an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

a new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications in the present work first a new fractional numerical differentiation formula called the l1 2 formula to approximate the caputo fractional derivative of order 0 1 is developed it is established by means of the quadratic interpolation approximation using three points t j 2 f t j 2 t j 1 f t j 1 and t j f t j for the integrand f t on each small interval t j 1 t j j 2 while the linear interpolation approximation is applied on the first small interval t 0 t 1 as a result the new formula can be formally viewed as a modification of the classical l1 formula which is obtained by the piecewise linear approximation for f t both the computational efficiency and numerical accuracy of the new formula are superior to that of the l1 formula the coefficients and truncation errors of this formula are discussed in detail two test examples show the numerical accuracy of l1 2 formula second by the new formula two improved finite difference schemes with high order accuracy in time for solving the time fractional sub diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed respectively in addition the application of the new formula into solving fractional ordinary differential equations is also presented several numerical examples are computed the comparison with the corresponding results of finite difference methods by the l1 formula demonstrates that the new l1 2 formula is much more effective and more accurate than the l1 formula when solving time fractional differential equations numerically

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

one local retracts and banach operator pairs in metric spaces abstract in this paper we first introduce the concept of nr map and then use this concept to establish the existence of common fixed points for banach operator pairs in the context of uniformly convex geodesic metric spaces new proofs of main results theorems 2 1 and 3 5 of chen and li j chen z li banach operator pair and common fixed points for nonexpansive maps nonlinear anal 74 2011 3086 3090 are presented further we prove de marr s theorem for the family of symmetric banach operator pairs in metric spaces and r trees for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity

integrable representations for generalized virasoro toroidal lie algebras abstract in this paper we give the structure of irreducible integrable modules of generalized virasoro toroidal lie algebras with finite dimensional weight spaces and non zero central charge

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

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

invariant approximations and r subweakly commuting maps the existence of invariant approximations for r subweakly commuting mappings is proved our results extend most of the known results to a new class of noncommuting mappings

a differential difference system related to toroidal lie algebra we present a novel differential difference system in 2 1 dimensional space time one discrete two continuum arising from bogoyavlensky s 2 1 dimensional korteweg de vries hierarchy our method is based on the bilinear identity of the hierarchy which is related to the vertex operator representation of the toroidal lie algebra tor2

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

a box type scheme for fractional sub diffusion equation with neumann boundary conditions combining order reduction approach and l1 discretization a box type scheme is presented for solving a class of fractional sub diffusion equation with neumann boundary conditions a new inner product and corresponding norm with a sobolev embedding inequality are introduced a novel technique is applied in the proof of both stability and convergence the global convergence order in maximum norm is o t 2 a h 2 the accuracy and efficiency of the scheme are checked by two numerical tests

numerical solution of non linear fourth order fractional sub diffusion wave equation with time delay abstract in this paper we constructed a linearized compact difference scheme for fourth order non linear fractional sub diffusion equation with time delay and variable coefficients the primary purpose of our work is to use the idea of the l 2 1 formula for temporal dimension and compact linear operator for spatial dimension the proposed method is unconditionally stable and convergent to the analytical solution with the order of convergence o 2 h 4 where and h are temporal and spatial lengths respectively numerical experimentation is carried out to show the efficiency and accuracy of the proposed scheme

banach operator pairs and common fixed points in hyperconvex metric spaces the purpose of this paper is to establish demarr s well known theorem for an arbitrary family of symmetric banach operator pairs in hyperconvex metric spaces without the compactness assumption we also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings as an application several results on the invariant best approximation are proved

two extensions of 1d toda hierarchy the extended toda hierarchy of carlet dubrovin and zhang is reconsidered in light of a 2 1d extension of the 1d toda hierarchy constructed by ogawa these two extensions of the 1d toda hierarchy turn out to have a very similar structure and the former may be thought of as a kind of dimensional reduction of the latter in particular this explains an origin of the mysterious structure of the bilinear formalism proposed by milanov

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

banach operator pairs common fixed points invariant approximations and nonexpansive multimaps abstract common fixed point results are proved for i nonexpansive maps ciric type maps and nonexpansive multimaps invariant approximation results are obtained for these types of maps as applications our results extend or generalize several known results including the recent results of chen and li j chen z li common fixed points for banach operators pairs in best approximation j math anal appl 336 2007 1466 1475 in particular we show that the results of chen and li on banach operator pairs are particular cases of the results of al thagafi and shahzad m a al thagafi n shahzad noncommuting selfmaps and invariant approximations nonlinear anal 64 2006 2778 2786

common fixed points for banach operator pairs in best approximation the notion of banach operator pairs is introduced as a new class of noncommuting maps some common fixed point theorems for banach operator pairs and the existence of the common fixed points of best approximation are presented these results are proved without the assumption of linearity or affinity for either f or g which shows that the concept about banach operator pairs is potentially useful in the study of common fixed points

magnetic hydrodynamics with asymmetric stress tensor in this paper we study equations of magnetic hydrodynamics with a stress tensor we interpret this system as the generalized euler equation associated with an abelian extension of the lie algebra of vector fields with a nontrivial 2 cocycle we use the lie algebra approach to prove the energy conservation law and the conservation of cross helicity

integrable representations for toroidal extended affine lie algebras abstract let g be any untwisted affine kac moody algebra any fixed complex number and g the corresponding toroidal extended affine lie algebra of nullity two for any k tuple 1 k of weights of g and k tuple a a 1 a k of distinct non zero complex numbers we construct a class of modules v a for the extended affine lie algebra g we prove that the g module v a is completely reducible we also prove that the g module v a is integrable when all weights i in are dominant thus we obtain a new class of irreducible integrable weight modules for the toroidal extended affine lie algebra g

common fixed points in best approximation for banach operator pairs with ciric type i contractions abstract the common fixed point theorems similar to those of ciric lj b ciric on a common fixed point theorem of a gregus type publ inst math beograd n s 49 1991 174 178 lj b ciric on diviccaro fisher and sessa open questions arch math brno 29 1993 145 152 lj b ciric on a generalization of gregus fixed point theorem czechoslovak math j 50 2000 449 458 fisher and sessa b fisher s sessa on a fixed point theorem of gregus internat j math math sci 9 1986 23 28 jungck g jungck on a fixed point theorem of fisher and sessa internat j math math sci 13 1990 497 500 and mukherjee and verma r n mukherjee v verma a note on fixed point theorem of gregus math japon 33 1988 745 749 are proved for a banach operator pair as applications common fixed point and approximation results for banach operator pair satisfying ciric type contractive conditions are obtained without the assumption of linearity or affinity of either t or i our results unify and generalize various known results to a more general class of noncommuting mappings

an extension of the korteweg de vries hierarchy arising from a representation of a toroidal lie algebra abstract in this article we show how to construct hierarchies of partial differential equations from the vertex operator representations of toroidal lie algebras in the smallest example rank 2 toroidal cover of sl2 we obtain an extension of the kdv hierarchy we use the action of the corresponding infinite dimensional group to construct solutions for these non linear pdes

extended affine weyl groups abstract in this paper we study the weyl groups of reduced extended affine root systems the root systems of extended affine lie algebras we start by describing the extended affine weyl group as a semidirect product of a finite weyl group and a heisenberg like normal subgroup this provides a unique expression for the weyl group elements in terms of some naturally arisen transformations which is crucial in the further study of extended affine weyl groups we use this to give a presentation called a presentation by conjugation for an important subclass of extended affine weyl groups using a new notion called the index which is an invariant of the extended affine root systems we show that one of the important features of finite and affine root systems related to weyl group holds for the class of extended affine root systems

representations of the lie algebra of derivations for quantum torus abstract let der c q be the derivation lie algebra of the quantum torus c q we construct a functor from gld modules to der c q modules which generalizes the constructions obtained by shen scientia sinica a 29 1986 570 larsson internat j modern phys a 7 1992 6493 and rao j algebra 182 1996 401 we also give a complete description of the structure of the der c q modules when all the entries of the quantum torus matrix q qij d d are roots of unity

a compact adi method and its extrapolation for time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions a compact alternating direction implicit adi finite difference method is proposed for two dimensional time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions the unconditional stability and convergence of the method is proved the error estimates in the weighted l 2 l 2 and l l norms are obtained the proposed method has the fourth order spatial accuracy and the temporal accuracy of order min 2 1 min 2 1 where 0 1 0 1 is the order of the fractional derivative in order to further improve the temporal accuracy two richardson extrapolation algorithms are presented numerical results demonstrate the accuracy of the compact adi method and the high efficiency of the extrapolation algorithms

locally extended affine lie algebras abstract we propose a new simplified definition of extended affine lie algebras ealas for short and also discuss a general version of extended affine lie algebras called locally extended affine lie algebras local ealas for short we prove a conjecture by v kac for local ealas it turns out that the root system of a local eala becomes a locally finite version of an extended affine root system several examples of new ealas and local ealas are introduced and finally we classify local ealas of nullity 0 and show the connection to locally finite split simple lie algebras

best simultaneous approximations asymptotically nonexpansive mappings and variational inequalities in banach spaces abstract the existence of common fixed points is established for three mappings where t is either generalized f g nonexpansive or asymptotically f g nonexpansive on a nonempty subset of a banach space as applications the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained our results unify and substantially improve several recent results existing in the current literature

structure and representation for a class of infinite dimensional lie algebras abstract let a be a commutative associative algebra over the complex field c and g be the complexification of the real lie algebra so 3 for any fixed elements e 1 e 2 e 3 a we define a lie algebra l e 1 e 2 e 3 g a with lie bracket given by 1 2 when the associative algebra a is the laurent polynomial algebra c t 1 1 t 1 we determine its derivation lie algebra der l and universal central extension l we also give a vertex operator representation for the lie algebra l this new class of lie algebras includes the affine lie algebra a 1 1 and the toroidal lie algebras of type a 1 we note that in general this kind of lie algebras is not z graded

bosonic realization of toroidal lie algebras of classical types we use fermionic operators to construct toroidal lie algebras of classical types including in particular symplectic affine algebras which are first realized by fermions

a new difference scheme for the time fractional diffusion equation abstract in this paper we construct a new difference analog of the caputo fractional derivative called the l 2 1 formula the basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered stability of the suggested schemes and also their convergence in the grid l 2 norm with the rate equal to the order of the approximation error are proved the obtained results are supported by the numerical calculations carried out for some test problems

highest weight modules over pre exp polynomial lie algebras abstract in this paper we introduce pre exp polynomial lie algebras which include loop algebras virasoro like algebras and some quantum torus lie algebras we study highest weight representations of these z n 1 graded lie algebras more precisely we show that non graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite dimensional weight spaces or not and they have all finite dimensional weight spaces if and only if the highest weight is an exp polynomial highest weight we also show that non graded and graded highest weight verma modules with the same highest weight are simultaneously irreducible or not and we determine necessary and sufficient conditions for a verma module to be irreducible

noncommuting selfmaps and invariant approximations abstract common fixed point results are established for a new class of noncommuting selfmaps we apply them to obtain several invariant approximation results which unify extend and complement well known results

common fixed points for banach operator pairs with applications the existence of common fixed point results for a banach operator pair under certain generalized contractions is established the invariant best approximation results are proved as applications and the existence of solutions of variational inequalities is obtained we also study the solution of functional equations arising from dynamic programming 2007 elsevier ltd all rights reserved

banach operator pair and common fixed points for nonexpansive maps abstract let x be a normed space and t and i be two nonexpansive self maps on the closed convex subset c x it is proved that if i t is a banach operator pair and t c is compact then f i t 0 a family of nonexpansive maps is also investigated the well known de marr s fixed point theorem is extended to the noncommuting case by introducing a notion of banach operator family as an application of the theorem the problem for the common fixed point in invariant approximation for convex set is solved directly in a quite different way from the others before

irreducible representations of virasoro toroidal lie algebras toroidal lie algebras and their vertex operator representations were introduced in mey and a class of indecomposable modules were investigated in this work we extend the toroidal algebra by the virasoro algebra thus constructing a semi direct product algebra containing the toroidal algebra as an ideal and the virasoro algebra as a subalgebra with the use of vertex operators and certain oscillator representations of the virasoro algebra it is proved that the corresponding fock space gives rise to a class of irreducible modules for the virasoro toroidal algebra

extended diffeomorphism algebras and trajectories in jet space let the dro diffeomorphism reparametrization observer algebra dro n be the extension of diff n diff 1 by its four inequivalent virasoro like cocycles here diff n is the diffeomorphism algebra in n dimensional spacetime and diff 1 describes reparametrizations of trajectories in the space of tensor valued p jets dro n has a fock module for each p and each representation of gl n analogous representations for gauge algebras higher dimensional kac moody algebras are also given the reparametrization symmetry can be eliminated by a gauge fixing procedure resulting in previously discovered modules in this process two dro n cocycles transmute into anisotropic cocycles for diff n thus the fock modules of toroidal lie algebras and their derivation algebras are geometrically explained

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

irreducible representations of the lie algebra of the diffeomorphisms of ad dimensional torus in this paper we investigate the irreducibility of certain modules for the lie algebra of diffeomorphisms of torus t the lie algebra of diffeomorw 1 1 x phisms can be described as the derivations of a s c t t see 1 d w x w x rss we denote this lie algebra by der a larsson l1 constructed a functor from gl modules to der a modules in this paper we prove that d the image of a finite dimensional irreducible gl module is most often d irreducible the only exceptions are fundamental modules and the onedimensional modules in these cases we describe the sub modules and the quotients surprisingly the same class of modules image of finite dimen w x sional gl module are also constructed in e without any reference to d gl modules they are motivated by the vertex operator constructions in d w x em more details near the end of this introduction we will now describe our results in more detail let gl be the d lie algebras of d d matrices over complex numbers c let gl s sl d d c id where sl is the finite dimensional simple lie algebra of trace zero d matrices and id the identity matrix which is central it is well known that finite dimensional irreducible modules and dominant integral weights are in one one correspondence for a finite dimensional simple lie algebra so let v c be finite dimensional irreducible sl module corresponding to a d dominant integral weight c let id the central element act by a complex number b and denote the corresponding gl module by v c b d a d d larsson defined see 1 6 a functor f for a belonging to c rz from w x gl modules to der a modules in l1 these modules are weight modules d

integrable representations for the twisted full toroidal lie algebras the purpose of this work is to classify irreducible integrable modules of the twisted full toroidal lie algebra with finite dimensional weight spaces and non zero central charges there are three families of such modules two of which are classified

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

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