finite difference schemes for the parabolic equation in a variable depth environment with a rigid bottom boundary condition we consider a linear schrodinger type partial differential equation the parabolic equation of underwater acoustics in a layer of water bounded below by a rigid bottom of variable topography using a change of depth variable technique we transform the problem into one with horizontal bottom for which we establish an a priori h1 estimate and prove an optimal order error bound in the maximum norm for a crank nicolson type finite difference approximation of its solution we also consider the same problem with an alternative rigid bottom boundary condition due to abrahamsson and kreiss and prove again a priori h1 estimates and optimal order error bounds for a crank nicolson scheme

a note on the implicit function theorem for quasi linear eigenvalue problems abstract we consider the quasi linear eigenvalue problem p u g u subject to dirichlet boundary conditions on a bounded open set where g is a locally lipschitz continuous function imposing no further conditions on or g we show that for near zero the problem has a bounded solution which is unique in the class of all small solutions moreover this curve of solutions parameterized by depends continuously on the parameter

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

pseudodifferential operators on supermanifolds and the atiyah singer index theorem fermionic quantization or clifford algebra is combined with pseudodifferential operators to simplify the proof of the atiyah singer index theorem for the dirac operator on a spin manifold

existence of bounded solutions for a class of quasilinear elliptic systems on manifolds with boundary we consider nonlinear elliptic partial differential equations for quasilinear operators of the form a u div a x u u a0 x u u x subject to fully nonlinear boundary conditions involving boundary operators of the form for each 0 b u div b x u u b0 x u u u x the main goal of this paper is to give under suitable assumptions on a and b an explicit l estimate for bounded solutions of these elliptic boundary value problems then we establish the existence of at least one solution to such problems extending the author s previous work our methods rely on the definition of approximate problems deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit these methods can be applied to a large class of equations involving operators of leray lions type on suitable banach spaces for a general class of boundary operators b which are possibly of the same order as a as examples these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea level temperatures for relatively long time scales

finite difference solution to the parabolic wave equation in this article an implicit finite difference ifd scheme is introduced for solving underwater acoustic wave propagation problems this scheme is applied to the parabolic wave equation model for the acoustic problem with a free surface and arbitrary bottom and bottom boundary conditions the mathematical theory of the implicit finite difference method as applied to the parabolic equation is developed and a computer model to implement the ifd is introduced representative problems are presented to demonstrate the capability of the ifd model advantages and limitations of the ifd are discussed

difference schemes for the parabolic wave equation in ocean acoustics abstract we introduce and analyze a collection of difference schemes for the numerical solution of a model multidimensional equation of schrodinger type with applications to the three dimensional parabolic wave equation arising from the sound propagation in the ocean this collection of methods includes explicit and implicit schemes two level and three level schemes and real and complex schemes many of these are analogous to classical schemes for the heat equation and the wave equation but some schemes are unique to the schrodinger equation von neumann type stability results are given for all schemes numerical results arising from the application to an ocean acoustic problem are presented

a gaussian finite element method for description of sound diffraction a new method for solving diffraction problems is presented it is based on the use of gaussian diffraction theory comparison of the gaussian beam expansion and fourier series expansion reveals that the gaussian expansion is a more general and more powerful technique the new method combines the gaussian beam superposition technique wen and breazeale j acoust soc am 83 1752 1756 1988 and the numerical finite element solution to the parabolic equation huang j acoust soc am 84 1405 1413 1988 the new method is capable of solving for the sound field even in an inhomogeneous medium or in the presence of reflection as shown by computer modeling the source may be a gaussian source or a distributed source calculated results are compared with experimental results in a laboratory water tank

optimal schwarz waveform relaxation for the one dimensional wave equation we introduce a nonoverlapping variant of the schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension we derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains independently of the length of the time interval these optimal transmission conditions are in general nonlocal but we show that the nonlocality depends on the time interval under consideration and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed we show that convergence in two iterations can be achieved independently of the number of subdomains in that case the algorithm thus scales optimally with the number of subdomains provided the time windows are chosen appropriately for continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates we then introduce a finite volume discretization which permits computations on nonmatching grids and we prove convergence of the fully discrete schwarz waveform relaxation algorithm we finally illustrate our analysis with numerical experiments

solutions and positive solutions for superlinear robin problems we consider nonlinear nonhomogeneous robin problems with a p 1 superlinear reaction term which need not satisfy the ambrosetti rabinowitz condition we look for positive solutions and prove existence and multiplicity theorems for the particular case of the p laplacian we prove existence results under a different geometry near the origin we consider nonlinear nonhomogeneous robin problems with a p 1 superlinear reaction term which need not satisfy the ambrosetti rabinowitz condition we look for positive solutions and prove existence and multiplicity theorems for the particular case of the p laplacian we prove existence results under a different geometry near the origin

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

galerkin methods for parabolic and schrodinger equations with dynamical boundary conditions and applications to underwater acoustics in this paper we consider galerkin finite element methods that approximate the solutions of initial boundary value problems in one space dimension for parabolic and schrodinger evolution equations with dynamical boundary conditions error estimates of optimal rates of convergence in l 2 and h 1 are proved for the associated semidiscrete and fully discrete crank nicolson galerkin approximations the problem involving the schrodinger equation is motivated by considering the standard parabolic paraxial approximation to the helmholtz equation used in underwater acoustics to model long range sound propagation in the sea in the specific case of a domain with a rigid bottom of variable topography this model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations in the real parabolic case numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type

a stable explicit scheme for the ocean acoustic wave equation abstract a class of ocean acoustic wave propagation problems is represented by a parabolic equation of the schrodinger type using conventional explicit finite difference schemes e g the euler scheme to solve the parabolic wave equation is unstable thus important advantages of explicit schemes are completely missing this paper presents a conditionally stable explicit scheme by introducing an extra dissipative term this new explicit scheme is then applied to solve the ocean acoustic parabolic wave equation fully utilizing the advantages of explicit schemes the theoretical development the computational aspects and the advantages are discussed application of the scheme to a realistic ocean acoustic problem is included the solution obtained is compared with the unconditionally stable crank nicolson solution

on a nonlinear and non homogeneous problem without a r type condition in orlicz sobolev spaces using variational methods we prove some existence and multiplicity results for a class of nonlinear and non homogeneous problems without a r type condition in orlicz sobolev spaces

nonuniform depth grids in parabolic equation solutions the parabolic wave equation is solved using a finite difference solution in depth that involves a nonuniform grid the depth operator is discretized using galerkin s method with asymmetric hat functions examples are presented to illustrate that this approach can be used to improve efficiency for problems in ocean acoustics and seismo acoustics for shallow water problems accuracy is sensitive to the precise placement of the ocean bottom interface this issue is often addressed with the inefficient approach of using a fine grid spacing over all depth efficiency may be improved by using a relatively coarse grid with nonuniform sampling to precisely position the interface efficiency may also be improved by reducing the sampling in the sediment and in an absorbing layer that is used to truncate the computational domain nonuniform sampling may also be used to improve the implementation of a single scattering approximation for sloping fluid solid interfaces

an explicit finite difference scheme for solving the ocean acoustic parabolic wave equation abstract a conditionally stable explicit scheme is developed and applied to the ocean acoustic parabolic wave equation the scheme is leapfrogged in the range variable and centered differenced in the vertical variable the theoretical development the computational aspects and the advantage of the scheme are discussed the method is ideal for vector computers

existence of a positive solution and numerical solution for some elliptic superlinear problem abstract in this paper we prove the existence of positive solutions of an elliptic superlinear problem also we are interested here in getting results concerning the existence of positive solutions for the discrete formulation of our problem therefore in order to do it we employ the radial solutions of the elliptic superlinear problem obtaining a second order dynamic equation on time scales which encompasses discrete continuous and hybrid formulations of our problem this unified equation allows us to present numerical simulations which give us a more precise analysis and description concerning the behavior of the solution according to the parameters

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

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

implicit high order compact algorithm for computational acoustics accurate solution of the linearized multidimensional euler equations for aeroacoustics as a system of simple wave equations is demonstrated if organized this system has unambiguous easily implemented boundary conditions allowing waves of same group speeds to pass through numerical boundaries or comply with wall conditions thus the task of designing a complex multidimensional scheme with approximate boundary conditions reduces to the design of accurate schemes for the simple wave equation in particular an implicit compact finite difference scheme and a characteristically exact but numerically nth order accurate boundary condition are used this low dispersion scheme has a third order spatial accuracy for various types of nonuniform meshes fourth order accuracy on uniform meshes and by choice a temporal accuracy of second order for algorithmic simplicity as the crank nicolson scheme the robustness and accuracy of the scheme and the validity of the system decoupling are demonstrated through a series of numerical experiments and comparisons with published results including the recent institute for computer applications in science and engineering nasa langley research center benchmark problems of acoustic and convective wave propagation in cartesian and cylindrical domains and reflection at stationary and or moving boundaries

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

asymptotic hamiltonian dynamics the toda lattice the three wave interaction and the non holonomic chaplygin sleigh abstract in this paper we discuss asymptotic stability in energy preserving systems which have an almost poisson structure in particular we consider a class of poisson systems which includes the toda lattice in standard hamiltonian systems one of course does not expect asymptotic stability the key here is the structure of the phase space of the poisson or almost poisson systems and the nature of the equilibria our systems are in some sense generalizations of the integrable toda lattice system but are not integrable in general as a particular example we point out an interesting connection between three mechanical two degree of freedom systems that exhibit asymptotic stability two of them are classical hamiltonian systems while the third is a non holonomic system non holonomic systems are generally energy preserving but not hamiltonian but the system analyzed here turns out to have a phase space which is the union of hamiltonian ones we also discuss various higher dimensional examples

applications of energy methods to finite difference solutions of the parabolic wave equation abstract implicit finite difference techniques may be applied readily to solve acoustic wave propagation problems in diverse ocean environments for only the most simple cases however can the stability of such schemes be established in this paper the method of energy inequalities is applied to examine the stability and step size requirements for solving the parabolic approximation to the wave equation

exotic cohomology and index theory the main result of this paper is a version of the atiyah singer index theorem for dirac type operators on noncompact complete riemannian manifolds the statement of the theorem involves a novel cohomology theory for such manifolds this theory called exotic cohomology depends on the structure at infinity of a space more precisely it depends on the way that large bounded sets fit together for each cohomology class in this theory we define a higher index of a dirac type operator enjoying the stability and vanishing properties of the usual atiyah singer index these higher indices are analogous to the novikov higher signatures our main theorem will compute these higher indices in terms of standard topological invariants applications of this index theorem include a different approach to some of the results of gromov and lawson 10 on topological obstructions to positive scalar curvature the concept of index that we will use involves the a theory functors k0 and kx for operator algebras 3 14 suppose that b is an ideal in a unital algebra c and let t e c be invertible modulo b in the classical atiyah singer index theorem one takes c to be the bounded operators on the l space of some compact manifold b the compact operators and t an elliptic pseudodifferential operator of order zero then t has an index in the theory group kq b in the classical case this is just z and one recovers the usual fredholm index now let m be a complete riemannian manifold possibly noncompact in 18 i introduced an algebra m which is defined as follows sf m consists of all bounded operators a on l m that have a kernel representation

analysis of an implicit finite difference solution to an underwater wave propagation problem abstract an implicit finite difference scheme approximating a third order partial differential equation is examined the scheme is derived shown to be consistent and its stability properties are analyzed the partial differential equation is a parabolic approximating equation to the reduced wave equation