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

superfast solution of linear convolutional volterra equations using qtt approximation we address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular toeplitz matrices and the recently proposed qtt format the inverses of such matrices can be computed by the divide and conquer and modified bini s algorithms for which we present the versions with the qtt approximation we also present an efficient formula for the shift of vectors given in qtt format which is used in the divide and conquer algorithm as a result we reduce the complexity of inversion from the fast fourier level o n log n to the speed of superfast fourier transform i e o log 2 n the results of the paper are illustrated by numerical examples

a numerical method for solving a nonlinear 2 d optimal control problem with the classical diffusion equation this paper presents a numerical solution for solving a nonlinear 2 d optimal control problem 2dop the performance index of a nonlinear 2dop is described with a state and a control function furthermore dynamic constraint of the system is given by a classical diffusion equation it is preferred to use the ritz method for finding the numerical solution of the problem the method is based upon the legendre polynomial basis by using this method the given optimisation nonlinear 2dop reduces to the problem of solving a system of algebraic equations the benefit of the method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed moreover compared with the eigenfunction method the satisfactory results are obtained only in a small number of polynomials order this numerical approach is applicable and effective for such a kind of nonlinear 2dop the convergence of the method is extensively discussed and finally two illustrative example

a new numerical method for solving high order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional high order differential equations with variable coefficients it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results in particular fractional problems of the fourth sixth and eighth orders are considered the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of eigenvalue problems

a new spectral method for a class of linear boundary value problems in this paper we use bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear boundary value problems the proposed method has been applied to several bvps with dirichlet neumann and mixed types boundary conditions numerical examples provide favorable comparisons with other existing methods

leitmann s direct method for fractional optimization problems abstract based on a method introduced by leitmann g leitmann a note on absolute extrema of certain integrals int j non linear mech 2 1967 55 59 we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control

two routes to chaos in the fractional lorenz system with dimension continuously varying abstract the object of this paper is to reveal the relation between dynamics of the fractional system and its dimension defined as a sum of the orders of all involved derivatives we take the fractional lorenz system as example and regard one or three of its orders as bifurcation parameters in this framework we compute the corresponding bifurcation diagrams via an optimal poincare section technique developed by us and find there exist two routes to chaos when its dimension increases from some values to 3 one is the process of cascaded period doubling bifurcations and the other is a crisis boundary crisis which occurs in the evolution of chaotic transient behavior we would like to point out that our investigation is the first to find out that a fractional differential equations fdes system can evolve into chaos by the crisis furthermore we observe rich dynamical phenomena in these processes such as two stage cascaded period doubling bifurcations chaotic transients and the transition from coexistence of three attractors to mono existence of a chaotic attractor these are new and interesting findings for fdes systems which to our knowledge have not been described before

discretization of fractional order differentiator over paley wiener space the paley wiener space consists of functions whose fourier transform is compactly supported in the frequency domain in the context of signal processing such functions are also known as bandlimited signals which represent a large class of signals in signal processing an analog fractional order differentiator is representable by way of the cauchy integral formula special functions as well as the fourier laplace transformer while a digital differentiator of fractional order can be obtained through direct or indirect discretization techniques in this paper we present the design of a finite impulse response fir filter that discretizes the fractional order differentiator over functions in paley wiener space the proposed fir model has some meritorious properties that are preferred in applications the filter coefficients are independent of the signal samples it is capable of interpolating or extrapolating at an arbitrary point in the sampling domain it is adaptive to uniform or non uniform sampling scenarios we present explicit formulas on the matrices that lead to computing the filter coefficients a closed form formula on the error of approximation is derived to demonstrate the accuracy of the proposed discretization model of fractional derivatives numerical results also show that the proposed method is more computationally efficient than the well known methods such as the grunwald letnikov approximation

an approximation to solutions of linear ode by cubic interpolation we present a method of integration for non autonomous non homogeneous systems of linear ordinary differential equation ode which is based in both the cubic polynomial segmentary interpolation and the minimal square method this method is valid for nonhomogeneous ordinary linear second order differential equations in the neighborhood of regular and singular regular points we illustrate the method with the mathieu and bessel equations and two other equations that arise in the study of quantum systems with axial symmetry which are versions of the spheroidal wave equation

effective construction of a class of positive operators in hilbert space which do not admit triangular factorization to introduce the main notions of triangular factorization consider a hilbert space l 2 a b infty leq a b leq infty the orthogonal projectors p xi in l 2 a b are defined by the relations p xi f x f x for a x xi p xi f x 0 for xi x b f epsilon l 2 a b denote the identity operator by i

approximating the basset force by optimizing the method of van hinsberg et al in this work we put the method proposed by van hinsberg et al 29 to the test highlighting its accuracy and efficiency in a sequence of benchmarks of increasing complexity furthermore we explore the possibility of systematizing the way in which the method s free parameters are determined by generalizing the optimization problem that was considered originally finally we provide a list of worked out values ready for implementation in large scale particle laden flow simulations

fractional dynamics of populations abstract nature often presents complex dynamics which cannot be explained by means of ordinary models in this paper we establish an approach to certain fractional dynamic systems using only deterministic arguments the behavior of the trajectories of fractional non linear autonomous systems around the corresponding critical points in the phase space is studied in this work we arrive to several interesting conclusions for example we conclude that the order of fractional derivation is an excellent controller of the velocity how the mentioned trajectories approach to or away from the critical point such property could contribute to faithfully represent the anomalous reality of the competition among some species in cellular populations as cancer or hiv we use classical models which describe dynamics of certain populations in competition to give a justification of the possible interest of the corresponding fractional models in biological areas of research

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

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

differintegral interpolation from a bandlimited signal s samples the whittaker shannon cardinal series dictates that any l 2 bandlimited signal is defined everywhere by its sufficiently closely spaced sample values this paper derives those interpolation functions necessary for direct evaluation of such a signal s derivatives integrals and fractional derivatives directly from the sample values generation and recursion formulas for these interpolation functions are presented

a real distinct poles rational approximation of generalized mittag leffler functions and their inverses applications to fractional calculus abstract the generalized mittag leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives respectively however their computational complexities have made them difficult to deal with numerically we propose a real distinct pole rational approximation of the generalized mittag leffler function under some mild conditions this approximation is proven and empirically shown to be l acceptable due to the complete monotonicity property of the mittag leffler function we derive a rational approximation for the inverse generalized mittag leffler function these approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order several applications are presented such as complementary error function solution of fractional differential equations and the ultraslow diffusion model using the structural derivative

matched filtering for generalized stationary processes the methods for solving optimal filtering problems in the case of the classical stationary processes have been well known since the late 1940s practice often gives rise to what is not a classical stationary process but a generalized one and white noise is one simple example hence it is of interest to describe the system action on the generalized stationary processes and then to carry over filtering methods to them for arbitrary generalized stochastic processes this seems to be a challenging problem in this correspondence we identify a rather general class of s sub j generalized stationary processes for which the desired extension can be done for matched filters this class can be considered as a model of colored noise and it is wide enough to include white noise positive frequencies white noise as well as certain generalized processes occurring in practice namely when the smoothing effect gives rise to the situation in which the distribution of probabilities may not exist at some time instances one advantage of the suggested model is that it connects optimal filter design with inverting of integral operators the methods for the latter can be found in the extensive literature

spline algorithms for the hartree fock equation for the helium ground state spline algorithms are evaluated for the non linear integro differential equation describing the hartree fock approximation for the he 1s21s ground state the error in the energy decreases as h2k 2 where h is a grid parameter and k is the order of the spline it is shown that for higher order splines the method is fast and accurate and contrary to the conclusion reached by altenberger siczek and gilbert that spline methods are suitable for scf atomic structure calculations accuracy and timing studies are presented as well as comparisons with other accurate procedures

adjoint state method for fractional diffusion parameter identification fractional partial differential equations provide models for sub diffusion among which the fractal mobile immobile model fmim is often used to represent solute transport in complex media the fmim involves four parameters among which we have the order of an integro differential operator that accounts for the possibility for solutes to be sequestered during very long times to guess fmim parameters from experiments an accurate method consists in optimizing an objective function that measures how much model solutions deviate from data we show that solving an adjoint problem helps accurate computing of the gradient of such an objective function with respect to the parameters we illustrate the method by applying it on experimental data issued from tracing tests in porous media

integral fractional pseudospectral methods for solving fractional optimal control problems the main purpose of this work is to provide a unified framework and develop integral fractional pseudospectral methods for solving fractional optimal control problems as a generalization of conventional pseudospectral integration matrices fractional pseudospectral integration matrices fpims and their efficient and stable computation are the key to our new approach in order to achieve this goal we take a special and smart way to compute fpims the essential idea is to transform the fractional integral of lagrange interpolating polynomials through a change of variables into their jacobi weighted integral which can be calculated exactly using the jacobi gauss quadrature this together with the stable barycentric representation of lagrange interpolating polynomials and the explicit barycentric weights for the gauss flipped radau and radau type points corresponding to the jacobi polynomials leads to an exact efficient and stable scheme to compute fpims even at millions of jacobi type points numerical results on two benchmark optimal control problems demonstrate the performance of the proposed pseudospectral methods

new quadrature approach based on operational matrix for solving a class of fractional variational problems this manuscript presents a new numerical approach to approximate the solution of a class of fractional variational problems the presented approach is consisting of using the shifted legendre orthonormal polynomials as basis functions of the operational matrix of fractional derivatives described in the caputo sense and that of fractional integrals described in the sense of riemann liouville with the help of the legendre gauss quadrature formula together with the lagrange multipliers method for converting such fractional variational problems into easier problems that consist of solving an algebraic system in the unknown coefficients the convergence of the proposed method is analyzed finally in order to demonstrate the accuracy of the present method some test problems are introduced with their approximate solutions and comparisons with other numerical approaches

stability preservation analysis for frequency based methods in numerical simulation of fractional order systems in this paper the frequency domain based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation first the stability boundary curve is exactly determined for these methods then this boundary is analyzed and compared with an accurate ideal boundary in different frequency ranges also the critical regions in which the stability does not preserve are determined finally the analytical achievements are confirmed via some numerical illustrations

convergence of the grunwald letnikov scheme for time fractional diffusion using bivariate generating functions we prove convergence of the grunwald letnikov difference scheme for the fractional diffusion equation in one space dimension with and without central linear drift in the fourier laplace domain as the space and time steps tend to zero in a well scaled way this implies convergence in distribution weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle the difference schemes allow also interpretation as discrete random walks for fractional diffusion with central linear drift we show that in the fourier laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation

numerical solution of kdv equation using modified bernstein polynomials abstract here we present an algorithm for approximating numerical solution of korteweg de vries kdv equation in a modified b polynomial basis a set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable galerkin method is used to determine the expansion coefficients to construct initial trial functions for the time variable the system of equations is solved using fourth order runge kutta method the accuracy of the solutions is dependent on the size of the b polynomial basis set we have presented our numerical result with an exact analytical result excellent agreement is found between exact and approximate solutions this procedure has a potential to be used in more complex system of differential equations where no exact solution is available

numerical solutions of eighth order bvp by the galerkin residual technique with bernstein and legendre polynomials in this paper galerkin method with bernstein and legendre polynomials as basis functions is used for the numerical solutions of eighth order linear and nonlinear differential equations for two different cases of boundary conditions in this method the basis functions are transformed into a new set of basis functions which vanish at the boundary where the essential types of boundary conditions are defined and a matrix formulation is derived for solving the eighth order boundary value problems bvps the numerical results obtained by the proposed method reflect that the performance of the present method is reliable more efficient and converges monotonically to the exact solutions

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

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

regular article integration by cell algorithm for slater integrals in a spline basis an algorithm for evaluating slater integrals in a b spline basis is introduced based on the piecewise property of the b splines the algorithm divides the two dimensional r 1 r 2 region into a number of rectangular cells according to our chosen grid and implements the two dimensional integration over each individual cell using gaussian quadrature over the off diagonal cells the integrands are separable so that each two dimensional cell integral is reduced to a product of two one dimensional integrals furthermore the scaling invariance of the b splines in the logarithmic region of the chosen grid is fully exploited such that only some of the cell integrations need to be implemented the values of given slater integrals are obtained by assembling the cell integrals this algorithm significantly improves the efficiency and accuracy of the traditional method that relies on the solution of differential equations and renders the b spline method more effective when applied to multi electron atomic systems

partition functions of n 2 2 gauge theories on s 2 and vortices we apply localization techniques to compute the partition function of a two dimensional mathcal n 2 2 r symmetric theory of vector and chiral multiplets on s2 the path integral reduces to a sum over topological sectors of a matrix integral over the cartan subalgebra of the gauge group for gauge theories which would be completely higgsed in the presence of a fayet iliopoulos term in flat space the path integral alternatively reduces to the product of a vortex times an antivortex partition functions weighted by semiclassical factors and summed over isolated points on the higgs branch for applications we evaluate the partition function for some u n gauge theories showing equality of the path integrals for theories conjectured to be dual by hori and tong and deriving new expressions for vortex partition functions

modeling and numerical analysis of fractional order bloch equations this paper deals with the bloch equations which are a set of macroscopic equations that are used for modeling of nuclear magnetization as a function of time these equations were introduced by felix bloch in 1946 and they are used for a description of the nuclear magnetic resonance nmr this physical phenomenon is used in medicine chemistry physics and engineering to study complex material fractional order generalization of the bloch equations was presented by richard magin et al in 2008 as an opportunity to extend their use to describe a wider range of experimental situations involving heterogeneous porous or composite materials in this paper we describe numerical and simulation models created for matlab simulink of the classical and the fractional order bloch equations the behaviour and stability analysis of the bloch equations are presented as well

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

connection formulae for painleve v functions ii the d function bose gas problem abstract we continue our study of those painleve v equations which admit a one parameter family of solutions analytic at the origin and solve the global problem of connecting the 2 parameter expansion about t 0 with the 2 parameter expansion valid for t i a special case of our results is applied to the study of the one particle reduced density matrix of the one dimensional gas of impenetrable bosons we also use our results to obtain a 2 parameter connection formula for a class of painleve iii functions

numerical solution of some classes of integral equations using bernstein polynomials this paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using bernstein polynomials as basis the integral equations considered are fredholm integral equations of second kind a simple hypersingular integral equation and a hypersingular integral equation of second kind the method is explained with illustrative examples also the convergence of the method is established rigorously for each class of integral equations considered here

solutions of differential equations in a bernstein polynomial basis an algorithm for approximating solutions to differential equations in a modified new bernstein polynomial basis is introduced the algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the galerkin method to determine the expansion coefficients to construct a solution matrix formulation is used throughout the entire procedure however accuracy and efficiency are dependent on the size of the set of bernstein polynomials and the procedure is much simpler compared to the piecewise b spline method for solving differential equations a recursive definition of the bernstein polynomials and their derivatives are also presented the current procedure is implemented to solve three linear equations and one nonlinear equation and excellent agreement is found between the exact and approximate solutions in addition the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques this procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations

on the convergence of krylov subspace methods for matrix mittag leffler functions in this paper we analyze the convergence of some commonly used krylov subspace methods for computing the action of matrix mittag leffler functions as is well known such functions find application in the solution of fractional differential equations we illustrate the theoretical results by some numerical experiments

fractional derivative models for viscoelastic materials at finite deformations abstract fractional derivative models which are expressed by combining standard dashpots fractional dashpots and elastic springs in series or parallel are often utilized to account for the behaviors for viscoelastic materials even with the models extended to finite deformation the precise definition of objective fractional derivative remains challenging the proposed fractional derivative model is expressed by the combination of an elastic spring in series with two parallel fractional dashpots we extend the fractional derivative model to finite deformation through a new approach without defining an objective fractional derivative and assuming the decomposition of the deformation rate into the elastic and inelastic parts this proposed model can be reduced to the maxwell model for finite deformation such reduction results in a model that stands in between the two existing maxwell models in which the objective rate of the cauchy stress is taken as the material corotational rate and the relative corotational rate respectively the proposed model is applied to the simple shear deformation

legendre spectral collocation method for the solution of the model describing biological species living together in this article numerical techniques have been developed to solve a system of two nonlinear integro differential equations which arises in biological model this model has been solved by legendre spectral collocation method and bernstein polynomials the present methods reduce this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by newton s method also the results obtained by present methods have been compared with that of by some existing methods available in literature some illustrative examples have been discussed to show the applicability and accuracy of the present methods

bernstein series solution of a class of linear integro differential equations with weakly singular kernel abstract in this study a new collocation method based on the bernstein polynomials is introduced for the approximate solution of a class of linear volterra integro differential equations with weakly singular kernel if the exact solution is polynomial then the exact solution can be obtained if the exact solution is not a polynomial then an accurate solution can be obtained with a combination of choice in the number of nodes and the number of digits in the solver in addition the method is presented with error and stability analysis

a bethe ansatz type formula for the superconformal index inspired by recent work by closset kim and willett we derive a new formula for the superconformal or supersymmetric index of 4d mathcal n 1 theories such a formula is a finite sum over the solution set of certain transcendental equations that we dub bethe ansatz equations of a function evaluated at those solutions

use of modified bernstein polynomials to solve kdv burgers equation numerically abstract numerical solution of korteweg de veries burgers kdvb equation is presented using modified bernstein polynomials b polynomials over the spatial domain b polynomials are used to expand the desired solution requiring discretization with only the time variable galerkin method is used to determine the expansion coefficients to construct initial trial functions we use fourth order runge kutta method to solve the system of equations for the time variable the accuracy of the solutions is dependent on the size of the b polynomial basis set numerical results obtained using this method are compared with existing analytical results excellent agreement is found between the exact solution and approximate solution obtained by this method

increasing the efficiency of shooting methods for terminal value problems of fractional order shooting methods are a well established tool for the numerical solution of terminal value problems of fractional order however they can be computationally quite expensive because of their iterative nature in which a each single iteration may be costly and b the number of iterations can be large in this paper we propose algorithmic strategies for improving the efficiency of such methods our strategies are aimed at simultaneously reducing the cost of each iteration and reducing the number of required iterations

solving system of volterra fredholm integral equations with bernstein polynomials and hybrid bernstein block pulse functions this work approximates the unknown functions based on the bernstein polynomials and hybrid bernstein block pulse functions in conjunction with the collocation method for the numerical solution of system of fredholm volterra integral equations in both methods the system of integral equations is approximated by the gauss quadrature formula with respect to the legendre weight function the proposed methods reduce the system of integral equations to a system of algebraic equations that can be easily solved by any usual numerical methods moreover the convergence analysis of these algorithms will be shown by preparing some theorems numerical experiments are presented to show the superiority and efficiency of proposed methods in comparison with some other well known methods

on riemann and caputo fractional differences in this paper we define left and right caputo fractional sums and differences study some of their properties and then relate them to riemann liouville ones studied before by miller k s and ross b atici f m and eloe p w abdeljawad t and baleanu d and a few others also the discrete version of the q operator is used to relate the left and right caputo fractional differences a caputo fractional difference equation is solved the solution proposes discrete versions of mittag leffler functions

delayed feedback control of fractional order chaotic systems we study the possibility to stabilize unstable steady states and unstable periodic orbits in chaotic fractional order dynamical systems by the time delayed feedback method by performing a linear stability analysis we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parameterized by the feedback gain and the time delay an insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues we demonstrate that the method can also stabilize unstable periodic orbits for a suitable choice of the feedback gain providing that the time delay is chosen to coincide with the period of the target orbit in addition it is shown numerically that delayed feedback control with a sinusoidally modulated time delay significantly enlarges the stability region of steady states in comparison to the classical time delayed feedback scheme with a constant delay

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

tempered fractional calculus fractional derivatives and integrals are convolutions with a power law multiplying by an exponential factor leads to tempered fractional derivatives and integrals tempered fractional diffusion equations where the usual second derivative in space is replaced by a tempered fractional derivative govern the limits of random walk models with an exponentially tempered power law jump distribution the limiting tempered stable probability densities exhibit semi heavy tails which are commonly observed in finance tempered power law waiting times lead to tempered fractional time derivatives which have proven useful in geophysics the tempered fractional derivative or integral of a brownian motion called a tempered fractional brownian motion can exhibit semi long range dependence the increments of this process called tempered fractional gaussian noise provide a useful new stochastic model for wind speed data a tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations and it also provides a useful new correlation model in time series

numerical solution of 2d fractional optimal control problems by the spectral method along with bernstein operational matrix abstractthis paper presents an approximate method to solve a class of two dimensional fractional optimal control problems with nonlinear dynamical system to implement the new method by considering the initial boundary conditions the unknown state and control functions are approximated by the bernstein polynomials b polynomials basis using spectral ritz method then the problem is reduced to an unconstrained nonlinear optimisation problem meanwhile to reduce computational complexity a new fractional operational bernstein matrix generalised on an arbitrary interval is constructed and applied the choice of polynomial basis functions along with the ritz method provides good flexibility in which all the given initial and boundary conditions are imposed at last we extensively argue the convergence of the new method and several illustrative test problems are added to demonstrate the applicability and effectiveness of the new procedure moreover our achievements are compared with the previous results

a tau approach for solution of the space fractional diffusion equation fractional differentials provide more accurate models of systems under consideration in this paper approximation techniques based on the shifted legendre tau idea are presented to solve a class of initial boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain the fractional derivatives are described in the caputo sense the technique is derived by expanding the required approximate solution as the elements of shifted legendre polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from the computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use

solving generalized pantograph equations by shifted orthonormal bernstein polynomials abstract in this paper we introduce shifted orthonormal bernstein polynomials sobps and derive the operational matrices of integration and delays for these polynomials then we apply them to convert the pantograph equations to a system of linear equations an important property of this method is that the condition number of the coefficient matrix of the system is small which confirms that our method is stable error analysis and comparison with other methods are given to confirm the validity efficiency and applicability of the proposed method

nonlinear reaction with fractional dynamics in this paper we consider a system of three fractional differential equations describing a nonlinear reaction our analysis includes both analytical technique and numerical simulation this allows us to control the efficiency of the numerical method we combine the numerical approximation of fractional integral with finding zeros of a function of one variable the variable gives a desired solution the solution has a peak its position in time depends on the order of fractional derivative the two other solutions of this system have a completely monotonic character

the elliptic gamma function and sl 3 z z3 abstract the elliptic gamma function is a generalization of the euler gamma function and is associated to an elliptic curve its trigonometric and rational degenerations are the jackson q gamma function and the euler gamma function respectively the elliptic gamma function appears in baxter s formula for the free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qkzb equations in this paper the properties of this function are studied in particular we show that elliptic gamma functions are generalizations of automorphic forms of g sl 3 z z 3 associated to a non trivial class in h3 g z

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

connection formulae for painleve v functions we study those painleve v equations which admit a one parameter family of solutions analytic at the origin and solve the global problem of connecting the two parameter asymptotic expansion at t with the two parameter expansion about t 0 this connection is the one needed to study the correlation functions of the transverse ising chain at the critical field strength from these connection formulae we derive a one parameter connection formula for the general painleve ii equation and a two parameter connection formula for a class of painleve iii equations

systems based decomposition schemes for the approximate solution of multi term fractional differential equations we give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi term fractional differential equations using the caputo form of the fractional derivative the schemes we compare are based on conversion of the original problem into a system of equations we review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation

fractional diffusion equations by the kansa method this study makes the first attempt to apply the kansa method in the solution of the time fractional diffusion equations in which the multiquadrics and thin plate spline serve as the radial basis function in the discretization formulation the finite difference scheme and the kansa method are respectively used to discretize time fractional derivative and spatial derivative terms the numerical solutions of one and two dimensional cases are presented and discussed which agree well with the corresponding analytical solution

numerical solution of a singular integro differential equation this paper presents a method based on polynomial approximation using bernstein polynomial basis to obtain approximate numerical solution of a singular integro differential equation with cauchy kernel the numerical results obtained by the present method compares favorably with those obtained by various galerkin methods earlier in the literature also the convergence of the method is established rigorously for the studied integro differential equation

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order