Search

In this paper, a new method based on hybridization of Lucas and Fibonacci polynomials is developed for approximate solutions of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equations. Firstly time discretization is made by using finite difference approaches. After that unknown function and its derivatives are expanded to Lucas series. Based on these series expansion, differentiation matrices are derived by utilizing Fibonacci polynomials. By doing so, the solution of the mentioned equations is reduced to the solution of an algebraic system of equations. By solving this system of equations the Lucas series coefficients are obtained. Then substituting these coefficients into Lucas series expansion approximate solutions can be constructed successively. The main goal of this paper is to indicate that Lucas polynomial based method is appropriate for 1D and 2D nonlinear problems. Efficiency and performance of the proposed method are judged on six test problems which consists of the 1D and 2D version of mentioned equation by calculating L 2 and L ∞ error norms. Feasibility of the method is verified by obtained accurate results. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new inner product AB 2 multiplication algorithm and effective hardware architecture for exponentiation in finite fields GF(2 m ). Exponentiation is more efficiently implemented by applying AB 2 multiplication repeatedly rather than AB multiplication. Thus, efficient AB 2 multiplication algorithms and simple architectures are the key to implementing exponentiation. Accordingly, this paper proposes an efficient inner product multiplication algorithm based on an irreducible all one polynomial (AOP) and simple architecture, which has the same hardware equipment as Fenn''s AB multiplier. The proposed bit-serial multiplication algorithm and architecture are highly regular and simpler than those of previous works. [Copyright &y& Elsevier]

This paper introduces the notion of normal factorisation of polynomials and then presents a new numerical algorithm for the computation of this factorisation. This procedure avoids computation of roots of polynomials and it is based on the use of algorithms determining the greatest common divisor of polynomials. In this paper, the general aspects of the algorithm are discussed and a symbolic implementation is given. The theoretical algorithm provides the basis for a numerical procedure, the general aspects of which are also discussed here. The advantage of such a factorisation is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots. For such polynomials, robust numerical techniques based on finding roots may then be used, if it is desired to work out the usual irreducible factorisation. A detailed description of the implementation of the algorithm is presented. The problem considered here is an integral part of computations for algebraic control problems. [Copyright &y& Elsevier]

Abstract: We describe a method that computes the number of roots of a polynomial inside a region bounded by the curve , with an analysis of its computational cost. It is based on the number of roots being the same as the winding number of . While the usual methods for computing the winding number involve numerical integration, in this paper we use a geometrical construction. We show its correctness without referring to global information about (like its Lipschitz constant on ). The analysis of its cost is based on the distance from the roots to , expressed using a condition number suitably defined. The method can be used in a divide-and-conquer root-finding algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we develop new numeric modified Adomian decomposition algorithms by using the Wazwaz–El-Sayed modified decomposition recursion scheme, and investigate their practicality and efficiency for several nonlinear examples. We show how we can conveniently generate higher-order numeric algorithms at will by this new approach, including, by using examples, 12th-order and 20th-order numeric algorithms. Furthermore, we show how we can achieve a much larger effective region of convergence using these new discrete solutions. We also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th-order numeric modified decomposition algorithm over the classic 4th-order Runge–Kutta algorithm by example. The efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by Duan for generation of the Adomian polynomials to high orders. [Copyright &y& Elsevier]

Abstract: The problem of economic dispatch with multiple fuel units has been widely addressed via different techniques using approximate methods due to the exponential complexity of full enumeration in the underlying combinatory problem. A method has recently been outlined by Min et al. (2008), that allows the problem to be solved in an exact way in polynomial time. In this paper, we present an alternative technique and take this idea further, studying and comparing two algorithms of polynomial complexity: basic recurrence and divide-and-conquer. Moreover, we provide the exact solution to the problem by Lin and Viviani (1984), that constitutes the traditional test for all approximate methods and present a comprehensive survey of several heuristic approaches. [Copyright &y& Elsevier]

Abstract: This paper deals with an inverse problem of determining a source term in the one-dimensional fractional advection–dispersion equation (FADE) with a Dirichlet boundary condition on a finite domain, using final observations. On the basis of the shifted Grünwald formula, a finite difference scheme for the forward problem of the FADE is given, by means of which the source magnitude depending upon the space variable is reconstructed numerically by applying an optimal perturbation regularization algorithm. Numerical inversions with noisy data are carried out for the unknowns taking three functional forms: polynomials, trigonometric functions and index functions. The reconstruction results show that the inversion algorithm is efficient for the inverse problem of determining source terms in a FADE, and the algorithm is also stable for additional data having random noises. [Copyright &y& Elsevier]

Abstract: This paper explores the problems of generalized center conditions and integrability of resonant infinity for a complex polynomial differential system. The method is based on converting resonant infinity into an elementary singular point by a homeomorphic transformation. The calculation of generalized singular point quantities is an effective way of finding necessary conditions for integrable systems. A new recursive algorithm for computing generalized singular point quantities at the origin of the transformed system is derived. At the same time, a necessary and sufficient condition for resonant infinity to be a generalized complex center is presented. As an application of the new recursive algorithm, the generalized center conditions for resonant infinity for a class of cubic systems are discussed. To the best of our knowledge, this is the first time that the generalized center problem has been considered for resonant infinity. [Copyright &y& Elsevier]

Abstract: In this paper Haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration. This approach is the generalization and improvement of the methods (Siraj-ul-Islam et al. (2010) ) where the numerical methods are only applicable to the integrals with constant limits. Apart from generalization of the methods , the new approach has two major advantages over the classical methods based on quadrature rule: (i) No need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically (ii) Mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n. The new methods are more efficient. The novel methods are compared with existing methods and applied to a number of benchmark problems. Accuracy of the methods are measured in terms of absolute errors. [Copyright &y& Elsevier]

Abstract: In this paper, a scheduling problem for automated guided vehicles in container terminals is defined and formulated as a Minimum Cost Flow model. This problem is then solved by a novel algorithm, NSA+, which extended the standard Network Simplex Algorithm (NSA). Like NSA, NSA+ is a complete algorithm, which means that it guarantees optimality of the solution if it finds one within the time available. To complement NSA+, an incomplete algorithm Greedy Vehicle Search (GVS) is designed and implemented. The NSA+ and GVS are compared and contrasted to evaluate their relative strength and weakness. With polynomial time complexity, NSA+ can be used to solve very large problems, as verified in our experiments. Should the problem be too large for NSA+, or the time available for computation is too short (as it would be in dynamic scheduling), GVS complements NSA+. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents algorithms for decomposing any zero-dimensional polynomial set into simple sets over an arbitrary finite field, with an associated ideal or zero decomposition. As a key ingredient of these algorithms, we generalize the squarefree decomposition approach for univariate polynomials over a finite field to that over the field product determined by a simple set. As a subprocedure of the generalized squarefree decomposition approach, a method is proposed to extract the th root of any element in the field product. Experiments with a preliminary implementation show the effectiveness of our algorithms. [Copyright &y& Elsevier]

Abstract: This paper deals with single-machine scheduling problems with decreasing linear deterioration, i.e., jobs whose processing times are a decreasing function of their starting times. In addition, the jobs are related by parallel chains and a series–parallel graph precedence constraints, respectively. It is shown that for the problems of minimization of the makespan, polynomial algorithms exist. [Copyright &y& Elsevier]

Abstract: In this paper, we study the properties of regular systems and improve the efficiency of the regular decomposition method RegSer implemented in Epsilon. We define a weaker concept which retains most properties of regular system. It can be shown that from a weak regular system one can also define a regular set and vice versa. We present an algorithm RecurWeakRegSer to decompose a given polynomial system into weak regular systems. When , the output of RecurWeakRegSer() often contains fewer components than that of RegSer(). This is one advantage of RecurWeakRegSer. Another one is that RecurWeakRegSer is more efficient than RegSer. This was shown by experiments that we carried out. Since it is an essential step in RegSer to compute subresultant polynomial remainder sequences (PRS), and there is some weakness in the implementation, we implement a new version of subresultant algorithm using the optimization strategy of Ducos so that the efficiency of RegSer can be improved. [Copyright &y& Elsevier]

Abstract: The piecewise algebraic variety, as a set of the common zeros of multivariate splines, is a kind of generalization of the classical algebraic variety. In this paper, we present an algorithm for isolating the zeros of the zero-dimensional piecewise algebraic variety which is primarily based on the interval zeros of univariate interval polynomials. Numerical example illustrates that the proposed algorithm is flexible. [Copyright &y& Elsevier]

Abstract: In this paper, we focus on a class of dual-rate sampled-data systems in which all the inputs are available at each instant while only scarce outputs can be measured ( being an integer more than unity). To estimate the parameters of such dual-rate systems, we derive a mathematical model by using the polynomial transformation technique, and apply the extended least squares algorithm to identify the dual-rate systems directly from the available input–output data . Then, we study the convergence properties of the algorithm in details. Finally, we give an example to test and illustrate the algorithm involved. [Copyright &y& Elsevier]

In this paper, we introduce a new algorithm for applying the Adomian decomposition method to nonlinear differential and partial differential equations. The proposed algorithm does not require the explicit evaluation of Adomian polynomials to approximate the nonlinearity term. The numerical experiments show that the proposed algorithm is more accurate at larger values for time, though at the cost of more computations. [Copyright &y& Elsevier]

Abstract: The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of this algorithms is based on the approximate calculation of higher derivatives using well-known finite-difference technique for the partial differential equations. The approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval integration. This property offers different facility for adaptive error control. This paper describes several explicit Taylor series algorithms with numerical derivatives and their implicit extension and examines its consistency and stability properties. The implicit extension based on a collocation term added to the explicit truncated Taylor series and the approximate solution obtained as a continuously differentiable piecewise polynomials function. Some numerical test results is presented to prove the efficiency of these new-old algorithm. [Copyright &y& Elsevier]

Abstract: One of the main problems dealing with iterative methods for solving polynomial systemsis the initialization of the iteration. This paper provides an algorithm to initialize the search of solutions of polynomial systems. [Copyright &y& Elsevier]

In the self-stabilizing algorithmic paradigm for distributed computation, each node has only a local view of the system, yet in a finite amount of time, the system converges to a global state satisfying some desired property. In this paper, we present polynomial time self-stabilizing algorithms for finding a dominating bipartition, a maximal independent set, and a minimal dominating set in any graph. [Copyright &y& Elsevier]

In this paper, we analyse the stability of parallel algorithms for the evaluation of polynomials written as a finite series of orthogonal polynomials. The basic part of the computation is the solution of a triangular tridiagonal linear system. This fact allows us to present a more detailed analysis. The theoretical results show that the parallel algorithms are almost as stable as their sequential counterparts for practical applications. Extensive numerical experiments confirm the theoretical conclusions. [Copyright &y& Elsevier]