Showing papers in "Computers & Mathematics With Applications in 2011"
TL;DR: It is shown that a soft topological space gives a parametrized family of topological spaces and it is established that the converse does not hold.
Abstract: In the present paper we introduce soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms are introduced and their basic properties are investigated. It is shown that a soft topological space gives a parametrized family of topological spaces. Furthermore, with the help of an example it is established that the converse does not hold. The soft subspaces of a soft topological space are defined and inherent concepts as well as the characterization of soft open and soft closed sets in soft subspaces are investigated. Finally, soft T"i-spaces and notions of soft normal and soft regular spaces are discussed in detail. A sufficient condition for a soft topological space to be a soft T"1-space is also presented.
832 citations
TL;DR: This paper defines left and right Caputo fractional sums and differences, study some of their properties and then relate them to Riemann-Liouville ones studied before by Miller K. S. and Ross B. and Atici F.M.
Abstract: In this paper, we define left and right Caputo fractional sums and differences, study some of their properties and then relate them to Riemann-Liouville ones studied before by Miller K. S. and Ross B., Atici F.M. and Eloe P. W., Abdeljawad T. and Baleanu D., and a few others. Also, the discrete version of the Q-operator is used to relate the left and right Caputo fractional differences. A Caputo fractional difference equation is solved. The solution proposes discrete versions of Mittag-Leffler functions.
475 citations
TL;DR: A linear superposition principle of exponential traveled waves is analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves.
Abstract: A linear superposition principle of exponential traveling waves is analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves. Applications are made for the 3+1 dimensional KP, Jimbo-Miwa and BKP equations, thereby presenting their particular N-wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated N-wave solutions, and a few illustrative examples are presented, together with an algorithm using weights.
374 citations
TL;DR: The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.
Abstract: In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations. This method is called the homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He's polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.
312 citations
TL;DR: The time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on are studied.
Abstract: In this paper, we study the time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order 1+@b@?[1,2] and the finite difference scheme for the time Caputo derivative with order @a@?(0,1) (for subdiffusion) and (1,2) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0<@a<1 is slower than that for @a=1, but the diffusion velocity for 1<@a<2 is faster than that for @a=1. For the spatial diffusion, we have a similar observation.
298 citations
TL;DR: The presented model for small resolutions is investigated to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid and it is concluded that details of the membrane mesh, as tessellation method and resolution, play only a minor role.
Abstract: The deformation of an initially spherical capsule, freely suspended in simple shear flow, can be computed analytically in the limit of small deformations [D. Barthes-Biesel, J.M. Rallison, The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech. 113 (1981) 251-267]. Those analytic approximations are used to study the influence of the mesh tessellation method, the spatial resolution, and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice Boltzmann finite element method. For the description of the capsule membrane, a finite element method and the Skalak constitutive model [R. Skalak, A. Tozeren, R.P. Zarda, S. Chien, Strain energy function of red blood cell membranes, Biophys. J. 13 (1973) 245-264] have been employed. Our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid. We come to the conclusion that details of the membrane mesh, as tessellation method and resolution, play only a minor role. The hydrodynamic resolution, i.e., the width of the discrete delta function, can significantly influence the accuracy of the simulations. The discretization of the delta function introduces an artificial length scale, which effectively changes the radius and the deformability of the capsule. We discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy.
287 citations
TL;DR: The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs and the proposed algorithms are applied to solve two types ofFDEs, linear and nonlinear, subject to initial or boundary conditions.
Abstract: We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs.
287 citations
TL;DR: The properties of soft open, soft nbd and soft closure are investigated and the properties ofsoft interior, soft exterior and soft boundary are defined which are fundamental for further research on soft topology and will strengthen the foundations of the theory of soft topological spaces.
Abstract: For dealing with uncertainty researchers introduced the concept of soft sets. Shabir and Naz (2011) [28], defined several basic notions on soft topology and studied many properties. In this paper, we continue investigating the properties of soft open (closed), soft nbd and soft closure. We also define and discuss the properties of soft interior, soft exterior and soft boundary which are fundamental for further research on soft topology and will strengthen the foundations of the theory of soft topological spaces.
283 citations
TL;DR: The asymptotic stability and the absolute stability of these methods are proved and error representations and estimates for the truncation, propagation and global error are derived.
Abstract: This paper is devoted to the numerical treatment of fractional differential equations. Based on the Grunwald-Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given.
270 citations
TL;DR: This concept of ''memory-dependent derivative'', which is simply defined in an integral form of a common derivative with a kernel function on a slipping interval, is better than the fractional one for reflecting the memory effect.
Abstract: Enlightened by the Caputo type of fractional derivative, here we bring forth a concept of ''memory-dependent derivative'', which is simply defined in an integral form of a common derivative with a kernel function on a slipping interval. In case the time delay tends to zero it tends to the common derivative. High order derivatives also accord with the first order one. Comparatively, the form of kernel function for the fractional type is fixed, yet that of the memory-dependent type can be chosen freely according to the necessity of applications. So this kind of definition is better than the fractional one for reflecting the memory effect (instantaneous change rate depends on the past state). Its definition is more intuitionistic for understanding the physical meaning and the corresponding memory-dependent differential equation has more expressive force.
262 citations
TL;DR: It is proved that certain De Morgan's law hold in soft set theory with respect to different operations on soft sets and the notion of restricted symmetric difference of soft sets is defined and investigated.
Abstract: Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, first we prove that certain De Morgan's law hold in soft set theory with respect to different operations on soft sets. Then, we discuss the basic properties of operations on soft sets such as intersection, extended intersection, restricted union and restricted difference. Moreover, we illustrate their interconnections between each other. Also we define the notion of restricted symmetric difference of soft sets and investigate its properties. The main purpose of this paper is to extend the theoretical aspect of operations on soft sets.
TL;DR: It is found that an approximate time delay can enhance or suppress the emergence of chaos, and the lowest orders for chaos to exist in the delayed fractional order financial systems are determined, respectively.
Abstract: In this paper, a delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations. A great variety of interesting dynamical behaviors of such a system including single-periodic, multiple-periodic, and chaotic motions are displayed. In particular, the effect of time delay on the chaotic behavior is investigated, it is found that an approximate time delay can enhance or suppress the emergence of chaos. Meanwhile, corresponding to different values of delay, the lowest orders for chaos to exist in the delayed fractional order financial systems are determined, respectively.
TL;DR: In this paper, the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzysoft set, interior fuzzy soft set, fuzzy soft basis, and fuzzy soft subspace topology are studied.
Abstract: In this paper, the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzy soft set, interior fuzzy soft set, fuzzy soft basis, fuzzy soft subspace topology are studied.
TL;DR: The existence and uniqueness of solution to this problem are proved by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnosel'skii theorem.
Abstract: In this paper, we consider a discrete fractional boundary value problem of the form -@D^@ny(t)=f(t+@n-1,y(t+@n-1)), y(@n-2)=g(y), y(@n+b)=0, where f:[@n-1,...,@n+b-1]"N"""@n"""-"""2xR->R is continuous, g:C([@n-2,@n+b]"N"""@n"""-"""2,R) is a given functional, and 1<@n@?2. We give a representation for the solution to this problem. Finally, we prove the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnosel'skii theorem.
TL;DR: In this paper, an approximation technique based on the shifted Legendre-tau idea is presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain.
Abstract: Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.
TL;DR: properties of soft separation axioms defined in Shabir and Naz (2011) are investigated and it is shown that if a soft topological space (X,@t,E) is soft T"1 and soft regular (i.e. asoft T"3-space), then (x,E), it is soft closed for each [email protected]?X (their Theorem 3.21).
Abstract: Shabir and Naz (2011) [12] introduced and studied the notions of soft topological spaces, soft interior, soft closure and soft separation axioms. But we found that some results are incorrect (see their Remark 3.23). So the purpose of this note is, first, to point out some errors in Remark 4 and Example 9 of Shabir and Naz (2011) [12], and second, to investigate properties of soft separation axioms defined in Shabir and Naz (2011) [12]. In particular, we investigate the soft regular spaces and some properties of them. We show that if a soft topological space (X,@t,E) is soft T"1 and soft regular (i.e. a soft T"3-space), then (x,E) is soft closed for each [email protected]?X (their Theorem 3.21).
TL;DR: The controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems in a Banach space has been established by using the theory of fractionalist calculus, fixed point technique and a new concept called (@a,u)-resolvent family is introduced.
Abstract: In this work, the controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems in a Banach space has been established by using the theory of fractional calculus, fixed point technique and also we introduced a new concept called (@a,u)-resolvent family. As an application that illustrates the abstract results, an example is given.
TL;DR: The Cartesian product, composition, union and join is defined on interval-valued fuzzy graphs and some properties of self-complementary and self-weak complementary interval- valued fuzzy complete graphs are presented.
Abstract: We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of interval-valued fuzzy complete graphs and present some properties of self-complementary and self-weak complementary interval-valued fuzzy complete graphs.
TL;DR: Using the semigroup theory, the fractional power theory and fixed point strategy, a new set of sufficient conditions are formulated which guarantees the approximate controllability of semilinear fractional differential systems.
Abstract: Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of control systems governed by the semilinear fractional differential equations in Hilbert spaces. By using the semigroup theory, the fractional power theory and fixed point strategy, a new set of sufficient conditions are formulated which guarantees the approximate controllability of semilinear fractional differential systems. The results are established under the assumption that the associated linear system is approximately controllable. Further, we extend the result to study the approximate controllability of fractional systems with nonlocal conditions. An example is provided to illustrate the application of the obtained theory.
TL;DR: By the critical point theory, a new approach is provided to study the existence of solutions to the following fractional boundary value problem: {ddt(12"0D"t^-^@b(u^'(t))+12"tD"T^-#@b (u^’(t)))+@?F(t,u(t)=0,a.e. t@?[0,T],u(0)=u
Abstract: In this paper, by the critical point theory, a new approach is provided to study the existence of solutions to the following fractional boundary value problem: {ddt(12"0D"t^-^@b(u^'(t))+12"tD"T^-^@b(u^'(t)))+@?F(t,u(t))=0,a.e. t@?[0,T],u(0)=u(T)=0, where "0D"t^-^@b and "tD"T^-^@b are the left and right Riemann-Liouville fractional integrals of order 0@?@b R is a given function and @?F(t,x) is the gradient of F at x. Our interest in this problem arises from the fractional advection-dispersion equation (see Section 2). The variational structure is established and various criteria on the existence of solutions are obtained.
TL;DR: In this paper, the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0 is developed and the first implementations of this theory are described.
Abstract: In this paper, we develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0
TL;DR: Using a modified renormalization tensor, a new SPH scheme is presented for approximating second derivatives that has the property of first-order consistency and is compared with the best available schemes when applied to a 2D heat equation.
Abstract: Several schemes for discretization of first and second derivatives are available in Smoothed Particle Hydrodynamics (SPH). Here, four schemes for approximation of the first derivative and three schemes for the second derivative are examined using a theoretical analysis based on Taylor series expansion both for regular and irregular particle distributions. Estimation of terms in the truncation errors shows that only the renormalized (the first-order consistent) scheme has acceptable convergence properties to approximate the first derivative. None of the second derivative schemes has the first-order consistency. Therefore, they converge only when the particle spacing decreases much faster than the smoothing length of the kernel function. In addition, using a modified renormalization tensor, a new SPH scheme is presented for approximating second derivatives that has the property of first-order consistency. To assess the computational performance of the proposed scheme, it is compared with the best available schemes when applied to a 2D heat equation. The numerical results show at least one order of magnitude improvement in accuracy when the new scheme is used. In addition, the new scheme has higher-order convergence rate on regular particle arrangements even for the case of only four particles in the neighborhood of each particle.
TL;DR: This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs) based upon the Legendre orthonormal polynomial basis, and achieves the solution of the FOCP.
Abstract: This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann-Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique.
TL;DR: It is pointed out that there exists an equivalence relation between the @a-shortest path in an uncertain network and the shortest path in a corresponding deterministic network, which leads to an effective algorithm to find the@a- shortest path and the most shortest path.
Abstract: Uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths. With help from the operational law of uncertainty theory, this paper gives the uncertainty distribution of the shortest path length. Also, it investigates solutions to the @a-shortest path and the most shortest path in an uncertain network. It points out that there exists an equivalence relation between the @a-shortest path in an uncertain network and the shortest path in a corresponding deterministic network, which leads to an effective algorithm to find the @a-shortest path and the most shortest path. Roughly speaking, this algorithm can be broken down into two parts: constructing a deterministic network and then invoking the Dijkstra algorithm.
TL;DR: A framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations using negligible error compared to the exact solution is developed.
Abstract: In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.
TL;DR: An algorithm is presented for the construction of Runge-Kutta pairs of orders 5 and 4 based only in the first column simplifying assumption and the result is a pair that outperforms other known pairs in the bibliography when tested to the standard set of problems of DETEST.
Abstract: Among the most popular methods for the solution of the Initial Value Problem are the Runge-Kutta pairs of orders 5 and 4. These methods can be derived solving a system of nonlinear equations for its coefficients. To achieve this, we usually admit various simplifying assumptions. The most common of them are the so-called row simplifying assumptions. Here we neglect them and present an algorithm for the construction of Runge-Kutta pairs of orders 5 and 4 based only in the first column simplifying assumption. The result is a pair that outperforms other known pairs in the bibliography when tested to the standard set of problems of DETEST. A cost free fourth order formula is also derived for handling dense output.
TL;DR: Some topological properties of the cone b -metric spaces are established and some fixed point existence results for multivalued mappings defined on such spaces are proved.
Abstract: In this paper we establish some topological properties of the cone b-metric spaces and then improve some recent results about KKM mappings in the setting of a cone b-metric space. We also prove some fixed point existence results for multivalued mappings defined on such spaces.
TL;DR: Some coupled fixed point theorems involving a (@j,@f)-weakly contractive condition for mapping having the mixed monotone property in ordered partial metric spaces are proved.
Abstract: In this paper, we prove some coupled fixed point theorems involving a (@j,@f)-weakly contractive condition for mapping having the mixed monotone property in ordered partial metric spaces. These results are analogous to theorems of Van Luong and Xuan Thuan (2011) [10] on the class of ordered partial metric spaces. Also, an application is given to support our results.
TL;DR: A way to solve the fractional differential equations using the Riemann-Liouville fractional integral for repeated fractional integration and the generalized block pulse operational matrices of differentiation are proposed.
Abstract: The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.
TL;DR: A collection of all soft sets with respect to new operations give rise to four idempotent monoids, which with the help of these monoids can study semiring (hemiring) structures of soft sets.
Abstract: Recently new operations have been defined for soft sets. In this paper, we study some important properties associated with these new operations. A collection of all soft sets with respect to new operations give rise to four idempotent monoids. Then with the help of these monoids we can study semiring (hemiring) structures of soft sets. Some of these semirings (hemirings) are actually lattices. Finally, we show that soft sets with a fixed set of parameters are MV algebras and BCK algebras.