on unifying the concepts of scale instrumentation and stochastics in the development of multiphase transport theory a generalized theory of multiphase transport is presented which combines the concepts of scale instrumentation stochastics and time series with the development of transport equations by defining the filtering process as a convolution of a measure p with a field property we are able to exploit the fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment an ideal instrument is defined which filters out high frequency noise corresponding to short distances and yet does not alter the structure of low frequencies using ideal instruments we successively filter out lower frequency noise in a multiscale multiphase environment formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process an equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed power spectra are developed which relate spectra on different scales to the measuring instrument used by successively applying filtering theorems a hierarchy of multiscale transport equations is developed filtered properties in the transport equations are mass averages different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases the concept of a wide sense stationary ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

two splitting positive definite mixed finite element methods for parabolic integro differential equations abstract two novel mixed finite element procedures are established for parabolic integro differential equations which can be split into two independent symmetric positive definite sub schemes and do not need to solve a coupled system of equations without requiring the lbb consistency condition the convergence analysis shows that both methods lead to the optimal order l 2 norm error estimate for u and optimal h div norm error estimate for a numerical example is presented to illustrate the theoretical analysis

convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered compact difference is used and the integrals are approximated by the composite simpson scheme convergence of the two compact schemes are derived using the energy method compact difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered in this paper fourth order compact difference is used to approximate the second order spatial derivative and the integrals in the boundary conditions are approximated by the composite simpson quadrature formula the backward euler and crank nicolson schemes are presented as the fully discrete schemes error estimates in the discrete h1 and l norms are given by the energy method showing both schemes are fourth order accurate in space and they have first order and second order accuracy in time respectively numerical results are provided to confirm the theoretical results

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

conformal and lie superalgebras motivated from free fermionic fields in this paper we construct six families of conformal superalgebras of infinite type motivated from free quadratic fermonic fields with derivatives and we prove their simplicity the lie superalgebras generated by these conformal superalgebras are proven to be simple except for a few special cases in the general linear superalgebras and the type q lie superalgebras in which these lie superalgebras have a one dimensional centre and the quotient lie superalgebras modulo the centre are simple certain natural central extensions of these families of conformal superalgebras are also given moreover we prove that these conformal superalgebras are generated by their finite dimensional subspaces of minimal weight in a certain sense it is shown that a conformal superalgebra is simple if and only if its generated lie superalgebra does not contain a proper nontrivial ideal with a one variable structure

numerical algorithms for diffusion reaction problems with non classical conditions abstract parabolic equations with nonlocal boundary conditions have been given considerable attention in recent years in this paper new high order algorithms for the linear diffusion reaction problem are derived the convergence of the new schemes is studied and numerical examples are given to show the efficiency of the new methods to solve linear and nonlinear diffusion reaction equations with these non classical conditions

asymptotically one dimensional diffusions on the sierpinski gasket and the abc gaskets diffusion processes on the sierpinski gasket and theabc gaskets are constructed as limits of random walks in terms of the associated renormalization group the present method uses the inverse trajectories which converge to unstable fixed points corresponding to the random walks on one dimensional chains in particular non degenerate fixed points are unnecessary for the construction a limit theorem related to the discrete time multi type non stationary branching processes is applied

stability of the weighted splitting finite difference scheme for a two dimensional parabolic equation with two nonlocal integral conditions nonlocal conditions arise in mathematical models of various physical chemical or biological processes therefore interest in developing computational techniques for the numerical solution of partial differential equations pdes with various types of nonlocal conditions has been growing fast we construct and analyse a weighted splitting finite difference scheme for a two dimensional parabolic equation with nonlocal integral conditions the main attention is paid to the stability of the method we apply the stability analysis technique which is based on the investigation of the spectral structure of the transition matrix of a finite difference scheme we demonstrate that depending on the parameters of the finite difference scheme and nonlocal conditions the proposed method can be stable or unstable the results of numerical experiments with several test problems are also presented and they validate theoretical results

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

nonlocal reactive transport with physical and chemical heterogeneity localization errors the origin of nonlocality in macroscale models for subsurface chemical transport is illustrated it is argued that media that are either nonperiodic e g media with evolving heterogeneity or periodic viewed on a scale wherein a unit cell is discernible must display some nonlocality in the mean a metaphysical argument suggests that owing to the scarcity of information on natural scales of heterogeneity and on scales of observation associated with an instrument window constitutive theories for the mean concentration should at the outset of any modeling effort always be considered nonlocal the intuitive appeal to nonlocality is reinforced with an analytical derivation of the constitutive theory for a conservative tracer without appeal to any mathematical approximations deng et al 1993 present a first order nonlocal eulerian theory for transport of a conservative solute in an infinite nondeforming domain under steady flow conditions hu et al this issue extended these results to account for nonequilibrium linear sorption with random partition coefficient kd but deterministic constant reaction rate kr these theories are localized herein and comparisons are made between the fully nonlocal fnl nonlocal in time nlt and fully localized fl theories for conservative transport there is little difference between the first order fl and fnl models for spatial moments up to and including the third however for conservative transport the first order nlt model differs significantly from the fnl model in the third spatial moments for reactive transport all spatial moments differ between the fnl and fl models the second transverse horizontal and third longitudinal horizontal moments for the nlt model differ from the fnl model these results suggest that localized first order transport models for conservative tracers are reasonable if only lower order moments are desired however when the chemical reacts with its environment the localization approximation can lead to significant error in all moments and a fnl model will in general be required for accurate simulation

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

weak homogenization of anisotropic diffusion on pre sierpinski carpets we study a kind of restoration of isotropy on the pre sierpinski let and be the effective resistances in the x and y directions respectively of the sierpinski at the nth stage of its construction if it is made of anisotropic material whose anisotropy is parametrized by the ratio of resistances for a unit square we prove that isotropy is weakly restored asymptotically in the sense that for all sufficiently large n the ratio is bounded by positive constants independent of r the ratio decays exponentially fast when r 1 furthermore it is proved that the effective resistances asymptotically grow exponentially with an exponent equal to that found by barlow and bass for the isotropic case r 1

equivalence of bicontinuum and second order transport in heterogeneous soils and aquifers the mathematical equivalence of a closed second order transport equation for reactive solutes in saturated heterogeneous porous media e g soils and aquifers and a two region mobile mobile formulation is demonstrated in two ways 1 by averaging the bicontinuum equations and 2 by transforming the second order equation to canonical form the derivation of this equivalence is limited to media with heterogeneities in a plane orthogonal to the flow direction which represents a bidirectional generalization of the stratified case results indicate how the parameters of the bicontinuum formulation can be obtained from statistical descriptions of this heterogeneity and establish a stochastic interpretation for the bicontinuum equations one dimensional analytical solutions of the equivalent transport equations are compared to a three dimensional numerical simulation of transport in a spatially autocorrelated lognormal random conductivity field results indicate that the upscaled equations provide good approximations of the dynamics of the mean resident concentration the flux averaged concentration and their variances

influence of solitons in the initial state on chaos in the driven damped sine gordon system abstract the appearance of chaos in the a c driven damped sine gordon equation is studied numerically several transitions from periodic to chaotic behaviour are investigated in detail for flat initial conditions spatial structures breather kink in the initial conditions smooth out many of these transitions and give rise to an interesting symbiosis of time and spatial intermittency this symbiosis appears to be due to the competition between the background tendency towards chaos and the system s preference to maintain a spatial pattern the way that this competition is relieved is also found to depend very strongly on symmetry in the initial conditions

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

eulerian lagrangian theory of transport in space time nonstationary velocity fields exact nonlocal formalism by conditional moments and weak approximation a unified eulerian lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally v x t f x t where f x t is a random function including sources and or the time derivative of head solute concentration satisfies locally the eulerian equation c x t t j x t g x t where j x t is advective solute flux and g x t is a random source independent of f x t we consider the prediction of c x t and j x t by means of their unbiased ensemble moments c x t and j x t conditioned as implied by the subscript on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate x t of v x t these predictors satisfy c x t v t j x t g x t where j x t x t c x t q x t and q x t is a dispersive flux we show that q is given exactly by three space time convolution integrals of conditional lagrangian kernels with q with c and with c for a broad class of v x t fields including fractals this implies that q x t is generally nonlocal and non fickian rendering c x t non gaussian the direct contribution of random variations in f to q depends on c rather than on c we elucidate the nature of the above kernels discuss conditions under which the convolution of and c becomes pseudo fickian with a lagrangian dispersion tensor similar to that derived in 1921 by taylor recall a 1952 result by batchelor which yields an exact expression for c at early time use the latter to conclude that linearizations which predict that c bifurcates at early time when the probability density function of v is unimodal cannot be correct propose instead a weak approximation which leads to a nonlinear integro differential equation for c due to an instantaneous point source and which improves with the quantity and quality of hydraulic data demonstrate that the weak approximation is analogous to the direct interaction closure of turbulence theory offer non fickian and pseudo fickian weak approximations for the second conditional moment of the concentration prediction error demonstrate that it yields the so called two particle covariance as a special case conclude that the conditional variance of c does not become infinite merely as a consequence of disregarding local dispersion and discuss how to estimate explicitly the cumulative release of a contaminant across a compliance surface together with the associated estimation error

spatial characterization of a hydrogeochemically heterogeneous aquifer using partitioning tracers optimal estimation of aquifer parameters in this paper a distributed parameter extended kalman filter is developed to estimate spatially distributed residual saturation of nonaqueous phase liquids napl and darcy flux and to predict site specific movement of a partitioning tracer plume in a three dimensional heterogeneous aquifer the napl and darcy flux fields are assumed to be time invariant spatial random fields with known spatial correlation structures the extended kalman filter consists of two components 1 a moment propagation algorithm that uses a set of coupled stochastic partial differential equations derived from the advection dispersion retardation equation to predict how the moments propagate through space between measurement times zhang and graham this issue and 2 a moment update algorithm which updates the moments at measurement times because the optimal kalman filtering algorithm is computationally intensive for three dimensional problems a simplified sequential conditioning algorithm is developed that first conditions the darcy flux field using only nonpartitioning tracer measurements and then conditions the napl field using only partitioning tracer measurements the extended kalman filter is applied to a three dimensional domain patterned after the interwell partitioning tracer tests conducted at hill air force base by annable et al 1998 two different cross correlation scenarios and two different sampling networks are investigated

on diffusion in fractal porous media generalized hydrodynamics and irreversible thermodynamics are used to derive the relation between a general wave vector and frequency dependent diffusion tensor and the power spectral density the results are valid for porous media with evolving heterogeneities an integrodifferential transport equation valid in porous media with evolving heterogeneities is presented

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

infinite dimensional algebras and a trigonometric basis for the classical lie algebras this paper explores features of the infinite dimensional algebras that have been previously introduced in particular it is shown that the classical simple lie algebras an bn cn dn may be expressed in an egalitarian basis with trigonometric structure constants the transformation to the standard cartan weyl basis and the particularly transparent n limit that this formulation allows is provided

finite difference methods for solution of nonlocal boundary value problems the article is devoted to the construction and investigation of various types of difference schemes relevant to nonlocal boundary value problems stated for multidimensional elliptic equations suitable tools for performing the investigation are developed including difference analogue of green s formula and inequalities corresponding to nonlocal boundary conditions

fem ids for a second order strongly damped wave equation with memory abstract a linear triangular finite element method implicit difference scheme fem ids for the solution of second order strongly damped wave equation sdwe with memory on domain with interfaces is proposed sufficient conditions that guarantee the existence of a unique solution are given the finite element discretization is such that the arbitrary but smooth interface is first approximated by a polygon with a boundary whose vertices all lie on the interface with the interface being at distance from the approximate interface linear interpolation operator incorporating the influence of isproved this together with ritz volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme the scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method

a nonlinear nonlocal mixed problem for a second order pseudoparabolic equation we study a nonlocal mixed problem for a nonlinear pseudoparabolic equation which can for example model the heat conduction involving a certain thermodynamic temperature and a conductive temperature we prove the existence uniqueness and continuous dependence of a strong solution of the posed problem we first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense the technique of deriving the a priori estimate is based on constructing a suitable multiplicator from the resulted energy estimate it is possible to establish the solvability of the linear problem then by applying an iterative process based on the obtained results for the linear problem we establish the existence uniqueness and continuous dependence of the weak solution of the nonlinear problem

a note on head and velocity covariances in three dimensional flow through heterogeneous anisotropic porous media a procedure for the characterization of the head and velocity fields for saturated flow in heterogeneous statistically anisotropic formations is developed the fields are characterized through a series of spatial covariances as well as the velocity head velocity log conductivity and head log conductivity cross covariances all these functions are given in terms of the parameters which characterize the input log conductivity field and the average head garadient a general procedure is outlined followed by explicit derivation of head and velocity covariances for heterogeneous statistically anisotropic formations closed form analytic results are obtained for the statistically isotropic formations while the results for the anisotropic ones required taking quadrature over analytic expressions the results may be used for transport predictions and for conditioning on measurements and may be particularly useful for testing of numerical codes

on a singular two dimensional nonlinear evolution equation with nonlocal conditions abstract this paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with bessel operator we prove the existence and uniqueness of the weak solution of the given nonlinear problem we start by solving the associated linear problem after writing this latter in its operator form we establish an a priori bound from which we deduce the uniqueness of the strong solution for the solvability of the associated linear problem we prove that the range of the operator generated by the considered problem is dense on the basis of the obtained results of the linear problem we apply an iterative process to establish the existence and uniqueness of the nonlinear problem

exact effective transport dynamics in a one dimensional random environment we study effective transport under linear equilibrium adsorption characterized by a spatially random retardation factor in a stochastic framework we present a methodology to quantify explicitly the impact of spatial disorder on effective transport dynamics we derive an exact effective transport equation which is equivalent to transport under linear kinetic adsorption characterized by a spectrum of adsorption times the distribution of adsorption times is given explicitly in terms of the spatial disorder distribution furthermore we find that effective transport is formally equivalent to a decoupled continuous time random walk this observation and the explicit nature of the presented results allow for a mapping of the static disorder distribution on the transition time distribution

anisotropic random walks and asymptotically one dimensional diffusion on abc gaskets asymptotically one dimensional diffusion processes are studied on the class of fractals calledabc gaskets the class is a set of certain variants of the sierpinski gasket containing infinitely many fractals without any nondegenerate fixed point of renoramalization maps while the standard method of constructing diffusions on the sierpinski gasket and on nested fractals relies on the existence of a nondegenerate fixed point and hence it is not applicable to allabc gaskets the asymptotically one dimensional diffusion is constructed on anyabc gasket by means of an unstable degenerate fixed point to this end the generating functions for numbers of steps of anisotropic random walks on theabc gaskets are analyzed along the line of the authors previous studies in addition a general stategy of handling random walk sequences with more than one parameter for the construction of asymptotically one dimensional diffusion is proposed

symmetry and integrability of non singlet sectors in matrix quantum mechanics we study the non singlet sectors of matrix quantum mechanics mqm through an operator algebra which generates the spectrum the algebra is a nonlinear extension of the w algebra where the nonlinearity comes from the angular part of the matrix which can not be neglected in the non singlet sector the algebra contains an infinite set of commuting generators which can be regarded as the conserved currents of mqm we derive the spectrum and the eigenfunctions of these conserved quantities by a group theoretical method an interesting feature of the spectrum of these charges in the non singlet sectors is that they are identical to those of the singlet sector except for the multiplicities we also derive the explicit form of these commuting charges in terms of the eigenvalues of the matrix and show that the interaction terms which are typical in calogero sutherland system appear finally we discuss the bosonization and rewrite the commuting charges in terms of a free boson together with a finite number of extra degrees of freedom for the non singlet sectors

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions abstract this paper is concerned with some dynamical property of a reaction diffusion equation with nonlocal boundary condition under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function the asymptotic behavior of the time dependent solution is characterized in relation to a finite or an infinite set of constant steady state solutions this characterization is determined solely by the initial function and it leads to the stability and instability of the various steady state solutions in the case of finite constant steady state solutions the time dependent solution blows up in finite time when the initial function in greater than the largest constant solution also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain

on the solvability of a class of nonlinear singular parabolic equation with integral boundary condition abstract in this paper the existence and uniqueness of a weak solution for nonlinear singular parabolic equation with integral boundary conditions is proved first the associated linear problem is solved after writing the linear problem on its operatorial form we establish an a priori bound from which we deduce the uniqueness of the strong solution for the solvability of the associated linear problem we prove that the range of the operator generated by the considered problem is dense on the basis of the obtained results of the linear problem we apply an iterative process to establish the existence and uniqueness of the weak solution of the nonlinear problem

harmonization and homogenization on fractals this paper suggests a direct approach to define the laplacian the spectral dimension of nested fractals and the pre sierpinski carpet conductivity we find a geometric construction of the harmonic functions on the gasket and therefore can describe effectively the dense set of functions having finite energy the paper is mostly aimed at the homogenization on the pre sierpinski gasket whose horizontal and nonhorizontal bonds have different conductivities a andb respectively we prove the convergence of the rescaled energies on the pre sierpinski gasket to a b e where e is the standard energy on the gasket with uniform conductivities we also find an explicit expression for the effective conductivity a b and deduce that its set of singularities turns out to be the julia set of a certain rational function a special section is devoted to the problem of the pre sierpinski carpet conductivity asymptotic behavior for this problem a new proof of barlow bass inequalities with sharper constants is given

a fast fourier transform stochastic analysis of the contaminant transport problem a three dimensional stochastic analysis of the contaminant transport problem is developed in the spirit of naff 1990 the new derivation is more general and simpler than previous analysis the fast fourier transformation is used extensively to obtain numerical estimates of the mean concentration and various spatial moments data from both the borden and cape cod experiments are used to test the methodology results are comparable to results obtained by other methods and to the experiments themselves

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

numerical solutions of reaction diffusion equations with nonlocal boundary conditions abstract the purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction diffusion equations with nonlocal boundary conditions using the finite difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time dependent and the steady state finite difference systems each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence comparison theorem for the corresponding finite difference system the existence comparison theorems are used to investigate the asymptotic behavior of the discrete time dependent solution in relation to the discrete maximal and minimal solutions of the steady state problem numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes

infinite dimensional grassmannian structure of two dimensional quantum gravity we study the infinite dimensional grassmannian structure of 2d quantum gravity coupled to minimal conformal matters and show that there exists a large symmetry thew 1 symmetry using this symmetry structure we prove that the square root of the partition function which is a function of thep reduced kp hierarchy satisfies the vacuum condition of thew 1 algebra we further show that this condition is reduced to the vacuum condition of thew p algebra when the redundant variables for thep reduction are eliminated this mechanism also gives a prescription for extracting thew p algebra from thew 1 algebra

influence of complex coefficients on the stability of difference scheme for parabolic equations with non local conditions abstract the stability of a finite difference scheme for schrodinger kuramoto tsuzuki and parabolic equations subject to non local conditions with complex coefficients is dealt with the stability conditions which have to be met by complex coefficients in non local conditions have been determined the main result of this study is that complex coefficients together with non local conditions cause new effects on the stability of difference scheme numerical experiment has revealed additional regularities in the stability conditions

infinite dimensional lie superalgebras and hook schur functions making use of a howe duality involving the infinite dimensional lie superalgebra and the finite dimensional group gll of cw3 we derive a character formula for a certain class of irreducible quasi finite representations of in terms of hook schur functions we use the reduction procedure of to to derive a character formula for a certain class of level 1 highest weight irreducible representations of the affine lie superalgebra associated to the finite dimensional lie superalgebra these modules turn out to form the complete set of integrable modules of level 1 we also show that the characters of all integrable level 1 highest weight irreducible modules may be written as a sum of products of hook schur functions

numerical algorithm for parabolic problems with non classical conditions the parabolic problems with non classical conditions are discussed in a reproducing kernel space in this paper a reproducing kernel space is constructed in which the non classical conditions of the parabolic problems are satisfied based on the reproducing kernel space a new technique for solving the non classical parabolic problems is presented some examples are displayed to demonstrate the validity and applicability of the proposed method

the two level finite difference schemes for the heat equation with nonlocal initial condition abstract in this paper the two level finite difference schemes for the one dimensional heat equation with a nonlocal initial condition are analyzed as the main result we obtain conditions for the numerical stability of the schemes in addition we revise the stability conditions obtained in 21 for the crank nicolson scheme we present several numerical examples that confirm the theoretical results within linear as well as nonlinear problems in some particular cases it is shown that for small regions of the time step size values the explicit ftcs scheme is stable while certain implicit methods such as crank nicolson scheme are not

inertial effects in dispersion in porous media 1 in this work we develop the macroscale transport equation for dispersion of a nonreactive chemical species with a particular focus on the influence of inertial contributions at moderate reynolds numbers our starting point is the continuum level description of transport written at the subpore scale volume averaging is used to upscale these equations to develop the macroscale solute balance that applies at the darcy scale we develop a fully transient version of the ancillary closure problem that predicts the total dispersion tensor and we solve the closure using finite fourier transforms the result of this effort is a nonlocal macroscale transport equation where the nonlocal dispersion depends upon the microscale geometry of the pore space and the physical characteristics of the fluid both the longitudinal and transverse components of the total dispersion tensor are computed for a simple three dimensional unit cell the computational results indicate that a simple three dimensional periodic unit cell is able to capture the correct behavior for the longitudinal dispersion in the range 101 pep 2 5 105 although the magnitude of the longitudinal dispersion coefficient is underpredicted by up to a factor of about 4 for the transverse dispersion coefficient the simple unit cell provides much less satisfactory results when compared with experimental data the inertial effects for the longitudinal dispersion coefficient were relatively small but for the transverse dispersion coefficient inertial effects were predicted to increase the transverse dispersion coefficient up to 40 times that which would be predicted for stokes flow

a random field model for anomalous diffusion in heterogeneous porous media heterogeneity as it occurs in porous media is characterized in terms of a scaling exponent or fractal dimension a feature of primary interest for two phase flow is the mixing length this paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length the analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem in order to determine the mixing exponents the scaling behavior changes from long length scale dominated to short length scale dominated at a critical value of the scaling exponent of the rock heterogeneity the long length scale dominated diffusion is anomalous

macroscopic balance equations in soils and aquifers the case of space and time dependent instrumental response macroscopic balance equations for mass and linear momentum in soils and aquifers are derived in the relativist concept under the condition that all physical variables are associated with a single instrumental weighting function that depends on both position and time the resulting balance equations differ in concept from expressions obtained recently in the representative elementary volume rev approach by exhibiting additional terms that depend on the time and space derivatives of the instrumental weighting function these terms indicate that mass and linear momentum balance at the macroscopic scale are not preserved unless the weighting function has space time invariance properties over the period and spatial domain of measurement

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

classification of quasifinite modules over the lie algebras of weyl type abstract for a nondegenerate additive subgroup of the n dimensional vector space f n over an algebraically closed field f of characteristic zero there is an associative algebra and a lie algebra of weyl type w n spanned by all differential operators ud1m1 dnmn for u f the group algebra and m1 mn 0 where d1 dn are degree operators in this paper it is proved that an irreducible quasifinite w z 1 module is either a highest or lowest weight module or else a module of the intermediate series furthermore a classification of uniformly bounded w z 1 modules is completely given it is also proved that an irreducible quasifinite w n module is a module of the intermediate series and a complete classification of quasifinite w n modules is also given if is not isomorphic to z

a new approach to the self dual yang mills equations inspired by sato s new theory for soliton equations we find a new approach to the self dual yang mills equations we first establish a correspondence of solutions between the self dual yang mills equations and a new system of equations with infinitely many unknown functions it then turns out that the latter equations can be easily solved by a simple explicit procedure this leads to an explicit description of a very broad class of solutions to the self dual yang mills equations and also to a construction of transformations acting on these solutions

character and determinant formulae of quasifinite representation of thew1 algebra we diagonalize the hilbert space of some subclass of the quasifinite module of thew1 algebra states are classified according to their eigenvalues for infinitely many commuting charges and the young diagrams the parameter dependence of their norms is explicitly derived the full character formulae of the degenerate representations are given as summation of the bilinear combinations of the schur polynomials

asymptotic properties of the eulerian truncation approximation analysis of the perfectly stratified transport problem it is well known that stochastic groundwater problems are difficult to solve without making approximations of one kind or another a popular approximation used in both subsurface flow and transport applications is the so called eulerian truncation this approximation which relies on a perturbation expansion of the governing equation neglects certain terms involving products of small fluctuations several recent publications e g dagan and neuman 1991 neuman and orr 1993 neuman 1993 question the validity of eulerian truncation arguing that it is inconsistent because neglected terms are of the same order as those retained in this paper we analyze the asymptotic properties of the eulerian truncation solution using a taylor series approach which clearly reveals the properties of different order approximations we show that the eulerian truncation does in fact generate an asymptotic ensemble mean expansion for the perfectly stratified example considered by dagan and neuman 1991 moreover we show that the eulerian truncation solution to this problem is identical to the well known cumulant expansion approximation which is also based on an asymptotic expansion

the method of lines for solution of the one dimensional wave equation subject to an integral conservation condition hyperbolic partial differential equations with an integral condition serve as models in many branches of physics and technology recently much attention has been expended in studying these equations and there has been a considerable mathematical interest in them in this work the solution of the one dimensional nonlocal hyperbolic equation is presented by the method of lines the method of lines mol is a general way of viewing a partial differential equation as a system of ordinary differential equations the partial derivatives with respect to the space variables are discretized to obtain a system of odes in the time variable and then a proper initial value software can be used to solve this ode system we propose two forms of mol for solving the described problem several numerical examples and also some comparisons with finite difference methods will be investigated to confirm the efficiency of this procedure

ensemble averaged equations for reactive transport in porous media under unsteady flow conditions we present a method for deriving the ensemble averaged reactive solute transport equation for unsteady non divergence free flow field conditions our approach uses a cumulant expansion lie group theory and time ordered exponentials to develop the ensemble averaged transport equation the cumulant expansion is in powers of a c where measures the magnitude of the perturbations of the transport and reaction operators and c is the correlation time of these perturbations because the cumulant expansion avoids secular terms terms in powers of time the problem can be closed by rationally truncating the expansion the truncated terms can be shown to be of lower order than those terms that are kept provided that a particular constraint in terms of the kubo number is met the use of lie group theory allows one to automatically combine the eulerian and lagrangian approaches a particular time ordered exponential that arises in the analysis can be interpreted as a translation operator that possesses a well defined algebra these translation operators appear in the second order covariance terms of the cumulant expansion and their effect is to shift one of the terms of the covariance functions relative to the other along the trajectory formed by the ensemble averaged velocity field this approach does not require neglecting the local dispersion tensor and has the advantage that no integral transformations are conducted therefore all results are expressed in terms of real space variables

a numerical approach for a semilinear parabolic equation with a nonlocal boundary condition a semilinear reaction diffusion problem with a nonlocal boundary condition is studied this paper presents a new and very easy implementable numerical algorithm for computations this is based on a suitable linearization in time and on the principle of linear superposition any method for the space discretization fem was taken in this analysis can be chosen the derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition stability analysis has been performed and the optimal error estimates have been derived numerical results have been compared with other known techniques

laplace transform finite element method for non fickian flows abstract in this paper we consider numerical methods for integro differential problems based on time discretization via laplace transformation we focus our attention in models arising in the context of non fickian solute transport phenomena in porous media the mathematical models which describe the evolution of the solute concentrations are characterized by volterra equations we present and analyze an hybrid method which combines the laplace transformation with respect to the time variable with the finite element discretization in the spatial variables numerical results illustrate the performance of the method

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

quasifinite representations of classical lie subalgebras of w1 we show that there are precisely two up to conjugation anti involutions of the algebra of differential operators on the circle preserving the principal gradation we classify the irreducible quasifinite highest weight representations of the central extension d of the lie subalgebra of this algebra fixed by and find the unitary ones we realize them in terms of highest weight representations of the central extension of the lie algebra of infinite matrices with finitely many non zero diagonals over the algebra c u um 1 and its classical lie subalgebras ofb canddtypes character formulas forpositive primitiverepresentations of d including all the unitary ones are obtained we also realize a class of primitive representations of d in terms of free fields and establish a number of duality results between these primitive representations and finite dimensional irreducible representations of finite dimensional lie groups and supergroups we show that the vacuum modulevcof d carries a vertex algebra structure and establish a relationship betweenvcforc 12z and w algebras

the application of cubic trigonometric b spline to the numerical solution of the hyperbolic problems in this paper a collocation finite difference scheme based on new cubic trigonometric b spline is developed and analyzed for the numerical solution of a one dimensional hyperbolic equation wave equation with non local conservation condition the usual finite difference scheme is used to discretize the time derivative while a cubic trigonometric b spline is utilized as an interpolation function in the space dimension the scheme is shown to be unconditionally stable using the von neumann fourier method the accuracy of the proposed scheme is tested by using it for several test problems the numerical results are found to be in good agreement with known exact solutions and with existing schemes in literature

role of boundary conditions in convergence and nonlocality of solutions to stochastic flow problems in bounded domains steady flow through a heterogeneous porous medium in a bounded domain is investigated using a recursive perturbation scheme the effect of boundary conditions on a two dimensional flow with arbitrary variation of the mean flow allowing large gradients is investigated using analytical expressions for the head and velocity covariance functions boundary conditions were decomposed into deterministic and stochastic components two flow cases with the same zero order and different first order boundary conditions were analyzed boundary conditions are deterministic for the first case and random for the second significant differences between the two cases indicate random or absence of randomness processes must be modeled at boundaries first order solutions for the head and velocity covariance functions for a bounded rectangular domain are derived the resulting integral kernels involve greens functions and they are evaluated by numerical integration boundary conditions for higher order problems influence the absolute value and shape of the kernels and therefore of the head and velocity variance and covariance functions it is found that the validity of the perturbation scheme is dependent on the magnitude of the kernels and not only on the condition f2 1 where f is the variance of the log hydraulic conductivity assumed gaussian and weakly homogeneous in space the type of boundary conditions affects the values of the kernels and therefore determine the convergence limits for the problem milder head gradients also allow larger values of f2 nonlocality is also contingent on boundary type weakly homogeneous log fluctuating conductivity fields give rise to head and velocity covariances which are not weakly homogeneous the inhomogeneous fields are obtained from a linear filter of the solution to the problem without stochasticity higher gradient regions induce higher head and velocity variances in the presence of space varying gradients nonlocal effects are most important away from the boundaries lpar center of the domain small local head gradients result in small head and velocity gradients and therefore where observations are made in stagnation regions data should be analyzed taking into account this effect the results may be used to interpret experimental data for columns or data taken where conditions do not fit the average uniform flow assumption and when processes at the boundaries influence the flow and therefore the mixing of contaminants

numerical procedures for a boundary value problem with a non linear boundary condition numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial boundary value problem for the two dimensional parabolic partial differential equation pde with a non linear boundary condition this class of parabolic pdes plays a very important role in many branches of science and engineering the non linear condition is in the form of a double integral giving the specification of mass in the solution domain not only the problem has both neumann and dirichlet boundary conditions but the dirichlet boundary condition is in a non standard form while sharing some common features with the one dimensional models the solution of two dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of one dimensional case due to the structure of the boundary conditions a reduced one dimensional parabolic equation for the new unknown v y t email protected 1 0u x y t dx can be formulated the resulting problem has a non local boundary condition the new two dimensional parabolic pde with neumann s boundary conditions will be solved numerically by using the method of lines semi discretization approach the space derivatives in the pde are approximated by finite difference replacements the solution of the resulting system of first order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function the accuracy in time is controlled by choosing several subdiagonal pade approximants to replace this matrix exponential term numerical techniques are developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem from the literature the article concludes with the results of some numerical experiments

subalgebras of w 1 infinity and their quasifinite representations we propose a series of new subalgebras of the w1 infinity algebra parametrized by polynomials p omega and study their quasifinite representations we also investigate the relation between such subalgebras and the gl infinity algebra as an example we investigate the winfinity algebra which corresponds to the case p omega w presenting its free field realizations and kac determinants at lower levels

a q γ analog of the w1 algebra a q analog of the w1 algebra is introduced irreducible quasifinite highest weight modules of this algebra and r matrices acting on a tensor product of these modules are investigated a connection with the q deformed virasoro and wn algebras is also discussed

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

a galerkin method for two dimensional hyperbolic integro differential equation with purely integral conditions the present paper is devoted to the solution for two dimensional hyperbolic integro differential equations subject to purely integral conditions first it is demonstrated the existence and uniqueness of the solution under certain conditions for the numerical approach a galerkin method based on least squares is proposed the numerical examples illustrate the technique and show the efficiency of the proposed method

a monte carlo assessment of eulerian flow and transport perturbation models monte carlo studies of flow and transport in two dimensional synthetic conductivity fields are employed to evaluate first order flow and eulerian transport theories hydraulic conductivity is assumed to obey fractional brownian motion fbm statistics with infinite integral scale or to have an exponential covariance structure with finite integral scale the flow problem is solved via a block centered finite difference scheme and a random walk approach is employed to solve the transport equation for a conservative tracer the model is tested for mass conservation and convergence of computed statistics and found to yield accurate results it is then used to address several issues in the context of flow and transport the validity of the first order relation between the fluctuating velocity covariance and the fluctuating log conductivity is examined the simulations show that for exponential covariance this approximation is justified in the mean flow direction for log conductivity variance f2 of the order of unity however as f2 increases the relation for the transverse velocity component deviates from the fully nonlinear monte carlo results eulerian transport models neglect triplet correlation functions that appear in the nonlocal macroscopic flux the relative importance of the triplet correlation term for conservative chemicals is examined this term appears to be small relative to the convolution flux term in mildly heterogeneous media as f2 increases or the integral scale grows the triplet correlation becomes significant in purely convective transport the triplet correlation term is significant if the heterogeneity is evolving the exact nonlocal macroscale flux for the purely convective case significantly differs from that of the convective dispersive transport this is in agreement with recent theoretical analysis and numerical studies and it suggests that neglecting local scale dispersion may lead to large errors localization errors in the flux term are evaluated using monte carlo simulations the nonlocal in time model significantly differs from the fully nonlocal model for small variance and integral scale there is a slight difference between the fully localized flux and the fully nonlocal convolution flux this is also in agreement with recent theories that suggest that moments through the second for the two models are nearly identical for conservative tracers the fully localized model does not perform well in the purely convective cases

harmonic calculus on limits of networks and its application to dendrites abstract the main purpose of this paper is to give a general framework to analysis on fractals including finitely ramified self similar sets random fractals canter sets dendrites and the recurrent case of infinitely ramified self similar sets we study a limit of a sequence of electrical networks satisfying certain compatibility which is a generalization of the decimation invariance of random walks playing an important role in constructing diffusion processes on self similar sets as a limit we propose notions of finite resistance forms and effective resistance metrics also it is shown that a finite resistance form induces an effective resistance metric and vice versa a finite resistance form becomes a regular dirichlet form if the domain is dense in l 2 space in such a case the effective resistance metric is thought of as the intrinsic metric for the dirichlet form in the latter half of the paper we apply the general framework to dendrites which are topological spaces containing no loops a shortest path metric where all arcs are geodesics on a dendrite is shown to be an effective resistance metric and the corresponding finite resistance form turns out to be a local regular dirichlet form furthermore we establish a simple relation between the hausdorff dimensions and the spectral dimensions of dendrites with shortest path metrics

the operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers the operational meaning of the representative elementary volume rev concept on which current foundational theories of water movement through porous media are based is analyzed critically it is concluded that the rev concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable in its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments the effects of resolution and precision differences among experimental measuring devices now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers can be incorporated naturally within the relativist concept but are excluded a priori in the rev concept the relativist point of view unlike the rev concept has a clear operational meaning and in principle extends theoretical validity to all local measurements made on porous media it is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated anisotropic deformable soil or aquifer

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

monte carlo studies of flow and transport in fractal conductivity fields comparison with stochastic perturbation theory a monte carlo simulation of flow and transport is employed to study tracer migration in porous media with evolving scales of heterogeneity fractal media transport is studied with both conservative and reactive chemicals in media that possess physical as well as chemical heterogeneity linear kinetic equations are assumed to relate the sorbed phase and the aqueous phase concentrations the fluctuating log conductivity possesses the power law spectrum of a fractional brownian motion fbm chemical heterogeneity is represented as spatially varying reaction rates that also are assumed to obey fbm statistics and may be correlated to the conductivity field the model is based on a finite difference approximation to the flow problem and a random walk particle tracking approach for solving the solute transport equation the model is used to make comparisons with the nonlocal transport equations recently developed by deng et al 1993 and hu et al 1995 1997 the results presented herein support these nonlocal models for a wide range of heterogeneous systems however the infinite integral scale associated with the fractal conductivity has a significant effect on the prediction of the nonlocal theories this suggests that integral scale should play a role in stochastic eulerian perturbation theories the importance of the local scale dispersion depends to a great extent on the magnitude of the local dispersivities the effect of neglecting local dispersion decreases with the decrease in local dispersivity

a two grid block centered finite difference method for nonlinear non fickian flow model in this paper a two grid block centered finite difference scheme is introduced and analyzed to solve the nonlinear parabolic integro differential equation arising in modeling non fickian flow in porous media this method is considered where the nonlinear problem is solved only on a coarse grid of size h and a linear problem is solved on a fine grid of size h error estimates are established on non uniform rectangular grid which show that the discrete l l2 and l2 h1 errors are o t h 2 h 3 finally some numerical experiments are presented to show the efficiency of the two grid method and verify that the convergence rates are in agreement with the theoretical analysis

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

volume averaging for determining the effective dispersion tensor closure using periodic unit cells and comparison with ensemble averaging in this work we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions although this rectification is something that is generally believed to be true there has been very little research that explores this issue explicitly we show that under certain limiting but mild assumptions the closure problem provides a fourier series solution for the effective dispersion tensor when second order spatial stationarity is imposed on the velocity field the method yields a simple fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity this limiting result is identical to the quasi fickian forms that have been developed previously via ensemble averaging by deng et al 1993 and recently by fiori and dagan 2000 except in the definition of the averaging operation as a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and its volume averaged statistics this second objective is complimentary in many ways to recent research reported by rubin et al 1999 via ensemble averaging and by wang and kitanidis 1999 via volume averaging on the block averaged effective dispersion tensor the variability of the effective dispersion tensor from realization to realization is assessed by computing the volume averaged effective dispersion tensor for an ensemble of finite fields with the same ensemble statistics ensembles were generated using three different sizes of unit cells all three unit cell sizes yield similar results for the value of the mean effective dispersion tensor however the coefficient of variation depends strongly upon the size of the unit cell and our results are consistent with those developed by fiori 1998 from the ensemble averaging perspective this implies that in applications the actual value of the effective dispersion tensor may be significantly different than expected on the basis of unconditioned hydraulic conductivity statistics and this variation should be considered when applying macrodispersion to real world systems

the boundary penalty method for the diffusion equation subject to the specification of mass in this paper we concern with a new boundary penalty finite element method for the one dimensional time dependent diffusion equation subject to the specification of mass instead of solving the diffusion equation directly we will use the finite element method to solve a boundary penalty problem which is derived from the diffusion equation and analyse the error estimate of the semi discrete and fully discrete approximations when the penalty parameter e is sufficiently small the numerical experiments show that our method is very effective and supports our analysis