Showing papers in "Computers & Mathematics With Applications in 2009"
TL;DR: This paper points out that several assertions in a previous paper by Maji et al. are not true in general, and gives some new notions such as the restricted intersection, the restricted union, therestricted difference and the extended intersection of two soft sets.
Abstract: Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In this paper, we first point out that several assertions (Proposition 2.3 (iv)-(vi), Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by Maji et al. [P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562] are not true in general, by counterexamples. Furthermore, based on the analysis of several operations on soft sets introduced in the same paper, we give some new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets. Moreover, we improve the notion of complement of a soft set, and prove that certain De Morgan's laws hold in soft set theory with respect to these new definitions.
1,223 citations
TL;DR: The Schauder fixed point theorem is applied and an existence result is proved for the following system, where @a,@b,p,q,@h,@c satisfy certain conditions.
Abstract: This paper studies a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Applying the Schauder fixed point theorem, an existence result is proved for the following system D^@au(t)=f(t,v(t),D^pv(t)),D^@bv(t)=g(t,u(t),D^qu(t)),[email protected]?(0,1),u(0)=0,u(1)[email protected](@h),v(0)=0,v(1)[email protected](@h), where @a,@b,p,q,@h,@c satisfy certain conditions.
477 citations
TL;DR: The soft set theory, proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty by combining the interval-valued fuzzy set and soft set models.
Abstract: The soft set theory, proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. By combining the interval-valued fuzzy set and soft set models, the purpose of this paper is to introduce the concept of the interval-valued fuzzy soft set. The complement, ''AND'' and ''OR'' operations are defined on the interval-valued fuzzy soft sets. The DeMorgan's, associative and distribution laws of the interval-valued fuzzy soft sets are then proved. Finally, a decision problem is analyzed by the interval-valued fuzzy soft set. Some numerical examples are employed to substantiate the conceptual arguments.
430 citations
TL;DR: This paper introduces a new approach for ranking of trapezoidal fuzzy numbers based on the left and the right spreads at some @a-levels of Trapezoid fuzzy numbers.
Abstract: Ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems. Several strategies have been proposed for ranking of fuzzy numbers. Each of these techniques have been shown to produce non-intuitive results in certain cases. In this paper, we will introduce a new approach for ranking of trapezoidal fuzzy numbers based on the left and the right spreads at some @a-levels of trapezoidal fuzzy numbers. The calculation of the proposed method is far simpler and easier. Finally, some comparative examples are used to illustrate the advantage of the proposed method.
407 citations
TL;DR: A new scheme is proposed which allows us to obtain large density ratio and to reproduce the coexistence curve with high accuracy and the spurious currents at vapor-liquid interface are also greatly reduced.
Abstract: We investigate the use of various equations of state (EOS) in the single-component multiphase lattice Boltzmann model. Several EOS are explored: van der Waals, Carnahan-Starling and Kaplun-Meshalkin EOS [A.B. Kaplun, A.B. Meshalkin, Thermodynamic validation of the form of unified equation of state for liquid and gas, High Temperature 41 (3) (2003) 319-326]. The last one was modified in order to obtain the correct critical point. The Carnahan-Starling and modified Kaplun-Meshalkin EOS are in better agreement with the experimental data on coexistence curves than the van der Waals EOS. It is shown that the approximation of the gradient of special potential is crucial to obtain the correct coexistence curve, especially its low-density part. The correct method of incorporating the body forces into the lattice Boltzmann model is also very important. We propose a new scheme which allows us to obtain large density ratio (up to 10^9 in the stationary case) and to reproduce the coexistence curve with high accuracy. The spurious currents at vapor-liquid interface are also greatly reduced.
386 citations
TL;DR: In this paper, the concept of fuzzy soft group is introduced and in the meantime, some of their properties and structural characteristics are discussed and studied, including fuzzy soft function and fuzzy soft homomorphism.
Abstract: In this paper, the concept of fuzzy soft group is introduced and in the meantime, some of their properties and structural characteristics are discussed and studied. Furthermore, definitions of fuzzy soft function and fuzzy soft homomorphism are defined and the theorems of homomorphic image and homomorphic pre-image are given. After that, the definition of normal fuzzy soft group is given and some of its basic properties are studied.
261 citations
TL;DR: A general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics and it is revealed that the first method is very effective and convenient.
Abstract: Variational iteration method has been used to handle linear and nonlinear differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this work, a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the variational iteration method with those obtained by Adomian decomposition method reveals that the first method is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.
246 citations
TL;DR: The Analytical approximate solution of a fractional diffusion equation is deduced with the help of powerful Variational Iteration method by using an initial value, which accelerate the rapid convergence of the series solution.
Abstract: In the present paper the Analytical approximate solution of a fractional diffusion equation is deduced with the help of powerful Variational Iteration method. By using an initial value, the explicit solutions of the equation for different cases have been derived, which accelerate the rapid convergence of the series solution. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented graphically.
193 citations
TL;DR: The error analyses of the fractional Adams method for the fractionsal ordinary differential equations for the three cases (i)-(iii) are studied and numerical simulations are also included which are in line with the theoretical analysis.
Abstract: The generalized Adams-Bashforth-Moulton method, often simply called ''the fractional Adams method'', is a useful numerical algorithm for solving a fractional ordinary differential equation: D"*^@ay(t)=f(t,y(t)),y^(^k^)(0)=y"0^(^k^),k=0,1,...,n-1, where @a>0,[email protected][email protected]@? is the first integer not less than @a, and D"*^@ay(t) is the @ath-order fractional derivative of y(t) in the Caputo sense. Although error analyses for this fractional Adams method have been given for (a) 0 1, [email protected]?C^1^+^@?^@a^@?[0,T], (c) 0 1, [email protected]?C^3(G), there are still some unsolved problems-(i) the error estimates for @[email protected]?(0,1), [email protected]?C^3(G), (ii) the error estimates for @[email protected]?(0,1), [email protected]?C^2(G), (iii) the solution y(t) having some special forms. In this paper, we mainly study the error analyses of the fractional Adams method for the fractional ordinary differential equations for the three cases (i)-(iii). Numerical simulations are also included which are in line with the theoretical analysis.
189 citations
TL;DR: This work proves for generic steady solutions of the Lattice Boltzmann models that the variation of the numerical errors is set by specific combinations (called ''magic numbers'') of the relaxation rates associated with the symmetric and anti-symmetric collision moments, and confirms the governing role of the ''magic'' combinations for steady solution of the Stokes equation.
Abstract: We prove for generic steady solutions of the Lattice Boltzmann (LB) models that the variation of the numerical errors is set by specific combinations (called ''magic numbers'') of the relaxation rates associated with the symmetric and anti-symmetric collision moments. Given the governing dimensionless physical parameters, such as the Reynolds or Peclet numbers, and the geometry of the computational mesh, the numerical errors remain the same for any change of the transport coefficients only when the ''free'' (''kinetic'') anti-symmetric rates and the boundary rules are chosen properly. The single-relaxation-time (BGK) model has no free collision rate and yields viscosity dependent errors with any boundary scheme for hydrodynamic problems. The simplest and most efficient collision operator for invariant errors is the two-relaxation-times (TRT) model. As an example, this model is able to compute viscosity independent permeabilities for any porous structure. These properties are derived from steady recurrence equations, obtained through linear combinations of the LB evolution equations, in which the equilibrium and non-equilibrium components are directly interconnected via finite-difference link-wise central operators. The explicit dependency of the non-equilibrium solution on the relaxation rates is then obtained. This allows us, first, to confirm the governing role of the ''magic'' combinations for steady solutions of the Stokes equation, second, to extend this property to steady solutions of the Navier-Stokes and anisotropic advection-diffusion equations, third, to develop a parametrization analysis of the microscopic and macroscopic closure relations prescribed via link-wise boundary schemes.
188 citations
TL;DR: This paper is an elementary introduction to the concepts of the homotopy perturbation method and gives an intuitive grasp for the solution procedure throughout the paper.
Abstract: This paper is an elementary introduction to the concepts of the homotopy perturbation method. Particular attention is paid to giving an intuitive grasp for the solution procedure throughout the paper.
TL;DR: An operator-oriented characterization of L- fuzzy rough sets is presented, that is, L-fuzzy approximation operators are defined by axioms, and the relationship between L-magnitude rough sets and L-topological spaces is obtained.
Abstract: Rough set theory was developed by Pawlak as a formal tool for approximate reasoning about data Various fuzzy generalizations of rough approximations have been proposed in the literature As a further generalization of the notion of rough sets, L-fuzzy rough sets were proposed by Radzikowska and Kerre In this paper, we present an operator-oriented characterization of L-fuzzy rough sets, that is, L-fuzzy approximation operators are defined by axioms The methods of axiomatization of L-fuzzy upper and L-fuzzy lower set-theoretic operators guarantee the existence of corresponding L-fuzzy relations which produce the operators Moreover, the relationship between L-fuzzy rough sets and L-topological spaces is obtained The sufficient and necessary condition for the conjecture that an L-fuzzy interior (closure) operator derived from an L-fuzzy topological space can associate with an L-fuzzy reflexive and transitive relation such that the corresponding L-fuzzy lower (upper) approximation operator is the L-fuzzy interior (closure) operator is examined
TL;DR: A gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle is presented.
Abstract: In this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective.
TL;DR: A novel coverage control scheme based on elitist non-dominated sorting genetic algorithm (NSGA-II) is proposed in a heterogeneous sensor network and an ameliorated binary coding is addressed to represent both sensing radius adjustment and sensor selection.
Abstract: In this paper, the problem of maintaining sensing coverage by keeping a small number of active sensor nodes and a small amount of energy consumption in a wireless sensor network is studied As opposed to the uniform sensing model previously, we consider a large number of sensors with adjustable sensing radius that are randomly deployed to monitor a target area A novel coverage control scheme based on elitist non-dominated sorting genetic algorithm (NSGA-II) is proposed in a heterogeneous sensor network By devising a cluster-based architecture, the algorithm is applied in a distributed way Furthermore, an ameliorated binary coding is addressed to represent both sensing radius adjustment and sensor selection Numerical and simulation results validate that the procedure to find the optimal balance point among the maximum coverage rate, the least energy consumption, as well as the minimum number of active nodes is fast and effective
TL;DR: The Hermite-Hadamard-Fejer inequalities for an h-convex function are proved and applications on p-logarithmic mean and mean of the order p are obtained.
Abstract: In this paper we prove the Hermite-Hadamard-Fejer inequalities for an h-convex function and we point out the results for some special classes of functions. Also, some generalization of the Hermite-Hadamard inequalities and some properties of functions H and F which are naturally joined to the h-convex function are given. Finally, applications on p-logarithmic mean and mean of the order p are obtained.
TL;DR: A class of feed-forward neural network operators is introduced using these operators as approximation tools, and the upper bounds of errors, in uniform norm, approximating continuous functions, are estimated.
Abstract: The aim of this paper is to investigate the error which results from the method of approximation operators with logarithmic sigmoidal function. By means of the method of extending functions, a class of feed-forward neural network operators is introduced. Using these operators as approximation tools, the upper bounds of errors, in uniform norm, approximating continuous functions, are estimated. Also, a class of quasi-interpolation operators with logarithmic sigmoidal function is constructed for approximating continuous functions defined on the total real axis.
TL;DR: A novel coverage control scheme based on multi-objective genetic algorithm that can achieve balanced performance on different types of detection sensor models while maintaining high coverage rate is proposed.
Abstract: Due to the constrained energy and computational resources available to sensor nodes, the number of nodes deployed to cover the whole monitored area completely is often higher than if a deterministic procedure were used. Activating only the necessary number of sensor nodes at any particular moment is an efficient way to save the overall energy of the system. A novel coverage control scheme based on multi-objective genetic algorithm is proposed in this paper. The minimum number of sensors is selected in a densely deployed environment while preserving full coverage. As opposed to the binary detection sensor model in the previous work, a more precise detection model is applied in combination with the coverage control scheme. Simulation results show that our algorithm can achieve balanced performance on different types of detection sensor models while maintaining high coverage rate. With the same number of deployed sensors, our scheme compares favorably with the existing schemes.
TL;DR: This paper gives characterizations of (fuzzy) p-ideals in BCI-algebras, and provides relations between fuzzy p-Ideals and p-IDEalistic soft BCK/BCI- algebrs.
Abstract: Molodtsov [D. Molodtsov, Soft set theory-First results, Comput. Math. Appl. 37 (1999) 19-31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408-1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft p-ideals and p-idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) p-ideals in BCI-algebras. We provide relations between fuzzy p-ideals and p-idealistic soft BCI-algebras.
TL;DR: The notions of (transitive) soft d-algebras, soft edge d- algebrs, soft d^*-algeses,soft d-ideals, hard d^@?-idealistic, or d*-ideAListic, and d-Idealistic (d^@?, d**), or d+ideals (d**)-ideals are introduced and their related properties are surveyed.
Abstract: The notions of (transitive) soft d-algebras, soft edge d-algebras, soft d^*-algebras, soft d-ideals, soft d^@?-ideals, soft d^*-ideals, and d-idealistic (d^@?-idealistic, or d^*-idealistic) soft d-algebras are introduced. Also, their related properties are surveyed.
TL;DR: Two new two-step iterative methods for solving the system of nonlinear equations using quadrature formulas are suggested and it is proved that these new methods have cubic convergence.
Abstract: In this paper, we suggest and analyze two new two-step iterative methods for solving the system of nonlinear equations using quadrature formulas. We prove that these new methods have cubic convergence. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. These new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations.
TL;DR: A two-distribution function LBE method is proposed in which the incompressibility is enforced by the pressure evolution equation and, as long as the intermolecular force is expressed in the potential form, the incompressing method for binary fluids is able to eliminate parasitic currents.
Abstract: Discretization errors in the computation of the intermolecular force in the lattice Boltzmann equation (LBE) method cause parasitic currents. A slight imbalance in the interfacial stresses due to truncation errors initiates the parasitic currents. It was shown that these currents could be eliminated to round-off if the potential form of the intermolecular force for non-ideal gases was used with compact isotropic discretization. In the present work, a formulation of the intermolecular forces for binary fluids and the role of incompressibility on parasitic currents are examined. A two-distribution function LBE method is proposed in which the incompressibility is enforced by the pressure evolution equation. As long as the intermolecular force is expressed in the potential form, the incompressible LBE method for binary fluids is able to eliminate parasitic currents.
TL;DR: The cohesion degree of the neighborhood of an object and the coupling degree between neighborhoods of objects are defined based on the neighborhood-based rough set model and a new initialization method is proposed, and the corresponding time complexity is analyzed.
Abstract: As a simple clustering method, the traditional K-Means algorithm has been widely discussed and applied in pattern recognition and machine learning. However, the K-Means algorithm could not guarantee unique clustering result because initial cluster centers are chosen randomly. In this paper, the cohesion degree of the neighborhood of an object and the coupling degree between neighborhoods of objects are defined based on the neighborhood-based rough set model. Furthermore, a new initialization method is proposed, and the corresponding time complexity is analyzed as well. We study the influence of the three norms on clustering, and compare the clustering results of the K-means with the three different initialization methods. The experimental results illustrate the effectiveness of the proposed method.
TL;DR: This paper attempts to develop an efficient method based on particle swarm optimization (PSO) algorithm with swarm intelligence by comparing the results with genetic algorithm (GA) using four problems in the literature and an example of supply chain model.
Abstract: Bi-level linear programming is a technique for modeling decentralized decision. It consists of the upper-level and lower-level objectives. This paper attempts to develop an efficient method based on particle swarm optimization (PSO) algorithm with swarm intelligence. The performance of the proposed method is ascertained by comparing the results with genetic algorithm (GA) using four problems in the literature and an example of supply chain model. The results illustrate that the PSO algorithm outperforms GA in accuracy.
TL;DR: The purpose of this study is to introduce a modification of the homotopy perturbation method using Laplace transform and Pade approximation to obtain closed form solutions of nonlinear coupled systems of partial differential equations.
Abstract: The purpose of this study is to introduce a modification of the homotopy perturbation method using Laplace transform and Pade approximation to obtain closed form solutions of nonlinear coupled systems of partial differential equations. Two test examples are given; the coupled nonlinear system of Burger equations and the coupled nonlinear system in one dimensional thermoelasticity. The results obtained ensure that this modification is capable of solving a large number of nonlinear differential equations that have wide application in physics and engineering.
TL;DR: A fixed point theorem of Imdad and Kumar, for a pair of non-self maps, is extended to non-normal cone spaces to solve the inequality of the following type: For α ≥ 1, β ≥ 1 using LaSalle's inequality.
Abstract: In this paper we extend a fixed point theorem of Imdad and Kumar, for a pair of non-self maps, to non-normal cone spaces.
TL;DR: In this work, the fractional KdV-Burgers-Kuramoto equation is studied and variational iteration method and Adomian's decomposition method are applied to obtain its solution.
Abstract: In this work, the fractional KdV-Burgers-Kuramoto equation is studied. He's variational iteration method (VIM) and Adomian's decomposition method (ADM) are applied to obtain its solution. Comparison with HAM is made to highlight the significant features of the employed methods and their capability of handling completely integrable equations.
TL;DR: A novel algorithm based on Adomian decomposition for fractional differential equations is proposed and this derived computational method is used to find a smaller ''efficient dimension'' such that the fractional Lorenz equation is chaotic.
Abstract: In this paper, a novel algorithm based on Adomian decomposition for fractional differential equations is proposed. Comparing the present method with the fractional Adams method, we use this derived computational method to find a smaller ''efficient dimension'' such that the fractional Lorenz equation is chaotic. We also apply this new method to the time-fractional Burgers equation with initial and boundary value conditions. Numerical results and computer graphics show that the constructed numerical is efficient.
TL;DR: The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Maxwell fluid are determined using Fourier sine and Laplace transforms and are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions.
Abstract: The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Maxwell fluid are determined using Fourier sine and Laplace transforms. They are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions. The similar solutions for Maxwell and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general results. Graphical illustrations show that the velocity profiles corresponding to a generalized Maxwell fluid are going to that for an ordinary Maxwell fluid if @a->1.
TL;DR: A comparative study among He's homotopy perturbation method and three traditional methods for an analytic and approximate treatment of nonlinear integral and integro-differential equations.
Abstract: In this paper, we conduct a comparative study among He's homotopy perturbation method and three traditional methods for an analytic and approximate treatment of nonlinear integral and integro-differential equations The proper implementation of He's homotopy perturbation method can extremely minimize the size of work if compared to existing traditional techniques The analysis is accompanied by examples that demonstrate the comparison, and shows the pertinent features of the homotopy perturbation technique
TL;DR: The variational iteration method is applied to solve the generalized pantograph equation and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional.
Abstract: The variational iteration method is applied to solve the generalized pantograph equation. This technique provides a sequence of functions which converges to the exact solution of the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. Employing this technique, it is possible to find the exact solution or an approximate solution of the problem. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.