bifurcation and exact traveling wave solutions for dual power zakharov kuznetsov burgers equation with fractional temporal evolution abstract in this article we introduce the dual power zakharov kuznetsov burgers equation with fractional temporal evolution in the sense of modified riemann liouville derivative we investigate the dynamical behavior bifurcations and phase portrait analysis of the exact traveling wave solutions of the system with and without damping effect we apply the g g expansion method in context of fractional complex transformation and seek a variety of exact traveling wave solutions including solitary wave kink type wave breaking wave and periodic wave solutions of the equation furthermore the remarkable features of the traveling wave solutions and phase portraits of dynamical system are demonstrated through interesting figures

classical non markovian boltzmann equation the modeling of particle transport involves anomalous diffusion x2 t t with 1 with subdiffusive transport corresponding to 0 1 these anomalies give rise to fractional advection dispersion equations with memory in space and time the usual boltzmann equation with only isolated binary collisions is markovian and in particular the contributions of the three particle distribution function are neglected we show that the inclusion of higher order distribution functions give rise to an exact non markovian boltzmann equation with resulting transport equations for mass momentum and kinetic energy with memory in both time and space the two and the three particle distribution functions are considered under the assumption that the two and the three particle correlation functions are translationally invariant that allows us to obtain advection dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives

group analysis and exact solutions of the time fractional fokker planck equation abstract in this paper the lie symmetry analysis method is extended to deal with the nonlinear time fractional fokker planck fp equation with riemann liouville derivative the erdelyi kober fractional derivative which is depending on a parameter is used for the reduction of fp equation symmetry reduction is provided and some exact analytic solutions to the time fractional fp equation are investigated by virtue of the reduction method introduced by m c nucci

a new operator splitting method for american options under fractional black scholes models abstract a new operator splitting method is proposed for american options under time fractional black scholes models the fractional linear complementarity problem is split into two easy sub problems with the leading coefficients separated from the convolution sum and matched through a general correction step the method is implementation friendly in the sense that one can easily modify a fractional european solver to obtain the proposed method since the correction step is decoupled and is trivial to solve the method is validated through numerical experiments and demonstrated to be superior to the traditional approach the paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models

on invariant analysis of some time fractional nonlinear systems of partial differential equations i an investigation of lie point symmetries for systems of time fractional partial differential equations including ito system coupled burgers equations coupled korteweg de vries equations hirota satsuma coupled kdv equations and coupled nonlinear hirota equations has been done using the obtained symmetries each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving erdelyi kober fractional differential operator depending on a parameter

local fractional fokker planck equation we propose a new class of differential equations which we call local fractional differential equations they involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time a local fractional analog of the fokker planck equation has been derived starting from the chapman kolmogorov condition we solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise

thermal wave model of bioheat transfer with modified riemann liouville fractional derivative in this paper a new fractional thermal wave model of the bioheat transfer ftwmbt caused by spatial heating is built using taylor s series expansion of modified riemann liouville fractional derivatives a one dimensional analytical solution of the ftwmbt in a finite medium is obtained the ftwmbt in the case 1 interpolates the standard thermal wave model of bioheat transfer and the well known pennes bioheat equation 0 finally numerical results are presented graphically for various values of different parameters this study demonstrates that fractional models can provide a unified approach to examine the heat transfer in biological tissue

derivation and solutions of some fractional black scholes equations in coarse grained space and time application to merton s optimal portfolio abstract by using the new fractional taylor s series of fractional order f x h e h d x f x where e denotes the mittag leffler function and d x is the so called modified riemann liouville fractional derivative which we introduced recently to remove the effects of the non zero initial value of the function under consideration one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a usual gaussian white noise one can then derive two new families of fractional black scholes equations and one shows how one can obtain their solutions merton s optimal portfolio is once more considered and some new results are contributed with respect to the modeling on one hand and to the solution on the other hand finally one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics

on the solutions of nonlinear fractional klein gordon equation with modified riemann liouville derivative abstract in this paper fractional variational iteration method fvim is applied for finding approximate analytical solutions of nonlinear fractional klein gordon equation time fractional derivative is described in the riemann liouville sense a new application of fractional variational iteration method fvim is extended to derive analytical solutions in the form of a series for this equation the behavior of the solutions and the effects of different values of fractional order are indicated graphically it is shown that the solutions obtained by the fvim are reliable convenient and effective for strongly nonlinear partial equations with modified riemann liouville derivative

crystal energy functions via the charge in types a and c the ram yip formula for macdonald polynomials at t 0 provides a statistic which we call charge in types a and c it can be defined on tensor products of kashiwara nakashima single column crystals in this paper we prove that the charge is equal to the negative of the energy function on affine crystals the algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial r matrix

a bounded and efficient scheme for multidimensional problems with anomalous convection and diffusion abstract this work is motivated by a generalization of the well known burgers fisher and burgers huxley equations in multiple dimensions considering riesz fractional diffusion and convection initial boundary conditions which are positive and bounded are imposed on a closed and bounded rectangular domain in this manuscript we propose a finite difference method to approximate the positive and bounded solutions of the fractional model the methodology is a linear three steps crank nicolson technique which is based on the use of fractional centered differences the properties of fractional centered differences are employed to establish the existence and the uniqueness of solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness of the approximations we show in this work that the method is capable of preserving some of the constant solutions of the continuous model additionally we prove that our technique is a second order consistent stable and quadratically convergent scheme suitable bounds for the numerical solutions are also derived in this work finally some illustrative simulations show that the method is able to preserve the positivity and the boundedness of the numerical approximations in agreement with the analytic results proved in this work numerical comparisons provided in this work confirm the rate of convergence of the numerical technique

a wavelet operational method for solving fractional partial differential equations numerically fractional calculus is an extension of derivatives and integrals to non integer orders and a partial differential equation involving the fractional calculus operators is called the fractional pde they have many applications in science and engineering however not only the analytical solution existed for a limited number of cases but also the numerical methods are very complicated and difficult in this paper we newly establish the simulation method based on the operational matrices of the orthogonal functions we formulate the operational matrix of integration in a unified framework by using the operational matrix of integration we propose a new numerical method for linear fractional partial differential equation solving in the method we 1 use the haar wavelet 2 establish a lyapunov type matrix equation and 3 obtain the algebraic equations suitable for computer programming two examples are given to demonstrate the simplicity clarity and powerfulness of the new method

new exact solutions of nonlinear fractional acoustic wave equations in ultrasound in this paper new exact solutions of fractional nonlinear acoustic wave equations have been devised the travelling periodic wave solutions of fractional burgers hopf equation and khokhlov zabolotskaya kuznetsov kzk equation have obtained by first integral method nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible taking nonlinear effects into account the ultrasound beam analysis makes more accurate in these applications the burgers hopf equation is one of the extensively studied models in mathematical physics in addition the kzk parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3d diffraction sound beams in dissipative media in the present analysis these nonlinear equations have solved by first integral method as a result new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations the obtained results are presented graphically to demonstrate the efficiency of this proposed method

bgg reciprocity for current algebras abstract we study the category i gr of graded representations with finite dimensional graded pieces for the current algebra g c t where g is a simple lie algebra this category has many similarities with the category o of modules for g and in this paper we formulate and study an analogue of the famous bgg duality we recall the definition of the projective and simple objects in i gr which are indexed by dominant integral weights the role of the verma modules is played by a family of modules called the global weyl modules we show that in the case when g is of type s l 2 the projective module admits a flag in which the successive quotients are finite direct sums of global weyl modules the multiplicity with which a particular weyl module occurs in the flag is determined by the multiplicity of a jordan holder series for a closely associated family of modules called the local weyl modules we conjecture that the result remains true for arbitrary simple lie algebras we also prove some combinatorial product sum identities involving kostka polynomials which arise as a consequence of our theorem

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

lie symmetry analysis conservation laws and numerical approximations of time fractional fokker planck equations for special stochastic process in foreign exchange markets abstract in this paper the transition joint probability density function of the solution of the ornstein uhlenbeck process is presented by a deterministic parabolic time fractional pde fpde named time fractional fokker planck equation the article generally is divided to two parts theoretical and approximated analysis in the theoretical sections lie group method and invariant subspace method are applied for finding exact solutions and conservation laws of the considered equation in the next part by chebyshev wavelets s method the numerical solutions are driven then the usefulness of this approximated method is comparing with the exact solutions by some plotted graphs

on the fractional solution of the equation f x y f x f y and its application to fractional laplace s transform abstract it is shown that if the problem is defined in the setting of fractional calculus via fractional difference on non differentiable functions then the solution of the functional equation f x y f x f y is exactly defined as the solution of a linear fractional differential equation the dual or counterpart problem that is the fractional solution of the equation g xy g x g y is also considered and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized mittag leffler function which is nowhere differentiable this framework suggests a definition of fractional laplace s transform expressed in terms of generalized mittag leffler function and its main properties are outlined mainly inverse function and convolution one takes this opportunity to display a new fractional taylor s series for functions f x y of two variables x and y many open problems are stated which are directly related to the non differentiability of the functions so involved therefore the title on the fractional solution

fractional variational calculus for nondifferentiable functions we prove necessary optimality conditions in the class of continuous functions for variational problems defined with jumarie s modified riemann liouville derivative the fractional basic problem of the calculus of variations with free boundary conditions is considered as well as problems with isoperimetric and holonomic constraints

a nonlinear fractional sharma tasso olver equation due to its importance in scientific applications the fractional nonlinear sharma tasso olver equation has been studied by several authors from point of view of exact solutions in this paper new explicit and exact solutions are obtained by using the fractional complex transform and the improved extended tanh coth method the results can be compared with other methods used to solve the same model

derivation of a new mertons optimal problem presented by fractional stochastic stock price and its applications in this article a new model of mertons optimal problem is derived this derivation is based on stock price presented by fractional order stochastic differential equation an extension of hamiltonjacobibellman is used to transfer our proposed model to a fractional partial differential equation as an application of our proposed model two optimal problems are discussed and solved analytically

a fractional calculus of variations for multiple integrals with application to vibrating string we introduce a fractional theory of the calculus of variations for multiple integrals our approach uses the recent notions of riemann liouville fractional derivatives and integrals in the sense of jumarie the main results provide fractional versions of the theorems of green and gauss fractional euler lagrange equations and fractional natural boundary conditions as an application we discuss the fractional equation of motion of a vibrating string

the fractional quadratic form identity and hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy starting from a general isospectral problem of fractional order we propose a fractional quadratic form identity from which the hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy is derived the method can be generalized to other fractional integrable couplings

bmo and morrey campanato estimates for stochastic convolutions and schauder estimates for stochastic parabolic equations abstract in this paper we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t x 2 u t x d t g t x d t u 0 0 t 0 t x g for a random field u t x 0 t g u t x u t x r where 2 2 0 2 is the fractional laplacian t 0 is arbitrarily fixed g r d is a bounded domain g 0 t g r is a joint measurable coefficient and t t 0 is either a brownian motion or a levy process on a given filtered probability space f p f t t 0 t to this end we derive the bmo estimates and morrey campanato estimates respectively for stochastic singular integral operators arising from the equations concerned then by utilizing the embedding theory between the campanato space and the holder space we establish the controllability of the norm of the space c 2 d where 0 d 0 t g with all these in hand we are able to show that the q th order bmo quasi norm of the q 0 order derivative of the solution u is controlled by the norm of g under the condition that t is a levy process finally we derive the schauder estimate for the p moments of the solution of the above stochastic fractional heat equations driven by levy noise

lie symmetry analysis explicit solutions and conservation laws for the space time fractional nonlinear evolution equations abstract this paper studies the symmetry analysis explicit solutions convergence analysis and conservation laws cls for two different space time fractional nonlinear evolution equations with riemann liouville rl derivative the governing equations are reduced to nonlinear ordinary differential equation ode of fractional order using their lie point symmetries in the reduced equations the derivative is in erdelyi kober ek sense power series technique is applied to derive an explicit solutions for the reduced fractional odes the convergence of the obtained power series solutions is also presented moreover the new conservation theorem and the generalization of the noether operators are developed to construct the nonlocal cls for the equations some interesting figures for the obtained explicit solutions are presented

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified riemann liouville derivative and exterior derivatives of fractional orders example of the fractional kaup newell kn soliton equation hierarchy and hamiltonian system are considered which is a new integrable hierarchy and possesses hamiltonian structure

analysis of lie symmetries with conservation laws and solutions for the generalized 3 1 dimensional time fractional camassa holm kadomtsev petviashvili equation abstract in this paper under investigated is a generalized 3 1 dimensional camassa holm kadomtsev petviashvili gch kp equation which describes the role of dispersion in the formation of patterns in liquid drops with the help of the semi inverse method the euler lagrange equation and agrawal s method the time fractional gch kp equation is derived in the sense of riemann liouville fractional derivatives further the symmetry of the 3 1 dimensional time fractional gch kp equation is studied by fractional order symmetry meanwhile based on the new conservation theorem the conservation laws of 3 1 dimensional time fractional gch kp equation are constructed finally the solutions to the equation are given via a bilinear method and the radial basis functions rbfs meshless approach

breakthroughs in the theory of macdonald polynomials in 1998 i g macdonald 1 introduced a remarkable new basis for the space of symmetric functions the elements of this basis are denoted where is a partition and p q are two free parameters the s which are now called macdonald polynomials specialize to many of the well known bases for the symmetric functions by suitable choices of the parameters q and t in fact we can obtain in this manner the schur functions the hall littlewood symmetric functions the jack symmetric functions the zonal symmetric functions the zonal spherical functions and the elementary and monomial symmetric functions

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

a combinatorial formula for nonsymmetric macdonald polynomials we give a combinatorial formula for the nonsymmetric macdonald polynomials e x q t the formula generalizes our previous combinatorial interpretation of the integral form symmetric macdonald polynomials j x q t we prove the new formula by verifying that it satisfies a recurrence due to knop and sahi that characterizes the nonsymmetric macdonald polynomials

fuzzy min max neural networks i classification a supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described each fuzzy set is an aggregate union of fuzzy set hyperboxes a fuzzy set hyperbox is an n dimensional box defined by a min point and a max point with a corresponding membership function the min max points are determined using the fuzzy min max learning algorithm an expansion contraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining the use of a fuzzy set approach to pattern classification inherently provides a degree of membership information that is extremely useful in higher level decision making the relationship between fuzzy sets and pattern classification is described the fuzzy min max classifier neural network implementation is explained the learning and recall algorithms are outlined and several examples of operation demonstrate the strong qualities of this new neural network classifier

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

a probabilistic interpretation of the macdonald polynomials the two parameter macdonald polynomials are a central object of algebraic combinatorics and representation theory we give a markov chain on partitions of k with eigenfunctions the coefficients of the macdonald polynomials when expanded in the power sum polynomials the markov chain has stationary distribution a new two parameter family of measures on partitions the inverse of the macdonald weight rescaled the uniform distribution on cycles of permutations and the ewens sampling formula are special cases the markov chain is a version of the auxiliary variables algorithm of statistical physics properties of the macdonald polynomials allow a sharp analysis of the running time in natural cases a bounded number of steps suffice for arbitrarily large k

a combinatorial formula for macdonald polynomials abstract in this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types these formulas resemble the formulas of haglund haiman and loehr for macdonald polynomials of type gl n at q 0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded alcove walks and at q t 0 these formulas specialize to the formula for the weyl character in terms of the littelmann path model in the positively folded gallery form of gaussent and littelmann

derivation of an amplitude of information in the setting of a new family of fractional entropies by generalizing the basic functional equation f xy f x f y in the form f b xy f b x f b y b 1 one can derive a family of solutions which are exactly the inverse of the mittag leffler function referred to as mittag leffler logarithm or logarithm of fractional order this result provides a new family of generalized informational entropies which are indexed by a parameter clearly related to fractals via fractional calculus and which is quite relevant in the presence in defect of observation the relation with shannon s entropy renyi s entropy and tsallis entropy is clarified and it is shown that tsallis generalized logarithm has a significance in terms of fractional calculus the case b 2 looks like directly relevant to amplitude of probability in quantum mechanics and provides an approach to the definition of amplitude of informational entropy one examines the kind of result one can so obtain in applying the maximum entropy principle in the presence of uncertain definition or fuzzy definition the mittag leffler function would be more relevant than the gaussian normal law to some extent this new formulation could be fully supported by the derivation of a new family of fractional fisher information

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

an efficient computational algorithm for min max operations abstract this paper describes an efficient computational algorithm for evaluating min max operations and the derivatives of min max functions which are common computation problems in the training of many types of fuzzy neural networks analysis and simulation are used to assess the efficiency of the computational algorithm developed in this paper

simplest equation method for some time fractional partial differential equations with conformable derivative abstract the conformable fractional derivative was proposed by r khalil et al in 2014 which is natural and obeys the leibniz rule and chain rule based on the properties a class of time fractional partial differential equations can be reduced into odes using traveling wave transformation then the simplest equation method is applied to find exact solutions of some time fractional partial differential equations the exact solutions solitary wave solutions periodic function solutions rational function solutions of time fractional generalized burgers equation time fractional generalized kdv equation time fractional generalized sharma tasso olver fsto equation and time fractional fifth order kdv equation 3 1 dimensional time fractional kdv zakharov kuznetsov kdv zk equation are constructed this method presents a wide applicability to solve some nonlinear time fractional differential equations with conformable derivative

analytical treatment of differential equations with fractional coordinate derivatives this paper outlines a reliable strategy to use the homotopy perturbation method based on jumarie s derivative for solving fractional differential equations in this framework compact structures of fourth order fractional diffusion wave equations are considered as prototype examples moreover convergence of the proposed approach for these types of equations is investigated results show that the response expressions are mittag leffler stable

approximation by max min operators a general theory and its applications abstract in this study we obtain a general approximation theorem for max min operators including many significant applications we also study the error estimation in this approximation by using holder continuous functions the main motivation for this work is the paper by bede et al 2008 12 as a special case of our results we explain how to approximate nonnegative continuous functions of one and two variables by means of the max min shepard operators we also study the approximation by the max min bernstein operators furthermore to verify the theory we display graphical illustrations

matrix product formula for macdonald polynomials we derive a matrix product formula for symmetric macdonald polynomials our results are obtained by constructing polynomial solutions of deformed knizhnik zamolodchikov equations which arise by considering representations of the zamolodchikov faddeev and yang baxter algebras in terms of t deformed bosonic operators these solutions are generalised probabilities for particle configurations of the multi species asymmetric exclusion process and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions for weakly increasing compositions anti dominant weights these basis elements coincide with non symmetric macdonald polynomials our formulas imply a natural combinatorial interpretation in terms of solvable lattice models they also imply that normalisations of stationary states of multi species exclusion processes are obtained as macdonald polynomials at q 1

on combinatorial formulas for macdonald polynomials a recent breakthrough in the theory of type a macdonald polynomials is due to haglund haiman and loehr who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of young diagrams ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of so called alcove walks these originate in the work of gaussent littelmann and of the author with postnikov on discrete counterparts to the littelmann path model in this paper we relate the above developments by explaining how the ram yip formula compresses to a new formula which is similar to the haglund haiman loehr one but contains considerably fewer terms

idea of invariant subspace combined with elementary integral method for investigating exact solutions of time fractional npdes abstract in this paper inspired by the idea of invariant subspace method and combined with elementary integral method we introduced a novel approach for investigating exact solutions of a time fractional nonlinear partial differential equation npde based on hypothetical structure of solution of separated variable a time fractional npde defined by time and space variables can be reduced to a nonlinear ordinary differential equation node or nodes defined by space variable alone and then using the elementary integral method to solve the node or nodes different kinds of exact solutions of a time fractional npde are obtained finally as examples the time fractional hunter saxton equation and time fractional li olver equation were studied different kinds of exact solutions of these equations were obtained and their dynamical properties were illustrated

improved fractional sub equation method for 3 1 dimensional generalized fractional kdv zakharov kuznetsov equations in the present paper we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics namely the time fractional kdv zakharov kuznetsov kdv zk and space time fractional modified kdv zakharov kuznetsov mkdv zk equations by using improved fractional sub equation method as a result new types of exact analytical solutions are obtained the obtained results are shown graphically here the fractional derivative is described in the jumarie s modified riemann liouville sense

a review of definitions of fractional derivatives and other operators abstract given the increasing number of proposals and definitions of operators in the scope of fractional calculus it is important to introduce a systematic classification nonetheless many of the definitions that emerged in the literature can not be considered as fractional derivatives we analyze a list of expressions to have a general overview of the concept of fractional integrals derivatives moreover some formulae that do not involve the term fractional are also included due to their particular interest in the area

a littlewood richardson rule for macdonald polynomials macdonald polynomials are orthogonal polynomials associated to root systems and in the type a case the symmetric macdonald polynomials are a common generalization of schur functions macdonald spherical functions and jack polynomials we use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine hecke algebra from this we obtain a product formula for macdonald polynomials of general lie type

new solitary wave solutions of 2 1 dimensional space time fractional burgers equation and korteweg de vries equation abstract in this article via the improved fractional subequation method the fully analytical solutions of the 2 1 dimensional space time fractional burgers equation and korteweg de vries equation involving jumarie s modified riemann liouville derivative have been derived as a result with the aid of symbolic computation many types of new analytical solutions are obtained which include new solitary wave periodic wave and rational wave solutions the graphical representations show that these gained solutions have abundant structures

estimation of unknown function of a class of integral inequalities and applications in fractional integral equations in this paper we establish a class of iterated integral inequalities which includes a nonconstant term outside the integrals the upper bound of the embedded unknown function is estimated explicitly by adopting novel analysis techniques such as change of variable amplification method differential and integration the derived result can be applied in the study of qualitative properties of solutions of fractional integral equations

different methods for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the exp function method the g g expansion method and the generalized kudryashov method are used to construct exact solutions for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation this fractional equation can be turned into another nonlinear ordinary differential equation by fractional complex transformation and then these three methods are applied to solve it as a result some new exact solutions are obtained the three methods demonstrate power reliability and efficiency

efficient learning algorithms for three layer regular feedforward fuzzy neural networks a key step of using gradient descend methods to develop learning algorithms of a regular feedforward fuzzy neural network fnn is to differentiate max min functions which contain max spl or and min spl and operations the paper aims at several objectives first investigate further the differentiation of spl or spl and functions second employ general fuzzy numbers which include triangular and trapezoidal fuzzy numbers as special cases to define a three layer regular fnn the general fuzzy numbers related can be approximately determined by their corresponding finite level sets so we can approximately represent the input output i o relationship of the regular fnn as functions of the endpoints of all finite level sets third a fuzzy back propagation algorithm is presented and to speed up the convergence of the learning algorithm a fuzzy conjugate gradient algorithm for fuzzy weights and biases is developed furthermore the convergence of the algorithm is analyzed systematically finally some real simulations demonstrate the efficiency of our learning algorithms the regular fnn is applied to the approximate realization of fuzzy inference rules and fuzzy functions defined on given compact sets

the first integral method for some time fractional differential equations abstract in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the first integral method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations the power of this manageable method is presented by applying it to several examples this approach can also be applied to other nonlinear fractional differential equations

lie symmetry analysis of the time fractional kdv type equation abstract the lie symmetry analysis method is extended to deal with the time fractional kdv type equation it is shown that this equation can be reduced to an equation with the erdelyi kober fractional derivative this method can also be applied to symmetry classification of fractional equations with some arbitrary functions

on group invariant solutions of fractional order burgers poisson equation abstract in this paper symmetry property of fractional order space time burgers poisson fbp equation is investigated the equation is obtained by replacing first order time and space derivatives to the corresponding fractional derivatives of order and respectively in the classical burgers poisson equation we present lie symmetries and corresponding infinitesimal generators for the fbp equation by using fractional lie group method with the help of these infinitesimal generators some group invariant solutions are sought by reducing the order of the equation

holder continuous of the solutions to stochastic nonlocal heat equations abstract in this paper we aim to obtain the holder continuous of solutions to stochastic nonlocal heat equations by using campanato estimates and sobolev embedding theorem we first prove the holder continuous of the mild solution of stochastic nonlocal diffusion equations locally in r d in the sense that the solution u belongs to the space c d t l p then by using tail estimates we obtain the estimates of the mild solution in l p c d t lastly we prove the holder continuous of the mild solution on bounded domain

the fractional supertrace identity and its application to the super ablowitz kaup newell segur hierarchy in this paper a fractional supertrace identity on superalgebras is presented based on the fractional supertrace identity we can construct fractional super hamiltonian structures of the generalized zero curvature equations associated with lie superalgebras as an application we get the fractional super ablowitz kaup newell segur akns hierarchy and its super hamiltonian structure by using of fractional supertrace identity this method can be used to get more fractional super hierarchies

ls galleries the path model and mv cycles we give an interpretation of the path model of a representation 18 of a complex semisimple algebraic group g in terms of the geometry of its affine grassmannian in this setting the paths are replaced by ls galleries in the affine coxeter complex associated to the weyl group of g to explain the connection with geometry consider a demazure hansen bott samelson desingularization of the closure of an orbit g c t in the affine grassmannian the homology of has a basis given by bia lynicki birula cell s which are indexed by the t fixed points in now the points of can be identified with galleries of a fixed type in the affine tits building associated to g and the t fixed points correspond in this language to combinatorial galleries of a fixed type in the affine coxeter complex we determine those galleries such that the associated cell has a non empty intersection with g c t identified with an open subset of and we show that the closures of the strata associated to ls galleries are exactly the mv cycles 24 which form a basis of the representation v for the langland s dual group g

generalized weyl modules and nonsymmetric q whittaker functions abstract we introduce generalized global weyl modules and relate their graded characters to nonsymmetric macdonald polynomials and nonsymmetric q whittaker functions in particular we show that the series part of the nonsymmetric q whittaker function is a generating function for the graded characters of generalized global weyl modules

a fokker planck equation of fractional order with respect to time by combining the maximum entropy principle with some considerations related to derivatives of fractional order one is led to suggest a fokker planck of fractional order with respect to time which could be related to dynamical systems subject to fractional brownian motion the relation with the process associated with the equation p t 1 n 1 2np x2n is exhibited

the application of fractional derivatives in stochastic models driven by fractional brownian motion abstract in this paper in order to establish connection between fractional derivative and fractional brownian motion fbm we first prove the validity of the fractional taylor formula proposed by guy jumarie then by using the properties of this taylor formula we derive a fractional ito formula for h 1 2 1 which coincides in form with the one proposed by duncan for some special cases whose formula is based on the wick product lastly we apply this fractional ito formula to the option pricing problem when the underlying of the option contract is supposed to be driven by a geometric fractional brownian motion the case that the drift volatility and risk free interest rate are all dependent on t is also discussed

new approach to find exact solutions of time fractional kuramoto sivashinsky equation abstract in the present paper we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics viz time fractional kuramoto sivashinsky equation by a new proposed method via fractional complex transform as a result new types of exact analytical solutions are obtained here the fractional derivative is described in jumarie s modified riemann liouville sense

generalized einstein relation and the metzler klafter conjecture in a composite subdiffusive regime abstract in this paper a generalized diffusion model driven by the composite subdiffusive fractional brownian motion fbm is employed based on this stochastic process we derive a fractional fokker planck equation ffpe and obtain its solution it is proved that the generalized einstein relation ger and the metzler and klafter conjecture on the asymptotic behavior of stretched gaussian hold the ffpe in a composite subdiffusive regime