reconstructing state spaces from multivariate data using variable delays we study two methods for constructing a nonuniform embedding for multivariate data a nonuniform embedding is a state space reconstruction which is more flexible than the common delay coordinates with fixed delays since it contains variable delays using these methods we can extract causal relationships among many variables in a more suitable way we demonstrate that the proposed methods can give more precise predictions and simpler models than some previous methods

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

pair correlation and twin primes revisited we establish a connection between the conjectural two over two ratios formula for the riemann zeta function and a conjecture concerning correlations of a certain arithmetic function specifically we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result this casts a new light on the underpinnings of the ratios conjecture which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe

numerical solutions for fractional reaction diffusion equations fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation in a fractional diffusion equation the second derivative in the spatial variable is replaced by a fractional derivative of order less than two the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used fractional reaction diffusion equations combine the fractional diffusion with a classical reaction term in this paper we develop a practical method for numerical solution of fractional reaction diffusion equations based on operator splitting then we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions we also discuss applications to biology where the reaction term models species growth and the diffusion term accounts for movements

on the voronoi formula for gl n we prove a general vorono u i formula for cuspidal automorphic representations of rm gl n over number fields this generalizes recent work by miller schmid and goldfeld li on maass forms our method follows closely the adelic framework of integral representations of l functions the proof is flexible enough to allow ramification and we propose possible variants for example the assumption that the additive twist is trivial at places where the representation is ramified is sufficient to obtain an explicit final statement with hyper kloosterman sums

estimates for rankin selberg l functions and quantum unique ergodicity abstract subconvex bounds in the weight aspect for rankin selberg l functions associated to two cusp forms are established these bounds are applied to prove the equidistribution of mass conjecture for cm forms

block pulse series analysis of linear systems incorporating observers the paper considers the analysis of linear systems incorporating observers via block pulse functions and presents recursive algorithms convenient for computers compared with walsh function approaches the results are much simpler in form an example is given to illustrate the use of the method

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

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

nonuniform state space reconstruction and coupling detection we investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past current and future states the embedding scheme can be adapted for different purposes such as mixed modeling cross prediction and granger causality in particular we apply this method in order to detect and evaluate information transfer in coupled systems as a practical application we investigate in records of scalp epileptic eeg the information flow across brain areas

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

approximate solution of fractional vibration equation using jacobi polynomials an approximate method using jacobi polynomials is proposed for the approximate solutions of fractional vibration equation convergence and numerical stability of the proposed method are shown results are discussed for different cases of jacobi polynomials results are compared with existing analytical methods in this paper we present a method based on the jacobi polynomials for the approximate solution to fractional vibration equation fve of large membranes proposed method converts the fve into sylvester form of algebraic equations whose solution gives the approximate solution convergence analysis of the proposed method is given it is also shown that our approximate method is numerically stable numerical results are discussed for different values of wave velocities and fractional order involved in the fve these numerical results are shown through figures for particular cases of jacobi polynomials such as 1 legendre polynomial 2 chebyshev polynomial of second kind 3 chebyshev polynomial of third kind 4 chebyshev polynomial of fourth kind 5 gegenbauer polynomial the accuracy of the proposed method is proved by comparing results of our method and other exiting analytical methods comparison of results are presented in the form of tables for particular cases of fve and jacobi polynomials

laguerre operational matrices for fractional calculus and applications the laguerre operational matrices for the integration and differentiation of a laguerre vector whose elements are laguerre polynomials are generalized to fractional calculus for investigating distributed systems the generalized operational matrices corresponding to 8 1 8 8 82 1 and exp 8 8 1 are derived as examples comparison of the laguerre series approximate inversions of irrational laplace transforms with exact solutions is very satisfactory

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

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

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

on the design of discrete time optimal dynamic controllers using the partial state observer abstract this paper considers the problem of designing optimal dynamic controllers based on a partial state observer for discrete time linear time invariant systems with inaccessible state it is assumed that the initial system state is an unknown random vector with known mean and covariance the performance index is taken to be the expectation with respect to the initial state of the standard quadratic one for the discrete time regulator necessary and sufficient conditions for optimality are derived provided that the subspace which ensures the existence of a partial state observer is given a design algorithm is also obtained which can be directly solved for the parameters of the optimal observer and requires the solution of a matrix riccati equation of the observer order as in the continuous time case the separation property in the restricted sense is seen to hold finally an example is presented to illustrate the procedure

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

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

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

an additive problem in the fourier coefficients of cusp forms we establish an estimate on sums of shifted products of fourier coefficients coming from holomorphic or maass cusp forms of arbitrary level and nebentypus these sums are analogous to the binary additive divisor sum which has been studied extensively as an application we derive extending work of duke friedlander and iwaniec a subconvex estimate on the critical line for l functions associated to character twists of these cusp forms

existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay a class of nonlinear fractional order partial differential equations with delay email protected au x t t a a t u x t f t u x t 1 t u x t l t email protected 0 t 0 be investigated in this paper where cd a is the standard caputo s fractional derivative of order email protected email protected 1 and l is a positive integer number the function f is defined as f t u 1 u l rxrx xr r and email protected email protected is a m dimension space using lebesgue dominated convergence theorem leray schauder fixed point theorem and banach contraction mapping theorem we obtain some sufficient conditions for the existence of the solutions of the above fractional order partial differential equations

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

wavelet based quasilinearization method for semi linear parabolic initial boundary value problems finite difference free wavelets based quasilinearization methods developed for spibvp fisher huxley and newell whitehead segel equations solved very efficiently comparison study provided for quasilinearization based on haar and legendre wavelets proposed approach performs better than the methods in the recent literature in this paper numerical methods based on quasilinearization and haar and legendre wavelets to solve a class of semi linear parabolic initial boundary value problem spibvp have been presented the haar and legendre wavelet methods have been successfully combined with quasilinearization to solve spibvp efficiently the presented numerical scheme has been illustrated using appropriate examples including fisher equation and the obtained results show that the proposed numerical scheme is robust and easy to apply

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

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

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

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

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

a reliable numerical algorithm for fractional advection dispersion equation arising in contaminant transport through porous media abstract this article presents a reliable numerical approach for the fractional advection dispersion equation by making use of legendre scaling functions as a basis the fractional advection dispersion equation describes the anomalous transport in surface and subsurface water using two dimensional basis formed by legendre scaling functions we get operational matrix for fractional integrations and differentiations substituting these operational matrices in the equation leads linear algebraic equations whose solutions can be derived with the aid of sylvester s approach this in turn yields approximate solutions for advection dispersion equation convergence analysis of the proposed scheme is presented the potency and accuracy of the proposed numerical algorithm are shown by plotting error figures

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

multivariate phase space reconstruction by nearest neighbor embedding with different time delays a recently proposed nearest neighbor based selection of time delays for phase space reconstruction is extended to multivariate time series with an iterative selection of variables and time delays a case study of numerically generated solutions of the x and z coordinates of the lorenz system and an application to heart rate and respiration data are used for illustration

the spectral decomposition of shifted convolution sums let pi 1 pi 2 be cuspidal automorphic representations of pgl 2 r qf conductor 1 and hecke eigenvalues lambda pi 1 2 n and let h 0 be an integer for any smooth compactly supported weight functions w 1 w 2 r x c and any y 0 aspectral decomposition of the shifted convolution sum sigma m n h lambda pi 1 vertical bar m vertical bar lambda pi 2 vertical bar n vertical bar root vertical bar mn vertical bar w 1 m y w 2 n y is obtained as an application aspectral decomposition of the dirichlet series graphics is proved for rs 1 2 with polynomial growth on vertical lines in the s aspect and uniformity in the h aspect

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

the operational matrices of integration and differentiation for the fourier sine cosine and exponential series for the fourier sine cosine series basis vector varphi t and the fourier exponential series basis vector psi t a linear nonsingular transfor marion t is determined such that psi t t varphi t this result is then used to show that the operational matrices of integration p and q for varphi t and psi t respectively are related by the expression tp qt analogous results are derived for the corresponding operational matrices of differentiation d and r general expressions are derived for t p q d and r

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

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

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

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

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

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

a wavelet based approach for vibration analysis of framed structures in this paper an explicit time integration scheme is proposed for structural vibration analysis by using wavelet functions initially the differential equation of vibration governing sdof single degree of freedom systems has been solved by wavelet operators and later the proposed approach has been generalized for mdof multi degrees of freedom systems for this purpose two different types of wavelet functions have been exemplified including complex chebyshev wavelet functions and simple haar wavelet functions in the proposed approach a straightforward formulation has been derived from the numerical approximation of response through the wavelet definition emphasizing on frequency domain approximation a simple step by step algorithm has been implemented and improved to calculate the response of mdof systems moreover stability and accuracy of results have been evaluated the effectiveness of the proposed approach is demonstrated using three examples compared with some of the existing numerical integration schemes such as family of newmark b wilson q and central difference method in all the procedures computation time involved has also been considered finally it is concluded that the vibration analysis of structures is improved by lesser computation time and high accuracy of proposed approach particularly in large scaled systems

the twisted fourth moment of the riemann zeta function we compute the asymptotics of the fourth moment of the riemann zeta func tion times an arbitrary dirichlet polynomial of length t 1 11 the study of the moments of the riemann zeta function has a long and distinguished history starting with the work of hardy and littlewood in 1918 and continuing to the present day one motivation for understanding moments is that they yield information about the maximum size of the zeta function the lindelof hypothesis another application is to zero density estimates which in turn have consequences for primes in short intervals however they have become an interesting topic in their own right very few rigorous results are known just the second and fourth power moments indeed it is only recently that a believable conjecture for higher powers has been made the twisted moments that is moments of the riemann zeta function times an arbitrary dirichlet polynomial are important too for example levinson s method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment in a series of papers duke friedlander and iwaniec used estimates for amplified moments of a family of l functions in order to deduce a subconvexity bound for an individual member of the family of course there are far easier methods to give a subconvexity bound for zeta but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general in this paper we prove an asymptotic formula for the twisted fourth moment of the riemann zeta function where we may take a dirichlet polynomial of length up to t 1 11 e

mollification of the fourth moment of automorphic l functions and arithmetic applications we compute the asymptotics of twisted fourth power moments of modular l functions of large prime level near the critical line this allows us to prove some new non vanishing results on the central values of automorphic l functions in particular those obtained by base change from gl 2 q to gl 2 k for k a cyclic field of low degree

legendre polynomials approximation to dynamic linear state equations with initial or boundary value conditions the state equations of the linear dynamic system with initial or boundary value conditions are solved by an effective shifted legendre polynomials approximation a powerful calculation algorithm is proposed to integrate the state equation with very accurate results as time tends to infinity a recursive formula is developed to calculate the expansion coefficients of the state functions for saving computer time and minimizing the computational error the proposed method is very suitable to solve a class of problems with peculiar control functions such as periodic functions or piecewise discontinuous functions several illustrative examples are given satisfactory computational results are obtained

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

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

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

haar wavelet collocation method for three dimensional elliptic partial differential equations a new collocation method based on haar wavelet is presented for numerical solution of three dimensional elliptic partial differential equations with dirichlet boundary conditions an important advantage of the method is that it can be applied to both linear as well as nonlinear problems the algorithm based on this new method is simple and can be easily implemented in any programming language experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results the proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations the numerical experiments confirm the accuracy and diverse applicability of the method

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

nearest neighbor embedding with different time delays a nearest neighbor based selection of time delays for phase space reconstruction is proposed and compared to the standard use of time delayed mutual information the possibility of using different time delays for consecutive dimensions is considered a case study of numerically generated solutions of the lorenz system is used for illustration the effect of contamination with various levels of additive gaussian white noise is discussed

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

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

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

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

laguerre series approach to the analysis of a linear control system incorporating observers application of the laguerre polynomial expansion to the analysis of a linear optimal control system incorporating observers is studied the method simplifies the system of equations into the successive solution of a set of linear algebraic recursive equations compared with other orthogonal polynomials the present method is simpler in form and more convenient for use with a digital computer a numerical example is given to illustrate the use of this method

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

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

jacobi collocation method for the approximate solution of some fractional order riccati differential equations with variable coefficients abstract this paper presents a computational method for the approximate solution of arbitrary order non linear fractional riccati differential equations with variable coefficients proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the jacobi polynomials convergence analysis of the proposed method is provided numerical results for different fractional orders of the riccati differential equations are discussed figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of jacobi polynomials such as the legendre polynomials the chebyshev polynomials of the second kind the chebyshev polynomials of the third kind the chebyshev polynomial of the fourth kind and the gegenbauer or ultraspherical polynomials numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods it is observed that the results from the proposed method are more accurate maximum absolute error and the root mean square error tables are given for all five kinds of polynomials for comparison purposes

haar wavelet approximation for the solution of cubic nonlinear schrodinger equations abstract in this study haar wavelet collocation method is used for the numerical solution of 1d and 2d cubic nonlinear schrodinger equations with initial and dirichlet boundary conditions the space derivatives are estimated through haar wavelet collocation method whereas for time derivative we have used crank nicolson scheme the proposed method is implemented upon several test problems and the numerical results of these test problems establish that the proposed method is accurate

optimal reconstruction of dynamical systems a noise amplification approach in this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements these statistics can be evaluated on any reconstructed attractor thereby allowing a direct comparison among different approaches uniform or nonuniform delay vectors pca legendre coordinates etc it can also be used to select the most appropriate parameters of a reconstruction strategy in the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function its definition is based on theoretical arguments on noise amplification the complexity of the reconstructed attractor and a direct measure of local stretch which constitutes an irrelevance measure the proposed method is demonstrated on synthetic and experimental time series

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

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