# Showing papers in "Computers & Mathematics With Applications in 2010"

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TL;DR: The decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases.

Abstract: Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions. With the definitions of Mittag-Leffler stability and generalized Mittag-Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.

1,200 citations

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TL;DR: Three areas of bioengineering research (bioelectrodes, biomechanics, bioimaging) are described where fractional calculus is being applied to build new mathematical models that predict macroscale behavior from microscale observations and measurements.

Abstract: Fractional (non-integer order) calculus can provide a concise model for the description of the dynamic events that occur in biological tissues. Such a description is important for gaining an understanding of the underlying multiscale processes that occur when, for example, tissues are electrically stimulated or mechanically stressed. The mathematics of fractional calculus has been applied successfully in physics, chemistry, and materials science to describe dielectrics, electrodes and viscoelastic materials over extended ranges of time and frequency. In heat and mass transfer, for example, the half-order fractional integral is the natural mathematical connection between thermal or material gradients and the diffusion of heat or ions. Since the material properties of tissue arise from the nanoscale and microscale architecture of subcellular, cellular, and extracellular networks, the challenge for the bioengineer is to develop new dynamic models that predict macroscale behavior from microscale observations and measurements. In this paper we describe three areas of bioengineering research (bioelectrodes, biomechanics, bioimaging) where fractional calculus is being applied to build these new mathematical models.

732 citations

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TL;DR: The main aim is to generalize the Legendre operational matrix to the fractional calculus and reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem.

Abstract: Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

704 citations

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TL;DR: By using the fractional power of operators and some fixed point theorems, a class of fractional neutral evolution equations with non local conditions with nonlocal conditions is discussed and various criteria on the existence and uniqueness of mild solutions are obtained.

Abstract: In this paper, by using the fractional power of operators and some fixed point theorems, we discuss a class of fractional neutral evolution equations with nonlocal conditions and obtain various criteria on the existence and uniqueness of mild solutions In the end, we give an example to illustrate the applications of the abstract results

557 citations

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TL;DR: A stream-cipher algorithm based on one-time keys and robust chaotic maps, in order to get high security and improve the dynamical degradation, and is suitable for application in color image encryption.

Abstract: We designed a stream-cipher algorithm based on one-time keys and robust chaotic maps, in order to get high security and improve the dynamical degradation. We utilized the piecewise linear chaotic map as the generator of a pseudo-random key stream sequence. The initial conditions were generated by the true random number generators, the MD5 of the mouse positions. We applied the algorithm to encrypt the color image, and got the satisfactory level security by two measures: NPCR and UACI. When the collision of MD5 had been found, we combined the algorithm with the traditional cycle encryption to ensure higher security. The ciphered image is robust against noise, and makes known attack unfeasible. It is suitable for application in color image encryption.

490 citations

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TL;DR: This work defines soft matrices and their operations which are more functional to make theoretical studies in the soft set theory and constructs a soft max-min decision making method which can be successfully applied to the problems that contain uncertainties.

Abstract: In this work, we define soft matrices and their operations which are more functional to make theoretical studies in the soft set theory We then define products of soft matrices and their properties We finally construct a soft max-min decision making method which can be successfully applied to the problems that contain uncertainties

433 citations

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TL;DR: The fundamental solution of the fractal derivative equation for anomalous diffusion is derived, which characterizes a clear power law, and this new model is compared with the corresponding fractional derivative model in terms of computational efficiency, diffusion velocity, and heavy tail property.

Abstract: This paper makes an attempt to develop a fractal derivative model of anomalous diffusion. We also derive the fundamental solution of the fractal derivative equation for anomalous diffusion, which characterizes a clear power law. This new model is compared with the corresponding fractional derivative model in terms of computational efficiency, diffusion velocity, and heavy tail property. The merits and distinctions of these two models of anomalous diffusion are then summarized.

394 citations

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TL;DR: Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov's method is a tedious task, and through a direct stability domain characterization, three LMI stability analysis conditions are proposed.

Abstract: After an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives, also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov's method is a tedious task. If the fractional order @n is such that 0<@n<1, the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability analysis conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI framework. These conditions are applied to the gain margin computation of a CRONE suspension.

391 citations

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TL;DR: Application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown and some of their properties are studied.

Abstract: In this paper, we define generalised fuzzy soft sets and study some of their properties. Application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown.

380 citations

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TL;DR: Initial concepts of soft rings are introduced and some operations on soft sets are defined in this paper.

Abstract: Molodtsov (1999) introduced the concept of soft sets in [1]. Then, Maji et al. (2003) defined some operations on soft sets in [2]. Aktas and Ca@?man (2007) defined the notion of soft groups in [3]. Finally, soft semirings are defined by Feng et al. (2008) in [5]. In this paper, we have introduced initial concepts of soft rings.

348 citations

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TL;DR: The concept of a triangular IFN (TIFN) is introduced as a special case of the IFN and a new methodology for ranking TIFNs is developed on the basis of a ratio of the value index to the ambiguity index and applied to multiattribute decision making problems in which the ratings of alternatives on attributes are expressed with TIFN.

Abstract: The concept of an intuitionistic fuzzy number (IFN) is of importance for quantifying an ill-known quantity, and the ranking of IFNs is a very difficult problem. The aim of this paper is to introduce the concept of a triangular IFN (TIFN) as a special case of the IFN and develop a new methodology for ranking TIFNs. Firstly the concepts of TIFNs and cut sets as well as arithmetical operations are introduced. Then the values and ambiguities of the membership function and the non-membership function for a TIFN are defined. A new ranking method is developed on the basis of the concept of a ratio of the value index to the ambiguity index and applied to multiattribute decision making problems in which the ratings of alternatives on attributes are expressed with TIFNs. The validity and applicability of the proposed method, as well as analysis of the comparison with other methods, are illustrated with a real example.

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TL;DR: The notion of vague soft set is introduced which is an extension to the soft set and the basic properties of vaguesoft sets are presented and discussed.

Abstract: Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainty. However, it is difficult to be used to represent the vagueness of problem parameters. In this paper, we introduce the notion of vague soft set which is an extension to the soft set. The basic properties of vague soft sets are presented and discussed.

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TL;DR: The concepts of soft set relations are introduced as a sub soft set of the Cartesian product of the soft sets and many related concepts such as equivalent soft set relation, partition, composition, function etc are discussed.

Abstract: The traditional soft set is a mapping from a parameter to the crisp subset of universe. Molodtsov introduced the theory of soft sets as a generalized tool for modeling complex systems involving uncertain or not clearly defined objects. In this paper the concepts of soft set relations are introduced as a sub soft set of the Cartesian product of the soft sets and many related concepts such as equivalent soft set relation, partition, composition, function etc. are discussed.

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TL;DR: In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed and the complement, ''and'', ''or'', union, intersection, necessity and possibility operations are defined on the interval -valued intuitionism fuzzy soft sets.

Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed. Our interval-valued intuitionistic fuzzy soft set theory is a combination of an interval-valued intuitionistic fuzzy set theory and a soft set theory. In other words, our interval-valued intuitionistic fuzzy soft set theory is an interval-valued fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory. The complement, ''and'', ''or'', union, intersection, necessity and possibility operations are defined on the interval-valued intuitionistic fuzzy soft sets. The basic properties of the interval-valued intuitionistic fuzzy soft sets are also presented and discussed.

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TL;DR: This paper obtains the existence and multiplicity results of positive solutions of fractional order derivative of D 0 + α by using some fixed point theorems.

Abstract: In this paper, we are concerned with the nonlinear differential equation of fractional order D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 t 1 , 1 α ≤ 2 , where D 0 + α is the standard Riemann–Liouville fractional order derivative, subject to the boundary conditions u ( 0 ) = 0 , D 0 + β u ( 1 ) = a D 0 + β u ( ξ ) . We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.

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TL;DR: Under certain conditions, the formal solution of the initial-boundary-value problem for the generalized time-fractional diffusion equation that turns out to be a classical solution under some additional conditions is shown.

Abstract: In this paper, some uniqueness and existence results for the solutions of the initial-boundary-value problems for the generalized time-fractional diffusion equation over an open bounded domain Gx(0,T),[email protected]?R^n are given. To establish the uniqueness of the solution, a maximum principle for the generalized time-fractional diffusion equation is used. In turn, the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is considered in the paper, too. Another important consequence of the maximum principle is the continuous dependence of the solution on the problem data. To show the existence of the solution, the Fourier method of the variable separation is used to construct a formal solution. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional diffusion equation that turns out to be a classical solution under some additional conditions.

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TL;DR: The initial value problem is discussed for a class of fractional neutral functional differential equations and the criteria on existence are obtained and the criterion on existence is obtained.

Abstract: In this paper, the initial value problem is discussed for a class of fractional neutral functional differential equations and the criteria on existence are obtained.

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TL;DR: The effect of system size is studied and Darcy's law from the linear dependence of the flux on the body force exerted is validated and the values of the permeability measurements as a function of porosity tend to concentrate in a narrower region of the porosity, as the system size of the computational sub-sample increases.

Abstract: We present results of lattice-Boltzmann simulations to calculate flow in realistic porous media. Two examples are given for lattice-Boltzmann simulations in two- and three-dimensional (2D and 3D) rock samples. First, we show lattice-Boltzmann simulation results of the flow in quasi-two-dimensional micromodels. The third dimension was taken into account using an effective viscous drag force. In this case, we consider a 2D micromodel of Berea sandstone. We calculate the flow field and permeability of the micromodel and find excellent agreement with Microparticle Image Velocimetry (@m-PIV) experiments. Then, we use a particle tracking algorithm to calculate the dispersion of tracer particles in the Berea geometry, using the lattice-Boltzmann flow field. Second, we use lattice-Boltzmann simulations to calculate the flow in Bentheimer sandstone. The data set used in this study was obtained using X-ray microtomography (XMT). First, we consider a single phase flow. We systematically study the effect of system size and validate Darcy's law from the linear dependence of the flux on the body force exerted. We observe that the values of the permeability measurements as a function of porosity tend to concentrate in a narrower region of the porosity, as the system size of the computational sub-sample increases. Finally, we compute relative permeabilities for binary immiscible fluids in the XMT rock sample.

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Hohai University

^{1}TL;DR: This study makes the first attempt to apply the Kansa method in the solution of the time fractional diffusion equations, in which the MultiQuadrics and thin plate spline serve as the radial basis function.

Abstract: This study makes the first attempt to apply the Kansa method in the solution of the time fractional diffusion equations, in which the MultiQuadrics and thin plate spline serve as the radial basis function. In the discretization formulation, the finite difference scheme and the Kansa method are respectively used to discretize time fractional derivative and spatial derivative terms. The numerical solutions of one- and two-dimensional cases are presented and discussed, which agree well with the corresponding analytical solution.

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TL;DR: Artificial neural network (ANN) is proposed as a method to improve accurate RUL prediction of bearing failure and shows that better performance is achieved in order to predict bearing failure.

Abstract: Accurate remaining useful life (RUL) prediction of machines is important for condition based maintenance (CBM) to improve the reliability and cost of maintenance. This paper proposes artificial neural network (ANN) as a method to improve accurate RUL prediction of bearing failure. For this purpose, ANN model uses time and fitted measurements Weibull hazard rates of root mean square (RMS) and kurtosis from its present and previous points as input. Meanwhile, the normalized life percentage is selected as output. By doing that, the noise of a degradation signal from a target bearing can be minimized and the accuracy of prognosis system can be improved. The ANN RUL prediction uses FeedForward Neural Network (FFNN) with Levenberg Marquardt of training algorithm. The results from the proposed method shows that better performance is achieved in order to predict bearing failure.

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TL;DR: Several benchmark numerical optimization problems, constrained and unconstrained, are presented here to demonstrate the effectiveness and robustness of the proposed Hunting Search (HuS) algorithm and indicate that the proposed method is a powerful search and optimization technique.

Abstract: A novel optimization algorithm is presented, inspired by group hunting of animals such as lions, wolves, and dolphins. Although these hunters have differences in the way of hunting, they are common in that all of them look for a prey in a group. The hunters encircle the prey and gradually tighten the ring of siege until they catch the prey. In addition, each member of the group corrects its position based on its own position and the position of other members. If the prey escapes from the ring, hunters reorganize the group to siege the prey again. Several benchmark numerical optimization problems, constrained and unconstrained, are presented here to demonstrate the effectiveness and robustness of the proposed Hunting Search (HuS) algorithm. The results indicate that the proposed method is a powerful search and optimization technique. It yields better solutions compared to those obtained by some current algorithms when applied to continuous problems.

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TL;DR: Using the new fractional Taylor’s series, two new families of fractional Black–Scholes equations are derived, and some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics are made.

Abstract: By using the new fractional Taylor’s series of fractional order f ( x + h ) = E α ( h α D x α ) f ( x ) where E α ( . ) denotes the Mittag–Leffler function, and D x α is the so-called modified Riemann–Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration, one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a (usual) Gaussian white noise. One can then derive two new families of fractional Black–Scholes equations, and one shows how one can obtain their solutions. Merton’s optimal portfolio is once more considered and some new results are contributed, with respect to the modeling on one hand, and to the solution on the other hand. Finally, one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics.

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TL;DR: The existence of positive solutions to the singular boundary value problem for fractional differential equation is considered and a fixed point theorem for the mixed monotone operator is relied on.

Abstract: In this paper, we consider the existence of positive solutions to the singular boundary value problem for fractional differential equation. Our analysis relies on a fixed point theorem for the mixed monotone operator.

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TL;DR: A reliable new algorithm of DTM is proposed, namely multi-step DTM, which will increase the interval of convergence for the series solution, and is applied to Lotka-Volterra, Chen and Lorenz systems.

Abstract: The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi-step DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

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TL;DR: This paper develops a general purpose Lattice Boltzmann code that runs entirely on a single GPU and shows that simple precision floating point arithmetic is sufficient for LBM computation in comparison to double precision.

Abstract: Graphics Processing Units (GPUs), originally developed for computer games, now provide computational power for scientific applications. In this paper, we develop a general purpose Lattice Boltzmann code that runs entirely on a single GPU. The results show that: (1) simple precision floating point arithmetic is sufficient for LBM computation in comparison to double precision; (2) the implementation of LBM on GPUs allows us to achieve up to about one billion lattice update per second using single precision floating point; (3) GPUs provide an inexpensive alternative to large clusters for fluid dynamics prediction.

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TL;DR: In this manuscript, some results of Lakshmikantham and Ciric (2009) in [5] are extended to the class of cone metric spaces.

Abstract: The notion of coupled fixed point is introduced by Bhaskar and Lakshmikantham (2006) in [13]. In this manuscript, some results of Lakshmikantham and Ciric (2009) in [5] are extended to the class of cone metric spaces.

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TL;DR: This work presents a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions, and reports quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50.

Abstract: We present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new.

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TL;DR: Some new inequalities of Simpson's type based on s-convexity are established and some applications to special means of real numbers are given.

Abstract: In this paper, we establish some new inequalities of Simpson's type based on s-convexity. Some applications to special means of real numbers are also given.

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TL;DR: The methodology developed here allows us to solve a very large class of FOCPs by converting them into a general, rational form of optimal control problem (OCP), and the fractional differentiation operator used in the FOCP is approximated using Oustaloup's approximation into a state-space realization form.

Abstract: In this article, we discuss fractional order optimal control problems (FOCPs) and their solutions by means of rational approximation. The methodology developed here allows us to solve a very large class of FOCPs (linear/nonlinear, time-invariant/time-variant, SISO/MIMO, state/input constrained, free terminal conditions etc.) by converting them into a general, rational form of optimal control problem (OCP). The fractional differentiation operator used in the FOCP is approximated using Oustaloup's approximation into a state-space realization form. The original problem is then reformulated to fit the definition used in general-purpose optimal control problem (OCP) solvers such as RIOTS_95, a solver created as a Matlab toolbox. Illustrative examples from the literature are reproduced to demonstrate the effectiveness of the proposed methodology and a free final time OCP is also solved.

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TL;DR: This work introduces an easily verifiable hypothesis, which allows for immediate applications of a general comparison type result from [V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal].

Abstract: We present two global existence results for an initial value problem associated to a large class of fractional differential equations. Our approach differs substantially from the techniques employed in the recent literature. By introducing an easily verifiable hypothesis, we allow for immediate applications of a general comparison type result from [V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008), 2677-2682].