method is demonstrated by applying it to a variety of selected equations.

91 citations


Journal ArticleDOI
TL;DR: The two-dimensional Kadomtsev–Petviashivilli-modified equal width equation is investigated, using the tanh method and the sine–cosine method to obtain a variety of exact solutions.
Abstract: The two-dimensional Kadomtsev–Petviashivilli-modified equal width equation is investigated. The strategy here consists of using the tanh method and the sine–cosine method to obtain a variety of exact solutions of this equation. Both schemes work well and reveal distinct exact solutions. The obtained solutions include solitary wave solutions and periodic solutions.

90 citations


Journal ArticleDOI
TL;DR: The generalization of the differential transformation method to n-dimensional case in order to solve partial differential equations (PDEs) and applies the results to a few initial boundary-value problems to illustrate the proposed method.
Abstract: This paper presents the generalization of the differential transformation method to n-dimensional case in order to solve partial differential equations (PDEs). A distinctive practical feature of this method is its ability to solve especially nonlinear differential equations efficiently. We apply our results to a few initial boundary-value problems to illustrate the proposed method.

83 citations


Journal ArticleDOI
TL;DR: A class of numerical methods proposed for solving general third-order ordinary differential equations directly by collocation at the grid points and at an off grid point yields a particular discrete scheme as a special case of the method.
Abstract: A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x n+j , i = 0(1)k and at an off grid point x = x n+u , where k is the step number of the method and u is an arbitrary rational number in (x n , x n+k ). A predictor of order 2k − 1 is also proposed to cater for y n+k in the main method. Taylor series expansion is employed for the calculation of y n+1, y n+2, y n+u and their higher derivatives. Evaluation of the resulting method at x = x n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.

75 citations


Journal ArticleDOI
TL;DR: A collocation method for the GEW equation, which is classified as a nonlinear PDE using quadratic B-splines at midpoints as element shape functions using a Maxwellian initial condition pulse is presented.
Abstract: We consider solitary wave solutions of the generalized equal width (GEW) wave equation u t + ϵu p u x − δu xxt = 0. This paper presents a collocation method for the GEW equation, which is classified as a nonlinear PDE using quadratic B-splines at midpoints as element shape functions. In this research, the scheme of the equation under investigation is found to be unconditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested methods that is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied.

64 citations


Journal ArticleDOI
TL;DR: It is shown that if a fuzzy preference relation satisfies reciprocal, transitive and comparable, it needs only O(n) comparisons of fuzzy numbers to rank n fuzzy numbers with fuzzy preference relations, which is more competitive than traditional methods that needs to calculate (n(n − 1)/2 preference relations.
Abstract: In this article, we propose a new property called ‘comparable’ for fuzzy preference relation. We show that if a fuzzy preference relation satisfies reciprocal, transitive and comparable, it needs only O(n) comparisons of fuzzy numbers to rank n fuzzy numbers with fuzzy preference relation, which is more competitive than traditional methods that needs to calculate (n(n − 1))/2 preference relations.

59 citations


Journal ArticleDOI
TL;DR: It is shown that this parameter can be used to measure the vulnerability of networks and obtain some Nordhaus–Gaddum type results for the rupture degree.
Abstract: We introduce a new graph parameter, the rupture degree. The rupture degree for a complete graph K n is defined as 1−n, and the rupture degree for an incomplete connected graph G is defined by r(G)=max{ω(G−X)−|X|−m(G−X):X⊂V(G), ω(G−X)>1}, where ω(G−X) is the number of components of G−X and m(G−X) is the order of a largest component of G−X. It is shown that this parameter can be used to measure the vulnerability of networks. Rupture degrees of several specific classes of graphs are determined. Formulas for the rupture degree of join graphs and some bounds of the rupture degree are given. We also obtain some Nordhaus–Gaddum type results for the rupture degree.

55 citations


Journal ArticleDOI
TL;DR: This paper discusses the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations, and proves that the schemes converge almost first-order uniformly with respect to small parameters.
Abstract: In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.

53 citations


Journal ArticleDOI
TL;DR: A comparative study of the differential transformation for solving systems of linear or non-linear ordinary differential equations (ODEs) and applies the results to some initial value problems to demonstrate the ability of the method to solve systems of differential equations.
Abstract: We present a comparative study of the differential transformation for solving systems of linear or non-linear ordinary differential equations (ODEs). A remarkable practical feature of this method is its ability to solve the system of linear or non-linear differential equations efficiently. This method also enables us to control the truncation error by adjusting the step size used in the numerical scheme. We apply our results to some initial value problems to demonstrate the ability of the method to solve systems of differential equations.

Journal ArticleDOI
TL;DR: An implementation of the Haar wavelet to the optimal control of linear singularly perturbed systems is presented and the slow and fast trajectories with respect to a quadratic cost function are calculated.
Abstract: In this paper we present an implementation of the Haar wavelet to the optimal control of linear singularly perturbed systems. The approximated composite control and the slow and fast trajectories with respect to a quadratic cost function are calculated by solving only the linear algebraic equations. The results are illustrated with a simple example.

Journal ArticleDOI
TL;DR: The tanh and sine–cosine methods are used to handle the two-dimensional ZK-modified equal-width equation (ZK-MEW) and reveal a number of useful features of the methods applied.
Abstract: The tanh and sine–cosine methods are used to handle the two-dimensional ZK-modified equal-width equation (ZK-MEW). The two methods work well to obtain exact solutions of different physical structures; solitary wave solutions and periodic solutions are also obtained. The framework presented here reveals a number of useful features of the methods applied.

Journal ArticleDOI
TL;DR: Two related global but linear smoothers that help the convergence of multigrid methods are presented and the Krylov acceleration technique is combined with the proposed multigrids method to improve performance.
Abstract: Fast solution of the non-linear partial differential equations (PDEs) arising from image restoration is of practical importance. The standard multigrid methods do not work well, because of the highly discontinuous coefficients of the underlying non-linear PDEs. We present two related global but linear smoothers that help the convergence of multigrid methods. Furthermore, the Krylov acceleration technique is combined with the proposed multigrid method to improve performance. Numerical experiments are shown.

Journal ArticleDOI
TL;DR: A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients with unknown Taylor coefficients, and the Taylor polynomial approach is obtained.
Abstract: A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.

Journal ArticleDOI
TL;DR: The Taylor matrix method is introduced, based on first taking the truncated Taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation so that the resultant matrix equation can be solved and the unknown Taylor coefficients can be found approximately.
Abstract: The purpose of this study is to give a Taylor polynomial approximation for the solution of mth-order linear difference equations with variable coefficients under the mixed conditions about any poin

Journal ArticleDOI
TL;DR: The behaviour of the roots of the multiple Changhee q-Bernoulli polynomials for values of the index n is discussed by using computer.
Abstract: In this article, we will discuss the behaviour of the roots of the multiple Changhee q-Bernoulli polynomials for values of the index n by using computer.

Journal ArticleDOI
TL;DR: Evaluation of the proposed multiderivative collocation method for direct solution of the general initial-value problems of ordinary differential equations gives a particular discrete scheme as a special case of the method.
Abstract: This article discusses a multiderivative collocation method for direct solution of the general initial-value problems of ordinary differential equations of the form y ( n )(x) = f(x, y, y′, y′, …, y n −1), y(a) = y 0, y i (a) = y i , i = 1, 2, 3. To ensure the symmetry of the method, collocation of the differential system has been taken at the selected grid points. Furthermore, a predictor for the calculation of the value of y n + k and its derivatives that appear in the main method is developed. Taylor series expansion is used to calculate the values of y n + i , i = 1, 2, 3 and their derivatives which also appear in the main method. The interval of periodicity and the error constant of the method at x = x n + k are calculated. Evaluation of the proposed method at x = x n + k gives a particular discrete scheme as a special case of the method. Finally, the efficiency of the method is tested on non-stiff initial-value problems.

Journal ArticleDOI
TL;DR: A general factorization theorem on subgraphs of Cayley digraphs by their automorphism groups is derived and the applications of these results to well-known interconnection networks are discussed.
Abstract: We consider the relationships between Cayley digraphs and their coset graphs with respect to subgroups and obtain some general results on homomorphism and broadcasting between them. We also derive a general factorization theorem on subgraphs of Cayley digraphs by their automorphism groups. We discuss the applications of these results to well-known interconnection networks such as the butterfly network, the de Bruijn network, the cube-connected cycles network and the shuffle-exchange network.

Journal ArticleDOI
TL;DR: The objective is to determine an optimal replenishment number, lot-size of a two-warehouse inventory system for deteriorating items removing the impractical assumption regarding the storage capacity of RW.
Abstract: The purpose of this research is to discuss an application of real-coded Genetic Algorithm (RCGA) for mixed integer non-linear programming in a two-warehouses inventory control problem. Our objective is to determine an optimal replenishment number, lot-size of a two-warehouse (owned and rented warehouse (RW)) inventory system for deteriorating items removing the impractical assumption regarding the storage capacity of RW. The model is formulated with infinite replenishment, finite planning horizon, linearly time dependent demand (increasing) and partially backlogged shortages. The mathematical formulation of the problem indicates that the model is a constrained non-linear mixed integer problem with one integer and one non-integer variables. To solve this problem, we develop a RCGA with ranking selection, whole arithmetic crossover and mutation (uniform mutation for integer variable and non-uniform for non-integer variable). The proposed model has been solved using this RCGA and illustrated with four numeri...

Journal ArticleDOI
TL;DR: The solution of the one-phase Stefan problem is presented, based on the Adomian decomposition method and optimalization, which consists of finding the distribution of temperature in the domain and the position of the moving interface.
Abstract: The solution of the one-phase Stefan problem is presented. This problem consists of finding the distribution of temperature in the domain and the position of the moving interface (freezing front). The proposed solution is based on the Adomian decomposition method and optimalization. The validity of the approach is verified by comparing the results obtained with the analytical solution.

Journal ArticleDOI
TL;DR: A combination of Fourier and block-pulse functions are used, to solve the linear integro differential equation, and the integral equation is converted to a system of linear equations.
Abstract: We use a combination of Fourier and block-pulse functions on the interval [0, 1], to solve the linear integro differential equation. We convert the integral equation to a system of linear equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Journal ArticleDOI
TL;DR: This work demonstrates the use of the modified decomposition method for the analytic treatment of non-linear Fredholm integral equations, non- linear Volterra integral equations and systems ofnon-linear integral equations.
Abstract: We demonstrate the use of the modified decomposition method for the analytic treatment of non-linear Fredholm integral equations, non-linear Volterra integral equations and systems of non-linear integral equations. The proper implementation of the modified method can dramatically reduce the amount of work required and may provide the exact solution using only a few iterations. The analysis is accompanied by numerical illustrations that show the pertinent features of the technique.

Journal ArticleDOI
TL;DR: A third-order-accurate variable-mesh two-parameter alternating group explicit (TAGE) iteration method for the numerical solution of the two-point singular boundary value problem subject to boundary conditions u(0)=A, u(1)=B, where A and B are finite constants.
Abstract: We propose a third-order-accurate variable-mesh two-parameter alternating group explicit (TAGE) iteration method for the numerical solution of the two-point singular boundary value problem subject to boundary conditions u(0)=A, u(1)=B, where A and B are finite constants. We also discuss a Newton–TAGE iteration method for the third-order numerical solution of a two-point non-linear boundary value problem. The proposed method is applicable to singular and non-singular problems and is suitable for use on parallel computers. The convergence analysis is briefly discussed. Computational results are provided to illustrate the proposed TAGE iterative methods.

Journal ArticleDOI
TL;DR: A second-order method is developed for the numerical solution of a non-linear, third-order, boundary-value (BV) problem that arises from a four-point recurrence relation involving exponential terms, these being replaced by Padé approximants.
Abstract: A second-order method is developed for the numerical solution of a non-linear, third-order, boundary-value (BV) problem. The method arises from a four-point recurrence relation involving exponential terms, these being replaced by Pade approximants. The convergence of the method is discussed. The method is tested on a sandwich beam problem to demonstrate its usefulness.

Journal ArticleDOI
TL;DR: Three convergent difference schemes, using spline in tension for the difference solution of singularly perturbed two-point singular boundary value problems are reported, applicable to problems both in non-singular and singular cases.
Abstract: In this article, we report three convergent difference schemes, using spline in tension for the difference solution of singularly perturbed two-point singular boundary value problems. The proposed methods are O(h 2) accurate and applicable to problems both in non-singular and singular cases. Convergence theory of a proposed difference method is analyzed and numerical experiments carried out on the new schemes supplement the analytical results.

Journal ArticleDOI
TL;DR: This paper adapts the fault-tree methodology of reliability engineering to the quantification of security exposure of computer systems and handles the doubly stochastic problem of estimating the uncertainty in the top event probability by using an analytic exact formula relating the variance of the topevent probability to the variances of the basic event probabilities.
Abstract: Quantitative assessment of the effect of security breaches on a computer system can be based on the following: specification of all foreseeable types of basic events and estimation of their probabilities of occurrence over a stated period of time; observation of the various types of security measures employed by the system; definition of the undesired top events resulting from security breaches, and estimation of the system’s vulnerability to each of these events as the cost incurred by the system if that event took place; mathematical modelling of the logical relations between the aforementioned entities. In this paper we adapt the fault-tree methodology of reliability engineering to the quantification of security exposure of computer systems. In this new context, a fault tree can be described as a logic diagram whose input represents breach events at various system levels, and whose vertices represent logic operations or gates. The root or output of the fault tree can be any of the undesired top events....

Journal ArticleDOI
TL;DR: The fuzzy system derived from collected data is considered by the fuzzy grey controlled variable to derive a fuzzy grey GM(1, 1) model to forecast the extrapolative values under the fuzzy system.
Abstract: Grey GM(1, 1) forecasting model is a kind of short-term forecasting method which has been successfully applied in management and engineering problems with as little as four data. However, when a new system is constructed, the system is uncertain and variable so that the collected data is usually of fuzzy type, which could not be applied to grey GM(1, 1) model forecast. In order to cope with such problem, the fuzzy system derived from collected data is considered by the fuzzy grey controlled variable to derive a fuzzy grey GM(1, 1) model to forecast the extrapolative values under the fuzzy system. Finally, an example is described for illustration.

Journal ArticleDOI
TL;DR: Combination of the harmonic and arithmetic means of the Runge–Kutta formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations.
Abstract: The derivation of a composite method for solving stiff ordinary differential equations is discussed. Combination of the harmonic and arithmetic means of the Runge–Kutta formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations. The numerical results and the A-stability of this new formula are examined.

Journal ArticleDOI
TL;DR: An extension of the Greville's partitioning method for computing the Moore–Penrose inverse of two-variable rational and polynomial complex matrices is proposed and a corresponding effective algorithm is developed, applicable in the case when there are only few degrees in A(s 1, s 2) corresponding to non-zero coefficient matrices.
Abstract: We propose an extension of the Greville’s partitioning method for computing the Moore–Penrose inverse of two-variable rational and polynomial complex matrices. In addition, we developed corresponding effective algorithm, applicable in the case when there are only few degrees in A(s 1, s 2) corresponding to non-zero coefficient matrices. These algorithms are implemented using symbolic and functional possibilities of the package MATHEMATICA.