lower and upper bounds for the allocation problem and other nonlinear optimization problems we demonstrate the impossibility of strongly polynomial algorithms for the allocation problem in the comparison model and in the algebraic tree computation model that follow from lower bound results consequently there are no strongly polynomial algorithms for nonlinear concave separable optimization over a totally unimodular constraint matrix this is in contrast to the case when the objective is linear we present scaling based algorithms that use a greedy algorithm as a subroutine the algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem in that the complexity meets the lower bound derived from the comparison model for other extensions of the allocation problem the scaling based algorithms presented here are the fastest known these algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables

fast and explicit wentzel kramers brillouin mode sum for the bottom interacting field including leaky modes a fast and explicit method is presented for calculating the bottom interacting field by means of a mode sum using the wentzel kramers brillouin approximation for the eigenfunctions and the tindle weston formula for eigenvalue spacing both trapped and leaky modes are automatically included requiring no numerical inversions for either eigenfunctions or eigenvalues examples are provided which demonstrate the speed and accuracy of the approach by comparison with reference solutions the importance of leaky modes especially in range dependent environments with variable water depth is demonstrated a simple criterion for neglecting leaky modes is developed

computationally efficient parabolic equation solutions to seismo acoustic problems involving thin or low shear elastic layers shallow water environments typically include sediments containing thin or low shear layers numerical treatments of these types of layers require finer depth grid spacing than is needed elsewhere in the domain thin layers require finer grids to fully sample effects due to elasticity within the layer as shear wave speeds approach zero the governing system becomes singular and fine grid spacing becomes necessary to obtain converged solutions in this paper a seismo acoustic parabolic equation solution is derived utilizing modified difference formulas using galerkin s method to allow for variable grid spacing in depth propagation results are shown for environments containing thin layers and low shear layers

nonlocal boundary conditions for finite difference parabolic equation solvers theoretical numerical models of underwater sound propagation usually incorporate a downgoing radiation condition on the transverse component of the acoustic field most one way or parabolic equation pe solvers approximate this radiation condition by appending an absorbing layer to the computational mesh and setting the field to zero at the base of this layer papadakis et al j acoust soc am 92 2030 2038 1992 replaced such approximate treatments with a nonlocal boundary condition nlbc that exactly transforms the semi infinite pe problem to an equivalent one in a bounded domain papadakis approach requires the evaluation of a spectral wave number integral whose integrand is inversely proportional to the impedance of the subbottom medium in this paper an alternate procedure is analyzed for obtaining nlbcs directly from the z space crank nicolson solvers for both the tappert and claerbout pes formulas for the field at range r r are derived in terms of the known field at the previously c

a finite method for the solution of a multi resource allocation problem with concave return functions it has recently been recognized that the problem of allocating continuous resources to several activities each with a concave return function has some elements in common with the transportation problem this has inspired a search for solution algorithms for the allocation problem in which the approach used is similar to that applied to the transportation problem this paper describes such an algorithm the algorithm terminates and produces the exact optimal solution a fortran implementation of the algorithm solves a 12 resources 100 activities problem in less than 10 central processor seconds on a cyber 173 computer

uniqueness in weighted lebesgue spaces for a class of fractional parabolic and elliptic equations abstract we investigate uniqueness in suitable weighted lebesgue spaces of solutions to a class of fractional parabolic and elliptic equations

the greedy procedure for resource allocation problems necessary and sufficient conditions for optimality in many resource allocation problems the objective is to allocate discrete resource units to a set of activities so as to maximize a concave objective function subject to upper bounds on the total amounts allotted to certain groups of activities if the constraints determine a polymatroid and the objective is linear it is well known that the greedy procedure results in an optimal solution in this paper we extend this result to objectives that are weakly concave a property generalizing separable concavity we exhibit large classes of models for which the set of feasible solutions is a polymatroid and for which efficient implementations of the greedy procedure can be given

a mapping approach for handling sloping interfaces a mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested at each range the medium is rigidly translated vertically so that a sloping interface becomes horizontal to simplify the approach the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected the effects of these terms can be approximately accounted for by applying a leading order correction to the phase the mapping introduces variations in topography which are relatively easy to handle for the case of a pressure release boundary condition the accuracy of the approach is demonstrated for problems involving fluid sediments the approach should also be accurate for problems involving elastic sediments and should be useful for solving three dimensional problems involving variable topography

a wide angle and high mach number parabolic equation various parabolic equations for advected acoustic waves have been derived based on the assumptions of small mach number and narrow propagation angles which are of limited validity in atmospheric acoustics a parabolic equation solution that does not require these assumptions is derived in the weak shear limit which is appropriate for frequencies of about 0 1 hz and above for atmospheric acoustics when the variables are scaled appropriately in this limit terms involving derivatives of the sound speed density and wind speed are small but can have significant cumulative effects to obtain a solution that is valid at large distances from the source it is necessary to account for linear terms in the first derivatives of these quantities a d pierce j acoust soc am 87 2292 2299 1990 this approach is used to obtain a scalar wave equation for advected waves since this equation contains two depth operators that do not commute with each other it does not readily factor into outgoing and incomin

the joint replenishment problem with general joint cost structures we consider inventory systems with several distinct items demands occur at constant item specific rates the items are interdependent because of jointly incurred fixed procurement costs the joint cost structure reflects general economies of scale merely assuming a monotonicity and concavity submodularity property under a power of two policy each item is replenished with constant reorder intervals which are power of two multiples of some fixed or variable base planning period our main results include a proof that depending upon whether the base planning period is fixed or variable the best among all power of two policies has an average cost which comes within either 6 or 2 of an easily computable lower bound for the minimum cost value we also derive two efficient algorithms to compute an optimal power of two policy the proposed algorithms generate as a by product a specific cost allocation of the joint cost structure to the individual items with this specific allocation the problem with separable costs is in fact equivalent to the original problem with nonseparable joint costs in the sense that the two problems share the same sets of optimal power of two policies with identical associated long run average costs

the one way wave equation and its invariance properties the harmonic wave equation in inhomogeneous media is exactly split into coupled first order equations with respect to a principal direction of propagation according to the bremmer scheme the resulting one way wave equation is shown not to conserve energy flux for dimensions two and three against the general belief in one way wave propagation or parabolic equation literature conservation of energy flux is only ensured in the high frequency limit on the other hand a simple invariant is found that may be seen as a generalization of the snell law to arbitrary non stratified media similarly the reciprocity property is not fully ensured in general and the time reversal symmetry is ensured for propagating fields besides in the one way wave equation the additional term to the standard parabolic equation is shown to strengthen mode coupling the analysis encompasses the evanescent waves

an improved numerical scheme for the sine gordon equation in 2 1 dimensions a rational approximant of order 4 which is applied to a three time level recurrence relation is used to transform the initial boundary value problem associated with the two dimensional sine gordon sg equation arising in the josephson junctions problem the resulting non linear system which is analyzed for stability is solved using an appropriate predictor corrector p c scheme in which an explicit scheme of order 2 is used as predictor for the implementation of the corrector in order to avoid extended matrix evaluations an auxiliary vector was successfully introduced in this p c scheme a modification in the corrector has been proposed according to which the already evaluated corrected values are considered the behavior of this p c scheme is tested numerically on line and ring solitons known from the bibliography regarding the sg equation and conclusions for both undamped and damped problems are derived copyright 2008 john wiley sons ltd

solution techniques for some allocation problems this paper presents methods for solving allocation problems that can be stated as convex knapsack problems with generalized upper bounds such bounds may express upper limits on the total amount allocated to each of several subsets of activities in addition our model arises as a subproblem in more complex mathematical programs we therefore emphasize efficient procedures to recover optimality when minor changes in the parameters occur from one problem instance to the next these considerations lead us to propose novel data structures for such problems also we introduce an approximation method to solve certain equations which arise during the procedures

comparison and regularity results for the fractional laplacian via symmetrization methods abstract in this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional laplacian this allows to get sharp estimates for the solutions obtained by comparing them with solutions of suitable radial problems furthermore we use such result to prove a priori estimates for solutions in terms of the data providing several regularity results which extend the well known ones for the classical laplacian

well posedness of a fractional porous medium equation on an evolving surface abstract we investigate the existence uniqueness and l 1 contractivity of weak solutions to a porous medium equation with fractional diffusion on an evolving hypersurface to settle the existence we reformulate the equation as a local problem on a semi infinite cylinder regularise the porous medium nonlinearity and truncate the cylinder then we pass to the limit first in the truncation parameter and then in the nonlinearity and the identification of limits is done using the theory of subdifferentials of convex functionals in order to facilitate all of this we begin by studying in the setting of closed riemannian manifolds and sobolev spaces the fractional laplace beltrami operator which can be seen as the dirichlet to neumann map of a harmonic extension problem a truncated harmonic extension problem will also be examined and convergence results to the solution of the harmonic extension will be given for a technical reason we will also consider some related extension problems on evolving hypersurfaces which will provide us with the minimal time regularity required on the harmonic extensions in order to properly formulate the moving domain problem this functional analytic theory is of course independent of the fractional porous medium equation and will be of use generally in the analysis of fractional elliptic and parabolic problems on manifolds

range dependent matched field inversion of swellex 96 data using the downhill simplex algorithm matched field inversion mfi techniques have been applied for effective and efficient estimation of geoacoustic parameters of the ocean bottom this paper presents a new tomographic mfi method for use in range dependent environments the mfi correlates modeled data with measured data and uses a search algorithm to determine model parameter values that maximize the correlator in the present method a parabolic equation propagation model is used to compute replica fields to account for mode coupling in the environment the search algorithm is a two stage hybrid method that combines an initial global component and a final local component the first stage employs a random search to determine n 1 parameter sets with the best correlations where n is the number of parameters being determined in the next stage the n 1 sets are used as inputs to the local downhill simplex algorithm the algorithm is shown to perform well for simulated vertical line array data for an environment representative of the swellex 96 experimental site the inversion technique is then applied to measured data obtained from a radial track in swellex 96 the geometric parameters of the experiment and dominant geoacoustic parameters were successfully recovered for data with relatively low signal to noise ratio

fast algorithms for bipartite network flow we discuss network flow in a bipartite graph g with node sets s and t we show that karzanov s algorithm runs in time o s 2 t on g and that the mpm algorithm can be modified to run in the same bound for the common case that the degree of each node in t is bounded by a constant we show that a modified version of the mpm algorithm runs in time o s 3 s t on g the importance of these results comes from the common occurrence of problems that can be modeled and solved using network flow on bipartite graphs where s ll t in many applications involving bounded degree on the nodes in t t theta s 2 and our results reduce the best bounds on bipartite flow from o s 6 to o s 3 we use our results on bipartite flow to obtain the fastest known bounds on several combinatorial problems and mention several additional problems involving bipartite flow where s is much smaller than t

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

inversion of acoustic data using a combination of genetic algorithms and the gauss newton approach an inversion procedure for obtaining speeds attenuation densities and thicknesses for a layered medium is described the inversion is carried out using the least squares technique and the forward modeling is based on safari the optimization is a hybrid method combining the global genetic algorithms and the local gauss newton method this is done by taking several gradient steps between each update of the object function for each individual in the population the gradients for the gauss newton method are computed analytically this makes the computation faster and more stable than computing the gradients by numerical differentiation the combination of a global and a local method makes the hybrid method faster and it gets closer to the global minimum than a pure global method examples based on both real and synthetic data in wave number frequency and range frequency domains show that the method works well

optimal flows in networks with multiple sources and sinks with applications to oil and gas lease investment programs in the classical maximal flow problem the objective is to maximize the supply to a single sink in a capacitated network in this paper we consider general capacitated networks with multiple sinks the objective is to optimize a general concave preference relation on the set of feasible supply vectors we show that an optimal solution can be obtained by a marginal allocation procedure an efficient implementation results in an adaptation of the augmenting path algorithm we also discuss an application of the procedure for an investment company that deals in oil and gas ventures

adjacency on polymatroids this paper characterizes adjacency for extreme points of a polymatroid two extreme points of a polymatroid are adjacent if and only if they either differ in exactly one component or differ in exactly two components with the two components satisfying a certain ordering relation a polynomial algorithm generates and lists all extreme points adjacent to a given extreme point of a polymatroid similar results hold for the core of a convex game

a survey on the continuous nonlinear resource allocation problem our problem of interest consists of minimizing a separable convex and differentiable function over a convex set defined by bounds on the variables and an explicit constraint described by a separable convex function applications are abundant and vary from equilibrium problems in the engineering and economic sciences through resource allocation and balancing problems in manufacturing statistics military operations research and production and financial economics to subproblems in algorithms for a variety of more complex optimization models this paper surveys the history and applications of the problem as well as algorithmic approaches to its solution the most common techniques are based on finding the optimal value of the lagrange multiplier for the explicit constraint most often through the use of a type of line search procedure we analyze the most relevant references especially regarding their originality and numerical findings summarizing with remarks on possible extensions and future research 2006 elsevier b v all rights reserved

existence results for a coupled system of caputo type sequential fractional differential equations with nonlocal integral boundary conditions this paper is concerned with the existence and uniqueness of solutions for a coupled system of caputo type sequential fractional differential equations supplemented with nonlocal riemann liouville integral boundary conditions the existence of solutions is derived by applying leray schauder s alternative while the uniqueness of solution is established via banach s contraction principle an illustrative example is also included the paper concludes with some interesting observations

the fractional laplacian in power weighted l p spaces integration by parts formulas and self adjointness abstract we consider the fractional laplacian operator s let s 0 1 on euclidean space and investigate the validity of the classical integration by parts formula that connects the l 2 r d scalar product between a function and its fractional laplacian to the nonlocal norm of the fractional sobolev space h s r d more precisely we focus on functions belonging to some weighted l 2 space whose fractional laplacian belongs to another weighted l 2 space we prove and disprove the validity of the integration by parts formula depending on the behaviour of the weight x at infinity the latter is assumed to be like a power both near the origin and at infinity the two powers being possibly different our results have direct consequences for the self adjointness of the linear operator formally given by 1 s the generality of the techniques developed allows us to deal with weighted l p spaces as well

optimal power of two replenishment strategies in capacitated general production distribution networks in this paper we develop a model for a capacitated production distribution network of general but acyclic topology with a general bill of materials as considered in mrp material requirement planning or drp distribution requirement planning systems this model assumes stationary deterministic demand rates and a standard stationary cost structure it is a generalization of the uncapacitated model treated in the seminal papers of maxwell and muckstadt 1985 and roundy 1986 the capacity constraints consist of bounds on the frequency with which individual items can or need to be replenished we derive a pair of simple and efficient algorithms capable of determining an optimal power of two policy these algorithms consist of a limited number of maximum flow computations in networks closely related to the production distribution network the complexity of these algorithms even when applied to the uncapacitated model compares favorably with that of the existing alternative solution methods

existence results for liouville caputo type fractional differential equations with nonlocal multi point and sub strips boundary conditions abstract this paper is concerned with the existence of solutions for a new class of boundary value problems involving nonlinear liouville caputo type fractional differential equations of arbitrary order and nonlocal multi point and sub strips boundary conditions the nonlinearity depends on the unknown function as well as its lower order fractional derivative the existence results are obtained with the aid of schauder type fixed point theorem and krasnoselskii s fixed point theorem while the uniqueness of solutions is established by means of classical contraction mapping principle we also discuss some variants of the given problem

discrete non local boundary conditions for split step pade approximations of the one way helmholtz equation this paper deals with the efficient numerical solution of the two dimensional one way helmholtz equation posed on an unbounded domain in this case one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of the so called discrete transparent boundary conditions for state of the art parabolic equation methods namely a split step discretization of the high order parabolic approximation and the split step pade algorithm of collins finally several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach

a higher order on surface radiation condition derived from an analytic representation of a dirichlet to neumann map on surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles an analytic representation of the dirichlet to neumann map for a circle is derived and used to construct a higher order on surface radiation condition for a generally convex perfectly conducting body in two dimensions this approach is based on a hankel function in which a tangential operator appears in the index in the high frequency limit this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems treating the scattered field propagation angle relative to the surface normal and the surface curvature as independent parameters the representation is fit to a rational function to provide an accurate and efficient on surface radiation condition that is tested for various examples

on stochastic completeness of jump processes we prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete we use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete where the cubic volume growth is sharp

a single scattering correction for large contrasts in elastic layers the single scattering solution is implemented in a formulation that makes it possible to accurately handle solid solid interfaces with the parabolic equation method problems involving large contrasts across sloping stratigraphy can be handled by subdividing a vertical interface into a series of two or more scattering problems the approach can handle complex layering and is applicable to a large class of seismic problems the solution of the scattering problem is based on an iteration formula which has improved convergence in the new formulation and the transverse operator of the parabolic wave equation which is implemented efficiently in terms of banded matrices accurate solutions can often be obtained by using only one iteration

two algorithms for maximizing a separable concave function over a polymatroid feasible region abstract in this paper we present two new algorithms for maximizing a separable concave function on a polymatroid both algorithms apply to the discrete as well as the continuous version of the problem the application of these algorithms to several types of polymatroids is discussed and we show that the decomposition algorithm runs in polynomial time in the discrete version for network and generalized symmetric polymatroids and the bottom up algorithm in the discrete version runs in polynomial time when the polymatroid is given as an explicit list of constraints

dispersive waves and pml performance for large stencil differencing in the parabolic equation the discrete reflection absorption performance of perfectly matched layer pml boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the pml region most previous work analyzing pml reflection performance has assumed use of low order second or fourth small stencil three or five point spatial discretizations whose dispersion is poor outside a narrow low wavenumber band large stencil differencing schemes regardless of order are easily created with physically realistic wide band dispersive characteristics but the effect their adoption has on pml performance has not been analyzed previously in the context of a semi discrete approximation to the parabolic paraxial equation this work derives the analytic form of the discrete pml reflection coefficient for an arbitrary order arbitrary stencil interior differencing scheme coupled to an arbitrary order arbitrary stencil pml region this result is then used to show that the absorption performance of the pml regions is largely insensitive to changes in interior differencing as a consequence the benefits of superior dispersion characteristics provided by use of large stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the pml

a scaled mapping parabolic equation for sloping range dependent environments parabolic equation solutions use various techniques for approximating range dependent interfaces one is a mapping approach m d collins et al j acoust soc am 107 1937 1942 2000 where at each range the domain is vertically translated so that sloping bathymetry becomes horizontal and range dependence is transferred to the upper surface in this paper a scaled mapping is suggested where the domain is vertically distorted so that both the bathymetry and upper surface are horizontal accuracy is demonstrated for problems involving fluid sediments generalizations of the approach should be useful for environments with layer thicknesses that vary with range

the flattened surface parabolic equation a simple surface flattening transformation is applied to the problem of underwater acoustic scattering from an irregular sea surface the flattened surface elliptic wave equation is factored to obtain a parabolic equation that has the same high angle capability as recent parabolic equation models the flattened surface parabolic equation contains an additional surface slope term but includes forward scattering from the irregular sea surface the transformation creates additional volume irregularities and irregularities in the water sediment interface that the range and depth mesh need to resolve the surface flattening transformation is easy to implement in a parabolic equation model it compares very well with results obtained using boundary integral equations an example is given of scattering at 400 hz from a sea surface with wave heights of 3 m or nearly a wavelength

a three dimensional parabolic equation model of sound propagation using higher order operator splitting and pade approximants an alternating direction implicit adi three dimensional fluid parabolic equation solution method with enhanced accuracy is presented the method uses a square root helmholtz operator splitting algorithm that retains cross multiplied operator terms that have been previously neglected with these higher order cross terms the valid angular range of the parabolic equation solution is improved the method is tested for accuracy against an image solution in an idealized wedge problem computational efficiency improvements resulting from the adi discretization are also discussed

estimating parameter uncertainties in matched field inversion by a neighborhood approximation algorithm in bayesian inversion the solution is characterized by its posterior probability density ppd a fast gibbs sampler fgs has been developed to estimate the multi dimensional integrals of the ppd which requires solving the forward models many times and leads to intensive computation for multi frequency or range dependent inversion cases this paper presents an alternative approach based on a neighborhood approximation bayes nab algorithm for lower dimension geoacoustic inversion the nab can approximate the ppd very well for higher dimensional problems and sensitive parameters however the nab algorithm has difficulty estimating the ppd accurately with limited model samples according to the preliminary ppd estimation from the nab this paper developed a multi step inversion scheme which adjusts the parameter search intervals flexibly in order to improve the approximation accuracy of the nab and obtain more complete parameter uncertainties the prominent feature of the nab is to approximate the ppd by incorporating all models for which the forward problem has been solved into the appraisal stage comparison of the fgs and nab for noisy synthetic benchmark test cases and mediterranean real data indicates that the nab provides reasonable estimates of the ppd moments while requiring significantly less computation time

inversion of seismoacoustic data using genetic algorithms and a posteriori probability distributions the goal of many underwater acoustic modeling problems is to find the physical parameters of the environment with the increase in computer power and the development of advanced numerical models it is now feasible to carry out multiparameter inversion the inversion is posed as an optimization problem which is solved by a directed monte carlo search using genetic algorithms the genetic algorithm presented in this paper is formulated by steady state reproduction without duplicates for the selection of parents the object function is scaled according to a boltzmann distribution with a temperature equal to the fitness of one of the members in the population the inversion would be incomplete if not followed by an analysis of the uncertainties of the result when using genetic algorithms the response from many environmental parameter sets has to be computed in order to estimate the solution the many samples of the models are used to estimate the a posteriori probabilities of the model parameters t

a comparison of transparent boundary conditions for the fresnel equation abstract we consider two numerical transparent boundary conditions that have been previously introduced in the literature the first condition bpp was proposed by baskakov and popov 1991 wave motion 14 121 128 and papadakis et al 1992 j acoust soc am 92 2030 2038 while the second sdy is that of schmidt and deuflhard 1995 comput math appl 29 53 76 and schmidt and yevick 1997 j comput phys 134 96 107 the latter procedure is explicitly tailored to the form of the underlying numerical propagation scheme and is therefore unconditionally stable and highly precise here we present a new derivation of the sdy approach as a result of this analysis we obtain a simple modification of the bpp method that guarantees accuracy and stability for long propagation step lengths

greene approximation wide angle parabolic equation for radio propagation this paper presents the greene second order parabolic equation pe solution to model radio propagation over irregular terrain this solution is implemented by finite difference and shift map technique the second order finite difference approach performs well for slopes up to about 15 and discontinuous slope changes up to about 30 which is better than can be achieved using the split step fourier approach the results prove that the greene pe solution has greater propagation angles than does the claerbout pe solution thus the greene second order solution can certainly give better results for large angle propagation over complicated terrain boundaries besides we treat the important problem of accuracy of the different approximations of the pe based propagation models and derive the general pe solutions with respect to the terrain slope for the different approximations it is useful to have collected in one paper the most important pe approximations for radio propagation

rational operators for filtering the parabolic equation pe method has been applied to approximate the wave equation construct radiation conditions solve scattering problems construct initial conditions and derive energy flux conditions a new application of the pe method filters based on rational approximations of depth operators is described and tested the rational approximation is designed to act as the identity operator on the desired part of the spectrum of the depth separated wave equation and to annihilate other parts of the spectrum this approach does not require explicit knowledge of the spectrum the applications of the filter include directly solving eigenvalue problems annihilating components of the wave number spectrum generating initial conditions at the source range and eliminating gibbs oscillations that arise in energy conserving pe solutions

discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics this paper is concerned with transparent boundary conditions tbcs for wide angle parabolic equations wapes in the application to underwater acoustics assuming cylindrical symmetry existing discretizations of these tbcs introduce slight numerical reflections at this artificial boundary and also render the overall crank nicolson finite difference method only conditionally stable here a novel discrete tbc is derived from the fully discretized whole space problem that is reflection free and yields an unconditionally stable scheme while we shall assume a uniform discretization in range the interior depth discretization i e in the water column may be nonuniform and we shall discuss strategies for the best exterior discretization i e in the sea bottom the superiority of the new discrete tbc over existing discretizations is illustrated on several benchmark problems in the literature different wapes or wape and the standard parabolic equation have been coupled in the water and the sea bottom we analyze under which conditions this yields a hybrid model that is conservative for the acoustic field

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

continuity of the temperature in boundary heat control problems abstract motivated by the boundary heat control problems formulated in the book of duvaut and lions we study a boundary stefan problem and a boundary porous media problem we prove continuity of the solution with the appropriate modulus we also extend the results to the fractional order case and to the anomalous diffusion problems

transparent boundary conditions for split step pade approximations of the one way helmholtz equation in this paper we generalize the nonlocal discrete transparent boundary condition introduced by f schmidt and p deuflhard 1995 comput math appl 29 53 76 and by f schmidt and d yevick 1997 j comput phys 134 96 107 to propagation methods based on arbitrary pade approximations of the two dimensional one way helmholtz equation our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition which then yields an unconditionally stable propagation method

complex pade approximants for wide angle acoustic propagators procedures for constructing wide angle propagators that attenuate evanescent modes in parabolic equation pe models of underwater sound propagation are examined in this context two new split step pade approximations to exponentiated square root operators are proposed the first method involves a propagator derived from a pade approximant of a square root operator that has been rotated in the complex plane while the propagator of the second method is obtained from a complex coefficient rational approximation of the square root operator numerical evidence confirms that these proposed methods yield improved behavior compared to previously suggested techniques in selected areas of the spectral domain

exact discrete nonlocal boundary conditions for high order pade parabolic equations the nonlocal boundary conditions nlbcs for high order finite difference parabolic equations pes are obtained by z transformation of the discrete pe in a homogeneous medium the considered nlbcs include the free space radiation condition possibly with a density jump at the nlbc interface the nlbc at an arbitrary impedance interface and the nlbcs for sources and the starting field beyond the nlbc interface the derivation is presented for the multiterm pade pe model owwe one way wave equation but the developed technique is applicable to a broad class of finite difference pes the obtained nlbcs are exact for the given finite difference scheme they are not limited by the order of pade approximation or by the pe steps in range and depth the nlbc convolution coefficients are calculated by the numerical inverse z transformation the accuracy and performance of the algorithm are analyzed for several benchmark problems the solution is robust for the range steps over 25 wavelengths and or the approximations up to the tenth order the nlbcs are faster and more accurate for large steps because fewer previous range steps contribute to the convolution precomputation of the nlbc coefficients may be required in time demanding applications the results are compared with earlier proposed nlbcs for high order pes

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

on the cauchy problem for a general fractional porous medium equation with variable density abstract we study the well posedness of the cauchy problem for a fractional porous medium equation with a varying density 0 we establish existence of weak energy solutions uniqueness and nonuniqueness is studied as well according to the behavior of at infinity

scalar electromagnetic propagation modelling using parabolic equations and the split step pade approximation this paper presents a solution of 2d electromagnetic wave propagation problems in complicated terrestrial domains scalar 2d parabolic approximations are derived from maxwell s equations for both vertical and horizontal polarization the parabolic equations are then solved using a new technique involving pade rational function approximations of the macroscopic operator this method allows for larger than normal range stepping speeding up computational time significantly the pade approximation and the numerical implementation are fully discussed the discontinuity at the earth s surface is handled directly by using classical continuity conditions and deriving exact interface conditions for linking the fields in the atmosphere to those in the terrain the interface conditions are then implemented using the concept of virtual points preliminary benchmark tests show the interface treatment to work well finally several example runs are presented illustrating results

asymptotic behavior of solutions for nonlinear elliptic problems with the fractional laplacian abstract in this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional laplacians and the critical exponents this work can be seen as a nonlocal analog of the results of han 1991 24 and rey 1990 35

wave number sampling at short range the self starter is an efficient approach for solving geoacoustic inverse problems involving a vertical array of receivers located on the order of ten wavelengths from a source r j cederberg and m d collins j acoust soc am 22 102 109 1997 with this approach accurate solutions can be obtained by sampling the wave number spectrum at approximately ten points there have been unsubstantiated claims that the spectral solution provides similar efficiency when implemented with an approach that involves perturbing the integration contour off the real line f b jensen et al computational ocean acoustics american institute of physics new york 1994 pp 231 240 it is demonstrated that this quadrature scheme breaks down at short ranges and that the self starter solution is about an order of magnitude faster these conclusions are based on comparisons of the rational approximations associated with the self starter and the quadrature scheme comparisons of acoustic fields for particular proble

a higher order tangent linear parabolic equation solution of three dimensional sound propagation a higher order square root operator splitting algorithm is employed to derive a tangent linear solution for the three dimensional parabolic wave equation due to small variations of the sound speed in the medium the solution shown in this paper unifies other solutions obtained from less accurate approximations examples of three dimensional acoustic ducts are presented to demonstrate the accuracy of the solution future work on the applications of associated adjoint models for acoustic inversions is proposed and discussed