a numerical approach to the generalized nonlinear fractional fokker planck equation in this paper we propose a fully discrete galerkin finite element method to solve the generalized nonlinear fractional fokker planck equation which has a multi fractional spatial operator characteristic that describes the levy flight in the time direction we use the finite difference method and in the spatial direction we use the fractional finite element method in the framework of the fractional sobolev spaces we derive a fully discrete scheme for the considered equation we prove the existence and uniqueness of the discrete solution and give the error estimates the numerical examples are also included which support the theoretical analysis

finite element method for nonlinear riesz space fractional diffusion equations on irregular domains in this paper we consider two dimensional riesz space fractional diffusion equations with nonlinear source term on convex domains applying galerkin finite element method in space and backward difference method in time we present a fully discrete scheme to solve riesz space fractional diffusion equations our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes which can help us to deal with space fractional terms on any convex domain the stability and convergence of the scheme are also discussed numerical examples are given to verify accuracy and stability of our scheme a finite element method is developed to solve 2d space fractional diffusion equations with nonlinear source term we develop a finite element method for fpdes on irregular domain we obtain explicit expressions for fractional derivatives of shape functions we give the details on how to compute fractional stiffness matrix fractional fitzhugh nagumo model is solved on circular domain

a fourth order compact adi scheme for two dimensional nonlinear space fractional schrodinger equation in this paper a novel compact operator is derived for the approximation of the riesz derivative with order alpha in 1 2 the compact operator is proved with fourth order accuracy combining the compact operator in space discretization a linearized difference scheme is proposed for a two dimensional nonlinear space fractional schrodinger equation it is proved that the difference scheme is uniquely solvable stable and convergent with order o tau 2 h 4 where tau is the time step size h max h 1 h 2 and h 1 h 2 are space grid sizes in the x direction and the y direction respectively based on the linearized difference scheme a compact alternating direction implicit scheme is presented and analyzed numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly

almost compact fuzzy sets in fuzzy topological spaces abstract the study of almost compact fuzzy topological spaces was initiated by concilio and gerla 4 which was followed by further investigations of the same in 11 the present paper aspires for an extension of the same concept to fuzzy sets in a fuzzy topological space we characterize almost compact fuzzy sets in terms of the concept of fuzzy filterbase some properties of such sets are established using the concepts of fuzzy ds closure and fuzzy ds cluster points of fuzzy filterbase

a note on compactness in a fuzzy setting abstract the concept of filter basis in a fuzzy setting is defined and investigated fuzzy filter bases are then used to introduce the notion of compactness in fuzzy setting the expected basic properties of compactness are explored

analysis of spectral projectors in one dimensional domains in this paper we analyze a class of projection operators with values in a subspace of polynomials these projection operators are related to the hubert spaces involved in the numerical analysis of spectral methods they are in the first part of the paper the standard sobolev spaces and in the second part some weighted sobolev spaces the weight of which is related to the orthogonality relation satisfied by the chebyshev polynomials these results are used to study the approximation of a model fourth order problem

secondary instability eof reduction and the transition to baroclinic chaos abstract high resolution numerical simulations illustrate how an initial baroclinic instability may become chaotic as the degree of supercritically is increased the route to chaos is via bifurcations to periodic and quasi periodic states but the spatial structures involved are much more complex than those assumed in previous low order models the fundamental participants in the transition can be educed by studying the symmetry breaking secondary instabilities of fixed points of the primary baroclinic instability problem these unstable disturbances to the equilibrium way baroclinic flow serve to eradicate the fundamental zonal currents reduced low dimensional models that replicate the full 10 4 mode computer simulations can be formulated by projecting the governing equations onto the eofs empirical orothogonal functions of the large scale calculations the successful constructions of such models using relatively few eofs appears to be related to the fact that the eofs are almost identical to the secondary instabilities of the primary baroclinic wave state and these secondary instabilities are in turn the primary participants in the supercritical periodic and quasiperiodic states

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

crank nicolson adi galerkin finite element method for two dimensional fractional fitzhugh nagumo monodomain model in this paper a two dimensional fractional fitzhugh nagumo monodomain model 2d ffhnmm with zero dirichlet boundary condition is considered the model consists of a coupled two dimensional space fractional nonlinear reaction diffusion model 2d sfnrdm and an ordinary differential equation the 2d sfnrdm and ordinary differential equation are decoupled at each time step a new crank nicolson alternating direction implicit adi galerkin finite element method for the 2d sfnrdm is developed the stability and convergence of the numerical method are discussed finally some numerical examples on 2d sfnrdm and 2d ffhnmm are given for verification of our theoretical analysis

on some weaker forms of fuzzy continuous and fuzzy open mappings on fuzzy topology spaces abstract the purpose of the present paper is to study some weaker forms of fuzzy continuous and fuzzy open maps between fuzzy topological spaces along with the introduction and investigations of a new class of non fuzzy continuous maps this latter type of mappings is seen to be independent of the other non continuous functions discussed here all these functions are characterized and certain properties concerning them are proved finally a comparative study as to their mutual dependence and interrelations under different conditions is taken up

error analysis of a finite difference method on graded meshes for a time fractional diffusion equation a reaction diffusion problem with a caputo time derivative of order alpha in 0 1 is considered the solution of such a problem is shown in general to have a weak singularity near the initial time t 0 and sharp pointwise bounds on certain derivatives of this solution are derived a new analysis of a standard finite difference method for the problem is given taking into account this initial singularity this analysis encompasses both uniform meshes and meshes that are graded in time and includes new stability and consistency bounds the final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme so one can choose an optimal mesh grading numerical results are presented that confirm the sharpness of the error analysis

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

crank nicolson difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative in this paper the crank nicolson cn difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative is studied the existence of this difference solution is proved by the brouwer fixed point theorem the stability and convergence of the cn scheme are discussed in the l 2 norm when the fractional order is two all those results are in accord with the difference scheme developed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also presented

on some near fuzzy continuous functions between fuzzy topological spaces abstract the concepts of fuzzy almost strongly continuous fuzzy continuous fuzzy weekly continuous and fuzzy weakly continuous are introduced and studied in the light of the concept of q coincidence in a fuzzy setting together with the concept of q neighbourhoods the notions of fuzzy neighbourhoods and fuzzy neighbourhoods are defined to investigate the characterizations and properties of these functions finally a comparative study regarding the mutual interrelations among these maps along with fuzzy continuous maps is made

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

irresolute and lmost open functions between fuzzy topological spaces abstract the aim of this paper is to introduce and characterize the concept of fuzzy irresolute functions which is strictly stronger than the concept of fuzzy semi continuous functions and independent of the concept of fuzzy continuous functions the independence of the two definitions of fuzzy almost open functions as introduced in 5 and 3 and some of their characterizations are also established

analysis of a time stepping scheme for time fractional diffusion problems with nonsmooth data this paper establishes the convergence of a time stepping scheme for the time fractional diffusion problems with nonsmooth data we first analyze the regularity of the time fractional diffusion pro

on nearly compact and 0 rigid fuzzy sets in fuzzy topological spaces abstract this paper intends to initiate the study of nearly compact and rigid fuzzy sets in a fuzzy topological space these two classes of fuzzy sets are seen to be properly contained in the class of fuzzy compact sets and properly contain the class of fuzzy almost compact sets several characterizations and certain other properties of these two types of fuzzy sets are established by using the concepts of fuzzy closures and closures of fuzzy sets and fuzzy filterbases

chaotic dynamics of coherent structures abstract a variety of chaotic flows evolving in relatively high dimensional spaces are considered it is shown through the use of an optimal choice of basis functions which are consequence of the karhunen loeve procedure that an accurate description can be given in a relatively low dimensional space particular examples of this procedure which are presented are the ginzburg landau equation turbulent convection in an unbounded domain and turbulent convection in a bounded domain

a novel compact adi scheme for two dimensional riesz space fractional nonlinear reaction diffusion equations abstract this paper is concerned with the construction and analysis of a novel linearized compact adi scheme for the two dimensional riesz space fractional nonlinear reaction diffusion equations convergence of the proposed scheme is proved the highlight is that the time discretization is achieved by applying a second order one step and linearized method the time discretization requires only one starting value which is sharp contrast to the extrapolated crank nicolson method or the usual second order linearized schemes numerical examples on several fractional models are presented to confirm our theoretical results

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

application of dynamical system theory to coherent structures in the wall region abstract in an attempt to apply dynamical system approaches to turbulence we derive closed sets of nonlinear ordinary differential equations for the wall region this is achieved by galerkin projection onto statistical coherent structures using the proper orthogonal decomposition lumley 1 which converges optimally fast in the quadratic mean sense application of this definition to the instantaneous velocity field of the wall region shows the presence of streamwise rolls a severe truncation of the dynamical system 10 dimensions is studied to investigate the time behavior of these rolls the energy transfer to unresolved modes is adjusted by a heisenberg parameter for large values of the parameter large loss the solution is steady and represents streamwise rolls having the experimentally observed cross stream spacing for lower values of the parameter intermittency appears due to the presence of a heteroclinic attracting orbit in the phase space the behavior of the streamwise rolls consists in a long time quasi steady state followed by a growing oscillation and a sudden burst during which the rolls lose their identify subsequently the rolls reform and the process is repeated these dynamics appear to minic the bursting behavior observed in the wall region for lower parameter values the regime is much more complex apparently chaotic the fluctuating pressure term appears as a forcing term at the upper boundary of the integration domain although its amplitude is very small it acts as a trigger for bursts and equilibrates the mean time between the events which would otherwise increase with time as the trajectory is attracted closer and closer to the heteroclinic cycle the whole mechanism is qualitatively the same when the wall region is artificially thickened which is done by applying stretching transformations this is in agreement with experimental results concerning drag reduced flows

a model for the transition to baroclinic chaos abstract the transition to chaos in a 2 layer baroclinic fluid with a slightly different viscosity in each layer is investigated a low order model is constructed by truncating the quasi geostrophic partial differential equations to include one azimuthal wave in each layer while retaining a general zonal flow correction when the viscosities are equal the model reduces to a modified form of the lorenz equations and displays a cusped return map for slightly different viscosities the transition to chaos follows the feigenbaum scenario the return map in the chaotic regime is nearly parabolic and highly contracted multiple attractors are found for the same control parameters the predicted route to chaos is similar to that observed in laboratory experiments although significant quantitative differences remain

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

almost compactness in fuzzy topological spaces abstract we introduce and study almost compactness for fuzzy topological spaces we show that the almost continuous image of an almost compact fuzzy topological space is almost compact moreover we show that generally almost compactness for fuzzy topological spaces is not product invariant but if x and y are almost fuzzy topological spaces and x is product related to y then their fuzzy topological product is almost compact

a semi alternating direction method for a 2 d fractional fitzhugh nagumo monodomain model on an approximate irregular domain a fitzhugh nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue in this paper we consider a two dimensional fractional fitzhugh nagumo monodomain model on an irregular domain the model consists of a coupled riesz space fractional nonlinear reaction diffusion model and an ordinary differential equation describing the ionic fluxes as a function of the membrane potential second we use a decoupling technique and focus on solving the riesz space fractional nonlinear reaction diffusion model a novel spatially second order accurate semi implicit alternating direction method siadm for this model on an approximate irregular domain is proposed third stability and convergence of the siadm are proved finally some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two dimensional fractional fitzhugh nagumo model on both an approximate circular and an approximate irregular domain

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

a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation in this paper a new alternating direction implicit galerkin legendre spectral method for the two dimensional riesz space fractional nonlinear reaction diffusion equation is developed the temporal component is discretized by the crank nicolson method the detailed implementation of the method is presented the stability and convergence analysis is strictly proven which shows that the derived method is stable and convergent of order 2 in time an optimal error estimate in space is also obtained by introducing a new orthogonal projector the present method is extended to solve the fractional fitzhugh nagumo model numerical results are provided to verify the theoretical analysis

some stronger forms of fuzzy continuous mapping on fuzzy topological spaces abstract the paper is devoted to the introduction and study of the concepts of certain classes of functions between fuzzy topological spaces each of these functions presents a stronger form of the fuzzy continuous functions these maps have been characterized and investigated mainly in the light of the notions of q neighbourhoods quasi coincidence fuzzy closure and ds neighbourhoods they have also been compared as to their mutual dependence as also with regard to their interactions with certain other fuzzy non continuous functions

numerical analysis of nonlinear subdiffusion equations we present a general framework for the rigorous numerical analysis of time fractional nonlinear parabolic partial differential equations with a fractional derivative of order alpha in 0 1 in time it relies on three technical tools a fractional version of the discrete gronwall type inequality discrete maximal regularity and regularity theory of nonlinear equations we establish a general criterion for showing the fractional discrete gronwall inequality and verify it for the l1 scheme and convolution quadrature generated by backward difference formulas further we provide a complete solution theory e g existence uniqueness and regularity for a time fractional diffusion equation with a lipschitz nonlinear source term together with the known results of discrete maximal regularity we derive pointwise l 2 omega norm error estimates for semidiscrete galerkin finite element solutions and fully discrete solutions which are of order o h 2 up to a logarithmic factor and o tau alpha

reduced models of atmospheric low frequency variability parameter estimation and comparative performance abstract low frequency variability lfv of the atmosphere refers to its behavior on time scales of 10 100 days longer than the life cycle of a mid latitude cyclone but shorter than a season this behavior is still poorly understood and hard to predict the present study compares various model reduction strategies that help in deriving simplified models of lfv three distinct strategies are applied here to reduce a fairly realistic high dimensional quasi geostrophic 3 level qg3 atmospheric model to lower dimensions i an empirical dynamical method which retains only a few components in the projection of the full qg3 model equations onto a specified basis and finds the linear deterministic and the stochastic corrections empirically as in selten 1995 5 ii a purely dynamics based technique employing the stochastic mode reduction strategy of majda et al 2001 62 and iii a purely empirical multi level regression procedure which specifies the functional form of the reduced model and finds the model coefficients by multiple polynomial regression as in kravtsov et al 2005 3 the empirical dynamical and dynamical reduced models were further improved by sequential parameter estimation and benchmarked against multi level regression models the extended kalman filter was used for the parameter estimation overall the reduced models perform better when more statistical information is used in the model construction thus the purely empirical stochastic models with quadratic nonlinearity and additive noise reproduce very well the linear properties of the full qg3 model s lfv i e its autocorrelations and spectra as well as the nonlinear properties i e the persistent flow regimes that induce non gaussian features in the model s probability density function the empirical dynamical models capture the basic statistical properties of the full model s lfv such as the variance and integral correlation time scales of the leading lfv modes as well as some of the regime behavior features but fail to reproduce the detailed structure of autocorrelations and distort the statistics of the regimes dynamical models that use data assimilation corrections do capture the linear statistics to a degree comparable with that of empirical dynamical models but do much less well on the full qg3 model s nonlinear dynamics these results are discussed in terms of their implications for a better understanding and prediction of lfv

runge kutta approximation of quasi linear parabolic equations we study the convergence properties of implicit runge kutta meth ods applied to time discretization of parabolic equations with time or solution dependent operator error bounds are derived in the energy norm the con vergence analysis uses two different approaches the first technically simpler approach relies on energy estimates and requires algebraic stability of the runge kutta method the second one is based on estimates for linear time invariant equations and uses fourier and perturbation techniques it applies to a 9 stable runge kutta methods and yields the precise temporal order of conver gence this order is noninteger in general and depends on the type of boundary conditions