fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation abstract in this paper we mainly investigate the fractional spectral collocation discretization of optimal control problem governed by a space fractional diffusion equation existence and uniqueness of the solution to optimal control problem is proved the continuous first order optimality condition is derived the eigenfunctions of two classes of fractional strum liouville problems are used as basis functions to approximate state variable and adjoint state variable respectively the fractional spectral collocation scheme for the control problem is constructed based on first optimize then discretize approach note that the solutions of fractional differential equations are usually singular near the boundary a generalized fractional spectral collocation scheme for the control problem is proposed based on first optimize then discretize approach a projected gradient algorithm is designed based on the discrete optimality condition numerical experiments are carried out to verify the effectiveness of the proposed numerical schemes and algorithm

on synchronization of networks of wilson cowan oscillators with diffusive coupling we investigate the problem of synchronization in a network of homogeneous wilson cowan oscillators with diffusive coupling such networks can be used to model the behavior of populations of neurons in cortical tissue referred to as neural mass models a new approach is proposed to address conditions for local synchronization for this type of neural mass models by analyzing the linearized model around a limit cycle we study synchronization within a network with direct coupling we use both analytical and numerical approaches to link the presence or absence of synchronized behavior to the location of eigenvalues of the laplacian matrix for the analytical part we apply two time scale averaging and the chetaev theorem while for the remaining part we use a recently proposed numerical approach sufficient conditions are established to highlight the effect of network topology on synchronous behavior when the interconnection is undirected these conditions are utilized to address points that have been previously reported in the literature through simulations synchronization might persist or vanish in the presence of perturbation in the interconnection gains simulation results confirm and illustrate our results

fractional langevin model of memory in financial markets the separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after decades of statistical analysis of financial time series certain stylized facts have emerged including the statistics of stock price fluctuations having fat tails and their linear correlations in time being exceedingly short lived on the other hand the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law one explanation of this long term memory is that it is a consequence of the time scale separation between microscopic and macroscopic economic variables we propose a fractional langevin equation as a dynamical model of the observed memory in financial time series

on the time fractional schrodinger equation theoretical analysis and numerical solution by matrix mittag leffler functions abstract this paper presents a deep analysis of a time dependent schrodinger equation with fractional time derivative after the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations the main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations possibly with high frequency as it is typical with schrodinger equations and a persisting memory as a consequence of the fractional order moreover an accurate spatial discretization gives rise to systems of large to very large size involving a noteworthy computational complexity by means of a theoretical analysis the exact solution is split into two or three terms depending on the order of the fractional derivative thus to face the numerical computation by different and suitably selected methods direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step by step strategy in conjunction with matrix function for the oscillating term in both cases krylov subspace methods are employed for the computation of matrix functions and convergence results are presented

weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients a class of weighted finite difference methods wfdms for solving a class of initial boundary value problems of space fractional partial differential equations with variable coefficients is presented their stability and convergence properties are considered it is proven that the wfdms are unconditionally stable for email protected email protected 12 and conditionally stable for 12

the global analysis on the spectral collocation method for time fractional schrodinger equation abstract in this paper a spectral collocation method is proposed and analyzed for solving the time fractional schrodinger equation the space derivative is discretized using the collocation method and the time fractional derivative using grunwald letnikov formulation the stability and convergence of the full discretization scheme are analyzed based on the z transform the global behavior of the finite difference spectral collocation method is derived numerical examples show a good agreement with the theoretical analysis

subordinated advection dispersion equation for contaminant transport a mathematical method called subordination broadens the applicability of the classical advection dispersion equation for contaminant transport in this method the time variable is randomized to represent the operational time experienced by different particles in a highly heterogeneous aquifer the operational time captures the fractal properties of the medium this leads to a simple parsimonious model of contaminant transport that exhibits many of the features heavy tails skewness and non fickian growth rate typically seen in real aquifers we employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives applied to a one dimensional approximation of the made 2 data set the model shows excellent agreement

optimal error estimates of spectral petrov galerkin and collocation methods for initial value problems of fractional differential equations we present optimal error estimates for spectral petrov galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval we also develop laguerre spectral petrov galerkin methods and collocation methods for fractional equations on the half line numerical results confirm the error estimates

error estimates of a spectral petrov galerkin method for two sided fractional reaction diffusion equations abstract we study regularity and the spectral method for two sided fractional diffusion equations with a reaction term we show that the regularity of the solution in weighted sobolev spaces can be greatly improved compared to that in standard sobolev spaces with this regularity we prove an optimal error estimate for the spectral petrov galerkin method numerical results are presented to verify our theoretical convergence orders

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

an efficient chebyshev tau method for solving the space fractional diffusion equations fractional diffusion equations fdes have recently been paid much attention finding accurate and efficient methods for solving fdes has become an active research undertaking in this paper an efficient method based on the shifted chebyshev tau idea is presented to solve an initial boundary value problem for the fdes the method is derived by expanding the required approximate solution as the elements of shifted chebyshev polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from a computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and only a small number of shifted chebyshev polynomials is needed

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

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

growth of correlation in the hopfield model introducing a finite correlation 0 between any two learned patterns others remaining uncorrelated we observe in a numerical simulation that the hopfield model stores these two patterns with correlation f such that f 0 for any loading capacity the patterns are memorized perfectly with f 0 up to 0 05 for finite correlations 0 not exceeding a value c where c decreases continuously to zero at 0 05

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

on legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations the variational iteration method and the homotopy analysis method as alternative methods have been widely used to handle linear and nonlinear models the main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently this paper deals with the numerical solutions of nonlinear fractional differential equations where the fractional derivatives are considered in caputo sense the main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations in these algorithms legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work the proposed algorithms are capable of reducing the size of calculations improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals numerical examples are examined to show the efficiency of the algorithms

spectral and discontinuous spectral element methods for fractional delay equations we first develop a spectrally accurate petrov galerkin spectral method for fractional delay differential equations fddes this scheme is developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently presented in m zayernouri and g e karniadakis j comput phys 252 2013 pp 495 517 specifically we obtain solutions to fddes in terms of new nonpolynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of the first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of the second kind fslp ii we prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the pg spectral method in contrast to standard nondelay fractional differential equations the delay character of fddes might induce solutions which are either nonsmooth or piecewise smooth in order to effectively trea

a finite difference scheme for semilinear space fractional diffusion equations with time delay a linearized quasi compact finite difference scheme is proposed for semilinear space fractional diffusion equations with a fixed time delay the nonlinear source term is discretized and linearized by taylor s expansion to obtain a second order discretization in time the space fractional derivatives are approximated by a weighted shifted grunwald letnikov formula which is of fourth order approximation under some smoothness assumptions of the exact solution under the local lipschitz conditions the solvability and convergence of the scheme are proved in the discrete maximum norm by the energy method numerical examples verify the theoretical predictions and illustrate the validity of the proposed scheme

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

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

fractional differential equations and the schrodinger equation in a previous paper we defined following a previous work of kolvankar and gangal a notion of a derivative 0 a 1 in this article we study a differential equations associated to our fractional calculus we then discuss a fundamental problem concerning the schrodinger equation in the framework of nottale s scale relativity theory

a stable numerical method for multidimensional time fractional schrodinger equations in this paper the stability analysis is presented for a first order difference scheme applied to a nonhomogeneous time fractional schrodinger differential equation based on the z transform method stability theorems are proved for the abstract case the stability results are applied on initial boundary value problems for multidimensional time fractional schrodinger differential equations theoretical findings are validated by numerical experiments on one and two dimensional time fractional schrodinger differential equations

design and analysis of a dissipative scheme to solve a generalized multi dimensional higgs boson equation in the de sitter space time abstract in this work we design a numerically efficient finite difference technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with riesz fractional derivatives of orders in 0 1 1 2 which considers a generalized potential and a time dependent diffusion factor an energy integral for the mathematical model is readily available and we propose an explicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties as the continuous model a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the 3 1 dimensional de sitter space time and find results qualitatively similar to those available in the literature the present work is the first paper to report on a hamiltonian discretization of the higgs boson equation both fractional and non fractional in the de sitter space time and its numerical analysis more precisely the present manuscript is the first paper of the literature in which a dissipation preserving scheme to solve the multi dimensional fractional higgs boson equation in the de sitter space time is proposed and thoroughly analyzed indeed it is worth pointing out that previous efforts used techniques based on the runge kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed

a spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion wave equations in this paper an efficient and accurate spectral numerical method is presented for solving second fourth order fractional diffusion wave equations and fractional wave equations with damping the proposed method is based on jacobi tau spectral procedure together with the jacobi operational matrix for fractional integrals described in the riemann liouville sense the main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought for spectral approximations the validity and effectiveness of the method are demonstrated by solving five numerical examples numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier an efficient numerical scheme for time fractional diffusion wave equations is proposed a time space jacobi tau approximation is developed for such equations several tau spectral methods can be achieved as special cases the validity and applicability of the proposed method are demonstrated accurate numerical results are obtained by selecting limited collocation nodes

synchronization of diffusively coupled limit cycle oscillators we develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems we examine two classes of systems reaction diffusion pdes with neumann boundary conditions and compartmental odes where compartments are interconnected through diffusion terms with adjacent compartments in both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle we provide two time scale averaging methods for certifying stability of spatially homogeneous time periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion we highlight cases where diffusion stabilizes or destabilizes such trajectories

the analytical solution and numerical solution of the fractional diffusion wave equation with damping fractional partial differential equations have been applied to many problems in physics finance and engineering numerical methods and error estimates of these equations are currently a very active area of research in this paper we consider a fractional diffusionwave equation with damping we derive the analytical solution for the equation using the method of separation of variables an implicit difference approximation is constructed stability and convergence are proved by the energy method finally two numerical examples are presented to show the effectiveness of this approximation

jacobi gauss lobatto collocation method for the numerical solution of 1 1 nonlinear schrodinger equations a jacobi gauss lobatto collocation j gl c method used in combination with the implicit runge kutta method of fourth order is proposed as a numerical algorithm for the approximation of solutions to nonlinear schrodinger equations nlse with initial boundary data in 1 1 dimensions our procedure is implemented in two successive steps in the first one the j gl c is employed for approximating the functional dependence on the spatial variable using n 1 nodes of the jacobi gauss lobatto interpolation which depends upon two general jacobi parameters the resulting equations together with the two point boundary conditions induce a system of 2 n 1 first order ordinary differential equations odes in time in the second step the implicit runge kutta method of fourth order is applied to solve this temporal system the proposed j gl c method used in combination with the implicit runge kutta method of fourth order is employed to obtain highly accurate numerical approximations to four types of nlse including the attractive and repulsive nlse and a gross pitaevskii equation with space periodic potential the numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method indeed for relatively few nodes used the absolute error in our numerical solutions is sufficiently small

a constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation abstract this paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations an algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed several illustrative examples with linear and nonlinear fractional differential equations are given to demonstrate the effectiveness of the method

pair correlated patterns in hopfield model of neural networks we study the retrieval properties of the hopfield model of neural networks when the memorized patterns are statistically correlated in pairs there is a finite correlationk between the memories of each pair but memories of different pairs are uncorrelated the analysis is restricted to the case of an arbitrary but finite number of memories in the thermodynamic limit we find that there are two retrieval regimes for 0 t 1 k the system recognizes the stored patterns and for 1 k t 1 k the system is able to recognize pairs but it is not able to distinguish between its two patterns

fractional adams bashforth moulton methods we first formulate a fractional class of explicit adams bashforth a b and implicit adams moulton a m methods of first and second order accuracy for the time integration of d t 0 c u x t g t u 0 1 where d t 0 c denotes the fractional derivative in the caputo sense in this fractional setting and in contrast to the standard adams methods an extra history load term emerges and the associated weight coefficients are dependent however when 1 the developed schemes reduce to the well known a b and a m methods with standard coefficients hence in terms of scientific computing our approach constitutes a minimal modification of the existing adams libraries next we develop an implicit explicit imex splitting scheme for linear and nonlinear fractional pdes of a general advection reaction diffusion type and we apply our scheme to the time space fractional keller segel chemotaxis system in this context we evaluate the nonlinear advection term explicitly employing the fractional a b method in the prediction step and we treat the corresponding diffusion term implicitly in the correction step using the fractional a m scheme moreover we perform the corresponding spatial discretization by employing an efficient and spectrally accurate fractional spectral collocation method our numerical experiments exhibit the efficiency of the proposed imex scheme in solving nonlinear fractional pdes

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

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

on finite difference methods for fourth order fractional diffusion wave and subdiffusion systems abstract in this paper firstly the finite difference method is explored for the fourth order fractional diffusion wave system the method is proved to be uniquely solvable stable and convergent in l norm by the energy method then we examine a subdiffusion system and present the numerical analysis using a different method numerical experiments are provided to demonstrate the accuracy and efficiency of the proposed schemes

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay in this paper a delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations a great variety of interesting dynamical behaviors of such a system including single periodic multiple periodic and chaotic motions are displayed in particular the effect of time delay on the chaotic behavior is investigated it is found that an approximate time delay can enhance or suppress the emergence of chaos meanwhile corresponding to different values of delay the lowest orders for chaos to exist in the delayed fractional order financial systems are determined respectively

sharp decay rates for wave equations with a fractional damping via new method in the fourier space abstract we study the cauchy problem for damped wave equations with a fractional damping u t in r n we derive more sharp decay estimates of the total energy based on the energy method in the fourier space combined with the haraux komornik inequality especially in the case when 0 1 2 the rate of decay of the total energy becomes almost optimal the method in this paper can be applied to other equations and in particular it seems to be quite effective in the case of frictional dissipation i e when 0

algorithm for some anomalously diffusive hyperbolic systems in molecular dynamics theoretical analysis and pattern formation abstract departing from a two dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions we investigate a general form of that model using a finite difference approach the model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms the presence of two dimensional diffusive terms consisting of fractional operators of the riesz type is considered here using spatial differentiation orders in the set 0 1 1 2 we impose initial conditions on a closed and bounded rectangle and a finite difference methodology based on the use of fractional centered differences is proposed among the most important results of this work we prove the existence and the uniqueness of the solutions of the numerical method and establish analytically the second order consistency of our scheme moreover the discrete energy method is employed to prove the stability and the quadratic convergence of the technique some numerical simulations obtained through our method show the presence of turing patterns and wave instabilities in agreement with some reports found in the literature on superdiffusive hyperbolic activator inhibitor systems we show numerically that the presence of turing patterns is independent of the size of the spatial domain as a new application we show that turing patterns are also present in subdiffusive scenarios

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

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

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

an element free based solution for nonlinear schrodinger equations using the icvmls ritz method an improved complex variable moving least squares ritz icvmls ritz method is proposed for predicting numerical solution of the two dimensional nonlinear schrodinger equation in this element free solution procedure the icvmls approximation is employed to reduce the number of unknown coefficients in the trial function it follows by the ritz procedure to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear schrodinger equation the central differencing scheme and newton s algorithm are adopted to solve the nonlinear equation system numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element free icvmls ritz method by comparing the computed results with the available analytical solutions

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

analysis and numerical methods for fractional differential equations with delay in this paper we consider fractional differential equations with delay we focus on linear equations we summarise existence and uniqueness theory based on the method of steps and we give a theorem on the propagation of derivative discontinuities we discuss the dependence of the solution on the parameters of the equation and conclude with a numerical treatment and examples based on the adaptation of a fractional backward difference method to the delay case

convergence analysis of space time jacobi spectral collocation method for solving time fractional schrodinger equations abstract in this paper the space time jacobi spectral collocation method jsc method is used to solve the time fractional nonlinear schr o dinger equations subject to the appropriate initial and boundary conditions at first the considered problem is transformed into the associated system of nonlinear volterra integro partial differential equations pdes with weakly singular kernels by the definition and related properties of fractional derivative and integral operators therefore by collocating the associated system of integro pdes in both of the space and time variables together with approximating the existing integral in the equation using the jacobi gauss type quadrature formula then the problem is reduced to a set of nonlinear algebraic equations we can consider solving the system by some robust iterative solvers in order to support the convergence of the proposed method we provided some numerical examples and calculated their l norm and weighted l2 norm at the end of the article

asymptotic properties of fractional delay differential equations abstract in this paper we study the asymptotic properties of d dimensional linear fractional differential equations with time delay we present necessary and sufficient conditions for asymptotic stability of equations of this type using the inverse laplace transform method and prove polynomial decay of stable solutions two examples illustrate the obtained analytical results

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

on the solution of a generalized higgs boson equation in the de sitter space time through an efficient and hamiltonian scheme abstract the present work is the first paper in the literature to report on a hamiltonian discretization of the fractional higgs boson equation in the de sitter space time and its theoretical analysis more precisely we design herein a numerically efficient finite difference hamiltonian technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with generalized potential and riesz space fractional derivatives of orders in 1 2 an energy integral for the model is readily available and we propose a nonlinear implicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties in the discrete scenario a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the de sitter space time and find results qualitatively similar to those available in the literature for the sake of convenience we provide the matlab code of an alternative linear discretization of the method presented in this work this linear implicit approach is thoroughly analyzed also

fractional spectral and pseudo spectral methods in unbounded domains theory and applications abstract this paper is intended to provide exponentially accurate galerkin petrov galerkin and pseudo spectral methods for fractional differential equations on a semi infinite interval we start our discussion by introducing two new non classical lagrange basis functions nlbfs 1 and nlbfs 2 which are based on the two new families of the associated laguerre polynomials galfs 1 and galfs 2 obtained recently by the authors in 28 with respect to the nlbfs 1 and nlbfs 2 two new non classical interpolants based on the associated laguerre gauss and laguerre gauss radau points are introduced and then fractional pseudo spectral differentiation and integration matrices are derived convergence and stability of the new interpolants are proved in detail several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives and integrals of some functions moreover the pseudo spectral galerkin and petrov galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones some useful comments from the numerical point of view on galerkin and petrov galerkin methods are listed at the end

nodal discontinuous galerkin methods for fractional diffusion equations on 2d domain with triangular meshes this paper as the sequel to previous work develops numerical schemes for fractional diffusion equations on a two dimensional finite domain with triangular meshes we adopt the nodal discontinuous galerkin methods for the full spatial discretization by the use of high order nodal basis employing multivariate lagrange polynomials defined on the triangles stability analysis and error estimates are provided which shows that if polynomials of degree n are used the methods are n 1 th order accurate for general triangulations finally the performed numerical experiments confirm the optimal order of convergence

highly accurate numerical schemes for multi dimensional space variable order fractional schrdinger equations as a natural generalization of the fractional schrdinger equation the variable order fractional schrdinger equation has been exploited to study fractional quantum phenomena in this paper we develop an exponentially accurate jacobigausslobatto collocation jgl c method to solve the variable order fractional schrdinger equations in one dimension 1d and two dimensions 2d in this method the aforementioned problem is reduced to a system of ordinary differential equations odes in the time variable as a result we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations one based on the implicit rungekutta irk method of fourth order and the other based on jacobigaussradau collocation jgr c method the validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1d and 2d the convergence of the methods is graphically analyzed the results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable order nonlinear partial differential equations

corrigendum to on eigenvalues of laplacian matrix for a class of directed signed graphs linear algebra appl 523 2017 281 306 abstract this note corrects an error in the results of subsection 3 1 in authors paper on eigenvalues of laplacian matrix for a class of directed signed graphs which appeared in linear algebra and its applications 523 2017 281 306

the use of finite difference element approaches for solving the time fractional subdiffusion equation in this paper two finite difference element approaches for the time fractional subdiffusion equation with dirichlet boundary conditions are developed in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method the two methods are unconditionally stable and convergent of order o tau q h r 1 in the l 2 norm where q 2 beta or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth beta 0 beta 1 is the order of the fractional derivative tau and h are the step sizes in time and space respectively and r is the degree of the polynomial space the corresponding schemes for the subdiffusion equation with neumann boundary conditions are presented as well where the stability and convergence are shown numerical examples are provided to verify the theoretical analysis comparisons between the algorithms derived in this paper and the existing algorithms are given

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

on spectral properties of signed laplacians with connections to eventual positivity signed graphs have appeared in a broad variety of applications ranging from social networks to biological networks from distributed control and computation to power systems in this paper we investigate spectral properties of signed laplacians for undirected signed graphs we find conditions on the negative weights under which a signed laplacian is positive semidefinite via the kron reduction and multiport network theory for signed laplacians that are indefinite we characterize their inertias with the same framework furthermore we build connections between signed laplacians generalized m matrices and eventually exponentially positive matrices

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

split step alternating direction implicit difference scheme for the fractional schrodinger equation in two dimensions in this paper we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space fractional schrodinger equation with the riesz fractional derivative the scheme is constructed by combining the split step method for handling the nonlinearity with the alternating direction implicit adi method for resolving the multi dimensions difficulty the riesz space fractional derivative is approximated by the second order accurate fractional centered difference based on matrix analysis we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems the unconditional convergence is proved rigorously in the linear case numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme

a fully spectral collocation approximation for multi dimensional fractional schrodinger equations a shifted legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one and two dimensional time fractional schrodinger equations tfses subject to initial boundary and non local conditions the first step depends mainly on shifted legendre gauss lobatto collocation sl gl c method for spatial discretization an expansion in a series of shifted legendre polynomials for the approximate solution and its spatial derivatives occurring in the tfse is investigated in addition the legendre gauss lobatto quadrature rule is established to treat the nonlocal conservation conditions thereby the expansion coefficients are then determined by reducing the tfse with its nonlocal conditions to a system of fractional differential equations sfdes for these coefficients the second step is to propose a shifted legendre gauss radau collocation sl gr c scheme for temporal discretization to reduce such system into a system of algebraic equations which is far easier to be solved the proposed collocation scheme both in temporal and spatial discretizations is successfully extended to solve the two dimensional tfse numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms by selecting relatively limited legendre gauss lobatto and gauss radau collocation nodes we are able to get very accurate approximations demonstrating the utility and high accuracy of the new approach over other numerical methods

laplacian matrices of general complex weighted directed graphs abstract we introduce the concept of general complex weighted directed graphs where each edge is assigned a complex number necessary and sufficient conditions for the laplacian matrix to be singular nonsingular are derived our results give the relationship between the laplacian matrix and the structure of its corresponding directed graph compared with the existing results our main contribution is that our results are established without the restriction that the adjacency matrix is hermitian

numerical solution of volterra fredholm integral equations using legendre collocation method a numerical method for solving the volterra fredholm integral equations is presented the method is based upon shifted legendre polynomials approximation the properties of shifted legendre polynomials are first presented these properties together with the shifted gauss legendre nodes are then utilized to reduce the volterra fredholm integral equations to the solution of a matrix equation an estimation of the error is given illustrative examples are included to demonstrate the validity and applicability of the technique

a fast second order implicit scheme for non linear time space fractional diffusion equation with time delay and drift term abstract in this paper a second order accurate implicit scheme based on the l2 1 formula for temporal variable and the fractional centered difference formula for spatial discretization is established to solve a class of time space fractional diffusion equations with time drift term and non linear delayed source function the stability of this scheme is proved rigorously by the discrete energy method under several auxiliary assumptions then we theoretically and numerically show that the proposed scheme converges in the l2 norm with the order o t 2 h 2 with time step t and mesh size h moreover it finds that the discreted linear systems are symmetric toeplitz systems in order to solve these systems efficiently the conjugate gradient method with suitable circulant preconditioners is designed in each iterative step the storage requirements and the computational complexity of the resulting equations are o n and o n log n respectively where n is the number of grid nodes numerical experiments are carried out to demonstrate the effectiveness of our proposed circulant preconditioners

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

analytical and numerical methods for the stability analysis of linear fractional delay differential equations in this paper several analytical and numerical approaches are presented for the stability analysis of linear fractional order delay differential equations the main focus of interest is asymptotic stability but bounded input bounded output bibo stability is also discussed the applicability of the laplace transform method for stability analysis is first investigated jointly with the corresponding characteristic equation which is broadly used in bibo stability analysis moreover it is shown that a different characteristic equation involving the one parameter mittag leffler function may be obtained using the well known method of steps which provides a necessary condition for asymptotic stability stability criteria based on the argument principle are also obtained the stability regions obtained using the two methods are evaluated numerically and comparison results are presented several key problems are highlighted

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

storing correlated patterns in hopfield s model the authors study the retrieval properties of hopfield s model 1982 when there is a finite correlation between two of the memorised patterns

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

fractional spectral collocation methods for linear and nonlinear variable order fpdes while several high order methods have been developed for fractional pdes fpdes with fixed order there are no such methods for fpdes with field variable order these equations allow multiphysics simulations seamlessly e g from diffusion to sub diffusion or from wave dynamics transitioning to diffusion by simply varying the fractional order as a function of space or time we develop an exponentially accurate fractional spectral collocation method for solving linear nonlinear fpdes with field variable order following the spectral theory developed in 1 for fractional sturm liouville eigenproblems we introduce a new family of interpolants called left right sided and central fractional lagrange interpolants we employ the fractional derivatives of left right sided riemann liouville and riesz type and obtain the corresponding fractional differentiation matrices by collocating the field variable fractional orders we solve several fpdes including time and space fractional advection equation time and space fractional advection diffusion equation and finally the space fractional burgers equation to demonstrate the performance of the method in addition we develop a spectral penalty method for enforcing inhomogeneous initial conditions our numerical results confirm the exponential like convergence of the proposed fractional collocation methods

a new notion of effective resistance for directed graphs part i definition and properties the graphical notion of effective resistance has found wide ranging applications in many areas of pure mathematics applied mathematics and control theory by the nature of its construction effective resistance can only be computed in undirected graphs and yet in several areas of its application directed graphs arise as naturally or more naturally than undirected ones in part i of this work we propose a generalization of effective resistance to directed graphs that preserves its control theoretic properties in relation to consensus type dynamics we proceed to analyze the dependence of our algebraic definition on the structural properties of the graph and the relationship between our construction and a graphical distance the results make possible the calculation of effective resistance between any two nodes in any directed graph and provide a solid foundation for the application of effective resistance to problems involving directed graphs

comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations in this article the homotopy perturbation method proposed by j h he is adopted for solving linear fractional partial differential equations the fractional derivatives are described in the caputo sense comparison of the results obtained by the homotopy perturbation method with those obtained by the variational iteration method reveals that the present methods are very effective and convenient

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

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

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

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

an efficient jacobi pseudospectral approximation for nonlinear complex generalized zakharov system in this paper we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized zakharov system gzs subject to initial boundary conditions the jacobi pseudospectral approximation is investigated for spatial approximation of the gzs it possesses the spectral accuracy in space the jacobi gauss lobatto quadrature rule is established to treat the boundary conditions and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable this scheme has the advantage of allowing us to obtain the spectral solution in terms of the jacobi parameters and which therefore means that the algorithm holds a number of collocation methods as special cases finally two illustrative examples are implemented to assess the efficiency and high accuracy of the jacobi pseudo spectral scheme

on eigenvalues of laplacian matrix for a class of directed signed graphs abstract the eigenvalues of the laplacian matrix for a class of directed graphs with both positive and negative weights are studied the laplacian matrix naturally arises in a wide range of applications involving networks first a class of directed signed graphs is studied in which one pair of nodes either connected or not is perturbed with negative weights a necessary and sufficient condition is proposed to attain the following objective for the perturbed graph the real parts of the non zero eigenvalues of its laplacian matrix are positive under certain assumption on the unperturbed graph it is established that the objective is achieved if and only if the magnitudes of the added negative weights are smaller than an easily computable upper bound this upper bound is shown to depend on the topology of the unperturbed graph it is also pointed out that the obtained condition can be applied in a recursive manner to deal with multiple edges with negative weights secondly for directed graphs a subset of pairs of nodes are identified where if any of the pairs is connected by an edge with infinitesimal negative weight the resulting laplacian matrix will have at least one eigenvalue with negative real part illustrative examples are presented to show the applicability of our results