Showing papers in "International Journal of Computer Mathematics in 2009"
TL;DR: New unconditionally stable implicit difference schemes for the numerical solution of multi-dimensional telegraphic equations subject to appropriate initial and Dirichlet boundary conditions are discussed.
Abstract: New unconditionally stable implicit difference schemes for the numerical solution of multi-dimensional telegraphic equations subject to appropriate initial and Dirichlet boundary conditions are discussed. Alternating direction implicit methods are used to solve two and three space dimensional problems. The resulting system of algebraic equations is solved using a tri-diagonal solver. Numerical results are presented to demonstrate the utility of the proposed methods.
123 citations
TL;DR: A linear multistep method of order 5 that is self-starting for the direct solution of the general second-order initial value problem (IVP) without the need for either predictors or starting values from other methods is proposed.
Abstract: In this paper, we propose a linear multistep method of order 5 that is self-starting for the direct solution of the general second-order initial value problem (IVP). The method is derived by the interpolation and collocation of the assumed approximate solution and its second derivative at x=xn+j, j=1, 2,..., r-1, and x=xn+j, j=1, 2,..., s-1, respectively, where r and s are the number of interpolation and collocation points, respectively. The interpolation and collocation procedures lead to a system of (r+s) equations involving (r+s) unknown coefficients, which are determined by the matrix inversion approach. The resulting coefficients are used to construct the approximate solution from which multiple finite difference methods (MFDMs) are obtained and simultaneously applied to provide a direct solution to IVPs. In particular, the method is implemented without the need for either predictors or starting values from other methods. Numerical examples are given to illustrate the efficiency of the method.
67 citations
TL;DR: It is shown, using a non-linear Lyapunov function and LaSalle invariance principle, that this endemic equilibrium is globally asymptotically stable for a special case of the avian-only system.
Abstract: A deterministic model for the transmission dynamics of avian influenza in birds (wild and domestic) and humans is developed. The model, which allows for the transmission of an avian strain and its mutant (assumed to be transmissible between humans), as well as the isolation of individuals with symptoms of any of the two strains, has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. Further, the model has a unique endemic equilibrium whenever this threshold quantity exceeds unity. It is shown, using a non-linear Lyapunov function and LaSalle invariance principle, that this endemic equilibrium is globally asymptotically stable for a special case of the avian-only system. Numerical simulations show that, on average, the isolation of individuals with the avian strain is more beneficial than isolating those with the mutant strain. Furthermore, disease burden increases with increasing mutation rate of the avian strain.
66 citations
TL;DR: It is found that chaos exists in the fractional-order Lorenz system of order less than 3.97, and the lowest order found to have chaos in this system is 2.97.
Abstract: In this article, we investigate the chaotic behaviours in the fractional-order Lorenz system. By utilizing the fractional calculus techniques, we found that chaos exists in the fractional-order Lorenz system of order less than 3. The lowest order we found to have chaos in this system is 2.97.
63 citations
TL;DR: Bézier curves with n shape parameters and triangular BéZier surfaces with 3n(n+1)/2 shape parameters are presented and the geometric properties of these curves and surfaces are discussed.
Abstract: Bezier curves with n shape parameters and triangular Bezier surfaces with 3n(n+1)/2 shape parameters are presented in this paper. The geometric significance of the shape parameters and the geometric properties of these curves and surfaces are discussed. The shapes of the curves and the surfaces can be modified intuitively, foreseeably and precisely by changing the values of the shape parameters.
62 citations
TL;DR: A new stochastic hybrid technique for constrained global optimization that is a combination of the electromagnetism-like (EM) mechanism with a random local search, which is a derivative-free procedure with high ability of producing a descent direction.
Abstract: In this paper, we present a new stochastic hybrid technique for constrained global optimization. It is a combination of the electromagnetism-like (EM) mechanism with a random local search, which is a derivative-free procedure with high ability of producing a descent direction. Since the original EM algorithm is specifically designed for solving bound constrained problems, the approach herein adopted for handling the inequality constraints of the problem relies on selective conditions that impose a sufficient reduction either in the constraints violation or in the objective function value, when comparing two points at a time. The hybrid EM method is tested on a set of benchmark engineering design problems and the numerical results demonstrate the effectiveness of the proposed approach. A comparison with results from other stochastic methods is also included.
59 citations
TL;DR: By using DTM, the method is extended for delay differential equations and is a reliable method that needs less work and does not require strong assumptions and linearization.
Abstract: Differential transform method (DTM) is extended for delay differential equations. By using DTM, we manage to obtain the numerical, analytical, and exact solutions of both linear and nonlinear equations. In comparison with the existing techniques, the DTM is a reliable method that needs less work and does not require strong assumptions and linearization.
51 citations
TL;DR: Operational matrices of integration and product based on Chebyshev wavelets are presented and a general procedure for forming these matrices is given.
Abstract: Operational matrices of integration and product based on Chebyshev wavelets are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. Numerical examples are given to demonstrate applicability of these matrices.
49 citations
TL;DR: An unconditionally stable high-order compact finite difference scheme that computes both the option price and the hedging delta ∂ V/∂ S simultaneously is proposed.
Abstract: In this paper, an unconditionally stable high-order compact finite difference scheme is proposed. The compact scheme is fourth-order accurate in both the temporal and spatial dimensions. The new method computes both the option price and the hedging delta ∂ V/∂ S simultaneously. Two numerical examples are presented to demonstrate the accuracy and efficiency of the proposed scheme.
46 citations
TL;DR: This article examines the use of cubic and PH quintic Bézier curve segments that have a single curvature extremum, and which have G 2 contact with the straight line segments to which they are joined, as fillets, and shows how the extreme circle of curvature can be determined.
Abstract: Fillets, also known as blend arcs, are used in CNC machining to round corners. Fillets are normally circular arcs, which have G 1 contact with the straight line segments to which they are joined. Recent advances in machining technology allow NURBS, including Pythagorean-hodograph (PH) curve segments, to be incorporated in CNC tool paths. This article examines the use of cubic and PH quintic Bezier curve segments that have a single curvature extremum, and which have G 2 contact with the straight line segments to which they are joined, as fillets. It is shown how the extreme circle of curvature can be determined. The point of curvature extremum and the corresponding value of the curvature can be changed by adjusting the joining points of the blending curve with the neighbouring straight lines. These blending curves can also be incorporated in computer-aided design packages for curve or surface design.
41 citations
TL;DR: This article focuses on the explanation of a new technique that combines the use of the principal components analysis (PCA) method with MARS, a mathematical method capable of predicting the BMD of post-menopausal women, taking into account only certain nutritional variables.
Abstract: In this work, the application of ‘multivariate adaptive regression splines’ (MARS) for modelling osteoporosis is described. This article focuses on the explanation of a new technique that combines the use of the principal components analysis (PCA) method with MARS. The use of this new technique allows for an easier management of large databases with a lower computational cost as the PCA allows the elimination of those variables that are redundant from the point of view of the phenomena under study. Osteoporosis is characterized by low ‘bone mineral density’ (BMD). This illness has a high-cost impact in all developed countries. The aim of this article is the development of a mathematical method capable of predicting the BMD of post-menopausal women, taking into account only certain nutritional variables. A nutritional habits and lifestyle questionnaire is drawn up. The variables obtained from this, together with the BMD of the patients calculated by densitometry, are processed using the ‘principal componen...
TL;DR: A C 1 interpolating scheme to deal with the problem of monotonicity is presented, which uses piecewise rational cubic function, in a most general form which involves four free parameters in its description in each interval which ensure themonotonicity of monotone data.
Abstract: We present a C1 interpolating scheme to deal with the problem of monotonicity (that is when data is monotone, the interpolant should also preserve the monotonicity). The scheme uses piecewise rational cubic function, in a most general form which involves four free parameters in its description in each interval. We derive data dependent sufficient conditions on these parameters which ensure the monotonicity of monotone data. These parameters are also used to provide freedom to the user to refine the appearance of the curves interactively.
TL;DR: Lower and upper bounds of γ k (C m ×C n ) for k=2 are established and in some cases, these bounds agree so that the exact 2-domination number is obtained.
Abstract: A k-dominating set for a graph G(V, E) is a set of vertices D⊆ V such that every vertex v∈V\ D is adjacent to at least k vertices in D. The k-domination number of G, denoted by γk(G), is the cardinality of a smallest k-dominating set of G. Here we establish lower and upper bounds of γk(Cm×Cn) for k=2. In some cases, these bounds agree so that the exact 2-domination number is obtained.
TL;DR: In this paper, an epidemiological model of competing strains of pathogens and hence differences in transmission for first versus secondary infection due to interaction of the strains with previously aquired immunities, as has been described for dengue fever, is known as antibody dependent enhancement (ADE).
Abstract: We analyse an epidemiological model of competing strains of pathogens and hence differences in transmission for first versus secondary infection due to interaction of the strains with previously aquired immunities, as has been described for dengue fever, is known as antibody dependent enhancement (ADE). These models show a rich variety of dynamics through bifurcations up to deterministic chaos. Including temporary cross-immunity even enlarges the parameter range of such chaotic attractors, and also gives rise to various coexisting attractors, which are difficult to identify by standard numerical bifurcation programs using continuation methods. A combination of techniques, including classical bifurcation plots and Lyapunov exponent spectra, has to be applied in comparison to get further insight into such dynamical structures. Here we present for the first time multi-parameter studies in a range of biologically plausible values for dengue. The multi-strain interaction with the immune system is expected to h...
TL;DR: Numerical experiments show that the SSOR- like method with a proper preconditioning matrix is better than SOR-like method presented by Golub et al.
Abstract: Saddle point problems arise in a wide variety of applications in computational and engineering. The aim of this paper is to present a SSOR-like iterative method for solving the saddle point problems. Here the convergence of this method is studied and specifically, the spectral radius and the optimal relaxation parameter of the iteration matrix are also investigated. Numerical experiments show that the SSOR-like method with a proper preconditioning matrix is better than SOR-like method presented by Golub et al. [G.H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), pp. 71–85].
TL;DR: A case-based planning and beliefs, desires, intentions (CBP–BDI) planning model which incorporates a novel artificial neural network to develop the intelligent environment that has been tested and analysed in this paper.
Abstract: This paper presents a case-based planning and beliefs, desires, intentions (CBP–BDI) planning model which incorporates a novel artificial neural network. The CBP–BDI model, which is integrated within an agent, is the core of a multi-agent system that allows managing the security in industrial environments. The BDI model integrates within a CBP engine of reasoning that incorporates artificial neural network-based techniques, and in this way it is possible to adapt past experiences to generate new plans. The proposed model uses self-organized maps to calculate optimum routes for the security guards. Besides, some technologies of ambient intelligence such as radio-frequency identification and Wi-Fi are used to develop the intelligent environment that has been tested and analysed in this paper.
TL;DR: The proposed cubic spline methods require 3-spatial grid points and are applicable to problems in both rectangular and polar coordinates and are compared with the corresponding successive over relaxation (SOR) and Newton-SOR iterative methods both in terms of accuracy and performance.
Abstract: In this article, we study the application of the alternating group explicit (AGE) and Newton-AGE iterative methods to a two-level implicit cubic spline formula of O(k 2+kh 2+h 4) for the solution of 1D quasi-linear parabolic equation u xx =φ (x, t, u, u x , u t ), 0 0 subject to appropriate initial and natural boundary conditions prescribed, where k>0 and h>0 are mesh sizes in t- and x-directions, respectively. The proposed cubic spline methods require 3-spatial grid points and are applicable to problems in both rectangular and polar coordinates. The convergence analysis at advanced time level is briefly discussed. The proposed methods are then compared with the corresponding successive over relaxation (SOR) and Newton-SOR iterative methods both in terms of accuracy and performance. This research work is dedicated to Late Prof. D.J. Evans
TL;DR: This paper applies a novel evolutionary optimization algorithm named quantum-behaved particle swarm optimization (QPSO) to estimate the parameters of chaotic systems, which can be formulated as a multimodal numerical optimization problem with high dimension from the viewpoint of optimization.
Abstract: This paper applies a novel evolutionary optimization algorithm named quantum-behaved particle swarm optimization (QPSO) to estimate the parameters of chaotic systems, which can be formulated as a multimodal numerical optimization problem with high dimension from the viewpoint of optimization. Moreover, in order to improve the performance of QPSO, an adaptive mechanism is introduced for the parameter beta of QPSO. Finally, numerical simulations are provided to show the effectiveness and efficiency of the modified QPSO method.
TL;DR: In this paper, an analytic method (eigenvalue–eigenvector method) for solving nth order fuzzy differential equations with fuzzy initial conditions is considered and it is shown that the solution of differential equation is a fuzzy number.
Abstract: In this paper, an analytic method (eigenvalue-eigenvector method) for solving nth order fuzzy differential equations with fuzzy initial conditions is considered. In this method, three cases are introduced, in each case, it is shown that the solution of differential equation is a fuzzy number. In addition, the method is illustrated by solving several numerical examples.
TL;DR: This paper studies the numerical solution of hybrid fuzzy differential equations by using Adams–Bashforth, Adams–Moulton and predictor-corrector methods and states the convergence and stability of the proposed methods.
Abstract: In this paper, we study the numerical solution of hybrid fuzzy differential equations by using Adams-Bashforth, Adams-Moulton and predictor-corrector methods. Predictor-corrector is obtained by combining Adams-Bashforth and Adams-Moulton methods. In addition, we state the convergence and stability of the proposed methods. Examples are presented to illustrate the computational aspects of these methods.
TL;DR: The adaptive FD6G2 method is superior to both a standard and an adaptive second-order FD method and is combined with space- and time-adaptivity to further enhance the method.
Abstract: In this paper, we develop a highly accurate adaptive finite difference (FD) discretization for the Black–Scholes equation. The final condition is discontinuous in the first derivative yielding that...
TL;DR: The aim of this paper is to extend the concept of inverse of a matrix with fuzzy numbers as its elements, which may be used to model uncertain and imprecise aspects of real-world problems.
Abstract: The aim of this paper is to extend the concept of inverse of a matrix with fuzzy numbers as its elements, which may be used to model uncertain and imprecise aspects of real-world problems. We pursue two main ideas based on employing real scenarios and arithmetic operators. In each case, exact and inexact strategies are provided. In the first idea, we give some necessary and sufficient conditions for invertibility of fuzzy matrices based on regularity of their scenarios. And then Zadeh's extension principle and interpolation on Rohn's approach for inverting interval matrices are followed to compute fuzzy inverse. In the second idea, Dubois and Prade's arithmetic operators will be employed for the same purpose. But with respect to the inherent difficulties which are derived from the positivity restriction on spreads of fuzzy numbers, the concept of ϵ-inverse of a fuzzy matrix and its relaxation are generalized and some useful theorems will be revealed. Finally fuzzifying the defuzzified version of the origi...
TL;DR: This note establishes topological relationship between the hyper-stars and three known classes of networks, namely, hypercubes, tori and odd graphs, via embedding.
Abstract: Hypercubes and star graphs are two of the most fundamental classes of interconnection networks. The class of hyper-stars was introduced as a hybrid of these two classes. In this note, we establish topological relationship between the hyper-stars and three known classes of networks, namely, hypercubes, tori and odd graphs, via embedding.
TL;DR: This paper considers the linear heat equation arisen from the Burgers's equation using the Hopf–Cole transformation and gives an explicit expression for computing exp(α A)y for some vector y.
Abstract: In this paper, we consider the linear heat equation arisen from the Burgers's equation using the Hopf–Cole transformation. Discretization of this equation with respect to the space variable results in a linear system of ordinary differential equations. The solution of this system involves in computing exp(α A)y for some vector y, where A is a large special tridiagonal matrix and α is a positive real number. We give an explicit expression for computing exp(α A)y. Finally, some numerical experiments are given to show the efficiency of the method.
TL;DR: Drawings are given and numerical models to evaluate the stability of the slope protection systems are presented and a reliable model of the interaction of the flexible contour beam with the cable network enables the achievement of more efficient solutions in the design analysis.
Abstract: This work studies the analysis of the resistant capacity of cable nets for the stabilization of slopes Two tests have been carried out, one with a distributed longitudinal load and the other with distributed transversal load, in order to simulate in situ the working conditions of these systems Tensile tests were also carried out on the cable elements of the network in order to obtain the non-linear mechanical properties On the one hand, the proposed numerical procedure uses the finite element method (FEM) and it takes into account the material and geometrical non-linearities due to the geometrical change in the cable net substructure On the other hand, the contour (boundary) beam is modelled by linear beam elements with contact between them The numerical results of the longitudinal and cross tests were simulated for different geometrical configurations, generating convergent results with respect to strain and resistance, which permit the extrapolation from tested cable nets to untested simulated cable nets, maintaining a basic configuration of constant parameters The laboratory tests only provide information about the strain and maximum resistance, but they do not establish a relationship between the values of stresses of each net element These data have been obtained through the computational simulation by FEM A reliable model of the interaction of the flexible contour beam with the cable network enables the achievement of more efficient solutions in the design analysis Finally, we compare the structural behaviour of the numerical and experimental results by means of the equivalent elastic modulus and the equivalent Poisson ratio Excellent agreement between the predicted results by FEM and test observations was found Besides, conclusions and suggested procedures of calculation applied on the cable networks are given and numerical models to evaluate the stability of the slope protection systems are presented
TL;DR: In this paper, the authors extend a method for approximate balanced reduced order model derivation for finite dimensional linear systems developed by Rowley (Int. Chaos Appl. Sci. Bifur. Eng. 15(3) (2005), pp. 997-1013) to infinite dimensional systems.
Abstract: We extend a method for approximate balanced reduced order model derivation for finite dimensional linear systems developed by Rowley (Int. J. Bifur. Chaos Appl. Sci. Eng. 15(3) (2005), pp. 997-1013) to infinite dimensional systems. The algorithm is related to standard balanced truncation, but includes aspects of the proper orthogonal decomposition in its computational approach. The method can be also applied to nonlinear systems. Numerical results are presented for a convection diffusion system.
TL;DR: This work presents a multidisciplinary study that identifies and applies unsupervised connectionist models in conjunction with modelling systems and shows that the most appropriate model to control these industrial tasks is the Box–Jenkins algorithm, which calculates the function of a linear system from its input and output samples.
Abstract: Real-world processes may be improved through a combination of artificial intelligence and identification techniques. This work presents a multidisciplinary study that identifies and applies unsupervised connectionist models in conjunction with modelling systems. This particular industrial problem is defined by a data set relayed through sensors situated on a robotic drill used in the construction of industrial storage centres. The first step entails determination of the most relevant structures in the data set with the application of the connectionist architectures. The second step combines the results of the first one to identify a model for the optimal working conditions of the drilling robot that is based on low-order models such as black box that approximate the optimal form of the model. Finally, it is shown that the most appropriate model to control these industrial tasks is the Box–Jenkins algorithm, which calculates the function of a linear system from its input and output samples.
TL;DR: This article gives a guideline for an efficient option price calculation of high-dimensional American-style options with the LS algorithm, and proposes an optimal selection of basis functions and a random number generator to guarantee stable results.
Abstract: Several methods for valuing high-dimensional American-style options were proposed in the last years. Longstaff and Schwartz (LS) have suggested a regression-based Monte Carlo approach, namely the least squares Monte Carlo method. This article is devoted to an efficient implementation of this algorithm. First, we suggest a code for faster runs. Regression-based Monte Carlo methods are sensitive to the choice of basis functions for pricing high-dimensional American-style options and, like all Monte Carlo methods, to the underlying random number generator. For this reason, we secondly propose an optimal selection of basis functions and a random number generator to guarantee stable results. Our basis depends on the payoff of the high-dimensional option and consists of only three functions. We give a guideline for an efficient option price calculation of high-dimensional American-style options with the LS algorithm, and we test it in examples with up to 10 dimensions.
TL;DR: A decomposition method for different types of Emden–Fowler-like equations is extended, and some special cases of the equation are solved as examples, to illustrate the reliableness of the method.
Abstract: In this paper, singular initial value problems are investigated. We extend a decomposition method for different types of Emden-Fowler-like equations. Some special cases of the equation are solved as examples, to illustrate the reliableness of the method. The solutions are constructed in the form of a convergent series.
TL;DR: This paper proves the existence of a smooth solution of the boundary value system by the method of upper and lower solutions, and construction of monotonic sequences of lower and upper solutions that converge to true solutions, respectively.
Abstract: In this paper, we study a double barrier option when the underlying asset price follows a regime-switching exponential mean-reverting process. Our method is a combination of analysis of a deterministic boundary value problem with a probabilistic approach. In this setting, the double barrier option prices satisfy a system of m linear second-order differential equations with variable coefficients and with Dirichlet boundary conditions, where m is the number of regimes considered for the economy. We prove the existence of a smooth solution of the boundary value system by the method of upper and lower solutions; we proceed to construct monotonic sequences of upper and lower solutions that converge to true solutions, respectively. The uniqueness of the solution is established by applying Dynkin's formula. This proof by construction also provides a numerical procedure to compute approximate option values. An important feature of the proposed numerical method is that the true option values are bracketed by the upper and the lower solutions. Examples are provided to illustrate the method.