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 the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

mode coupling and tagged particle correlation functions the stokes einstein law abstract in this paper a mode coupling theory for equilibrium tagged particle correlation functions in simple fluids is developed from molecular considerations the mode coupling formalism developed in a previous paper physica a 181 1992 89 to describe equilibrium fluctuations in simple liquids is generalized to include tagged particle correlations the mode coupling formalism and n ordering approximation scheme used in the previous paper allow a series for the generalized diffusion constant to be obtained which is exact in the thermodynamic limit the stokes einstein law for a slip brownian particle is then derived from this series in a systematic fashion the techniques applied here obviate the need to assume that the simple liquid is incompressible and allow a self consistent series of equations for the diffusion constant of the brownian particle and transport coefficients of the fluid to be obtained which are useful in the study of supercooled liquids

mode coupling and generalized hydrodynamics in this paper a general hydrodynamic mode coupling theory of equilibrium fluctuations in simple liquids is developed from molecular considerations the approach developed hem avoids the shortcomings of previous mode coupling theories by adopting a complete hierarchy of equations for the slow modes of a hydrodynamic system and solving it formally through the n ordering approximation scheme developed previously by machta and oppenheim a series is obtained from the hierarchy of equations which allows the generalized transport coeffcicnts to be obtained exactly in the thermodynamic limit up to arbitrary order in the wavevector frequency and mode coupling parameters a self consistent equation for the transport coefficients is formulated up to burnett order in the wavevector and all orders in the mode coupling parameter the results arc similar to phenomenological models used by previous researchers

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

analysis of switched diversity systems on generalized fading channels the performances of switched diversity systems operating on generalized nakagami fading channels are analyzed using a discrete time model the average bit error rate ber of binary noncoherent frequency shift keying ncfsk on slow nonselective nakagami fading channels is derived closed form expressions that can be used to determine optimum switching thresholds in a minimum error rate sense are also derived in addition the use of optimum fixed thresholds is considered it is found that a considerable amount of diversity gain can be obtained using an optimum fixed rather than adaptive switching threshold results are obtained for both independent and correlated nakagami fading branch signals the effects of fading severity and the correlation coefficient on both the ber and on the optimum switching threshold are investigated it is shown that useful diversity gain can be obtained with power correlation coefficients as high as 0 9 when the fading is strong the results for a rayleigh channel are obtained and presented as a special case of generalized fading model

unified analysis of switched diversity systems in independent and correlated fading channels the moment generating function mgf of the signal power at the output of dual branch switch and stay selection diversity ssd combiners is derived the first order derivative of the mgf with respect to the switching threshold is also derived these expressions are obtained for the general case of correlated fading and nonidentical diversity branches and hold for any common fading distributions e g rayleigh nakagami m rician nakagami q the mgf yields the performance bit or symbol error probability of a broad class of coherent differentially coherent and noncoherent digital modulation formats with ssd reception the optimum switching threshold in a minimum error rate sense is obtained by solving a nonlinear equation which is formed by using the first order derivative of the mgf this nonlinear equation can be simplified for several special cases for independent and identically distributed diversity branches the optimal switching threshold in closed form is derived for three generic forms of the conditional error probability for correlated rayleigh or nakagami m fading with identical branches the optimal switching threshold in closed form is derived for the noncoherent binary modulation formats we show previously published results as special cases of our unified expression selected numerical examples are presented and discussed

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

performance of two dual branch postdetection switch and stay combining schemes in correlated rayleigh and rician fading the performance of binary frequency shift keying bfsk and m ary frequency shift keying mfsk with dual branch postdetection switch and stay combining ssc in correlated rayleigh and rician fading is studied two postdetection ssc receivers are considered and the performances of noncoherent bfsk and mfsk are analyzed closed form expressions are derived for the average bit error rate ber of bfsk and mfsk with postdetection ssc in correlated rayleigh and rician fading optimum switching thresholds that minimize the average ber are obtained extensive monte carlo simulations are presented to test the validity of the analytical results the performance of dual branch postdetection ssc is compared with the performance of dual branch predetection ssc the effects of correlation fading parameter average fading power imbalance and switching threshold on the performances of postdetection ssc receivers are examined

signaling constellations for fading channels the performance of various coherent 8 ary and 16 ary modulations in additive white gaussian noise awgn and slowly fading channels are analyzed new expressions for the exact symbol error rates ser in fading with diversity combining are derived for any two dimensional signaling format having polygonal decision boundaries maximal ratio combining equal gain combining and selection combining are considered the ser formulas obtained make it possible for the first time to optimize parameters of various constellations precisely and to determine which constellation has the lowest probability of error for example a star constellation such as that specified in the ccitt v 29 standard can be improved by adjusting the amplitude ratios of the points in the constellation to save about 0 63 db power in awgn without sacrificing the phase error tolerance while maintaining the same error rate the sensitivity of each constellation to phase error is presented and comparisons are made six 8 ary signal sets and 11 16 ary signal sets are examined using the new symbol error probability formulas to determine best signal sets for fading channels

performance analysis of multibranch switched diversity systems we investigate the performance of multibranch switched diversity systems specifically we first derive generic formulas for the cumulative distribution function probability density function and moment generating function of the combined signal power for both switch and stay combining ssc and switch and examine combining sec schemes we then capitalize on these expressions to obtain closed form expressions for the outage probability and average error rate for various practical communication scenarios of interest as a byproduct of our analysis we prove that for ssc with identically distributed and uniformly correlated branches increasing the number of branches to more than two does not improve the performance but the performance can be different in the case the branches are not identically distributed and or not uniformly correlated we also show that in general the sec performance improves with additional branches the mathematical formalism is illustrated with a number of selected numerical examples

a general method for calculating error probabilities over fading channels signal fading is a ubiquitous problem in mobile and wireless communications in digital systems fading results in bit errors and evaluating the average error rate under fairly general fading models and multichannel reception is often required predominantly to date most researchers perform the averaging using the probability density function method or the moment generating function mgf method this paper presents a third method called the characteristic function chf method for calculating the average error rates and outage performance of a broad class of coherent differentially coherent and noncoherent communication systems with or without diversity reception in a myriad of fading environments unlike the mgf technique the proposed chf method based on parseval s theorem enables us to unify the average error rate analysis of different modulation formats and all commonly used predetection diversity techniques i e maximal ratio combining equal gain combining selection diversity and switched diversity within a single common framework the chf method also lends itself to the averaging of the conditional error probability involving the complementary incomplete gamma function and the confluent hypergeometric function over fading amplitudes which heretofore resisted to a simple form as an aside we show some previous results as special cases of our unified framework

brownian motion as a problem of eliminating fast variables abstract the hamilton equations for a brownian particle involve its mass as a large parameter as a consequence its motion is relatively slow and the fast motion of the surrounding fluid molecules can be eliminated by the standard method of eliminating fast variables the result is equivalent to the known langevin equation plus additional higher orders all coefficients are expressed in terms of correlation functions of the microscopic force on the particle

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

mode coupling in nonequilibrium granular flow systems in this paper we examine the role of mode coupling in nonequilibrium granular flow systems and derive mode coupling equations for equal time multi linear correlation functions in general nonequilibrium systems by applying n ordering analysis and projection operator techniques the hierarchy of equations obtained is simplified which permits the equal time multilinear averages to be calculated systematically the linear and equal time bilinear nonequilibrium correlation functions are calculated explicitly for a steady state system characterized by a small linear shear flow and uniform number and internal energy densities in this paper we examine the importance of mode coupling l in nonequilibrium systems and investigate the properties of equal time correlations of multilinear densities in steady state granular systems in recent years there has been intense study of the transport properties of granular fluid systems composed of sand or glass beads most of these studies have been semiphenomenological and involve the concept of the coefficient of restitution which is a measure of the inelasticity of collisions among the granular particles composing the fluid as examples of these studies we mention the analytical treatments of jenkins et al 2 and the simulations by walton et al 3 the inelasticity is essential for the treatment of these systems and for comparisons with experimental data 4 the paper is organized as follows using ideas developed to describe brownian motion dynamics 5 in section 2 we obtain a generalized fokkerplanck equation for the distribution function of the translational degrees of freedom of the granular system and thereby derive an effective time displacement operator which determines the time evolution of the distribution function in section 3 we construct a mode coupling theory for a general system whose dynamics are determined by a time displacement operator 0 by considering a generalization of the local equilibrium distribution to describe the

symmetric stable laws and stable like jump diffusions asymptotic expansions are obtained for finite dimensional symmetric stable distributions they are used to prove the existence of continuous transition probability densities of stable and stable like jump diffusions and to construct local multiplicative asymptotics and global two sided estimates for these densities as a consequence the distribution of the first passage times for stable jump diffusions is estimated and the integral test for the limsup behaviour of their sample paths as t to 0 is provided 1991 mathematics subject classification 60e07 60g17 60j35 47d07

derivation of the nonlinear hydrodynamic equations using multi mode techniques abstract a general mode mode coupling theory is developed for the microscopic mass energy and momentum densities of a simple classical fluid a projection operator method is employed to derive a generalized langevin equation that contains nonlinearities of all orders with both convective and dissipative terms a general nonequilibrium ensemble average which contains local equilibrium as a special case is employed to derive nonlinear transport equations that are nonlocal in both space and time the nonlinear euler and navier stokes equations are recovered using a factorization procedure based on an inverse system size approximation we show that in the context of mode mode coupling theory nonlinearities of all orders must be retained to derive the full nonlinear transport equations we also slow that the space and time dependent nonequilibrium pressure and transport coefficients are functions of the nonequilibrium mass and internal energy densities the thermodynamic closure relationships follow as a natural consequence of mode mode coupling theory for a system linearly displaced from equilibrium we demonstrate the role of the corrections to our factorization approximation in renormalizing the transport coefficients

mode coupling theory of hydrodynamics and steady state systems abstract we construct a formal mode coupling theory for hydrodynamic systems which includes contributions from all powers of the hydrodynamic variables this theory is applied to nonequilibrium steady state systems a generalization of the local equilibrium distribution is used to describe the nonequilibrium state this distribution independently constrains all moments of the hydrodynamic variables the infinite hierarchy of equations for the moments of the hydrodynamic variables is truncated using an inverse system size expansion explicit results are obtained for the time correlation functions of fluids with a linear temperature gradient or a linear shear these results agree with previous studies of these steady states

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

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

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