effects of small permanent charge and channel geometry on ionic flows via classical poisson nernst planck models in this work we examine effects of permanent charges on ionic flows through ion channels via a quasi one dimensional classical poisson nernst planck pnp model the geometry of the three dimensional channel is presented in this model to a certain extent which is crucial for the study in this paper two ion species one positively charged and one negatively charged are considered with a simple profile of permanent charges zeros at the two end regions and a constant q 0 over the middle region the classical pnp model can be viewed as a boundary value problem bvp of a singularly perturbed system the singular orbit of the bvp depends on q 0 in a regular way assuming q 0 is small a regular perturbation analysis is carried out for the singular orbit our analysis indicates that effects of permanent charges depend on a rich interplay between boundary conditions and the channel geometry furthermore interesting common features are revealed for q 0 0 only an average quantity of the channel

biased random walks on random combs we develop rigorous analytic techniques to study the behaviour of biased random walks on combs this enables us to calculate exactly the spectral dimension of random comb ensembles for any bias scenario in the teeth or spine two specific examples of random comb ensembles are discussed the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power law distribution of tooth lengths we also analyse transport properties along the spine for these probability measures

some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

error analysis of finite element method for poisson nernst planck equations in this paper we study the a priori error estimates of finite element method for the system of time dependent poisson nernst planck equations and for the first time we obtain its optimal error estimates in l h 1 and l 2 h 1 norms and suboptimal error estimates in l l 2 norm with linear element and optimal error estimates in l l 2 norm with quadratic or higher order element for both semi and fully discrete finite element approximations numerical experiments are also given to validate the theoretical results

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

mathematical studies of poisson nernst planck model for membrane channels finite ion size effects without electroneutrality boundary conditions abstract we study a quasi one dimensional steady state poisson nernst planck model with a local hard sphere potential for ionic flows of two oppositely charged ion species through a membrane channel of particular interest are qualitative properties of ionic flows in terms of individual fluxes j k without assuming electroneutrality boundary conditions for the first order approximation j k 1 in diameter of charged particle of j k our result shows that the quantities v j k 1 and v 2 j k 1 that play critical roles in characterizing ion size effects on ionic flows can be both negative depending sensitively on boundary concentrations and relative ion valences while they are always positive under electroneutrality conditions this new observation indicates that for fixed ion species opposite finite size effects on ionic flows can occur determined further by boundary concentrations numerical simulation is performed to detect some critical potentials closely related to ion size effects numerical simulation results are consistent with our analytical predictions

exchange lemmas for singular perturbation problems with certain turning points abstract in this work singular perturbation problems with a certain type of turning point are studied from a geometric point of view we first describe the delay phenomenon initially studied by pontryagin of dynamics near the slow manifold possessing the turning points based on the local properties of such turning points and the center manifold theory for general invariant manifolds developed by chow liu and yi we extend the well known exchange lemma first formulated by jones and kopell for problems with normally hyperbolic slow manifolds to problems with this type of turning point applications to singular boundary value problems with turning points are discussed

qualitative properties of steady state poisson nernst planck systems mathematical study we examine qualitative properties of solutions of self consistent poisson nernst planck systems including uniqueness in the case of vanishing permanent charge the predominant case studied our results unveil a rich structure inherent in these systems one that is determined by the boundary conditions and the signs of the oppositely charged carrier fluxes a particularly significant special case that of simple boundary conditions is shown to lead to uniqueness and to a complete characterization this case underlies the more complicated cases studied later a contraction mapping principle is included for completeness and allows for an arbitrary permanent charge distribution

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

1 2 order fractional fokker planck equation on comblike model from the generalized scheme of random walks on the comblike structure it is shown how a 1 2 order fractional fokker planck equation can be derived the operator method for the moments associated with the distribution function p x t is used to solve the resulting equation also the anomalous diffusion along the backbone of the structure has been considered

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

a poisson nernst planck model for biological ion channels an asymptotic analysis in a three dimensional narrow funnel we wish to predict ionic currents that flow through narrow protein channels of biological membranes in response to applied potential and concentration differences across the channel when some features of channel structure are known we propose to apply singular perturbation analysis to the coupled poisson nernst planck equations which are the basic continuum model of ionic permeation and semiconductor physics in semiconductor physics the problem is a singular perturbation because the ratio of the debye length to the width of the channel is a very small parameter that multiplies the laplacian term in the poisson equation in contrast to semiconductors the atomic scale geometry of narrow ion channels sometimes makes this ratio a large parameter which surprisingly renders the problem a singular perturbation in a different sense we construct boundary layers and match them asymptotically across the different regions of the channel to derive good approximations for fick s and ohm s laws our aim is to e

diffusion on a random comb distribution function of the survival probability we investigate the distribution functionq p describing the survival probability on a comb consisting of a backbone with lateral randomly disconnected infinite branches two different regimes are analyzed in some detail i at short times q p is shown to have a self similar structure devil s staircase ii at large times this function becomes smooth and tends toward a rather well defined unit step function the disorder averaged survival probability is expected to decrease ast 3 4 at large times whereas the relative fluctuations of the sample dependentp0 t display a very slow decay in time going to zero liket 1 8

discrete and continuous maximum principles for parabolic and elliptic operators the major qualitative properties of linear parabolic and elliptic operators pdes are the different maximum principles mps another important property is the stabilization property sp which connects these two types of operators pdes this means that under some assumptions the solution of the parabolic pde tends to an equilibrium state when t which is the solution of the corresponding elliptic pde to solve pdes we need to use some numerical methods and it is a natural requirement that these qualitative properties are preserved on the discrete level in this work we investigate this question when a two level discrete mesh operator is used as the discrete model of the parabolic operator which is a one step numerical procedure for solving the parabolic pde and a matrix as a discrete elliptic operator which is a linear algebraic system of equations for solving the elliptic pde we clarify the relation between the discrete parabolic maximum principle dpmp the discrete elliptic maximum principle demp and the discrete stabilization property dsp the main result is that the dpmp implies the dsp and the demp

analytic study of a model of diffusion on a random comblike structure abstract we present an analytic study of a model of diffusion on a random comblike structure in which no bias field does exist neither along the backbone nor along the branches for any given disordered structure our analytic treatment allows to compute in an exact manner the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone and of the particle position and dispersion the expressions of these quantities are shown to coincide asymptotically with those derived in simple mean field treatments the results for any given sample do not depend on the particular configuration self averaging property the behaviours strongly depend on the distribution of the lengths of the branches with an exponential distribution transport is normal while anomalous diffusion may take place for a power law distribution when long branches are present with a sufficiently high weight

density functional theory of charged hard sphere fluids an approximate electrostatic es excess free energy functional for charged hard sphere fluids is presented this functional is designed for systems with large density variations but may also be applied to systems without such variations based on the rosenfeld method of perturbation about a bulk homogeneous reference fluid y rosenfeld j chem phys 98 8126 1993 the new es functional replaces the reference fluid densities with a functional of the particle densities called the rfd functional the first order direct correlation function dcf in the particle densities is computed using as input the first and second order dcfs in rho i x the inhomogeneous densities defined by the rfd functional because this formulation imposes no a priori constraints on the form of the rfd functional it is valid for any choice of rho i x the rfd functional may be chosen 1 so that the input dcfs that is dcfs in rho i x may be approximated and 2 so the combination of rho i x and input dcfs yields a good estimate of the first order dcf in the particle densities in this way the general problem of finding the excess free energy functional has been replaced by the specific problem of choosing a rfd functional we present a particular rfd functional that together with bulk formulations for the input dcfs accurately reproduces the results of monte carlo simulations

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

qualitative properties of steady state poisson nernst planck systems perturbation and simulation study poisson nernst planck pnp systems are considered in the case of vanishing permanent charge a detailed case study based on natural categories described by system boundary conditions and flux is carried out via simulation and singular perturbation analysis our results confirm the rich structure inherent in these systems a natural quantity the quotient of the debye and characteristic length scales serves as the singular perturbation parameter the regions of validity are carefully analyzed by critical comparisons and contrasts between the simulation and the perturbation solution which can be represented in closed form

a bifurcation analysis of the one dimensional steady state semiconductor device equations this paper considers the one dimensional steady state semiconductor device equations modeling a thyristor these equations can be considered as a singularly perturbed boundary value problem the reduced problem is considered in order to derive an approximation to the current voltage curve a bifurcation equation is obtained that completely determines the qualitative structure of the current voltage characteristic a numeric example shows that the developed theory applies to data of realistic magnitude

qualitative properties of discrete nonlinear parabolic operators this paper is devoted to the qualitative properties of discretized parabolic operators such as nonnegativity and nonpositivity preservation maximum minimum principles and maximum norm contractivity in the linear case earlier papers of the authors farago and horvath in siam sci comput 28 2313 2336 2006 ima j numer anal 29 606 631 2009 have established the connections between the above qualitative properties and have given sufficient conditions for their validity the present paper extends the above results to nonlinear discretized parabolic operators also motivated by the authors recent paper farago and horvath in j math anal appl 448 473 497 2017 which has given related results on the continuous pde level a systematic study is presented ranging from general discrete mesh operators to proper finite element applications

multiple steady states in one dimensional electrodiffusion with local electroneutrality this paper concerns itself with construction and study of multiple steady states occurring in one dimensional ambipolar electrodiffusion with local electroneutrality constant and equal charge carrier mobilities in the absence of generation recombination piecewise constant fixed change density with four sign alterations is assumed multiple solution branches are constructed via a numerical and asymptotic solution of a system of nonlinear equations resulting from an explicit integration of the appropriate boundary value problem the parameter range in which multiplicity occurs is determined with the aid of asymptotic procedures existence of the appropriate multiple solutions follows from the newton kantorovich theorem

regular splittings and monotone iteration functions in this paper we introduce the set of so called monotone iteration functions mi functions belonging to a given function we prove necessary and sufficient conditions in order that a given mi function is in a precisely defined sense at least as fast as a second one regular splittings of a function which were initially introduced for linear functions by r s varga in 1960 are generating mi functions in a natural manner for linear functions every mi function is generated by a regular splitting for nonlinear functions however this is generally not the case

field induced drift and trapping in percolation networks the authors study the effects of a bias producing external field on a random walk on the infinite cluster in the percolation problem there are two competing physical effects drift and trapping which result in a drift velocity nu which rises and then falls as the field increases they study these effects on a one dimensional lattice with random length branches and on the diluted bethe lattice they calculate nu in the first model with a maximum allowed length for each branch in the limit that this cutoff length becomes infinite they find that the velocity vanishes identically above a finite critical value of the field for the bethe lattice they derive an upper bound on the critical field which varies as p pc 1 2 as the percolation concentration pc is approached in the one dimensional model they also investigate the anomalous regime in which the velocity vanishes they discuss the distribution of steady state times required to traverse n sites and find that it can be described in terms of a stable distribution of index x with superimposed oscillations the index x of the stable distribution is given by l xi where xi is a characteristic branch length and l is a bias induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch

singular perturbation solutions of steady state poisson nernst planck systems we study the poisson nernst planck pnp system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges assuming point charges and that the debye length is small relative to the domain size we derive an asymptotic formula for the steady state solution by matching outer and boundary layer solutions the case of two ionic species has been extensively studied the uniqueness of the solution has been proved and an explicit expression for the solution has been obtained however the case of three or more ions has received significantly less attention previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations by adopting a methodology that preserves the symmetries of the pnp system we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation due to the simple form of the transcendental equation it can be solved numerically in a straightforward manner our methodology thus provides a standard procedure for solving the pnp system and we illustrate this by solving some practical examples despite the fact that for three ions previous studies have indicated that multiple solutions may exist we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three ion case

a complete analysis of a classical poisson nernst planck model for ionic flow abstract in this work we examine the stationary one dimensional classical poisson nernst planck cpnp model for ionic flow a singularly perturbed boundary value problem bvp for the case of zero permanent charge we provide a complete answer concerning the existence and uniqueness of the bvp the analysis relies on a number of ingredients a geometric singular perturbation framework for a reduction to a singular bvp a reduction of the singular bvp to a matrix eigenvalue problem a relation between the matrix eigenvalues and zeros of a meromorphic function and an application of the cauchy argument principle for identifying zeros of the meromorphic function once the zeros of the meromorphic function in a stripe are determined an explicit solution of the singular bvp is available it is expected that this work would be useful for studies of other pnp systems

transport properties on a random comb we study the random walk of a particle in a random comb structure both in the presence of a biasing field and an the field free case we show that the mean field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk ctrw on the backbone of the comb with a definite waiting time density we find an exact expression for this central quantity the green function for the ctrw is then obtained its first and second moments determine the drift and diffusion at all times we show that the drift velocity v vanishes asymptotically for power law and stretched exponential distributions of branch lengths on the comb whatever be the biasing field strength for an exponential branch length distribution v is a nonmonotonic function of the bias increasing initially to a maximum and then decreasing to zero at a critical value in the field free case anomalous diffusion occurs for a range of power law distributions of the branch length the corresponding exponent for the mean square displacement is obtained as is the asymptotic form of the positional probability distribution for the random walk we show that normal diffusion occurs whenever the mean branch length is finite and present a simple formula for the effective diffusion constant these results are extended to regular nonrandom combs as well the physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time on a finite chain that controls the effective waiting time density of the ctrw

one dimensional steady state poisson nernst planck systems for ion channels with multiple ion species abstract the one dimensional poisson nernst planck pnp system is a basic model for ion flow through membrane channels if the debye length is much smaller than the characteristic radius of the channel the pnp system can be treated as a singularly perturbed system we provide a geometric framework for the study of the steady state pnp system involving multiple types of ion species with multiple regions of piecewise constant permanent charge special structures of this particular problem are revealed which together with the general framework allows one to reduce the existence and multiplicity of singular orbits to a system of nonlinear algebraic equations near each singular orbit an application of the exchange lemma from the geometric singular perturbation theory gives rise to the existence and local uniqueness of a solution of the singular boundary value problem a new phenomenon on multiplicity and spatial behavior of steady states involving three or more types of ion species is discovered in an example the phenomenon cannot occur when only two types of ion species are involved

conditions for strict inequality in comparisons of spectral radii of splittings of different matrices we present new comparison theorems for the spectral radii of matrices arising from splittings of different matrices under nonnegativity assumptions our focus is on establishing strict inequalities of the spectral radii without imposing strict inequalities of the matrices but we also obtain new results for nonstrict inequalities of the spectral radii we emphasize two different approaches one combinatorial and the other analytic and discuss their merits in the light of the results obtained we try to get fairly general results and indicate by counter examples that some of our hypotheses cannot be relaxed in certain directions

discrete maximum principle and adequate discretizations of linear parabolic problems in this paper we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with dirichlet type boundary condition first we formulate the qualitative properties for the differential equations and shed light on their relations then we show how the well known discretization schemes can be written in the form of a one step iterative process we give necessary and sufficient conditions of the main qualitative properties of these iterations we apply the results to the finite difference and galerkin finite element solutions of linear parabolic problems in our main result we show that the nonnegativity preservation property is equivalent to the maximum minimum principle and they imply the maximum norm contractivity in one two and three dimensions we list sufficient a priori conditions that ensure the required qualitative properties finally we demonstrate the above results on numerical examples

diffusion on random comblike structures field induced trapping effects an analytic study of diffusion in random comblike structures in which field induced trapping effects enter into play is presented in order to mimic at best a percolation cluster above pc the distribution of the lengths of the branches is taken as exponential the branches act as traps of efficiency growing with the applied bias the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone and of its position and dispersion are computed in an exact manner as a result when the field exceeds a certain critical value anomalous behaviours emerge characterized by exponents depending continuously on the field

qualitative analysis of one step iterative methods and consistent matrix splittings a qualitative analysis of one step iterative methods is presented with special regard to the connection between concavity preservation and time monotonicity we also analyze the relation of one step iterative methods to matrix splitting methods

geometric singular perturbation approach to steady state poisson nernst planck systems boundary value problems of a one dimensional steady state poisson nernst planck pnp system for ion flow through a narrow membrane channel are studied by assuming the ratio of the debye length to a characteristic length to be small the pnp system can be viewed as a singularly perturbed problem with multiple time scales and is analyzed using the newly developed geometric singular perturbation theory within the framework of dynamical systems the global behavior is first studied in terms of limiting fast and slow systems it is rather surprising that a complete set of integrals is discovered for the nonlinear limiting fast system this allows a detailed description of the boundary layers for the problem the slow system itself turns out to be a singularly perturbed one too which indicates that the singularly perturbed pnp system has three different time scales a singular orbit zeroth order approximation of the boundary value problem is identified based on the dynamics of limiting fast and slow

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

a note on comparison theorems for nonnegative matrices in a recent paper 4 csordas and varga have unified and extended earlier theorems of varga in 10 and wo nicki in 11 on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings the main purpose of this note is to show a connection between the csordas varga paper and a paper by beauwens 1 in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular monotonic norms already used in 1 play an important role in our work here

discrete maximum principle for higher order finite elements in 1d we formulate a sufficient condition on the mesh under which we prove the discrete maximum principle dmp for the one dimensional poisson equation with dirichlet boundary conditions discretized by the hp fem the dmp holds if a relative length of every element k in the mesh is bounded by a value h rel p 0 9 1 where p 1 is the polynomial degree of the element k the values h rel p are calculated for 1 p 100

comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same nonsingular matrix have been compared in the works of many authors in this paper new comparison theorems which extend song s results y song numer math 65 1993 245 252 for weak splittings of the same or different types are presented these new comparison theorems are derived by using nonnegativity in respect to a proper cone k n ew converses which partially solve an open question asked by v a miller m neumann numer math 47 1985 427 434 are also presented 1999 elsevier science inc all rights reserved

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

largest well posed spaces for the general diffusion system with nonlocal interactions abstract the authors derive a general diffusion gd system with nonlocal interactions of special structure via energetic variational approach and observe that there exist two critical values of s i e s 1 2 1 for the nonlocal interactions where s 1 2 reflects how strong nonlocal property we have and s 1 affects the linearization and choice of initial data spaces the authors also establish the global existence and uniqueness of mild solution

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

a finite element approximation theory for the drift diffusion semiconductor model two sided estimates are derived for the approximation of solutions to the drift diffusion steady state semiconductor device system which are identified with fixed points of gummel s solution map the approximations are defined in terms of fixed points of numerical finite element discretization maps by use of a calculus developed by krasnosel skii and his coworkers it is possible both to locate approximations near fixed points in an a priori manner as well as fixed points near approximations in an a posteriors manner these results thus establish a nonlinear approximation theory in the energy norm with rate keyed to what is possible in a standard linear theory this analysis provides a convergence theory for typical computational approaches in current use for semiconductor simulation

gevrey regularity of mild solutions to the parabolic elliptic system of drift diffusion type in critical besov spaces abstract this paper is devoted to studying analyticity of mild solutions to the parabolic elliptic system of drift diffusion type with small initial data in critical besov spaces more precisely by using multilinear singular integrals and fourier localization argument we show that global in time mild solutions are gevrey regular i e e t 1 v e t 1 w l t b p r 2 n p r n l t 1 b p r n p r n for all 1 p 2 n and 1 r where 1 is the fourier multiplier whose symbol is given by 1 i 1 n i as a corollary we obtain decay estimates of mild solutions in critical besov spaces without using cumbersome recursive estimation of higher order derivatives

numerical algorithms based on galerkin methods for the modeling of reactive interfaces in photoelectrochemical pec solar cells abstract this work concerns the numerical solution of a coupled system of self consistent reaction drift diffusion poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical pec solar cells with reactive semiconductor and electrolyte interfaces we present three numerical algorithms mainly based on a mixed finite element and a local discontinuous galerkin method for spatial discretization with carefully chosen numerical fluxes and implicit explicit time stepping techniques for solving the time dependent nonlinear systems of partial differential equations we perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model

exact results and self averaging properties for random random walks on a one dimensional infinite lattice we present new exact results for a one dimensional asymmetric disordered hopping model the lattice is taken infinite from the start and we do not resort to the periodization scheme used by derrida an explicit resummation allows for the calculation of the velocityv and the diffusion constantd which are found to coincide with those given by derrida and for demonstrating thatv is indeed a self averaging quantity the same property is established ford in the limiting case of a directed walk