fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

the geostatistical characteristics of the borden aquifer a complete reexamination of sudicky s 1986 field experiment for the geostatistical characterization of hydraulic conductivity at the borden aquifer in ontario canada is performed the sampled data reveal that a number of outliers low ln k values are present in the data base these low values cause difficulties in both variogram estimation and determining population statistics the analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate the classical cressie hawkins and squared median of the absolute deviations smad estimators are used to compute experimental variograms none of these estimators provides completely satisfactory variograms for the borden data with the exception of the classical estimator with outliers removed from the data set theoretical exponential variogram parameters are determined from nonlinear nl estimation differences are obtained between nl fits and those of sudicky 1986 for the classical screened estimated variogram nl fits produce an ln k variance of 0 24 nugget of 0 07 and integral scales of 5 1 m horizontal and 0 21 m vertical along a a for b b these values are 0 37 0 11 8 3 and 0 34 the fitted parameter set for b b data horizontal and vertical is statistically different than the parameter set determined for a a we also evaluate a probabilistic form of dagan s 1982 1987 equations relating geostatistical parameters to a tracer cloud s spreading moments the equations are evaluated using the parameter estimates and covariances determined from line a a as input with a velocity equal to 9 0 cm day the results are compared with actual values determined from the field test but evaluated by both freyberg 1986 and rajaram and gelhar 1988 the geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with dagan s stochastic theory for predicting the spread of a tracer cloud

multifractal anisotropic scaling of the hydraulic conductivity 1 we analyzed the scaling properties of the hydraulic conductivity k at three sites in northern america made borden and cape cod we found that k at all sites exhibits multifractality fractal and multiscaling in both the vertical and horizontal directions though the multiscaling was within a range smaller than that of the maximum distance between measurements in the vertical direction the k data for made borden and cape cod were multiscaling from 0 15 to 1 35 m 0 05 to 0 5 m and 0 15 to 0 9 m respectively they were multiscaling in the horizontal direction from 9 to 45 m 1 to 10 m and 1 to 17 m respectively the multiscaling was also anisotropic evidence of scaling was poorest for the horizontal direction of the made site and it spanned half an order of magnitude such results compel one to treat multifractality in the horizontal direction of made as supported by heuristic arguments rather than by pure statistical evaluation we fitted a multifractal model to the data and estimated its parameters we found the underlying statistics of all data to be non gaussian and the model capable of reproducing the probability distribution of k data especially the negative skewness of log k we also generated two dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model

on a class of nonlinear dispersive dissipative interactions abstract we study a prototype dissipative dispersive equation ut a um x un xxx uk xx a consts which represents a wide variety of interactions at the critical value k m n 2 which separates dispersive and dissipation dominated phenomena these effects are in a detailed balance and the patterns formed do not depend on the amplitude in particular when m n 2 k 1 the equation can be transformed into a form free of convection and dissipation making it accessible to analysis both bounded and unbounded oscillations as well as solitary waves are found a variety of exact solutions are presented with a notable example being a solitary doublet for n 1 and a 2 3 2 the problem may be mapped into a linear equation leading to rational periodic or aperiodic solutions among others

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

multifractal scaling of the intrinsic permeability existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability k as the variable of concern we treat k as a multifractal and investigate its scaling and fractality using measured horizontal k data from two locations in the united states the first data set was from a shoreline sandstone near coalinga california and the second was from an eolian sandstone goggin 1988 by applying spectral analyses and computing the scaling of moments of various orders using the double trace moment method lavallee 1991 lavallee et al 1992 we found that k is multiscaling i e scaling and multifractal we also found that the so called universal multifractal um schertzer and lovejoy 1987 model essentially a log levy multifractal was able to reproduce the multiscaling behavior reasonably well the um model has three parameters a s and h representing the multifractality index the codimension of the mean field and the distance to stationary multifractal respectively we found a 5 1 7 s 5 0 23 h 5 0 22 and a 5 1 6 s 5 0 11 h 5 0 075 for the shoreline and eolian data sets respectively the fact that a values were less than 2 indicates that the underlying statistics are non gaussian we generated stationary and nonstationary multifractals and illustrated the role of the um parameters on simulated fields studies that treated log k as the variable of concern have pointed out the necessity for large data records especially when the underlying distribution is levy stable our investigation revealed that even larger data records are required when treating k as a multifractal because log k is less intermittent or irregular than k

numerical solitons of generalized korteweg de vries equations abstract we propose a numerical method for finding solitary wave solutions of generalized korteweg de vries equations by solving the nonlinear eigenvalue problem on an unbounded domain the artificial boundary conditions are obtained to make the domain finite we specially discuss the soliton solutions of the k m n equation and kdv k m n equation furthermore for the mixed models of linear and nonlinear dispersion the collision behaviors of soliton soliton and soliton antisoliton are observed

on the existence and uniqueness of solutions to stochastic three dimensional lagrangian averaged navier stokes equations we prove the existence and uniqueness of solutions for a stochastic version of the threedimensional lagrangian averaged navier stokes equation in a bounded domain to this end we previously prove some existence and uniqueness results for an abstract stochastic equation and justify that our model falls within this framework

a fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

asymptotic behaviour of the three dimensional α navier stokes model with locally lipschitz delay forcing terms abstract we obtain some results on the existence and uniqueness and exponential stability of solutions for the three dimensional navier stokes model with delays when the forcing term containing the delay is sub linear and locally lipschitz continuous

a repository of equations with cosine sine compactons nonlinear evolution equations with cosine sine compacton solutions are reviewed including the rosenau hyman equation and generalizations of korteweg de vries camassa holm boussinesq benjamin bona mahony klein gordon and other equations each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined several equations claimed in the literature to be different among them are found to be equivalent

temporal scaling of hydraulic head fluctuations nonstationary spectral analyses and numerical simulations 1 temporal scaling of the hydraulic head time series h t was found in a previous analysis of hourly measured head data this issue is further investigated in this paper with nonstationary spectral analyses and numerical simulations the results show that temporal scaling may indeed exist in h t which fluctuates like a fractional brownian motion in most aquifers on the basis of a linear reservoir model with a white noise recharge input we show that the variance and covariance of h t are functions of time the head variance increases with time and approaches a constant limit as time progresses while the covariance decreases with the separation time interval for a fixed time and approaches the typical exponential covariance as time increases the spectra of the simulated h t using a one dimensional transient groundwater flow model with a white noise recharge in both homogeneous and heterogeneous aquifers are shown to be proportional to f where f is frequency and 1 84 or h 0 42 heterogeneity in the hydraulic conductivity may affect the fractal dimension of h t in highly permeable aquifers but not in the low permeable aquifer simulated in this study

comparison of fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab 1 it is not a compelling argument solely on the basis of a better fit to solute breakthrough curve btc data that a temporally nonlocal model is necessary to simulate transport in an advection dominated system one may counter that the classical advection dispersion equation ade is a valid model at some small scale and that the detailed hydraulic conductivity k data must be well represented then the nonlocality is only a result of upscaling and loss of information but is the nonlocal model demonstrably necessary at all scales we examine the experiment conducted by klise et al 2008 in which a 30 5 30 5 cm slab of relatively homogeneous cross bedded sandstone was exhaustively sampled for k the slab was sealed saturated with potassium iodide and x rayed 10 times while being flushed with fresh water the 8 649 air permeameter measurements were down and upscaled to make finer and coarser grids on which the velocity field was solved and the ade applied the optimized parameters in the ade were found to scale predictably most notably the longitudinal dispersivity which grew linearly with upscaling but at all levels of up and downscaling including the original k measurement scale of 0 33 cm the ade did not adequately represent the late time tails the temporally nonlocal time fractional ade t fade was applied and the optimized parameters and the immobile capacity did not depend on scale the better fit provided by the t fade in the late btc tails did not bring about a sacrificed fit elsewhere in the btc furthermore the optimized ade and t fade solutions do not converge at the smallest scale directly implying that the temporal nonlocality is a necessary model component we conclude that the logical inference if the ade is valid in heterogeneous material then there is tailing in the btc is not a proof that the reverse is true we provide a clear counterexample a corollary is that a mismatch between data and a discretized solution to the ade does not imply that more data will improve fits or predictive ability

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

multifractal analysis of flow velocity simulated with the lattice model approach in idealized three dimensional porous media 1 the strange attractor formalism has been used to check whether the flow velocity in idealized three dimensional porous media could be described by the multifractal approach the flow velocity magnitude was simulated with a lattice model in six media with different porosities the results obtained from the analysis of the mass exponent functions and the multifractal and generalized dimensions spectra show that the velocity of the simulated flows has a multifractal type of scaling that becomes stronger as the porosity decreases a correlation analysis has been performed showing the relationships between the statistical properties and the multifractal parameters of the velocity magnitude distribution

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

scaling of random fields by means of truncated power variograms and associated spectra an interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size we first demonstrate that the power semi variogram and associated spectra of random fields having homogeneous isotropic increments can be constructed as weighted integrals from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields modes we then analyze the effect of filtering out truncating high and low frequency modes from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes the integral scales of the lowest and highest frequency modes cutoffs are related respectively to the length scales of the sampling window domain and data support sample volume taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations the traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite in one and two dimensions or vanishing in one dimension integral scales our hierarchical theory allows bridging across scales at a specific locale by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger one may also venture we suspect with less predictive power to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites

asymptotic behaviour of the three dimensional α navier stokes model with delays we prove the existence and exponential stability of the stationary solutions for a three dimensional navier stokes model with delays instead of working directly with the model we establish the results for a general abstract delay model and then we apply them to our particular situation we use two techniques in the stability analysis the lyapunov and razumikhin approaches

existence and decay of solutions in full space to navier stokes equations with delays abstract we consider the navier stokes equations with delays in r n 2 n 4 we prove existence of weak solutions when the external forces contain some hereditary characteristics and uniqueness when n 2 moreover if the external forces satisfy a time decay condition we show that the solution decays at an algebraic rate

method of lines study of nonlinear dispersive waves in this study we consider partial differential equation problems describing nonlinear wave phenomena e g a fully nonlinear third order korteweg de vries kdv equation the fourth order boussinesq equation the fifth order kaup kupershmidt equation and an extended kdv5 equation first we develop a method of lines solution strategy using an adaptive mesh refinement algorithm based on the equidistribution principle and spatial regularization techniques on the resulting highly nonuniform spatial grids the computation of high order derivative terms appears particularly delicate and we focus attention on the selection of appropriate approximation techniques finally we solve several illustrative problems and compare our computational approach to conventional solution techniques

a finite difference scheme for the k 2 2 compacton equation the k 2 2 equation introduced by rosenau and hyman is a wave equation that possesses solutions compactons that vanish outside a bounded interval of the spatial axis a finite difference scheme for this equation is suggested that can successfully cope with compacton interactions leading to negative waves we show rigorously that in those interactions a loss of smoothness of the solution necessarily takes place

stochastic interpolation of aquifer properties using fractional levy motion this paper describes a new technique for conditional simulation of aquifer properties in unobserved regions the technique utilizes a recently introduced model for heterogeneity based on the levy stable family of probability distributions to achieve a higher degree of realism than is possible with gaussian based techniques specifically permeability and porosity variations are modeled using fractional levy motion a new algorithm for producing fractional levy motion conditioned to match known data is introduced two dimensional simulations using actual permeability measurements as input demonstrate that this algorithm produces random fields with the specified properties simulations using a subset of a detailed permeability map of an outcrop as input illustrate how the new method can be used to estimate important flow and transport properties and to quantify uncertainties in these estimates

self similar radiation from numerical rosenau hyman compactons the numerical simulation of compactons solitary waves with compact support is characterized by the presence of spurious phenomena as numerically induced radiation which is illustrated here using four numerical methods applied to the rosenau hyman k p p equation both forward and backward radiations are emitted from the compacton presenting a self similar shape which has been illustrated graphically by the proper scaling a grid refinement study shows that the amplitude of the radiations decreases as the grid size does confirming its numerical origin the front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters the amplitude of the radiations decreases exponentially in time being characterized by a nearly constant scaling exponent an ansatz for both the backward and forward radiations corresponding to a self similar function characterized by the scaling exponent is suggested by the present numerical results

multifractal analyses of hydraulic conductivity distributions the concept of a universal multifractal a generalization of a monofractal is a recently developed scaling model for natural phenomena characterized by irregularity presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity k which have a significant effect on contaminant transport in the subsurface structure function analyses of four k data sets show that some vertical variations of k display multifractal structures while others are consistent with monofractal behavior in order to make multifractal concepts more useful multifractal noise is introduced and defined as the increments of a multifractal it is concluded that the multifractal formalism of schertzer and lovejoy 1987 has provided a rather general approach for modeling in k variations in the vertical with the exception of horizontal variations of the borden data all results fell within the domain of universal multifractal behavior which includes the monofractal case parameters were well defined in an empirical sense and easy to calculate indicating a robust formalism results were consistent with the recent finding of liu and molz 1997 also non gaussian and scale variant behavior in hydraulic conductivity distributions submitted to water resources research 1997 hereinafter referred to as submitted paper that k variations display increasing heterogeneity at decreasing scales

attractors for 2d navier stokes models with delays abstract the existence of an attractor for a 2d navier stokes system with delay is proved the theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous however on some occasions the attractors may attract not only in the pullback sense but in the forward one as well also this formulation allows to treat in a unified way terms containing various classes of delay features constant variable distributed delays etc as a consequence some results for the autonomous model are deduced as particular cases of our general formulation

local discontinuous galerkin methods for nonlinear dispersive equations we develop local discontinuous galerkin dg methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions the so called compactons the schemes we present extend the previous works of yan and shu on approximating solutions for linear dispersive equations and for certain kdv type equations we present two classes of dg methods for approximating solutions of such pdes first we generate nonlinearly stable numerical schemes with a stability condition that is induced from a conservation law of the pde an alternative approach is based on constructing linearly stable schemes i e schemes that are linearly stable to small perturbations the numerical simulations we present verify the desired properties of the methods including their expected order of accuracy in particular we demonstrate the potential advantages of using dg methods over pseudospectral methods in situations where discontinuous fronts and rapid oscillations co exist in a solution

existence and decay of solutions to the dissipative quasi geostrophic equation with delays abstract we consider the dissipative quasi geostrophic equation with forcing and advective terms containing delays we prove the existence of weak solutions and algebraic decay of the energy provided the initial data 0 is in l p l 2 1 p 2 and the forcing term satisfies some mild conditions on its long time behavior

on the method of modified equations i asymptotic analysis of the euler forward difference method the method of modified equations is studied as a technique for the analysis of finite difference equations the non uniqueness of the modified equation of a difference method is stressed and three kinds of modified equations are introduced the first modified or equivalent equation is the natural pseudo differential operator associated to the original numerical method linear and nonlinear combinations of the equivalent equation and their derivatives yield the second modified or second equivalent equation and the third modified or simply modified equation respectively for linear problems with constant coefficients the three kinds of modified equations are equivalent among them and to the original difference scheme for nonlinear problems the three kinds of modified equations are asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step as small parameter yields exactly the same results in this paper both regular and multiple scales asymptotic techniques are used for the analysis of the euler forward difference method and the resulting asymptotic expansions are verified for several nonlinear autonomous ordinary differential equations it is shown that when the resulting asymptotic expansion is uniformly valid the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up even for large step sizes

effective saturated hydraulic conductivity of two dimensional random multifractal fields 0 991 p 0 0001 thisrelationship indicates that hkidecreases with increasing long range spatial correlation among the k values in the field furthermore as hki decreases it becomes increasingly dominated by flow channeling this is because high values of k become more and more clustered as p decreases this approach may prove useful for the prediction of hki from generalized dimensions estimated by multifractal analysis of field measurements of k the results may also be applicable to the design of sampling strategies for multiple small scale slug tests at a given resolution citation koirala s r e perfect r w gentry and j w kim 2008 effective saturated hydraulic conductivity of two dimensional random multifractal fields water resour res 44 w08410 doi 10 1029 2007wr006199

stability and dynamical properties of rosenau hyman compactons using pade approximants we present a systematic approach for calculating higher order derivatives of smooth functions on a uniform grid using pade approximants we illustrate our findings by deriving higher order approximations using traditional second order finite difference formulas as our starting point we employ these schemes to study the stability and dynamical properties of k 2 2 rosenau hyman compactons including the collision of two compactons and resultant shock formation our approach uses a differencing scheme involving only nearest and next to nearest neighbors on a uniform spatial grid the partial differential equation for the compactons involves first second and third partial derivatives in the spatial coordinate and we concentrate on four different fourth order methods which differ in the possibility of increasing the degree of accuracy or not of one of the spatial derivatives to sixth order a method designed to reduce round off errors was found to be the most accurate approximation in stability studies of single solitary waves even though all derivates are accurate only to fourth order simulating compacton scattering requires the addition of fourth derivatives related to artificial viscosity for those problems the different choices lead to different amounts of spurious radiation and we compare the virtues of the different choices

time stepping in petrov galerkin methods based on cubic b splines for compactons abstract four numerical methods with first to fourth order of accuracy have been developed for the time integration of the rosenau hyman k 2 2 equation the error in the solution and the invariants for the propagation of one compacton and the stability in collisions among compactons have been studied using these methods numerically induced radiation has also been characterized by means of wavefront velocity and wavefront amplitude showing that the self similarity of the radiation wavepackets observed in the numerical results is a consequence of the time stepping method among the four methods studied in this paper the best results in terms of accuracy computational cost and stability have been obtained by means of using the second order time integration method

numerical methods based on modified equations for nonlinear evolution equations with compactons abstract compactons are traveling wave solutions with compact support resulting from the balance of both nonlinearity and nonlinear dispersion numerical methods with second fourth sixth and eighth order approximations to the spatial derivatives obtained by means of the method of modified equations applied to the ismail taha finite difference scheme for the rosenau hyman equation are developed the whole set of methods is compared among them in accuracy invariant conservation and in compacton collisions the best method among those studied in terms of the tradeoff between accuracy and computational cost is determined

ensemble solute transport in two dimensional operator scaling random fields 1 motivated by field measurements of aquifer hydraulic conductivity k recent techniques were developed to construct anisotropic fractal random fields in which the scaling or self similarity parameter varies with direction and is defined by a matrix ensemble numerical results are analyzed for solute transport through these two dimensional operator scaling fractional brownian motion ln k fields both the longitudinal and transverse hurst coefficients as well as the radius of isotropy are important to both plume growth rates and the timing and duration of breakthrough it is possible to create operator scaling fractional brownian motion fields that have more continuity or stratification in the direction of transport the effects on a conservative solute plume are continually faster than fickian growth rates highly non gaussian shapes and a heavier tail early in the breakthrough curve contrary to some analytic stochastic theories for monofractal k fields the plume growth rates never exceed a mercado s 1967 purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance apparent superstratified growth must be the result of other demonstrable factors such as initial plume size

a general fractal model of flow and solute transport in randomly heterogeneous porous media 1 one of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous this paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media we describe the random field using fractional levy motion flm which more effectively represents a field with a high degree of variability understanding flow and solute transport in the flm field expands the applicability of stochastic theory following the work of v di federico and s p neuman 1997 and v di federico et al 1999 we derive truncated isotropic and anisotropic power variograms for the flm fields the results are analogous to but more general than those of fractional brownian motion fbm v di federico and s p neuman 1997 and v di federico et al 1999 for a random field when the levy index in flm is replaced by a constant of 2 fbm is recovered this study employs first order perturbation analyses of flow and solute transport in two dimensional flm fields for the stationary fbm and flm fields macrodispersion coefficients are compared the plume is found to grow slowest in the flm field but will show highest concentration the proposed fractal model is general in the sense that when the levy index of flm equals 2 the results of flow and solute transport for the commonly used fbm model are recovered when the upper cutoff of the truncated power variogram is close to the lower cutoff the results for the stationary exponential or gaussian model can be well approximated since the proposed general fractal model has broader applications than the stationary and fbm models it simulates flow and solute transport in more different scenarios to yield more accurate modeling results

navier stokes equations with delays some results on the existence and uniqueness of solutions to navier stokes equations when the external force contains some hereditary characteristics are proved