discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

dynamical behavior of reaction diffusion neural networks and their synchronization arising in modeling epileptic seizure a numerical simulation study abstract excessive synchronizations of neurons in the brain networks can be a reason for some episodic disorders such as epilepsy in this paper we simulate neural dynamic models and their synchronizations numerically by a new numerical algorithm in order to overcome the complexity of the problem a suitable combination of the legendre spectral element method and operator splitting technique is proposed to convert a complex system of partial differential equations into sparse linear algebraic systems the simulations are explored in different aspects such as accuracy computing the wavefront and angular speeds computing time the condition number of obtained algebraic systems to demonstrate the proposed scheme effectiveness the numerical simulations and comparisons confirm the method is suitable in terms of accuracy and speed

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

fractional burgers equation with nonlinear non locality we consider the viscous burgers equation with a fractional nonlinear term as a model involving non local nonlinearities in conservation laws which surprisingly has an analytical solution obtained by a fractional extension of the hopfcole transformation we use this model and its inviscid limit to develop stable spectral and discontinuous galerkin spectral element methods by employing the concept of spectral vanishing viscosity svv for the global spectral method svv is very effective and the computational cost is o n2 which is essentially the same as for the standard burgers equation we also develop a local discontinuous galerkin ldg spectral element method to improve the accuracy around discontinuities and we again stabilize the ldg method with the svv operator finally we solve numerically the inviscid fractional burgers equation both with the spectral and the spectral element ldg methods we study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters

supercritical elliptic problems on a perturbation of the ball abstract we examine the henon equation u x u p in r n with u 0 on where 0 we show there exists a sequence p k k n 2 n 2 p n with p 1 p 2 p 3 p k p n such that for any n 2 n 2 p p n which avoids p k k there exists a positive classical solution of the henon equation provided is a sufficiently small perturbation of the unit ball we also examine the lane emden fowler equation in the case of an exterior domain i e u u p in an exterior domain with u 0 on we show the existence of n 2 n 2 p 1 p 2 p 3 with p k such that if n 2 n 2 p which avoids p k k then there exists a positive fast decay classical solution provided is a sufficiently small perturbation of the exterior of the unit ball

a finite difference scheme for semilinear space fractional diffusion equations with time delay a linearized quasi compact finite difference scheme is proposed for semilinear space fractional diffusion equations with a fixed time delay the nonlinear source term is discretized and linearized by taylor s expansion to obtain a second order discretization in time the space fractional derivatives are approximated by a weighted shifted grunwald letnikov formula which is of fourth order approximation under some smoothness assumptions of the exact solution under the local lipschitz conditions the solvability and convergence of the scheme are proved in the discrete maximum norm by the energy method numerical examples verify the theoretical predictions and illustrate the validity of the proposed scheme

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

anomalous diffusion with ballistic scaling a new fractional derivative abstract anomalous diffusion with ballistic scaling is characterized by a linear spreading rate with respect to time that scales like pure advection ballistic scaling may be modeled with a symmetric riesz derivative if the spreading is symmetric however ballistic scaling coupled with a skewness is observed in many applications including hydrology nuclear physics viscoelasticity and acoustics the goal of this paper is to find a governing equation for anomalous diffusion with ballistic scaling and arbitrary skewness to address this problem we propose a new operator called the zolotarev derivative which is valid for all orders 0 2 the fourier symbol of this operator is related to the characteristic function of a stable random variable in the zolotarev m parameterization in the symmetric case the zolotarev derivative reduces to the well known riesz derivative for 1 the zolotarev derivative is a linear combination of riemann liouville fractional derivatives and a first derivative for 1 the zolotarev derivative is a non local operator that models ballistic anomalous diffusion we prove that this operator is continuous with respect to we derive generator caputo and riemann liouville forms of this operator and provide two examples the solutions of diffusion equations utilizing the zolotarev derivative with an impulse initial condition are shifted and scaled stable densities in the m parameterization

roughness in rings in this paper basic notions of the rough set theory will be given in fact the paper concerns a relationship between rough sets and ring theory we shall introduce the notion of rough subring resp ideal with respect to an ideal of a ring which is an extended notion of a subring resp ideal in a ring and we shall give some properties of the lower and the upper approximations in a ring

a roughness measure for fuzzy sets recently an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets m banerjee s k pal roughness of a fuzzy set information sciences 93 1996 235 246 essentially banerjee and pal s roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set in this paper we first remark that this measure of roughness has several undesirable properties and then propose a parameter free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set several interesting properties of this new measure are examined furthermore we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification

rough set approach to multi attribute decision analysis abstract we review the methodology of the rough set analysis of multi attribute decision problems rough set theory proved to be a useful tool for analysis of a vague description of decision situations it answers two basic questions related to multi attribute decision problems one about explanation of a decision situation and another about prescription of some decisions basing on analysis of a decision situation we define four classes of multi attribute decision problems depending on the structure of their representation its interpretation and the kind of questions related then we characterize the rough set methodology for each particular class of decision problems we use simple practical examples to illustrate this presentation a review of related literature is made throughout the paper

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

properties of measures of information in evidence and possibility theories abstract an overview of information measures recently introduced by several authors in the settting of shafer s theory of evidence is proposed new results pertaining to additivity and monotonicity properties of these measures of information are presented the interpretation of each measure of information as opposed to others is discussed the potential usefulness of measures of specificity or imprecision is suggested and a principle of minimal specificity is stated for the purpose of reconstructing a body of evidence from incomplete knowledge

fast algorithm based on tt m fe system for space fractional allen cahn equations with smooth and non smooth solutions abstract in this article a fast algorithm based on time two mesh tt m finite element fe scheme which aims at solving nonlinear problems quickly is considered to numerically solve the nonlinear space fractional allen cahn equations with smooth and non smooth solutions the implicit second order scheme containing both implicit crank nicolson scheme and second order backward difference method is applied to time direction a fast tt m method is used to increase the speed of calculation and the fe method is developed to approximate the spacial direction the tt m fe algorithm includes the following main computing steps firstly a nonlinear implicit second order fe scheme on the time coarse mesh c is solved by a nonlinear iterative method secondly based on the chosen initial iterative value a linearized fe system on time fine mesh c is solved where some useful coarse numerical solutions are found by the lagrange s interpolation formula the analysis for both stability and a priori error estimates are made in detail finally three numerical examples with smooth and non smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations from which it is easy to find that the computing time can be saved

fractional adams bashforth moulton methods we first formulate a fractional class of explicit adams bashforth a b and implicit adams moulton a m methods of first and second order accuracy for the time integration of d t 0 c u x t g t u 0 1 where d t 0 c denotes the fractional derivative in the caputo sense in this fractional setting and in contrast to the standard adams methods an extra history load term emerges and the associated weight coefficients are dependent however when 1 the developed schemes reduce to the well known a b and a m methods with standard coefficients hence in terms of scientific computing our approach constitutes a minimal modification of the existing adams libraries next we develop an implicit explicit imex splitting scheme for linear and nonlinear fractional pdes of a general advection reaction diffusion type and we apply our scheme to the time space fractional keller segel chemotaxis system in this context we evaluate the nonlinear advection term explicitly employing the fractional a b method in the prediction step and we treat the corresponding diffusion term implicitly in the correction step using the fractional a m scheme moreover we perform the corresponding spatial discretization by employing an efficient and spectrally accurate fractional spectral collocation method our numerical experiments exhibit the efficiency of the proposed imex scheme in solving nonlinear fractional pdes

crank nicolson galerkin spectral method for solving two dimensional time space distributed order weakly singular integro partial differential equation abstract the fractional pdes based upon the distributed order fractional derivative have several applications in physics the two dimensional time space distributed order weakly singular integro partial differential model is investigated by a combination of finite difference and galerkin spectral methods a second order finite difference formula is employed to approximate the temporal variable in this stage the stability and convergence of the semi discrete scheme are proved then the galerkin spectral method based on the modified jacobi polynomials is applied to discrete the space variable also in this step the error estimate of the full discrete scheme is studied finally two test problems have been presented to confirm the theoretical results

error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives abstract in the current investigation an error estimate has been proposed to solve the two dimensional weakly singular integro partial differential equation with space and time fractional derivatives based on the finite element finite difference scheme the time and space derivatives are based on the riemann liouville and riesz fractional derivatives respectively at first the temporal variable has been discretized by a second order difference scheme and then the space variable has been approximated by the finite element method fem the analytical study shows that the presented scheme is unconditionally stable and convergent finally some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique

rough sets and fuzzy sets some remarks on interrelations abstract this short paper refers to two remarks given in 3 and disputes them any rough set can be expressed by means of a membership function u 0 0 5 1 we show that this representation can be extended to union and intersection

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

fractional sturm liouville boundary value problems in unbounded domains recently zayernouri and karniadakis in 2013 78 investigated two classes of fractional sturm liouville eigenvalue problems on compact interval a b in more detail they were the first authors who not only obtained some explicit forms for the eigensolutions of these problems but also derived some useful spectral properties of the obtained eigensolutions until now to the best of our knowledge fractional sturm liouville eigenvalue problems on non compact interval such as 0 are not analyzed so our aim in this paper is to study these problems in detail to do so we study at first fractional sturm liouville operators fslos of the confluent hypergeometric differential equations of the first kind and then two special cases of fslos fslos 1 and fslos 2 are considered after this we obtain the analytical eigenfunctions for the cases and investigate the spectral properties of eigenfunctions and their corresponding eigenvalues also we derive two fractional types of the associated laguerre differential equations due to the non polynomial nature of the eigenfunctions obtained from the two fractional associated laguerre differential equations they are defined as generalized associated laguerre functions of the first and second kinds galfs 1 and galfs 2 furthermore we prove that these fractional sturm liouville operators are self adjoint and the obtained eigenvalues are all real the corresponding eigenfunctions are orthogonal with respect to the weight function associated to fslos 1 and fslos 2 and form two sets of non polynomial bases at the end two new quadrature rules and l 2 orthogonal projections with respect to and based on galfs 1 and galfs 2 are introduced the upper bounds of the truncation errors of these new orthogonal projections according to some prescribed norm are proved and then verified numerically with some test examples finally some fractional differential equations are provided and analyzed numerically

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

a petrov galerkin spectral element method for fractional elliptic problems abstract we develop a new c 0 continuous petrov galerkin spectral element method for one dimensional fractional elliptic problems of the form 0 d x u x u x f x 1 2 subject to homogeneous boundary conditions we employ the standard modal spectral element basis and the jacobi poly fractonomials as the test functions zayernouri and karniadakis 2013 we formulate a new procedure for assembling the global linear system from elemental local mass and stiffness matrices the petrov galerkin formulation requires performing elemental local construction of mass and stiffness matrices in the standard domain only once moreover we efficiently obtain the non local history stiffness matrices in which the non locality is presented analytically for uniform grids we also investigate two distinct choices of basis test functions i local basis test functions and ii local basis with global test functions we show that the former choice leads to a better conditioned system and accuracy while the latter results in ill conditioned linear systems and therefore that is not an efficient and a proper choice of test function we consider smooth and singular solutions where the singularity can occur at boundary points as well as in the interior domain we also construct two non uniform grids over the whole computational domain in order to capture singular solutions finally we perform a systematic numerical study of non local effects via full and partial history fading in order to further enhance the efficiency of the scheme

fractional spectral and pseudo spectral methods in unbounded domains theory and applications abstract this paper is intended to provide exponentially accurate galerkin petrov galerkin and pseudo spectral methods for fractional differential equations on a semi infinite interval we start our discussion by introducing two new non classical lagrange basis functions nlbfs 1 and nlbfs 2 which are based on the two new families of the associated laguerre polynomials galfs 1 and galfs 2 obtained recently by the authors in 28 with respect to the nlbfs 1 and nlbfs 2 two new non classical interpolants based on the associated laguerre gauss and laguerre gauss radau points are introduced and then fractional pseudo spectral differentiation and integration matrices are derived convergence and stability of the new interpolants are proved in detail several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives and integrals of some functions moreover the pseudo spectral galerkin and petrov galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones some useful comments from the numerical point of view on galerkin and petrov galerkin methods are listed at the end

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

nodal discontinuous galerkin methods for fractional diffusion equations on 2d domain with triangular meshes this paper as the sequel to previous work develops numerical schemes for fractional diffusion equations on a two dimensional finite domain with triangular meshes we adopt the nodal discontinuous galerkin methods for the full spatial discretization by the use of high order nodal basis employing multivariate lagrange polynomials defined on the triangles stability analysis and error estimates are provided which shows that if polynomials of degree n are used the methods are n 1 th order accurate for general triangulations finally the performed numerical experiments confirm the optimal order of convergence

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

a multi domain spectral method for time fractional differential equations this paper proposes an approach for high order time integration within a multi domain setting for time fractional differential equations since the kernel is singular or nearly singular two main difficulties arise after the domain decomposition how to properly account for the history memory part and how to perform the integration accurately to address these issues we propose a novel hybrid approach for the numerical integration based on the combination of three term recurrence relations of jacobi polynomials and high order gauss quadrature the different approximations used in the hybrid approach are justified theoretically and through numerical examples based on this we propose a new multi domain spectral method for high order accurate time integrations and study its stability properties by identifying the method as a generalized linear method numerical experiments confirm hp convergence for both time fractional differential equations and time fractional partial differential equations

rough approximation quality revisited in rough set theory the approximation quality is the traditional measure to evaluate the classification success of attributes in terms of a numerical evaluation of the dependency properties generated by these attributes in this paper we re interpret the classical in terms of a classic measure based on sets the marczewski steinhaus metric and also in terms of proportional reduction of errors pre measures we also exhibit infinitely many possibilities to define like statistics which are meaningful in situations different from the classical one and provide tools to ascertain the statistical significance of the proposed measures which are valid for any kind of sample copyright 2001 elsevier science b v

analysis of mixed finite element method mfem for solving the generalized fractional reaction diffusion equation on nonrectangular domains abstract in the current manuscript we consider a generalized fractional reaction diffusion equation the considered model is based on the time fractional derivative the developed scheme is based on two procedures at first we obtain a semi discrete scheme for the temporal direction the time fractional derivative is discretized by using a difference scheme with second order accuracy thus the final time discrete formula has second order accuracy in the temporal direction then we obtain a fully discrete scheme by applying the mixed finite element method mfem for the constructed numerical technique we prove the unconditional stability and also obtain an error bound for the full discrete scheme by using the energy method we employ some test problems to show the accuracy of the proposed technique

a spectral method of exponential convergence for singular solutions of the diffusion equation with general two sided fractional derivative an open problem in the numerical solution of fractional partial differential equations fpdes is how to obtain high order accuracy for singular solutions even for smooth right hand sides solutions of fpdes are singular here we consider the one dimensional diffusion equation with general two sided fractional derivative characterized by a parameter p in 0 1 for p 1 2 we recover the riesz fractional derivative while for p 1 0 we obtain the one sided fractional derivative we employ a petrov galerkin projection in a properly weighted sobolev space with two sided jacobi polyfracnomials as basis and test functions in particular we derive these two sided jacobi polyfractonomials as eigenfunctions of a sturm liouville problem with weights uniquely determined by the parameter p we provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term i e for smooth data corresponding to singular solutions we obtain exponential convergence wh

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

trapezoidal scheme for time space fractional diffusion equation with riesz derivative abstract in this paper a finite difference scheme is proposed to solve time space fractional diffusion equation which has second order accuracy in both time and space direction the time and space fractional derivatives are considered in the senses of caputo and riesz respectively first the centered difference approach is used to approximate the riesz fractional derivative in space then the obtained fractional ordinary differential equations are transformed into equivalent volterra integral equations and then the trapezoidal rule is utilized to approximate the volterra integral equations the stability and convergence of our scheme are proved via mathematical induction method finally numerical experiments are performed to confirm the high accuracy and efficiency of our scheme

rough variables with values in measurable spaces an x valued rough variable is introduced as a measurable function from a rough space to a measurable space x based on the concept of x valued rough variables some properties of x valued rough variables are studied furthermore the concept of convergence is presented and convergence of x valued rough variables is discussed

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

efficient spectral galerkin methods for fractional partial differential equations with variable coefficients efficient spectral galerkin algorithms are developed to solve multi dimensional fractional elliptic equations with variable coefficients in conserved form as well as non conserved form these algorithms are extensions of the spectral galerkin algorithms for usual elliptic pdes developed in 24 more precisely for separable fpdes we construct a direct method by using a matrix diagonalization approach while for non separable fpdes we employ a preconditioned bicgstab method with a suitable separable fpde with constant coefficients as preconditioner the cost of these algorithms is of o n d 1 flops where d is the space dimension we derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries we also present ample numerical results to validate the algorithms and error estimates

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

a finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation abstract an efficient numerical technique is proposed to solve one and two dimensional space fractional tempered fractional diffusion wave equations the space fractional is based on the riemann liouville fractional derivative at first the temporal direction is discretized using a second order accurate difference scheme then a classic galerkin finite element is employed to obtain a full discrete scheme furthermore for the time discrete and the full discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method finally two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique

a mass assignment theory of the probability of fuzzy events abstract mass assignment theory techniques for processing uncertainty in fril are reviewed the notion of the probability of a fuzzy event is introduced together with the t norm definition of conditional probabilities the latter is then shown to be probability possibility inconsistent an alternative theory of conditional probabilities based on mass assignments is presented together with a number of results illustrating some intuitive properties in particular the mass assignment theory of conditional probabilities is shown to be probability possibility consistent

multipulse like orbits for a singularly perturbed nearly integrable system abstract we use topological shooting arguments to study the existence of multipulse like orbits for a singularly perturbed 3 dimensional dynamical system established as a model problem for the 1 dimensional complex ginzburg landau equations our main result improves an earlier result obtained by doelman and hek

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

numerical solution of fractional integro differential equations by collocation method abstract this paper deals with the numerical solution of fractional integro differential equations of the type d q y t p t y t f t 0 t k t s y s d s t i 0 1 by polynomial spline functions we derive a system of equations that characterizing the numerical solution some numerical examples are also provided to illustrated our results

fractional spectral collocation methods for linear and nonlinear variable order fpdes while several high order methods have been developed for fractional pdes fpdes with fixed order there are no such methods for fpdes with field variable order these equations allow multiphysics simulations seamlessly e g from diffusion to sub diffusion or from wave dynamics transitioning to diffusion by simply varying the fractional order as a function of space or time we develop an exponentially accurate fractional spectral collocation method for solving linear nonlinear fpdes with field variable order following the spectral theory developed in 1 for fractional sturm liouville eigenproblems we introduce a new family of interpolants called left right sided and central fractional lagrange interpolants we employ the fractional derivatives of left right sided riemann liouville and riesz type and obtain the corresponding fractional differentiation matrices by collocating the field variable fractional orders we solve several fpdes including time and space fractional advection equation time and space fractional advection diffusion equation and finally the space fractional burgers equation to demonstrate the performance of the method in addition we develop a spectral penalty method for enforcing inhomogeneous initial conditions our numerical results confirm the exponential like convergence of the proposed fractional collocation methods

direct meshless local petrov galerkin dmlpg method for time fractional fourth order reaction diffusion problem on complex domains abstract a new numerical scheme has been developed based on the fast and efficient meshless local weak form i e direct meshless local petrov galerkin dmlpg method for solving the fractional fourth order partial differential equation on computational domains with complex shape the fractional derivative is the riemann liouville fractional derivative at first a finite difference scheme with the second order accuracy has been employed to discrete the time variable then the dmlpg technique is employed to achieve a full discrete scheme the time discrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the l 2 space also some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and non regular grid points

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

a high order schema for the numerical solution of the fractional ordinary differential equations in this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations fodes this technique is based on the so called block by block approach which is a common method for the integral equations in our approach the classical block by block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps while preserving the good stability property of the block by block schemes by using this new approach we are able to construct a high order schema for fodes of the order a a 0 the stability and convergence of the schema is rigorously established we prove that the numerical solution converges to the exact solution with order 3 a for 0 1 a series of numerical examples are provided to support the theoretical claims

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

alternating direction implicit spectral element method adi sem for solving multi dimensional generalized modified anomalous sub diffusion equation abstract the main aim of the current paper is to solve the multi dimensional generalized modified anomalous sub diffusion equation by using a new spectral element method at first the time variable has been discretized by a finite difference scheme with second order accuracy the stability and convergence of the time discrete scheme have been investigated we show that the time discrete scheme is unconditionally stable and the convergence order is o 2 in the temporal direction secondly the galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable the unconditional stability and convergence of the full discrete scheme have been proved by developing the proposed scheme we need to calculate one dimensional integrals for two dimensional problems and two dimensional integrals for three dimensional problems thus the used cpu time for the presented numerical procedure is lower than the two and three dimensional galerkin spectral element methods also the proposed method is suitable for computational domains obtained from the tensor product finally two examples are analyzed to check the theoretical results

numerical approximation of higher order time fractional telegraph equation by using a combination of a geometric approach and method of line we propose a simple and accurate numerical scheme for solving the time fractional telegraph tft equation within caputo type fractional derivative a fictitious coordinate is imposed onto the problem in order to transform the dependent variable u x t into a new variable with an extra dimension in the new space with the added fictitious dimension a combination of method of line and group preserving scheme gps is proposed to find the approximate solutions this method preserves the geometric structure of the problem power and accuracy of this method has been illustrated through some examples of tft equation

critical elliptic equations via a dynamical systems approach abstract in this work we consider the existence of positive solutions to various equations of the form u x 1 g x u u x p in b r u 0 on b r where b r is the open ball of radius r in r n centered at the origin and p n 2 n 2 we will generally assume g is nonnegative our approach will be to utilize some dynamical systems approaches

a fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation abstract we develop a fast solver for the spectral element method sem applied to the two sided fractional diffusion equation on uniform geometric and graded meshes by approximating the singular kernel with a degenerate kernel we construct a hierarchical matrix h matrix to represent the stiffness matrix of the sem and provide error estimates which we verify numerically we can solve efficiently the h matrix approximation problem using a hierarchical lu h lu decomposition method which reduces the computational cost to o r 2 n d log 2 n o r 3 n d log n where r is the rank of the h lu decomposition n d is the total number of degrees of freedom and n is the number of elements however we lose the high accuracy of the sem to prevent this we solve the corresponding preconditioned system by using the h matrix approximation problem as a preconditioner hence recovering the high order accuracy of the sem numerical results show that the condition number of the preconditioned system is independent of the polynomial degree p and grows with the number of elements but at modest values of the rank r here r is the rank of submatrices of the h matrix approximation it is below order 10 in our experiments which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system this reduction is higher in the two sided fractional derivative compared to one sided fractional derivative the corresponding cost is o r 2 n d log 2 n o r 3 n d log n o n d 2 moreover by using a structured mesh uniform or geometric mesh we can further reduce the computational cost to o r 2 n d log 2 n o r 3 n d log n o p 2 n log n for the preconditioned system furthermore we compare the present method with the adaptive cross approximation aca which is kernel independent and relies on the coefficient matrix the comparison shows that the accuracy using the present algorithm is better than the one using aca for the h matrix approximation problem and furthermore we have observed that the aca method may fail to converge in some cases