numerical solution of fractional differential equations with a collocation method based on muntz polynomials this paper presents a computational technique based on the collocation method and muntz polynomials for the solution of fractional differential equations an appropriate representation of the solution via the muntz polynomials reduces its numerical treatment to the solution of a system of algebraic equations the main advantage of the present method is its superior accuracy and exponential convergence consequently one can obtain good results even by using a small number of collocation points the accuracy and performance of the proposed method are examined by means of some numerical experiments

analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method the fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials motion of a large thin plate in a newtonian fluid creep and relaxation functions for viscoelastic materials the pi ld m controller for the control of dynamical systems etc phenomena in electromagnetics acoustics viscoelasticity electrochemistry and material science are also described by differential equations of fractional order the solution of the differential equation containing fractional derivative is much involved instead of application of the existing methods an attempt has been made in the present analysis to obtain the solution of nonlinear dynamic system containing fractional derivative ji zeng wang et al coiflets based method in the solution of nonlinear dynamic system containing fractional derivative in the fourth international conference on nonlinear mechanics icnm iv shanghai august 2002 pp 1304 1308 by the relatively new adomian decomposition method the results obtained by this method are then graphically represented and then compared with the exact solution a good agreement of the results is observed

numerical solution of volterra fredholm integral equations using the collocation method based on a special form of the muntz legendre polynomials abstract this paper presents a computational technique based on a special family of the muntz legendre polynomials to solve a class of volterra fredholm integral equations the relationship between the jacobi polynomials and muntz legendre polynomials in a particular state are expressed the proposed method reduces the integral equation into algebraic equations via the chebyshev gauss lobatto points so that the system matrix coefficients are obtained by the least squares approximation method the useful properties of the jacobi polynomials are exploited to analyze the approximation error the performance and accuracy of our method are examined with some illustrative examples

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

an approximate solution based on jacobi polynomials for time fractional convectiondiffusion equation operational matrix of shifted jacobi polynomials is considered solution of time fractional order convectiondiffusion problem is numerically estimated main problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded a system of nonlinear algebraic equations is achieved by approximating the known and unknown functions with the help of shifted jacobi functions problem can be used extensively in science and engineering as in oil reservoir simulations in this article we present a numerical method to numerically solve a time fractional convectiondiffusion equation our method is based on the operational matrices of shifted jacobi polynomials at first problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded then we approximate the known and unknown functions with the help of shifted jacobi functions a system of nonlinear algebraic equations is obtained finally the unknown coefficients are determined by mathematicatm we implemented the proposed method for several examples that they indicate the high accuracy method it should be noted that this method is generalizable to some appropriate problems

a two dimensional chebyshev wavelets approach for solving the fokker planck equations of time and space fractional derivatives type with variable coefficients abstract in the current study we consider the numerical solutions of the fokker planck equations of time and space fractional derivative type with variable coefficients the proposed method is based on the two dimensional chebyshev wavelet basis together with their corresponding operational matrices of fractional order integration the convergence analysis of the proposed method is rigorously established numerical tests are carried out to confirm the effectiveness and feasibility of the proposed scheme

an extension of the spectral tau method for numerical solution of multi order fractional differential equations with convergence analysis the main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations fdes based on a spectral tau method an extension of the operational approach of the tau method with the orthogonal polynomial bases is proposed to convert fdes to its matrix vector multiplication representation the fractional derivatives are described in the caputo sense the spectral rate of convergence for the proposed method is established in the l 2 norm we tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence

a new non standard finite difference method for analyzing the fractional navier stokes equations abstract in this paper the numerical solution of the fractional navier stokes equations fnse is derived by means of a new non standard finite difference method nsfdm the fractional differential operators are taken in sense of the riesz fractional derivative the pressure driven by the flow between two parallel plates is solved with the nsfdm and a modified trapezoidal quadrature rule of fractional order the stability and convergence of the proposed scheme are proved numerical examples illustrate the effectiveness of the proposed method

decomposition method for solving fractional riccati differential equations in this article we implement a relatively new analytical technique the adomian decomposition method for solving fractional riccati differential equations the fractional derivatives are described in the caputo sense in this scheme the solution takes the form of a convergent series with easily computable components the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of the solution the corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations some numerical examples are presented to illustrate the efficiency and reliability of the method

numerical solution of a class of fractional order integro differential algebraic equations using muntz jacobi tau method abstract this paper presents a spectral tau method based on muntz jacobi basis functions for approximating the solutions of fractional order volterra integro differential algebraic equations we examine the solvability of the considered equation and we show that the exact solutions suffer from a singularity at the origin numerical solvability and convergence analysis of the proposed approximate approach are also studied and the error bounds are obtained in a weighted l 2 norm some illustrative examples are presented to validate the theoretical results

the multi index mittag leffler functions as an important class of special functions of fractional calculus this paper is a short description of our recent results on an important class of the so called special functions of fractional calculus sf of fc which became important as solutions of fractional order or multi order differential and integral equations control systems and refined mathematical models of various physical chemical economical management bioengineering phenomena basically under sf of fc we mean the wright generalized hypergeometric function email protected q as a special case of the fox h function we have introduced and studied the multi index mittag leffler functions as their typical representatives including many interesting special cases that have already proven their usefulness in fc and its applications some new results are also presented and open problems are discussed

a novel class of fractionally orthogonal quasi polynomials and new fractional quadrature formulas a novel class of quasi polynomials orthogonal with respect to the fractional integration operator has been developed in this paper the related gaussian quadrature formulas for numerical evaluation of fractional order integrals have also been proposed by allowing the commensurate order of quasi polynomials to vary independently of the integration order a family of fractional quadrature formulas has been developed for each fractional integration order including novel quadrature formulas for numerical approximation of classical integer order integrals a distinct feature of the proposed quadratures is high computational efficiency and flexibility as will be demonstrated in the paper as auxiliary results the paper also presents methods for lagrangian and hermitean quasi polynomial interpolation and hermitean fractional quadratures the development is illustrated by numerical examples

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

stable multi domain spectral penalty methods for fractional partial differential equations we propose stable multi domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order first a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials the approximation order is analyzed and verified through numerical examples based on the discrete fractional derivative we introduce stable multi domain spectral penalty methods for solving fractional advection and diffusion equations the equations are discretized in each sub domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

fractional order legendre operational matrix of fractional integration for solving the riccati equation with fractional order this paper is devoted to both theoretical and numerical study of riccati equation with fractional order a formulation to the fractional order legendre operational matrix of fractional integration is constructed existence and uniqueness results for the considered problem are provided and proved the fractional derivative is described in the caputo sense some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm

some null solutions of the yang mills equations and cauchy riemann structures some new null solutions of the yang mills equations with congruences of twisting shear free and null geodesics are obtained these are test fields each of them is defined on a lorentzian manifold with a metric tensor adapted to the cauchy riemann geometry associated with the congruence some examples of solutions on minkowski space time are also given among them is a solution with a congruence of twisting shear free and null geodesics different from the robinson congruence

recovery of high order accuracy in jacobi spectral collocation methods for fractional terminal value problems with non smooth solutions abstract an open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high order accuracy for non smooth solutions the limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes in this paper we derive and analyze an exponentially accurate jacobi spectral collocation method for the numerical solution of nonlinear terminal value problems involving the caputo fractional derivative of rational order 0 1 the main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate we systematically describe the necessary steps in the implementation process additionally a comprehensive numerical analysis including error estimates under the l and weighted l 2 norms is derived the extensive numerical examples that accompany our analysis confirm our theoretical estimates as well as give additional insights into the convergence behavior of our method for problems with smooth and non smooth solutions

group graded rings and duality we give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend known results on skew group rings to corresponding results for large classes of group graded rings finally we modify the construction slightly to handle infinite groups introduction in the first section we give an alternate construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend results on skew group rings and their modules to corresponding results for large classes of groupgraded rings in the second section we modify the construction slightly to handle infinite groups and apply it to a theorem on prime and semiprime infinite crossed products i would like to thank my thesis advisor d s passman for his guidance and encouragement throughout the writing of this paper 1 finite group graded rings and the smash product let g be a multiplicative group an associative ring with identity is said to be g graded if 1 r e r x xec is a direct sum of additive subgroups r x with r x r y c r xy it follows that necessarily 1r e r 1 so that each r x is a unitary r l bimodule if r e r we write r x for the component of r in r x so that r exgcr x r is said to be strongly g graded if r x r y r xy for all x y e g throughout this section r is assumed to be g graded where g is a finite group with igi n let mc r denote the set of n x n matrices over r with the rows and columns indexed by the elements of g if a c mg r we write ax y for the entry in the x y position of a then if a 1 c mc r the matrix product a 8 is given by 2 x x y e xx zaz y z e c if u c g is any subset of g let r u ex eur x in particular r r g now suppose h c g is a subgroup of g we define r h c mc r by r h a e ms r ax y e r xhy l that is r h ex ygcr xhy l e x y where e x y is the matrix unit with 1r in the x y position and zeroes elsewhere note that r g mc r received by the editors september 27 1984 1980 mathematics subject classification primary 16a03 c 1985 american mathematical society 0002 9947 85 1 00 25 per page

duality theorems for graded algebras and coalgebras the concepts of smash product and smash coproduct have been proved to be extremely useful tools in the theory of graded rings and graded coalgebras and more recently in the theory of action and coactions of a hopf algebra on an algebra or on a coalgebra once these concepts were defined a natural question appeared how can the smash product resp the smash coproduct be related to a matrix algebra resp to a matrix w x coalgebra the answer is given by the duality theorems proved in 3 for w x graded rings and in 5 for graded coalgebras various duality theorems w x w x for algebras have been proved during the last ten years cf 2 or 10 the purpose of this paper is to study duality for the smash product resp the smash coproduct associated to a g set and a g graded ring resp a g graded coalgebra the mentioned smash product and smash coproduct w x w x w x have been constructed in 10 and 6 it is shown by an example in 12 that for a g graded ring r and a left g set a we do not have a duality

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

numerical methods for multi term fractional arbitrary orders differential equations our main concern here is to give a numerical scheme to solve a nonlinear multi term fractional arbitrary orders differential equation

numerical solution of fractional pantograph differential equations by using generalized fractional order bernoulli wavelet in the current study new functions called generalized fractional order bernoulli wavelet functions gfbwfs based on the bernoulli wavelets are defined to obtain the numerical solution of fractional order pantograph differential equations in a large interval for the concept of fractional derivative we will use caputo sense by using riemann liouville fractional integral operator first the generalized fractional order bernoulli wavelets are constructed then these functions and their properties are employed to derive the gfbwfs operational matrices of fractional integration and pantograph the operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations finally some examples are included for demonstrating the validity and applicability of our method

muntz systems and orthogonal muntz legendre polynomials the muntz legendre polynomials arise by orthogonalizing the muntz system xio x l with respect to lebesgue measure on 0 1 in this paper differential and integral recurrence formulae for the muntz legendre polynomials are obtained interlacing and lexicographical properties of their zeros are studied and the smallest and largest zeros are universally estimated via the zeros of laguerre polynomials the uniform convergence of the christoffel functions is proved equivalent to the nondenseness of the muntz space on 0 1 which implies that in this case the orthogonal muntz legendre polynomials tend to 0 uniformly on closed subintervals of 0 1 some inequalities for muntz polynomials are also investigated most notably a sharp l2 markov inequality is proved

numerical methods for fourth order fractional integro differential equations this paper outlines a reliable strategy for solving linear and nonlinear boundary value problems for fourth order fractional integro differential equations the fractional derivative is considered in the caputo sense the adomian decomposition method is found to be fast and accurate numerical examples are presented to illustrate the efficiency and reliability of the method

von neumann regularity of smash products associated with g set gradings abstract for a g graded ring r the problem of gr regularity i e von neumann regularity condition on homogeneous elements has been treated earlier see n st sescu and van oystaeyen 1982 11 once the concept of the smash product was introduced this problem has been resumed by several authors this paper is concerned with von neumann regularity of two smash products the smash product r g of the g graded ring r by the group g and the smash product r a of r by a finite left g set a the connections between the regularity of r a and the gr regularity of r are also investigated one consequence of our results is that the smash product r a is a von neumann regular ring if and only if the category g a r gr is regular in stenstrom s sense

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

morphisms of relative hopf modules smash products and duality the aim of this paper is to extend a result of h j schneider on endomorphism rings of relative hopf modules 17 theorem 3 2 to the case of projective hopf algebras with bijective antipode and to show the connection between this extension and several duality theorems for hopf algebras a consequence of this generalization is the following if h is a co frobenius hopf algebra over the field k a b is a hopf galois extension and m is a right a h hopf module then considering the usual ring embeddings

a fourth order compact solution of the two dimensional modified anomalous fractional sub diffusion equation with a nonlinear source term this work is concerned to the study of high order difference scheme for the solution of a two dimensional modified anomalous sub diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses the space fractional derivatives are described in the riemann liouville sense in the proposed scheme we discretize the space derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme we prove the stability and convergence of proposed scheme using the fourier analysis the convergence order of the proposed method is o t h x 4 h y 4 comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme

second order numerical methods for multi term fractional differential equations smooth and non smooth solutions abstract starting with the asymptotic expansion of the error equation of the shifted grunwald letnikov formula we derive a new modified weighted shifted grunwald letnikov wsgl formula by introducing appropriate correction terms we then apply one special case of the modified wsgl formula to solve multi term fractional ordinary and partial differential equations and we prove the linear stability and second order convergence for both smooth and non smooth solutions when the regularity of the solutions is known we show theoretically and numerically that numerical solutions with good accuracy can be obtained with only a few correction terms moreover the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions numerical simulations verify the theoretical results and demonstrate that the new wsgl formula leads to better performance compared to other known numerical approximations with similar resolution

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

stable evaluations of fractional derivative of the muntz legendre polynomials and application to fractional differential equations abstract the aim of this paper is to present efficient and stable methods to compute caputo fractional derivative cfd of the muntz legendre polynomials based on three term recurrence relations and gauss jacobi quadrature rules this approach with collocation method at chebyshev gauss lobatto points has been applied for solving linear and nonlinear fractional multi order differential equations fdes described in caputo sense the main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations in this work for the first time we present the new rates of convergence for projection error which are more accurate than the rate presented by shen and wang in shen and wang 2016 moreover we present convergence rate for spectral collocation method for linear fdes with initial value on a finite interval and endpoint singularities also we propose an error analysis for jacobi gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper finally the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples

wavelet approximation scheme for distributed order fractional differential equations abstract this paper is concerned with the study of wavelet approximation scheme based on legendre and chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations the operational matrix for distributed order fractional differential operator is derived for legendre and chebyshev wavelets basis furthermore the obtained operational matrix along with gauss legendre quadrature formula and standard tau method are utilized to reduce the distributed order linear differential equations into the system of linear algebraic equations for the better understanding of the method numerical algorithms are also provided for the considered problems in order to verify the desired accuracy of the proposed method five test examples are included and numerical experiments confirm the theoretical results and illustrate applicability and efficiency of the proposed method also the convergence analysis error bounds error estimate and numerical stability of presented method via legendre and chebyshev wavelets are investigated moreover comparison of the numerical results obtained by the proposed approach is provided with the results of existing method

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

block pulse operational matrix method for solving fractional partial differential equation in this paper we first introduce two dimensional block pulse functions and the block pulse operational matrices of the fractional order integration also the block pulse operational matrices of the fractional order differentiation are obtained then we present a computational method based on the above results for solving a class of fractional partial differential equations transforming the initial equation into a sylvester equation the error analysis of the method is given the method is computationally attractive and applications are demonstrated by some numerical examples moreover comparing the methodology with the known technique shows that our approach is more efficient and more convenient

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

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

iterated solutions of linear operator equations with the tau method the tau method produces polynomial approximations of solutions of differential equations the purpose of this paper is i to extend the recursive formulation of this method to general linear operator equations defined in a separable hilbert space and ii to develop an iterative refinement procedure which improves on the accuracy of tau approximations applications to fredholm integral equations demonstrate the effectiveness of this technique

numerical solutions for fractional differential equations by tau collocation method the main purpose of this paper is to provide an efficient numerical approach for multi order fractional differential equations based on a tau collocation method to do this multi order fractional differential equations transformed into a system of nonlinear algebraic equations in matrix form thus by solving this system unknown coefficients are obtained the fractional derivatives are described in the caputo sense the rate of convergence for the proposed method is established in the l w p norm some numerical example is also provided to illustrate our results the results reveal that the method is very effective and simple

further results on generalized multiple fractional part integrals for complex values in this paper the following multiple fractional part integrals i n 1 2 n 0 1 n j 1 n x j j s n 1 d x 1 d x n and j n 0 1 n s n s n 1 d x 1 d x n are studied for positive integer n and complex values of j j 1 2 n where u denotes the fractional part of u r s denotes the real part of s and s n x 1 x 2 x n it is proved that i 1 can be represented as a linear combination of the riemann zeta function the beta function and euler s constant as r 1 moreover i n 1 2 n can be expressed by i n 1 1 2 n 1 the beta function and the incomplete beta function for n 2 3 in addition the recurrence formula of j n n 2 3 is established and j n can be expressed by i 1 logarithmic function and some binomial coefficients

a generalized fractional order legendre wavelet tau method for solving fractional differential equations abstract a new fractional order wavelet basis as generalization of the classical legendre wavelet is defined the operational matrices both for derivative and fractional derivative in the sense of caputo for this fractional order wavelet are derived then a numerical scheme based on these operational matrices and the typical tau method is proposed for solving some nonlinear fractional differential equations illustrative examples show that the present wavelet tau method is numerically effective and convenient for solving fractional differential equations moreover the obtained results confirm that in comparison with the classical legendre wavelet method the fractional order wavelet basis is more efficient and accurate for solving fractional differential equations error analysis and convergence of the proposed wavelet method are also provided

solving a multi order fractional differential equation using adomian decomposition abstract an algorithm has been developed to convert the multi order fractional differential equation d y t f t y t d 1 y t d n y t y k 0 c k k 0 m where m d denotes caputo fractional derivative of order into a system of fractional differential equations further adomian decomposition method has been employed to solve the system of fractional differential equations some illustrative examples are presented

a class of shifted high order numerical methods for the fractional mobile immobile transport equations abstract in this article we apply the generalized bdf2 to the fractional mobile immobile transport equations for its temporal discretization and the finite element method in the spatial direction to derive the stability estimates and obtain the optimal error convergence rate some properties of the convolution weights are proved based on which we show the scheme is unconditionally stable with an error of o 2 h r 1 where and h represent the temporal and spatial mesh size respectively we conduct exhaustive numerical tests to further confirm our theoretical analysis and to overcome the initial singularity of the time fractional derivative we adopt the generalized bdf2 with starting parts in accordance with the framework of the shifted convolution quadrature scq

an effective scheme for solving system of fractional volterra fredholm integro differential equations based on the muntz legendre wavelets abstract in this study a class of wavelet techniques is used for finding approximate solutions of systems of fractional integro differential volterra fredholm fidvf equations based on the muntz legendre wavelets mlw for the suggested method operational matrices of the riemann liouville fractional rlf integral and caputo fractional cf derivative operators are obtained and used for converting the system of the integral equations into a system of linear or nonlinear algebraic equations using the lipschitz s condition for multivariate functions and the fixed point theorem the existence and uniqueness of the solution are shown and also convergence stability and error bound of the solution in interval 0 1 are investigated in this work at the end three examples are indicated and the results of the proposed method are compared with the first and second kind wavelet chebyshev methods

numerical solutions for fractional kdv burgers equation by adomian decomposition method abstract based on the symbolic computation system maple adomian decomposition method developed for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional kdv burgers equation the fractional derivatives are described in the caputo sense according to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional kdv burgers equation with time and space fractional derivatives finally the solutions of our model equation are calculated in the form of convergent series with easily computable components

fractional order legendre laguerre functions and their applications in fractional partial differential equations abstract in this paper we consider a new fractional function based on legendre and laguerre polynomials for solving a class of linear and nonlinear time space fractional partial differential equations with variable coefficients the concept of the fractional derivative is utilized in the caputo sense the idea of solving these problems is based on operational and pseudo operational matrices of integer and fractional order integration with collocation method we convert the problem to a system of algebraic equations by applying the operational matrices pseudo operational matrices and collocation method also we calculate the upper bound for the error of integral operational matrix of the fractional order we illustrated the efficiency and the applicability of the approach by considering several numerical examples in the format of table and graph we also describe the physical application of some examples

muntz legendre spectral collocation method for solving delay fractional optimal control problems abstract in this paper a numerical method is applied for solving delay fractional optimal control problems dfocps the fractional derivative is described in the caputo sense since the fractional derivative of muntz polynomials can be expressed in terms of the same polynomials those polynomials can accurately represent properties of fractional calculus in some situations such as in the frequency response based analysis of control systems containing a time delay it is necessary to substitute exponential function with an approximation in the form of a rational function the most common approximation is the pade approximation at the first step using pade approximation the delay problem is transformed to a non delay problem next using the operational matrix of the fractional derivative of muntz polynomials and pseudospectral method fractional optimal control problem focp is reduced to a nonlinear programming problem some numerical examples are given to illustrate the effectiveness of the proposed scheme

an efficient computational method for solving system of nonlinear generalized abel integral equations arising in astrophysics abstract in this paper we aim to solve nonlinear system of generalized abel integral equations arising in astrophysics the suggested approach is operational matrix technique by using legendre scaling functions as a basis convergence analysis of the suggested technique is provided numerical experiments are performed to show the effectiveness of the proposed scheme the results are shown through figures and tables

generalized legendre polynomials abstract orthogonal polynomials have been extensively investigated in this paper we construct a sequence of orthogonal dirichlet polynomials which generalize legendre polynomials we show that our new system satisfies a generalized rodrigues formula we also extend a well known result concerning the distribution of the zeros of legendre polynomials

on accurate product integration rules for linear fractional differential equations this paper addresses the numerical solution of linear fractional differential equations with a forcing term competitive and highly accurate product integration rules are derived by starting from an equivalent formulation in terms of a volterra integral equation with a generalized mittag leffler function in the kernel the error analysis is reported and aspects related to the computational complexity are treated numerical tests confirming the theoretical findings are presented

conformal geometry of flows in n dimensions flows generated by smooth vector fields are considered from the point of view of conformal geometry a flow is defined to be conformally geodesic if it preserves the distribution of vector spaces orthogonal to the lines of the flow it is shear free if moreover it preserves the conformal structure on these vector spaces differential equations characterizing such flows are derived for the general case of an n dimensional conformal space of arbitrary signature in the special case of null flows in spacetime one obtains a refined version of the theorem connecting null solutions of maxwell s equations with null flows that are geodesic and shear free

a tau method for the one dimensional parabolic inverse problem subject to temperature overspecification in this work an approximational technique based on shifted legendre tau ideas is presented for the one dimensional parabolic inverse problem with a control parameter the method consists of expanding the required approximate solution as the elements of a shifted legendre polynomial using the operational matrices we reduce the problem to a set of algebraic equations a numerical example is included to demonstrate the validity and applicability of the technique and a comparison is made with existing results the method is easy to implement and produces very accurate results

numerical solutions of the generalized kuramoto sivashinsky equation by chebyshev spectral collocation methods chebyshev spectral collocation methods known as el gendi method s e el gendi chebyshev solution of differential integral and integro differential equations comput j 12 1969 282 287 b mihaila i mihaila numerical approximation using chebyshev polynomial expansions el gendi s method revisited j phys a 35 2002 731 746 are extended to deal with the generalized kuramoto sivashinsky equation the problem is reduced to a system of ordinary differential equations that are solved by combinations of backward differential formula and appropriate explicit schemes implicit explicit bdf methods g akrivis y s smyrlis implicit explicit bdf methods for the kuramoto sivashinsky equation appl numer math 51 2004 151 169 good numerical results have been obtained and compared with the exact solutions

fractional differential systems a fuzzy solution based on operational matrix of shifted chebyshev polynomials and its applications in this paper a new formula of fuzzy caputo fractional order derivatives 0 v leq 1 in terms of shifted chebyshev polynomials is derived the proposed approach introduces a shifted chebyshev operational matrix in combination with a shifted chebyshev tau technique for the numerical solution of linear fuzzy fractional order differential equations the main advantage of the proposed approach is that it simplifies the problem alike in solving a system of fuzzy algebraic linear equations an approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors is derived furthermore we also discuss the convergence of the proposed method from the fuzzy perspective experimentally we show the strength of the proposed method in solving a variety of fractional differential equation models under uncertainty encountered in engineering and physical phenomena i e viscoelasticity oscillations and resistor capacitor rc circuits comparisons are also made with solutions obtained by the laguerre polynomials and the fractional euler method

document reranking by term distribution and maximal marginal relevance for chinese information retrieval in this paper we propose a document reranking method for chinese information retrieval the method is based on a term weighting scheme which integrates local and global distribution of terms as well as document frequency document positions and term length the weight scheme allows randomly setting a larger portion of the retrieved documents as relevance feedback and lifts off the worry that very fewer relevant documents appear in top retrieved documents it also helps to improve the performance of maximal marginal relevance mmr in document reranking the method was evaluated by map mean average precision a recall oriented measure significance tests showed that our method can get significant improvement against standard baselines and outperform relevant methods consistently

on the existence of nonclassical symmetries of partial differential equations relations between determining equations for nonclassical symmetries of bluman and cole and integrability conditions of differential equations supplemented by invariance conditions are studied it is shown that every system of equations with p independent variables admits p dimensional groups of such symmetries nonclassical symmetries of the eikonal equation in a curved space are investigated they are related to shear free congruences of null geodesics

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

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

numerical algorithm for calculating the generalized mittag leffler function a numerical algorithm for calculating the generalized mittag leffler mathrm e alpha beta z function for arbitrary complex argument z and real parameters alpha 0 and beta in mathbb r is presented the algorithm uses the taylor series the exponentially improved asymptotic series and integral representations to obtain optimal stability and accuracy of the algorithm special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm

numerical algorithm based on adomian decomposition for fractional differential equations in this paper a novel algorithm based on adomian decomposition for fractional differential equations is proposed comparing the present method with the fractional adams method we use this derived computational method to find a smaller efficient dimension such that the fractional lorenz equation is chaotic we also apply this new method to the time fractional burgers equation with initial and boundary value conditions numerical results and computer graphics show that the constructed numerical is efficient

solving a system of nonlinear fractional differential equations using adomian decomposition adomian decomposition method has been employed to obtain solutions of a system of nonlinear fractional differential equations dxiyi x ni x y1 yn yi k 0 cki 0 k i 1 i n and dxi denotes caputo fractional derivative some examples are solved as illustrations using symbolic computation

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

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

numerical solution of fractional differential equations using the generalized block pulse operational matrix the riemann liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration also the generalized block pulse operational matrices of differentiation are derived based on the above results we propose a way to solve the fractional differential equations the method is computationally attractive and applications are demonstrated through illustrative examples

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

a novel legendre operational matrix for distributed order fractional differential equations abstract in this paper for the first time the shifted legendre operational matrix of distributed order fractional derivative has been derived also this new operational matrix is used together with tau method for approximation of solutions of linear distributed order fractional differential equations and diffusion equations with distributed order in time moreover eight numerical examples are implemented in order to show the validity and reliability of the suggested methods

a new method based on legendre polynomials for solutions of the fractional two dimensional heat conduction equation abstract in this paper we develop a new scheme for numerical solutions of the fractional two dimensional heat conduction equation on a rectangular plane our main aim is to generalize the legendre operational matrices of derivatives and integrals to the three dimensional case by the use of these operational matrices we reduce the corresponding fractional order partial differential equations to a system of easily solvable algebraic equations the method is applied to solve several problems the results we obtain are compared with the exact solutions and we find that the error is negligible

theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear lane emden type equations abstract in this article a new method is generated to solve nonlinear lane emden type equations using legendre hermite and laguerre wavelets we are interested to note that these wavelets will give same solutions with good accuracy theorems on convergence analysis are stated and proved on the spaces which are created by legendre hermite and laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis the main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients a very high level of accuracy reflects the reliability of this scheme for such problems

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques

a wavelet method to solve nonlinear variable order time fractional 2d klein gordon equation abstract in this study we introduce the nonlinear variable order time fractional two dimensional 2d klein gordon equation by using the concept of variable order fractional derivatives the variable order fractional derivative operator is defined in the caputo type due to the useful properties of wavelets we propose an efficient semi discrete method based on the 2d legendre wavelets lws to numerically solve this equation in fact according to the proposed method the variable order time fractional derivative should be discretized in the first stage and then the solution of the problem expanded in terms of the 2d lws however the main objective of this study is to illustrate that the 2d lws can be a useful tool for solving the nonlinear variable order time fractional 2d klein gordon equation stability analysis of the presented method is investigated theoretically and numerically moreover the applicability and accuracy of the method are investigated by solving some numerical examples numerical results confirm the spectral accuracy of the established method