a new approach for the application of adomian decomposition method for the solution of fractional space diffusion equation with insulated ends abstract this paper presents the new approach for the solution of the space fractional diffusion equation defined in finite domain by adomian decomposition method adm a typical example of special interest with fractional space derivative of order 1 2 is considered in the present analysis and solved by adm after expressing the initial condition as fourier series the explicit solution of the space fractional diffusion equation has been presented in the closed form and then the numerical solution has been represented graphically the behaviour of adomian solutions and the effects of different values of are shown graphically

beyond the poisson renewal process a tutorial survey after sketching the basic principles of renewal theory and recalling the classical poisson process we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of mittag leffler and of wright type we compare these three processes with each other

analytical time domain green s functions for power law media frequency dependent loss and dispersion are typically modeled with a power law attenuation coefficient where the power law exponent ranges from 0 to 2 to facilitate analytical solution a fractional partial differential equation is derived that exactly describes power law attenuation and the szabo wave equation time domain wave equations for lossy media obeying a frequency power law j acoust soc am 96 491 500 1994 is an approximation to this equation this paper derives analytical time domain green s functions in power law media for exponents in this range to construct solutions stable law probability distributions are utilized for exponents equal to 0 1 3 1 2 2 3 3 2 and 2 the green s function is expressed in terms of dirac delta exponential airy hypergeometric and gaussian functions for exponents strictly less than 1 the green s functions are expressed as fox functions and are causal for exponents greater than or equal than 1 the green s functions are expressed as fox and

fox h functions in fractional diffusion the h functions introduced by fox in 1961 are special functions of a very general nature which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework in this paper we express the fundamental solutions of the cauchy problem for the space time fractional diffusion equation in terms of proper fox h functions based on their mellin barnes integral representations we pay attention to the particular cases of space fractional time fractional and neutral fractional diffusion

time fractional wave diffusion equation in an inhomogeneous half space we consider the fundamental solution of the time fractional wave diffusion equation in a three dimensional half space medium which contains an inhomogeneity in form of a plane parallel layer the corresponding green s function which is derived by means of the fourier and laplace transforms can be accurately and efficiently evaluated without recourse to the mittag leffler or the fox h function moreover it is shown that in the one dimensional case the fundamental solution in an inhomogeneous half space is no longer a probability density function in addition we consider the advection equation for the fractional laplacian and the caputo time fractional derivative of orders on a bounded domain simple algorithms for accurate evaluation of the m wright function and the mittag leffler function are enclosed at the end of this article

spectral analysis of fractional kinetic equations with random data we present a spectral representation of the mean square solution of the fractional kinetic equation also known as fractional diffusion equation with random initial condition gaussian and non gaussian limiting distributions of the renormalized solution of the fractional in time and in space kinetic equation are described in terms of multiple stochastic integral representations

fractional wave equation with a frictional memory kernel of mittag leffler type abstract in this paper we give an analytical treatment of a fractional wave equation with caputo time fractional derivative and frictional memory kernel of mittag leffler type this problem generalizes a recently solved problem 62 of a wave equation for a vibrating string in presence of a fractional friction with power law memory kernel such equations can be used in the context of modeling processes in complex and viscoelastic media

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

index theory of one dimensional quantum walks and cellular automata if a one dimensional quantum lattice system is subject to one step of a reversible discrete time dynamics it is intuitive that as much quantum information as moves into any given block of cells from the left has to exit that block to the right for two types of such systems namely quantum walks and cellular automata we make this intuition precise by defining an index a quantity that measures the net flow of quantum information through the system the index supplies a complete characterization of two properties of the discrete dynamics first two systems s1 s2 can be pieced together in the sense that there is a system s which acts like s1 in one region and like s2 in some other region if and only if s1 and s2 have the same index second the index labels connected components of such systems equality of the index is necessary and sufficient for the existence of a continuous deformation of s1 into s2 in the case of quantum walks the index is integer valued whereas for cellular automata it takes values in the group of positive rationals in both cases the map s mapsto rm ind s is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems systems with trivial index are precisely those which can be realized by partitioned unitaries and the prototypes of systems with non trivial index are shifts

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

salvatore pincherle the pioneer of the mellin barnes integrals the 1888 paper by salvatore pincherle professor of mathematics at the university of bologna on generalized hypergeometric functions is revisited we point out the pioneering contribution of the italian mathematician towards the mellin barnes integrals based on the duality principle between linear differential equations and linear difference equation with rational coefficients by extending the original arguments used by pincherle we also show how to formally derive the linear differential equation and the mellin barnes integral representation of the meijer g functions

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

fractional derivatives of products of airy functions abstract fractional derivatives of the products of airy functions are investigated d ai 2 x and d ai x bi x where ai x and bi x are the airy functions of the first and second type respectively they turn out to be linear combinations of d ai x and d gi x where gi x is the scorer function it is also proved that the wronskian w x of the system of half integrals d 1 2 ai x d 1 2 gi x and its hilbert transform w x h w x can be considered special functions in their own right since they are expressed in terms of ai 2 x and ai x bi x respectively various integral relations are established integral representations for d ai x a ai x a and its hilbert transform h d ai x a ai x a are derived

on the character of operational solutions of the time fractional diffusion equation the nonhomogeneous time fractional diffusion equation is analyzed in the frames of the mikusinski calculus the exact and the approximate solution of the considered problem are constructed and their character is analyzed

fractional schrodinger equation with noninteger dimensions the spatial and time dependent solutions of the schrodinger equation incorporating the fractional time derivative of distributed order and extending the spatial operator to noninteger dimensions are investigated they are obtained by using the green function approach in two situations the free case and in the presence of a harmonic potential the results obtained show an anomalous spreading of the wave packet which may be related to an anomalous diffusion process

analytical solution of a fractional diffusion equation by adomian decomposition method this paper presents an analytical solution of a fractional diffusion equation by adomian decomposition method by using an initial value the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically the present method performs extremely well in terms of efficiency and simplicity

numerical experiments with the osher upwind scheme for the euler equations the osher algorithm for solving the euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations a first order accurate version of the numerical method was derived by osher circa 1980 for the one dimensional non isentropic euler equations in cartesian coordinates in this paper the extension of the scheme to arbitrary two dimensional geometries is explained results are then presented for several example problems in one and two dimensions future work will include extension of the method to second order accuracy and the development of implicit time differencing for the osher algorithm

weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations abstract this paper is concerned with the existence and uniqueness of weighted pseudo almost automorphic mild solution to the semilinear fractional equation d t u t au t d t 1 f t u t bu t t r 1 2 where a is a linear densely defined operator of sectorial type on a complex banach space x and b is a bounded linear operator defined on x under the assumption of uniformly continuity on f we establish a composition of weighted pseudo almost automorphic in a general banach space and obtain existence results by means of banach contraction mapping the results obtained are utilized to study the existence and uniqueness of a weighted pseudo almost automorphic solution to fractional diffusion wave equation with dirichlet conditions

n dimensional fractional diffusion equation and green function approach spatially dependent diffusion coefficient and external force we investigate an n dimensional fractional diffusion equation with radial symmetry by using the green function approach we consider in our analysis the spatial dependence on the diffusion coefficient and the presence of an external force in particular we employ boundary conditions in a finite interval and after we extend it to a semi infinite interval we also show that a rich class of diffusive processes including normal and anomalous ones can be obtained from the solutions found here

optimal control of a nonhomogeneous dirichlet boundary fractional diffusion equation we study a nonhomogeneous dirichlet boundary fractional diffusion equation in a bounded domain the fractional time derivative is considered in the riemann liouville sense we first prove by transposition the existence and the uniqueness of the solution of the boundary fractional diffusion equation then under some appropriate assumptions on the closed convex set of the admissible controls we obtain a decoupled optimality system

anisotropic fractional diffusion equation abstract we analyze an anisotropic fractional diffusion equation that extends some known diffusion equations by considering a diffusion coefficient with spatial and time dependence the presence of external forces and time fractional derivatives we obtain new exact classes of solutions for a linear anisotropic fractional diffusion equation and investigate the time scaling behavior and an asymptotic solution for a nonlinear anisotropic fractional diffusion equation we connect the asymptotic solution obtained with the distribution that emerges from the nonextensive statistics to the nonlinear case we also verify different diffusive behavior for instance subdiffusion and superdiffusion in each direction

exact solutions of triple order time fractional differential equations for anomalous relaxation and diffusion i the accelerating case a b s t r a c t in recent years the interest around the study of anomalous relaxation and diffusion processes is increased due to their importance in several natural phenomena moreover a further generalization has been developed by introducing time fractional differentiation of distributed order which ranges between 0 and 1 we refer to accelerating processes when the driving power law has a changing in time exponent whose modulus tends from less than 1 to 1 and to decelerating processes when such an exponent modulus decreases in time moving away from the linear behaviour accelerating processes are modelled by a time fractional derivative in the riemann liouville sense while decelerating processes by a time fractional derivative in the caputo sense here the focus is on the accelerating case while the decelerating one is considered in the companion paper after a short reminder about the derivation of the fundamental solution for a general distribution of time derivative orders we consider in detail the triple order case for both accelerating relaxation and accelerating diffusion processes and the exact results are derived in terms of an infinite series of h functions the method adopted is new and it makes use of certain properties of the generalized mittag leffler function and the h function moreover it provides an elegant generalization of the method introduced by langlands 2006 t a m langlands physica a 367 2006 136 to study the double order case of accelerating diffusion processes

long range correlations in time series generated by time fractional diffusion a numerical study abstract time series models showing power law tails in autocorrelation functions are common in econometrics a special non markovian model for such kind of time series is provided by the random walk introduced by gorenflo et al as a discretization of time fractional diffusion the time series so obtained are analyzed here from a numerical point of view in terms of autocorrelations and covariance matrices

the wright functions as solutions of the time fractional diffusion equation we revisit the cauchy problem for the 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 email protected 0 2 by using the fourier laplace transforms the fundamentals solutions green functions are shown to be high transcendental functions of the wright type that can be interpreted as spatial probability density functions evolving in time with similarity properties we provide a general representation of these functions in terms of mellin barnes integrals useful for numerical computation

application of modified decomposition method for the analytical solution of space fractional diffusion equation abstract spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow finance and others areas of application this paper presents the analytical solutions of the space fractional diffusion equations by modified decomposition method mdm by using initial conditions the explicit solutions of the equations have been presented in the closed form the decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self cancelling noise phenomenon two examples the first one is one dimensional and the second one is two dimensional fractional diffusion equation are presented to show the application of the present technique the present method performs extremely well in terms of efficiency and simplicity

integrodifferential diffusion equation for continuous time random walk in this paper we present an integrodifferential diffusion equation for continuous time random walk that is valid for a generic waiting time probability density function using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential and a combination of power law and generalized mittag leffler function we show that for the case of the exponential waiting time probability density function a normal diffusion is generated and the probability density function is gaussian distribution in the case of the combination of a power law and generalized mittag leffler waiting probability density function we obtain the subdiffusive behavior for all the time regions from small to large times and probability density function is non gaussian distribution

fundamental solutions of the time fractional diffusion wave and parabolic dirac operators abstract in this paper we study the multidimensional time fractional diffusion wave equation where the time fractional derivative is in the caputo sense with order 0 2 applying operational techniques via fourier and mellin transforms we obtain an integral representation of the fundamental solution fs of the time fractional diffusion wave operator series representations of the fs are explicitly obtained for any dimension from these we derive the fs for the time fractional parabolic dirac operator in the form of integral and series representation fractional moments of arbitrary order 0 are also computed to illustrate our results we present and discuss some plots of the fs for some particular values of the dimension and of the fractional parameter

radiative transport equation for the mittag leffler path length distribution in this paper we consider the radiative transport equation for infinitely extended scattering media that are characterized by the mittag leffler path length distribution p l le tl which is a generalization of the usually assumed lambert beer law p l texp tl in this context we derive the infinite space green s function of the underlying fractional transport equation for the spherically symmetric medium as well as for the one dimensional string moreover simple analytical solutions are presented for the prediction of the radiation field in the single scattering approximation the resulting equations are compared with monte carlo simulations in the steady state and time domain showing within the stochastic nature of the simulations an excellent agreement

initial value boundary value problem for composite fractional relaxation equation we consider initial value boundary value problem for composite fractional relaxation equation involving caputo fractional derivative of order 0 1 we prove by means of change of variable that this problem is reduced to initial value boundary value problem for fractional diffusion equation involving riemann liouville fractional derivative of order 1 then by means of eigenfunctions expansions we establish the existence and uniqueness of solution

optimal control of fractional diffusion equation in this paper we apply the classical control theory to a fractional diffusion equation in a bounded domain the fractional time derivative is considered in a riemann liouville sense we first study the existence and the uniqueness of the solution of the fractional diffusion equation in a hilbert space then we show that the considered optimal control problem has a unique solution interpreting the euler lagrange first order optimality condition with an adjoint problem defined by means of right fractional caputo derivative we obtain an optimality system for the optimal control

solutions for a generalized fractional anomalous diffusion equation in this paper we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time dependent diffusion coefficient and an external force into account which subjects to the natural boundaries and the generic initial condition we obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the tsallis entropy

analytical study on the fractional anomalous diffusion in a half plane in this study anomalous diffusion in a half plane with a constant source and a perfect sink at each half of the boundary is considered the discontinuity of the boundary condition is erased by decomposing the solution into two parts a symmetric part and an antisymmetric part the symmetric part which has been studied extensively can be solved by an integral transform method green s function method or others to obtain the solution of the antisymmetric part a separable similarity solution is assumed and the erdelyi kober type fractional derivative is used by doing so the partial differential equation reduces to an ordinary one using the mellin transform method the solution of the antisymmetric part in terms of a fox h function is obtained some figures are given to show the characters of the diffusion process and the influences of different orders of fractional derivatives

fractional diffusion equation and green function approach exact solutions abstract we investigate the solutions of a fractional diffusion equation with radial symmetry by using the green function approach and by taking the n dimensional case into account in our analysis a spatial time dependent diffusion coefficient is considered i e d r t d t 1 r the presence of external forces f r k r e with e 1 and f r kr k r e is also taken into account in particular we discuss the results obtained by employing boundary conditions defined on a finite interval and afterwards the analysis is extended to a semi infinite interval finally we also discuss a rich class of diffusive processes that can be obtained from the results presented in this work

on a unified mixture distribution in this article a new mixture distribution associated with fox wright generalized hypergeometric function has been studied which generalizes many mixture distributions investigated earlier by many authors some basic functions associated with the probability density function of the mixture distribution such as kth moments characteristic function and factorial moments are derived some special cases are also pointed out

front propagation in anomalous diffusive media governed by time fractional diffusion in this paper a multi dimensional model is proposed to study the propagation of random fronts in media in which anomalous diffusion takes place the front position is obtained as the weighted mean of fronts calculated by means of the level set method using as weight function the probability density function which characterizes the anomalous diffusion process since anomalous diffusion is assumed to be governed by a time fractional diffusion equation its fundamental solution is the required probability density function it is shown that this fundamental solution can be expressed in the multi dimensional case in terms of the well known m wright mainardi function as in the one dimensional case making use of this representation for the practical purpose of numerical evaluation the propagation of random fronts in two dimensional subdiffusive media is discussed and investigated

optimal control of a fractional diffusion equation with state constraints this paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain the fractional time derivative is considered in the riemann liouville sense under a slater type condition we prove the existence a lagrange multiplier and a decoupled optimality system

grid independent upwind scheme for multidimensional flow recent advances in the development of a grid independent finite volume scheme for the euler equations of gas dynamics are described in the proposed method flowfield gradient data are reconstructed locally on a triangle using five elementary planar waves and an upwind numerical flux function for grid oblique waves is developed to model the effect of the passage of these waves on the data in a cell numerical examples for several two dimensional test problems are included

on a novel iterative method to compute polynomial approximations to bessel functions of the first kind and its connection to the solution of fractional diffusion diffusion wave problems we present an iterative method to obtain approximations to bessel functions of the first kind jp x p 1 via the repeated application of an integral operator to an initial seed function f0 x the class of seed functions f0 x leading to sets of increasingly accurate approximations fn x is considerably large and includes any polynomial when the operator is applied once to a polynomial of degree s it yields a polynomial of degree s 2 and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree we focus on the set of polynomial approximations generated from the seed function f0 x 1 this set of polynomials is useful not only for the computation of jp x but also from a physical point of view as it describes the long time decay modes of certain fractional diffusion and diffusion wave problems

fractional nonlinear diffusion equation solutions and anomalous diffusion abstract we devote this work to investigate the solutions of a generalized diffusion equation which contains spatial fractional derivatives and nonlinear terms the presence of external forces and absorbent terms is also considered the solutions found here can have a compact or long tail behavior and in particular for the last case in the asymptotic limit we relate these solutions to the levy or tsallis distributions in addition from the results presented here a rich class of diffusive processes including normal and anomalous ones can be obtained

fundamental solution of the tempered fractional diffusion equation in this paper we consider the space time fractional diffusion equation dt u x t k dx u x t 0 x r t 0 with the tempered riemann liouville derivative of order 0 1 in space and the caputo derivative of order 0 1 in time the fundamental solution which turns out to be a spatial probability density function is given in computable series form as well as in integral representation the spatial moments of the probability density function are determined explicitly for an arbitrary order n 0 moreover green s function of the untempered neutral fractional diffusion equation is analyzed in view of absolute and relative extreme points at the end of this article we point out a remarkably and important integral representation for accurate evaluation of the m wright mainardi function m x of order 0 1 and arguments x r0

short note on the emergence of fractional kinetics abstract in the present short note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non markovian master equations particle trajectories are assumed to be solely markovian and described by the continuous time random walk model but as a consequence of the complexity of the medium each trajectory is supposed to scale in time according to a particular random timescale the link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed in particular when a stationary distribution is considered the timescale distribution is uniquely determined as a function related to the fundamental solution of the space time fractional diffusion equation in contrast when the non stationary case is considered the timescale distribution is no longer unique two distributions are here computed one related to the m wright mainardi function which is green s function of the time fractional diffusion equation and another related to the mittag leffler function which is the solution of the fractional relaxation equation

the role of the fox wright functions in fractional sub diffusion of distributed order the fundamental solution of the fractional diffusion equation of distributed order in time usually adopted for modelling sub diffusion processes is obtained based on its mellin barnes integral representation such solution is proved to be related via a laplace type integral to the fox wright functions a series expansion is also provided in order to point out the distribution of time scales related to the distribution of the fractional orders the results of the time fractional diffusion equation of a single order are also recalled and then re obtained from the general theory

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena