approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

stability and convergence of a new explicit finite difference approximation for the variable order nonlinear fractional diffusion equation in this paper we consider the variable order nonlinear fractional diffusion equation u x t t b x t xr a x t u x t f u x t where xr a x t is a generalized riesz fractional derivative of variable order a x t 1 a x t 2 and the nonlinear reaction term f u x t satisfies the lipschitz condition f u 1 x t f u 2 x t

the number of runs in a string a run in a string is a nonextendable with the same minimal period periodic segment in a string the set of runs corresponds to the structure of internal periodicities in a string periodicities in strings were extensively studied and are important both in theory and practice combinatorics of words pattern matching computational biology let n be the maximal number of runs in a string of length n it has been shown that n o n the proof was very complicated and the constant coefficient in o n has not been given explicitly we demystify the proof of the linear upper bound for n and propose a new approach to the analysis of runs based on the properties of subperiods the periods of periodic parts of the runs we show that n n and there are at most o 67n runs with periods larger than 87 this supports the conjecture that the number of all runs is smaller than n we also give a completely new proof of the linear bound and discover several new interesting periodicity lemmas

on primary and secondary repetitions in words combinatorial properties of maximal repetitions runs in formal words are studied we classify all maximal repetitions in a word as primary and secondary where the set of all primary repetitions determines all the other repetitions in the word essential combinatorial properties of primary repetitions are established

how many runs can a string contain given a string x x 1 n a repetition of period p in x is a substring u r x i 1 i rp p u r 2 where neither u x i 1 i p nor x i 1 i r 1 p 1 is a repetition the maximum number of repetitions in any string x is well known to be q nlogn a run or maximal periodicity of period p in x is a substring u rt x i 1 i rp t of x where u r is a repetition t a proper prefix of u and no repetition of period p begins at position i of x or ends at position i rp t 1 in 2000 kolpakov and kucherov showed that the maximum number r n of runs in any string x 1 n is o n but their proof was nonconstructive and provided no specific constant of proportionality at the same time they presented experimental data to prompt the conjecture r n

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

anomalous subdiffusion with multispecies linear reaction dynamics we have introduced a set of coupled fractional reaction diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics the model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics the effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms one consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species the general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail we have carried out monte carlo simulations corresponding to these special cases and we find excellent agreement with theory

a fourier method and an extrapolation technique for stokes first problem for a heated generalized second grade fluid with fractional derivative in this article we consider stokes first problem for a heated generalized second grade fluid with fractional derivative sfp hgsgf implicit and explicit numerical approximation schemes for the sfp hgsgf are presented the stability and convergence of the numerical schemes are discussed using a fourier method in addition the solvability of the implicit numerical approximation scheme is also analyzed a richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed finally a numerical test is given the numerical results demonstrate the good performance of our theoretical analysis

solving linear and non linear space time fractional reaction diffusion equations by the adomian decomposition method in this paper we consider linear and non linear space time fractional reaction diffusion equations stfrde on a finite domain the equations are obtained from standard reaction diffusion equations by replacing a second order space derivative by a fractional derivative of order 1 2 and a first order time derivative by a fractional derivative of order 0 1 we use the adomian decomposition method to construct explicit solutions of the linear and non linear stfrde finally some examples are given

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

numerical analysis of the rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives australian research council dp0559807 national natural science foundation of china 10271098 nsf of china tianyuan fund for mathematics 10726061

numerical methods for solving a two dimensional variable order anomalous subdiffusion equation australian research council dp0559807 dp0986766 national natural science foundation of china 10271098 natural science foundation of fujian province 2009j01014

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

space efficient search for maximal repetitions we study here a problem of finding all maximal repetitions in a string of length n we show that the problem can be solved in time o n log n in the presence of constant extra space and general unbounded alphabets subsequently we show that in the model with a constant size alphabet the problem can be solved in time o n with a help of o n extra space previously best known algorithms require linear additional space in both models

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

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

some results on the number of periodic factors in words abstract the exponent of a word is the relation between its length and its minimal period a repetition subrepetition in a word is a factor of exponent not less than 2 less than 2 a gapped repeat in a word is a pair of identical factors with a positive gap between these factors in the paper a review of some recent results concerning the number of repetitions subrepetitions and gapped repeats in words of fixed length is presented