a fast finite difference method for distributed order space fractional partial differential equations on convex domains abstract fractional partial differential equations pdes provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long range spatial interactions which cannot be modeled accurately by classical second order diffusion equations however numerical methods for space fractional pdes usually generate dense or full stiffness matrices for which a direct solver requires o n 3 computations per time step and o n 2 memory where n is the number of unknowns the significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three dimensional space fractional diffusion equations computationally intractable fast numerical methods were previously developed for space fractional pdes on multidimensional rectangular domains without resorting to lossy compression but rather via the exploration of the tensor product form of the toeplitz like decompositions of the stiffness matrices in this paper we develop a fast finite difference method for distributed order space fractional pdes on a general convex domain in multiple space dimensions the fast method has an optimal order storage requirement and almost linear computational complexity without any lossy compression numerical experiments show the utility of the method

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

generalized langevin equation with a three parameter mittag leffler noise abstract the relaxation functions for a given generalized langevin equation in the presence of a three parameter mittag leffler noise are studied analytically the results are represented by three parameter mittag leffler functions exact results for the velocity and displacement correlation functions of a diffusing particle are obtained by using the laplace transform method the asymptotic behavior of the particle in the short and long time limits are found by using the tauberian theorems it is shown that for large times the particle motion is subdiffusive for 1 and superdiffusive for many previously obtained results are recovered due to the many parameters contained in the noise term the model considered in this work may be used to improve the description of data and to model anomalous diffusive processes in complex media

an analysis of numerical errors in large eddy simulations of turbulence the reliability of numerical simulations of turbulence depend on our ability to quantify and control discretization errors in the classical literature on error analysis typically simple linear equations are studied estimates of errors derived from such analyses depend on the assumption that each dependent variable can be characterized by a unique amplitude and scale of spatial variation that can be normalized to unity this assumption is not valid for strongly nonlinear problems such as turbulence where nonlinear interactions rapidly redistribute energy resulting in the appearance of a broad continuous spectrum of amplitudes in such situations the numerical error as well as the subgrid model can change with grid spacing in a complicated manner that cannot be inferred from the results of classical error analysis in this paper a formalism for analyzing errors in such nonlinear problems is developed in the context of finite difference approximations for the navier stokes equations when the flow is fully turbulent analytical expressions for the power spectra of these errors are derived by exploiting the joint normal approximation for turbulent velocity fields these results are applied to large eddy simulation of turbulence to obtain quantitative bounds on the magnitude of numerical errors an assessment of the significance of these errors in made by comparing their magnitudes with that of the nonlinear and subgrid terms one method of controlling the errors is suggested and its effectiveness evaluated through quantitative measures although explicit evaluations are presented only for large eddy simulation the expressions derived for the power spectra of errors are applicable to direct numerical simulation as well

pattern formation in a fractional reaction diffusion system abstract we investigate pattern formation in a fractional reaction diffusion system by the method of computer simulation of the model of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and space fractional derivatives we further compare the patterns obtained by computer simulation with those obtained by simulation of the similar system without fractional derivatives as a result we are able to show that nonlinearity plays the main role in structure formation and fractional derivative terms change the transient dynamics so when the order of time derivative increases and approaches the value of 1 5 the special structure formation switches to homogeneous oscillations in the case of space fractional derivatives the decrease of the order of these derivatives leads to more contrast dissipative structures the variational principle is used to find the approximate solution of such fractional reaction diffusion model in addition we provide a detailed analysis of the characteristic dissipative structures in the system under consideration

stability of quaternionic linear systems the main goal of this paper is to characterize stability and bounded input bounded output bibo stability of quaternionic dynamical systems after defining the quaternion skew field algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated having established these results the smith and the smith mcmillan forms of quaternionic matrices are introduced and studied finally all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework

stochastic solution of space time fractional diffusion equations classical and anomalous diffusion equations employ integer derivatives fractional derivatives and other pseudodifferential operators in space in this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are mittag leffler type this leads to explicit solutions for space time fractional diffusion equations with multiscaling space fractional derivatives and additional insight into the meaning of these equations

correlation functions for the fractional generalized langevin equation in the presence of internal and external noise we study generalized fractional langevin equations in the presence of a harmonic potential general expressions for the mean velocity and particle displacement the mean squared displacement position and velocity correlation functions as well as normalized displacement correlation function are derived we report exact results for the cases of internal and external friction that is when the driving noise is either internal and thus the fluctuation dissipation relation is fulfilled or when the noise is external the asymptotic behavior of the generalized stochastic oscillator is investigated and the case of high viscous damping overdamped limit is considered additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed in addition the cases of a constant external force and the force free case are obtained the validity of the generalized einstein relation for this process is discussed the considered fractional generalized langevin equation may be used to model anomalous diffusive processes including single file type diffusion

quaternionic differential operators motivated by a quaternionic formulation of quantum mechanics we discuss quaternionic and complex linear differential equations we touch only a few aspects of the mathematical theory namely the resolution of the second order differential equations with constant coefficients we overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients the resolution of the complex linear schrodinger equation in the presence of quaternionic potentials represents an interesting application of the mathematical material discussed in this paper

dynamics of reaction diffusion systems in a subdiffusive regime in this paper we examine the dynamics of reaction diffusion systems with fractional time derivatives it is shown that in these conditions diffusion is anomalous in the sense that the mean square displacement r 2 ensuremath sim t ensuremath gamma where ensuremath gamma l1 a situation known as subdiffusion we study the conditions for the appearance of a diffusion driven instability and show that the restrictive conditions for a turing instability are relaxed this implies that systems whose kinetics are not of the activator inhibitor kind can have a turing instability and a modulated final state we demonstrate our results with numerical calculations in two dimensions using a generic turing model

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

real linear quaternionic differential operators the renewed interest in searching for quaternionic deviations of standard complex quantum mechanics resulted in the last years in a better understanding of the quaternionic mathematical tools needed to solve quantum mechanical problems in particular a relevant progress has been achieved in solving eigenvalue problems and differential equations for quaternionic operators the practical methods recently proposed to solve quaternionic and complex linear second order differential equations with constant coefficients represent a fundamental starting point to discuss quaternionic potentials in quantum mechanics and study possible violations from complex theories nevertheless only for a restricted class of real linear quaternionic differential operators namely symmetric operators the solution of differential problems was given in this paper we study real linear quaternionic differential equations the proposed resolution s method is based on the jordan canonical form of real linear quaternionic matrices

on fractional diffusion and continuous time random walks abstract a continuous time random walk model is presented with long tailed waiting time density that approaches a gaussian distribution in the continuum limit this example shows that continuous time random walks with long time tails and diffusion equations with a fractional time derivative are in general not asymptotically equivalent

on hilfer s objection to the fractional time diffusion equation abstract hilfer physica a 329 2003 35 claims to give an example of a continuous time random walk ctrw model with long tailed waiting time probability density that approaches a gaussian behavior in the continuum limit rigorous limit theorems derived previously show however that in the limit of long time such a ctrw converges to a non gaussian behavior we discuss two types of continuum limits for the ctrw model the fractional continuum limit and the one introduced by hilfer we show that the fractional limit yields the correct long time behavior of the ctrw while hilfer s continuum limit does not we discuss a general approach to find a continuum limit of the ctrw process

representation of solutions and large time behavior for fully nonlocal diffusion equations abstract we study the cauchy problem for a nonlocal heat equation which is of fractional order both in space and time we prove four main theorems i a representation formula for classical solutions ii a quantitative decay rate at which the solution tends to the fundamental solution iii optimal l 2 decay of mild solutions in all dimensions iv l 2 decay of weak solutions via energy methods the first result relies on a delicate analysis of the definition of classical solutions after proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of ii next we use fourier analysis techniques to obtain the optimal decay rate for mild solutions here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions consequently the decay rate does not anymore improve when the dimension increases the theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis finally we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation our main idea is to show that the l 2 norm is actually a subsolution to a purely time fractional problem which allows us to use the known theory to obtain the result

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

analytic plane wave solutions for the quaternionic potential step by using the recent mathematical tools developed in quaternionic differential operator theory we solve the schrodinger equation in the presence of a quaternionic step potential the analytic solution for the stationary states allows one to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart a brief discussion on reflected and transmitted times performed by using the stationary phase method and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented the analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem up to now solved by numerical method

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

right eigenvalues for quaternionic matrices a topological approach abstract we apply the lefschetz fixed point theorem to show that every square matrix over the quaternions has right eigenvalues we classify them and discuss some of their properties such as an analogue of jordan canonical form and diagonalization of elements of the compact symplectic group sp n

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

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

quaternionic eigenvalue problem we discuss the right eigenvalue equation for h c and r linear quaternionic operators the possibility to introduce an isomorphism between these operators and real complex matrices allows us to translate the quaternionic problem into an equivalent real or complex counterpart interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

effects of operator splitting and low mach number correction in turbulent mixing transition simulations abstract transition and turbulence decay with the taylor green vortex have been effectively used to demonstrate emulation of high reynolds number r e physical dissipation through numerical convective effects of various non oscillatory finite volume algorithms for implicit large eddy simulation iles e g using the godunov based eulerian adaptive mesh refinement code xrage the inverse chevron shock tube experiment simulations have been also used to assess xrage based iles for shock driven turbulent mixing compared with available simulation and laboratory data the previous assessments are extended to evaluate new directionally unsplit high order algorithms in xrage including a correction to address the well known issue of excessive numerical diffusion of shock capturing e g godunov type schemes for low mach numbers the unsplit options for hydrodynamics in xrage are discussed in detail followed by fundamental tests with representative shock problems basic issues of transition to turbulence and turbulent mixing are discussed and results of simulations of high r e turbulent flow and mixing in canonical test cases are reported compared to the directional split cases and for each grid resolution considered unsplit results exhibit transition to turbulence with much higher effective r e and significantly more so with the low mach number correction

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

generalized langevin equation with tempered memory kernel abstract we study a generalized langevin equation for a free particle in presence of a truncated power law and mittag leffler memory kernel it is shown that in presence of truncation the particle from subdiffusive behavior in the short time limit turns to normal diffusion in the long time limit the case of harmonic oscillator is considered as well and the relaxation functions and the normalized displacement correlation function are represented in an exact form by considering external time dependent periodic force we obtain resonant behavior even in case of a free particle due to the influence of the environment on the particle movement additionally the double peak phenomenon in the imaginary part of the complex susceptibility is observed it is obtained that the truncation parameter has a huge influence on the behavior of these quantities and it is shown how the truncation parameter changes the critical frequencies the normalized displacement correlation function for a fractional generalized langevin equation is investigated as well all the results are exact and given in terms of the three parameter mittag leffler function and the prabhakar generalized integral operator which in the kernel contains a three parameter mittag leffler function such kind of truncated langevin equation motion can be of high relevance for the description of lateral diffusion of lipids and proteins in cell membranes

a priori bounds for degenerate and singular evolutionary partial integro differential equations we study quasilinear evolutionary partial integro differential equations of second order which include time fractional p laplace equations of time order less than one by means of suitable energy estimates and de giorgi s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems we also show that a maximum principle is valid for such equations