explicit solvability of a class of maliuzhinets equations with matrix coefficients a coupled system of maliuzhinets functional equations for a pair of unknown functions is studied it is shown that the system can be solved in an explicit form for a specific but non trivial class of its matrix coefficients two examples arising in electromagnetic diffraction theory in a wedge shaped region are briefly discussed

stochastic uncertainty analysis for unconfined flow systems 1 a new stochastic approach proposed by zhang and lu 2004 called the karhunen loeve decomposition based moment equation klme has been extended to solving nonlinear unconfined flow problems in randomly heterogeneous aquifers this approach is on the basis of an innovative combination of karhunen loeve decomposition polynomial expansion and perturbation methods the random log transformed hydraulic conductivity field lnks is first expanded into a series in terms of orthogonal gaussian standard random variables with their coefficients obtained as the eigenvalues and eigenfunctions of the covariance function of lnks next head h is decomposed as a perturbation expansion series h m where h m represents the mth order head term with respect to the standard deviation of lnks then h m is further expanded into a polynomial series of m products of orthogonal gaussian standard random variables whose coefficients are deterministic and solved sequentially from low to high expansion orders using modflow 2000 finally the statistics of head and flux are computed using simple algebraic operations on a series of numerical test results in 2 d and 3 d unconfined flow systems indicated that the klme approach is effective in estimating the mean and co variance of both heads and fluxes and requires much less computational effort as compared to the traditional monte carlo simulation technique

diffraction by an arbitrary angled dielectric wedge ii correction to physical optics solution for pt i see ibid vol 39 no 9 p 1272 81 1991 the error of the physical optics solution for the e polarized plane wave incidence in connection with diffraction by an arbitrary angled dielectric wedge is corrected by calculating the nonuniform current distributed along the dielectric interfaces two kinds of series expansions to the nonuniform current are employed one is an asymptotic expansion as the multipole line source located at the edge of the dielectric wedge since the correction field seems to be a cylindrical wave emanating from the edge in the far field region the other is arbitrary electric and magnetic surface currents expanded by infinite series of the bessel functions i e the neumann expansion of which fractional order is chosen to satisfy the edge condition near the edge of the dielectric wedge in the static limit both of the two different expansion coefficients for a wedge angle of 45 degrees relative dielectric constants 2 10 and 100 and the e polarized incident angle of 150 degrees are evaluated by solving the dual series equation numerically after finite truncation

arbitrarily oriented perfectly conducting wedge over a dielectric half space diffraction and total far field complex scattering targets are often made by structures constituted by wedges and penetrable substrates which may interact at near field in this paper we describe a complete procedure to study this problem with possible developments in radar technologies like gpr antenna development or electromagnetic compatibility tips near substrates the diffraction of an incident plane wave by a perfectly conducting pec wedge over a dielectric half space is studied using generalized wiener hopf equations gwhes and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf wh factorization to integral equations ies the mathematical aspects are described in a unified and consistent theory for angular and layered region problems the proposed procedure is valid for the general case and the paper focuses on e polarization at normal incidence the solutions are given in terms of geometrical uniform theory of diffraction gtd utd diffraction coefficients and total far fields for engineering applications this paper presents several numerical test cases that show the validity of the proposed method

numerical solutions to statistical moment equations of groundwater flow in nonstationary bounded heterogeneous media in this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers we derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions due to their mathematical complexity we solve the moment equations by the numerical technique of finite differences the numerical approach has flexibility in handling moderately irregular geometry different boundary conditions various trends in the mean log hydraulic conductivity spatial variabilities in the magnitude and direction of mean flow and different covariance functions all of which are important factors to consider for real world applications the effect of boundaries on the first two statistical moments involving head is strong and persistent for example in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain as in many other stochastic models the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small accounting for a spatially varying large scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem it justifies the small variance assumption for many aquifers

transformation techniques towards the factorization of non rational 2 2 matrix functions abstract for the wiener hopf factorization of 2 2 matrix functions g defined on a closed carleson curve transformations g ugv where u and v are invertible rational 2 2 matrix functions are important in the first part of this paper we establish a classification scheme for 2 2 matrix functions which is based on such transformations we determine invariants under these transformations and describe those matrix functions which can be transformed to triangular or daniele khrapkov form in the second part we consider special rational transformations and study the same problem for instance we consider transformations where u and v are rational matrix functions that are analytic and invertible on an open neighborhood of in the more complicated but for factorization theory important case where u and v are rational matrix functions that are analytic and invertible on an open neighborhood of the closure of the domain inside of or outside of respectively the answer is slightly different

higher order effects on flow and transport in randomly heterogeneous porous media advective solute transport in nonuniform geologic media is generally nonlocal and non fickian neuman 1993 cushman and ginn 1993 in statistically homogeneous log conductivity fields under uniform mean flow the transport is expected to become asymptotically local and fickian at late time during the earlier preasymptotic regime macrodispersivity a measure of the rate at which a plume spreads is expected to vary with solute residence time a first order linear in the natural log hydraulic conductivity variance 2 analysis of this variation has been performed by dagan 1984 1987 1988 he found that when local dispersion is neglected the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote however the transverse macrodispersivity first increases from zero to a peak value then decreases monotonically toward zero the first order asymptotic analyses by winter 1982 gelhar and axness 1983 and winter et al 1984 also yield zero transverse macrodispersivity when local dispersion is disregarded

on budaev and bogy s approach to diffraction by the 2d traction free elastic wedge several semianalytical approaches are now available for describing diffraction of a plane wave by the 2d two dimensional traction free isotropic elastic wedge in this paper we follow budaev and bogy who reformulated the original diffraction problem as a singular integral one this comprises two algebraic and two singular integral equations each integral equation involves two unknowns a function and a constant we discuss the underlying integral operators and develop a new semianalytical scheme for solving the integral equations we investigate the properties of the solution obtained and argue that it is the solution of the original diffraction problem we describe a comprehensive code verification and validation program

on the commutative factorization of n n matrix wiener hopf kernels with distinct eigenvalues in this article we present a method for factorizing n x n matrix wiener hopf kernels where n 2 and the factors commute we are motivated by a method posed by jones jones 1984a proc r soc a 393 185 192 to tackle a narrower class of matrix kernels however no matrix of jones form has yet been found to arise in physical wiener hopf models in contrast the technique proposed herein should find broad application to illustrate the approach we consider a 3 x 3 matrix kernel arising in a problem from elastostatics while this kernel is not of jones form we shall show how it can be factorized commutatively we discuss the essential difference between our method and that of jones and explain why our method is a generalization the majority of wiener hopf kernels that occur in canonical diffraction problems are however strictly non commutative for 2x2 matrices abrahams has shown that one can overcome this difficulty using pade approximants to rearrange a non commutative kernel into a partial commutative form an approximate factorization can then be derived by considering the dynamic analogue of antipov s model we show for the first time that abrahams pade approximant method can also be employed within a 3x3 commutative matrix form

optimal design for elisa and other forms of immunoassay in enzyme linked immunosorbent assay elisa as well as in many other kinds of immunoassay a log logistic or similar shaped calibration curve is fit using standards at a series of known levels and then used to transform the measured values for the unknowns into estimated concentrations the choice of the number of standards the concentration of the standards and the number of replicates of the standards and of the unknowns all affect the precision of the measurement this article develops an optima1 design paradigm for this type of problem and shows how optimal choices can be calculated so that the system achieves the maximum precision of which it is capable although exact calculation of optimal designs requires use of a computer program close approximations to the optimum can be derived from simple rules for hand calculation

on the solution of two coupled wiener hopf equations vector wiener hopi equations previously solved in closed form by using different techniques involve matrices having a particular structure a general procedure is given to factorize these matrices it leads to an approach which even if sometimes cumbersome is uniform and allows the solution of new problems in addition a new method is proposed which transforms simultaneous wiener hopf equations in the form of fredholm integral equation of the second kind

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

the wiener hopf technique for impenetrable wedges having arbitrary aperture angle diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the wiener hopf w h technique a system of functional equations called generalized wiener hopf equations gwhe is obtained only for certain values of the aperture angle are these equations recognizable as standard or classical wiener hopf equations cwhe however in all cases a mapping is found that reduces the gwhe to cwhe it means that the diffraction by an impenetrable wedge always reduces to a standard w h factorization the solution for the diffraction by a wedge with given face impedances the malyuzhinets problem is obtained in closed form by an explicit factorization of the kernel

wiener hopf solution for impenetrable wedges at skew incidence a new wiener hopf approach for the solution of impenetrable wedges at skew incidence is presented mathematical aspects are described in a unified and consistent theory for angular region problems solutions are obtained using analytical and numerical analytical approaches several numerical tests from the scientific literature validate the new technique and new solutions for anisotropic surface impedance wedges are solved at skew incidence the solutions are presented considering the geometrical and uniform theory of diffraction coefficients total fields and possible surface wave contributions

factorization in a torus and riemann hilbert problems abstract a new concept of meromorphic factorization for holder continuous functions defined on a contour that is the pullback of r or the unit circle in a riemann surface of genus 1 is introduced and studied and its relations with holomorphic factorization are discussed it is applied to study and solve some scalar riemann hilbert problems in and vectorial riemann hilbert problems in c including wiener hopf matrix factorization as well as to study some properties of a class of toeplitz operators with 2 2 matrix symbols

a method for designing nonlinear univariate calibration a method is proposed for designing nonlinear univariate calibration of measuring instruments the problem addressed is how to select a set of design points standards or calibrators to minimize the errors in the inverse predictions the curve parameters are assumed to vary randomly between calibrations with known expected value and known covariance matrix a design criterion is suggested for analytical procedures according to which the coefficient of variation and the area under the precision profile are minimized

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

some interior and exterior boundary value problems for the helmholtz equation in a quadrant boundary transmission problems for two dimensional helmholtz equations in a quadrant q 1 and its complement q c 1 respectively are considered in a sobolev space setting the first problem of a quadrant with dirichlet condition on one face and transmission condition on the other is solved in closed form for the case where all quadrants are occupied by the same medium unique sovlability can also be shown in the case of two different media up to exceptional cases of wave numbers while fredholm property holds in general in the second problem transmission conditions are prescribed on both faces these problems are reduced to equivalent integral equations in l 2 of wiener hopf hankel type which can be solved explicitly by obtaining canonical generalized factorizations of certain non rational 2 s 2 matrix valued symbols

a general multisegment artificial neural network architecture for the efficient evaluation of electromagnetic plane wave wedge diffraction a multisegment artificial neural network ann is proposed as an interpolation technique for the evaluation of the electromagnetic field diffracted at the edge of anisotropic impedance wedges under plane wave illumination at oblique incidence multisegmentation is needed as the high frequency wedge diffracted field is characterized by a number of discontinuities at the shadow boundaries of the geometrical optics and surface wave fields the proposed approach is applied as a test case to the problem of an anisotropic impedance right angled wedge illuminated by a skewly incident plane wave some exact analytical solutions valid for specific surface impedance tensors are used to obtain numerical data for the ann training phase as well as to show the interpolation capabilities of the implemented ann nevertheless the proposed ann structure is general and can be trained with data obtained from other available solutions analytical perturbative numerical valid for more general wedge configurations eventually leading to a single software tool encompassing all of them and providing accurate approximations of the wedge diffracted field in a relatively short time comparable to that of a closed form analytical solution

electromagnetic diffraction by arbitrary angle impedance wedges the problem of the diffraction of a plane electromagnetic wave incident at an oblique angle on a wedge of arbitrary angle with general tensor impedance boundary conditions is solved using a semi an

on higher order corrections to the flow velocity covariance tensor second order log fluctuating conductivity variance 2 corrections to the head and velocity covariance functions are derived for a lognormal stationary hydraulic conductivity field the fourier transform method proposed by deng et al 1993 is used extensively to obtain numerical estimates of these functions for an exponential log fluctuating conductivity covariance it is shown that the velocity covariance is insensitive to second order corrections in the head field the velocity covariance on the other hand is highly sensitive to second order corrections in the velocity when the log fluctuating conductivity variance approaches unity a closed expression is derived for a second order correction to the velocity variance when there is no second order correction to the head field the longitudinal second order correction to the velocity variance is 0 4 2 different from the first order approximation in isotropic media 1 5 2 different in a highly stratified formation and no different when the ratio of vertical to horizontal integral scales approaches infinity the second order corrections to the horizontal and vertical transverse velocity variances are 2 2 different from the first order approximations for both isotropic and anisotropic systems

probabilistic collocation method for flow in porous media comparisons with other stochastic methods 1 an efficient method for uncertainty analysis of flow in random porous media is explored in this study on the basis of combination of karhunen loeve expansion and probabilistic collocation method pcm the random log transformed hydraulic conductivity field is represented by the karhunen loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points the procedure is straightforward and analogous to the monte carlo method but the number of simulations required in pcm is significantly reduced steady state flows in saturated random porous media are simulated with the probabilistic collocation method and comparisons are made with other stochastic methods monte carlo method the traditional polynomial chaos expansion pce approach based on galerkin scheme and the moment equation approach based on karhunen loeve expansion klme this study reveals that pcm and klme are more efficient than the galerkin pce approach while the computational efforts are greatly reduced compared to the direct sampling monte carlo method the pcm and klme approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head

embedding formulae for diffraction by rational wedge and angular geometries embedding is the process of taking the far field directivity from a diffraction problem or problems involving line sources or multipoles placed at a sharp edge and then constructing the far field for the same geometry for more general incidence using only this canonical problem s thus far embedding has been limited to planar parallel scattering surfaces for instance collections of parallel cracks or slits it appeared that there was a fundamental limitation to embedding disallowing its use for angular structures in this article we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape a limitation is that the final formulae are for rational wedge angles

on the non commutative factorization of wiener hopf kernels of khrapkov type there are a great many problems in mathematical physics and engineering which when modelled mathematically are reduced to wiener hopf functional equations defined in some region of a complex plane in simple models this equation is scalar but in more complicated situations inherent coupling etc can often give rise to coupled systems of equations and so the resulting wiener hopf kernels are of matrix form the crucial step in the solution of any wiener hopf equation is to decompose the kernel which can be a quite general function of the complex transform variable into a product of two factors which are regular in overlapping half planes this can be accomplished explicitly for scalar kernels and may be generalized to a particular class of matrix kernels namely those which permit a commutative decomposition although this class is wide and therefore contains some valuable physical problems it is found that many such kernels yield factorization elements with exponentially large behaviour at infinity in their half planes of analyticity this prevents a later step in the wiener hopf procedure being carried through and therefore introduces a serious limitation to the usefulness of khrapkov s approach previous researchers have offered schemes to eliminate the exponential behaviour of the khrapkov factors but for reasons discussed herein these have problems in their successful implementation this article offers a new procedure for quenching the exponential growth which is based on a method proposed recently by the author for fully coupled kernels which do not have commutative factors the approach is two step first a commutative factorization is obtained which is algebraic at infinity as required but which has spurious branch cuts in the regions of intended analyticity second an approximate procedure is used to eliminate the offending branch points by introduction of pade approximants for a certain scalar function and by inclusion of a regularizing meromorphic matrix for one particular example kernel in which an exact factorization happens to be known the explicit approximate kernel factors thus derived are shown to yield very close agreement to the exact factors the application of this work to problems in the area of fluid structural interaction is discussed

on the solution of wiener hopf problems involving noncommutative matrix kernel decompositions many problems in physics and engineering with semi infinite boundaries or interfaces are exactly solvable by the wiener hopf technique it has been used successfully in a multitude of different disciplines when the wiener hopf functional equation contains a single scalar kernel for complex boundary value problems however the procedure often leads to coupled equations which therefore have a kernel of matrix form the key step in the technique is to decompose the kernel into a product of two functions one analytic in an upper region of a complex transform plane and the other analytic in an overlapping lower half plane this is straightforward for scalar kernels but no method has yet been proposed for general matrices in this article a new procedure is introduced whereby pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel as well as being simple to apply the use of approximants allows the accuracy of the factorization to be increased

an iterative wiener hopf method for triangular matrix functions with exponential factors this paper introduces a new method for constructing approximate solutions to a class of wiener hopf equations this is particularly useful since exact solutions of this class of wiener hopf equations currently cannot be obtained the proposed method could be considered as a generalization of the pole removal technique and schwarzschild s series the criteria for convergence is proved the error in the approximation is explicitly estimated and by a sufficient number of iterations it could be made arbitrarily small typically only a few iterations are required for practical purposes the theory is illustrated by numerical examples that demonstrate the advantages of the proposed procedure this method was motivated by and successfully applied to problems in acoustics

stochastic analysis of nonstationary subsurface solute transport 1 unconditional moments this paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media partial differential equations for three unconditional ensemble moments the concentration mean concentration covariance and velocity concentration cross covariance are derived by applying perturbation techniques to the governing transport equation for a conservative solute concentration uncertainty is assumed to be the result of unmodeled small scale fluctuations in a steady state velocity field the moment expressions which describe how each moment evolves over time and space resemble the classic deterministic advection dispersion equation and can be solved using similar methods a solution procedure based on a galerkin finite element algorithm is illustrated with a hypothetical two dimensional example for this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation the perturbation solution is shown to reproduce the statistics obtained from a monte carlo simulation quite well for a natural log conductivity standard deviation of 0 5 and moderately well for a natural log conductivity standard deviation of 1 0 the computational effort required for a perturbation solution is significantly less than that required for a monte carlo solution of acceptable accuracy sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases

numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using fdtd classical theories such as the uniform geometrical theory of diffraction utd utilize analytical expressions for diffraction coefficient for canonical problems such as the infinite perfectly conducting wedge we present a numerical approach to this problem using the finite difference time domain fdtd method we present results for the diffraction coefficient of the two dimensional 2 d infinite perfect electrical conductor pec wedge the 2 d infinite lossless dielectric wedge and the 2 d infinite lossy dielectric wedge for incident tm and te polarization and a 90 spl deg wedge angle we compare our fdtd results in the far field region for the infinite pec wedge to the well known analytical solutions obtained using the utd there is very good agreement between the fdtd and utd results the power of this approach using fdtd goes well beyond the simple problems dealt with in this paper it can in principle be extended to calculate the diffraction coefficients for a variety of shape and material discontinuities even in three dimensions

scattering by a semi infinite sandwich panel perforated on one side the scattering of sound waves by the edge of a sandwich panel which consists of two thin elastic plates containing a light interior cellular structure is analysed the wiener hopf method is used to examine the interaction effects of a semi infinite panel clamped to a semi infinite rigid screen one major difficulty is that the presence of two different velocity potentials on either side of the plane y 0 results in a matrix wiener hopf equation an approximate solution is given in the asymptotic limit of small values of a parameter tau which accounts for the perforations

design considerations for calibration this article examines the effects of the choice of design points on calibration in the presence of a simple linear regression model the distribution of the classical estimator is examined via modes median and moments and the asymptotic variance of this estimator is minimized the influence of the designs on confidence regions constructed via fieller s theorem is investigated through accuracy curves a numerical example is presented and some additional problems are suggested

wave diffraction by wedges having arbitrary aperture angle abstract the problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background in fact we may recognize significant research on this topic for more than one century despite this fact up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate sobolev space setting in the present paper we are considering the corresponding boundary value problems for the helmholtz equation with complex wave number admitting combinations of dirichlet and neumann boundary conditions the main ideas are based on a convenient combination of potential representation formulas associated with weighted mellin pseudo differential operators in appropriate sobolev spaces and a detailed fredholm analysis thus we prove that the problems have unique solutions with continuous dependence on the data which are represented by the single and double layer potentials where the densities are solutions of derived pseudo differential equations on the half line

asymmetric channel divider in stokes flow this article examines the classic problem of stokes flow into a divided channel with in contrast to previous literature the divider barrier asymmetrically placed with respect to the moving parallel channel walls the boundary value problem is reduced to a wiener hopf equation that is of matrix form and of a class for which no exact solution is known an explicit approximate solution in general accurate to any specified degree is obtained by a recent method which employs pade approximants numerical results exhibit the flows due to moving walls or various combinations of downstream pressure gradients

the effective boundary conditions for a perforated elastic sandwich panel in a compressible fluid the transmission of sound through a sandwich panel consisting of a honeycomb cellular structure bounded by two thin elastic plates is considerably reduced at a certain frequency if one or both of the plates is perforated so as to link a significant number of the cells to the exterior fluid this effect occurs at or near the helmholtz resonance frequency for the cells with the acoustic wavelength large compared with the cell dimensions an analysis is given for the problem of plane waves incident upon a plane sandwich plate of infinite extent using matched expansions for the cases of acoustically hard or acoustically transparent cell walls the compound panel is shown to be acoustically equivalent to that of a hypothetical surface with different normal velocities on either side and effective boundary conditions are derived with generalizations to deal with more complicated structures finite plates and more general incident fields

mode conversion corner singularities and matrix wiener hopf factorization in diffraction theory the diffraction of acoustic electromagnetic and elastic waves by an edge is a subject of continuing mathematical and physical importance the physical motivation for research is the vast number of technological and geophysical applications of the fundamental models of linear wave propagation in continuous media while the mathematical drive is to understand and devise new analytical and numerical techniques for their quantification this paper is concerned with certain physical aspects of diffraction which have proven to give the most troublesome of mathematical difficulties these are mode conversion that is when the incident wave is scattered among different modes supported by the propagating medium and edge or corner singularities when the diffracting obstacle creates a stress singularity it is shown by way of a particular example that the exact solution of diffraction problems involving these phenomena will in general give rise to a vector wiener hopf equation with a matrix kernel whose factorization is non commutative the main result of this paper is a new procedure motivated by a precise treatment of both the corner singularity and mode conversion for determining the factors in this case

stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a karhunen loeve based moment equation approach 1 a new approach has been developed for solving solute transport problems in randomly heterogeneous media using the karhunen loeve based moment equation klme technique proposed by zhang and lu 2004 the klme approach combines the karhunen loeve decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head flow velocity dispersion coefficient and solute concentration the equations obtained in this approach are sequential and their structure is formulated in the same form as the original governing equations such that any existing simulator such as modular three dimensional multispecies transport model for simulation of advection dispersion and chemical reactions of contaminants in groundwater systems mt3dms can be directly applied as the solver through a series of two dimensional examples the validity of the klme approach is evaluated against the classical monte carlo simulations results indicate that under the flow and transport conditions examined in this work the klme approach provides an accurate representation of the mean concentration for the concentration variance the accuracy of the klme approach is good when the conductivity variance is 0 5 as the conductivity variance increases up to 1 0 the mismatch on the concentration variance becomes large although the mean concentration can still be accurately reproduced by the klme approach our results also indicate that when the conductivity variance is relatively large neglecting the effects of the cross terms between velocity fluctuations and local dispersivities as done in some previous studies can produce noticeable errors and a rigorous treatment of the dispersion terms becomes more appropriate

generalised factorisation for a class of jones form matrix functions a systematic approach is proposed for the generalised factorisation of certain non rational n n matrix functions the first main result consists in a transformation of a meromorphic into a generalised factorisation by algebraic means it closes a gap between the classical wiener hopf procedure and the operator theoretic method of generalised factorisation secondly as examples we consider certain matrix functions of jones form or of n part form which are equivalent to each other in a sense the factorisation procedure is complete and explicit based only on the factorisation of scalar functions of rational matrix functions and upon linear algebra applications in elastodynamic diffraction theory are treated in detail and in a most effective way

electromagnetic fields in the presence of an infinite dielectric wedge electromagnetic fields excited by a line source in the presence of an infinite dielectric wedge with refractive index n are determined by application of the kontorovich lebedev transform singular integral equations for spectral functions are solved by perturbation procedure and the solution is obtained in the form of a neumann series in powers of 1kn k2 the devised numerical scheme permits evaluation of the higherorder terms and thus extends the perturbation solution to values of n not necessarily close to unity asymptotic approximations for the near and far fields inside and outside the dielectric wedge are derived the combination of the neumann type expansion of the transform functions with the representation of the field as a bessel function series extends solutions derived with the kontorovich lebedev method to the case of realvalued wavenumbers and arbitrarily positioned source and observer numerical results showing the influence of wedges with various values of dielectric and magnetic constants on the directivity of a line source are presented and verified through finite difference frequency domain simulations

a bayesian analysis of the linear calibration problem as part of a bayesian analysis of the linear calibration problem a posterior distribution is derived that is mathematically identical to the structural distribution obtained by kalotay 1971 but different from the bayesian distribution obtained by hoadley 1970 useful approximations are given although this posterior distribution has infinite variance no practical problems arise in using it the controversy concerning infinite variances and the use of inverse regression is discussed

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

scales of fluctuation and the propagation of uncertainty in random porous media the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the heterogeneity of the medium is modeled as a superposition of scales of heterogeneity that are statistically uncorrelated the stochastic process making up a hydraulic property is thus represented as a linear combination of normalized deterministic shapes multiplied by a set of uncorrelated random variables the karhunen loeve expansion is used to formalize this representation the uncertainty contained in each of these scales is then propagated through the system using the governing differential equation as a constraint the solution process will thus consist of the original scales used to represent the porous medium as well as various interaction between these scales the objective of this paper is to develop a procedure for quantifying the uncertainty in the field variables using the scales of fluctuation of the porous medium as the measuring stick

stochastic analysis of flow in a heterogeneous unsaturated saturated system 1 in this study we develop a stochastic model for transient unsaturated saturated flow in randomly heterogeneous media with the method of moment equations we first derive partial differential equations governing the statistical moments of the flow quantities by perturbation expansions and then implement these equations under general conditions with the method of finite differences the stochastic model developed is applicable to the entire domain of a bounded multidimensional unsaturated saturated system in the presence of random or deterministic recharge and sink source and in the presence of multiscale nonstationary medium features the unsaturated and saturated zones are coupled through the water table whose position is random in randomly heterogeneous porous media the presence of the water table renders the flow moments strongly nonstationary even in the absence of medium nonstationary features this finding is confirmed with monte carlo simulations the resulting first two moments of flow may be used to approximate confidence intervals for the flow quantities in addition the integrated stochastic flow model provides the prerequisite quantities for realistically studying contaminant transport in unsaturated saturated systems with a stochastic approach

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

electromagnetic scattering from a dielectric wedge and the single hypersingular integral equation electromagnetic fields scattered by a finite dielectric wedge are computed using a hypersingular integral equation hie the results are compared with those obtained previously using a singular integral equation sie and with the theory that predicts that the fields near the edge of the wedge behave like static fields the hie produces more consistent results than the sie probably because the unknown boundary function tends to a constant near the edge instead of diverging the two numerical methods agree reasonably well and these results agree only in part with the static field behavior

wiener hopf factorization for a group of exponentials of nilpotent matrices abstract a complete study of the generalized factorization for a group of 2 2 matrix functions of the form g i n where c r i denotes the 2 2 identity matrix and n represents a rational nilpotent matrix function is presented a closely related class involving the same matrix n is also studied the canonical and non canonical factorizations are considered and explicit formulas are obtained for the partial indices and the factors in such factorizations it is shown in particular that only one of the columns in the factors needs to be determined as a solution to a homogeneous linear riemann hilbert problem the other column being expressed in terms of the first necessary and sufficient conditions for existence of a canonical factorization within the same class are established as well as explicit formulas for the factors in this case

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined

a factorisation procedure for two by two matrix functions on the circle with two rationally independent entries the aim of this paper is the explicit canonical or standard factorisation of matrix functions with wiener algebra elements the present approach covers all regular 2 2 matrices where two entries are arbitrary and the remaining two are linear combinations of the former with rational coefficient functions it is based on the knowledge of how to factorise scalar functions and rational matrix functions in general one also needs the approximation of any scalar wiener algebra function with a rational function however this can be easily circumvented in many applications by intuitive manipulations with rational matrix functions

a stochastic analysis of steady state two phase flow in heterogeneous media 1 we present a novel approach to modeling stochastic multiphase flow problems for example nonaqueous phase liquid flow in a heterogeneous subsurface medium with random soil properties in particular with randomly heterogeneous intrinsic permeability and soil pore size distribution a stochastic numerical model for steady state water oil flow in a random soil property field is developed using the karhunen loeve moment equation klme approach and is numerically implemented an exponential model is adopted to define the constitutive relationship between phase relative permeability and capillary pressure the log transformed intrinsic permeability y x and soil pore size distribution b x are assumed to be gaussian random functions with a separable exponential covariance function the perturbation part of these two log transformed soil properties is then decomposed into an infinite series based on a set of orthogonal normal random variables xn the phase pressure capillary pressure and phase mobility are decomposed by polynomial expansions and the perturbation method combining these expansions of y x b x and dependent pressures the steady state water oil flow equations and corresponding boundary conditions are reformulated as a series of differential equations up to second order these differential equations are solved numerically and the solutions are directly used to construct moments of phase pressure and capillary pressure we demonstrate the validity of the proposed klme model by favorably comparing firstand second order approximations to monte carlo simulations the significant computational efficiency of the klme approach over monte carlo simulation is also illustrated

line source and point source scattering of acoustic waves by the junction of transmissive and soft hard half planes abstract firstly the analysis of a buyukaksoy g cinar a h serbest scattering of plane waves by the junction of transmissive and soft hard half planes zamp 55 2004 483 499 for the scattering of plane waves by the junction of transmissive and soft hard half planes is extended to the case of a line source the introduction of the line source changes the incident field and the method of solution requires a careful analysis in calculating the scattered field the graphical results are presented using mathematica we observe that the graphs of the plane wave situation a buyukaksoy g cinar a h serbest scattering of plane waves by the junction of transmissive and soft hard half planes zamp 55 2004 483 499 can be recovered by shifting the line source to a large distance subsequently the problem is further extended to the case of scattering due to a point source using the results obtained for a line source excitation the introduction of a point source three dimensions involves another variable which then requires the calculation of an additional integral appearing in the inverse transform

the wiener hopf solution of the isotropic penetrable wedge problem diffraction and total field the diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized wiener hopf equations and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf factorization to fredholm integral equations of second kind mathematical aspects are described in a unified and consistent theory for angular region problems the formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique the solutions consider engineering applications in terms of gtd utd diffraction coefficients and total fields

diffraction of an electric polarized wave by a dielectric wedge the problems of diffraction of an e polarized plane wave by an inhomogeneous imperfectly conducting dielectric wedge or a homogeneous dielectric wedge with vanishing conductivity are formulated as integral equations conditions for existence and uniqueness of the solutions of the integral equations are found the integral equations are solved by iterative methods an explicit expression for the far field is found the series expansion of the far field near the edge of a homogeneous wedge converges very fast for relative permittivities in the interval from one to ten the physical optical approximation of the far field is poor