z transform theory and the fdtd method in implementing the finite difference time domain fdtd method on materials which are dispersive or nonlinear the relationship between the flux density and the electric field can be the most complicated part of the problem because the fdtd method is a sampled time domain method this relationship can be can be looked upon as a digital filtering problem the z transform is typically used in digital filtering and signal processing problems the paper illustrates the use of the z transform in implementing the fdtd method where complicated dispersive or nonlinear materials are involved

evolution semigroups and sums of commuting operators a new approach to the admissibility theory of function spaces this paper is concerned with conditions for the admissibility of a translation invariant function space m with respect to a well posed linear evolution equation du dt au f t t r we propose a new approach to this problem by considering the sum of two commuting operators d dt dm and the operator of multiplication by a on m on the one hand the closure of this operator is the infinitesimal generator of the so called evolution semigroup associated with on the other hand the generator g of this semigroup relates a mild solution u of to the forcing term f by the rule gu f consequently various spectral criteria of the type dm a for the admissibility of the function space m with respect to can be proved in an elegant manner moreover they can be naturally extended to general classes of differential equations including higher order and abstract functional differential equations applications and examples are provided to illustrate the obtained results

a general method for fdtd modeling of wave propagation in arbitrary frequency dispersive media a general formulation is presented for finite difference time domain fdtd modeling of wave propagation in arbitrary frequency dispersive media two algorithmic approaches are outlined for incorporating dispersion into the fdtd time stepping equations the first employs a frequency dependent complex permittivity denoted form 1 and the second employs a frequency dependent complex conductivity denoted form 2 a pade representation is used in z transform space to represent the frequency dependent permittivity form 1 or conductivity form 2 this is a generalization over several previous methods employing either debye lorentz or drude models the coefficients of the pade model may be obtained through an optimization process leading directly to a finite difference representation of the dispersion relation without introducing discretization error stability criteria for the dispersive fdtd algorithms are given we show that several previously developed dispersive fdtd algorithms can be cast as special cases of our more general framework simulation results are presented for a one dimensional 1 d air muscle example considered previously in the literature and a three dimensional 3 d radiation problem in dispersive lossy soil using measured soil data

cusp development in hele shaw flow with a free surface free surface flow in a hele shaw cell is considered it has been known for some time that when the fluid region is contracting a finite time blow up can occur in which a cusp forms in the free surface the solution does not exist beyond the time of blow up examples are presented of this blow up and also of a new process in which a cusp of a different type forms in the free surface but after which the solution continues to exist after a transformation of the dependent variable the pressure the new cusps are related to permissible singularities in the free boundary of the obstacle problem

almost periodic solutions to some semilinear non autonomous thermoelastic plate equations abstract the paper is concerned with the existence of almost periodic solutions to the so called semilinear thermoelastic plate systems for that the strategy consists of seeing these systems as a particular case of the semilinear parabolic evolution equations x t a t x t f t x t t r where a t for t r is a family of sectorial linear operators on a banach space x satisfying the so called acquistapace terreni conditions and f is a function defined on a real interpolation space x for 0 1 under some reasonable assumptions it will be shown that has a unique almost periodic solution we then make use of the previous result to obtain the existence and uniqueness of an almost periodic solution to the thermoelastic plate systems

an exact solution to the one phase zero surface tension hele shaw free boundary problem abstract a family of explicit solutions are presented to the one phase zero surface tension hele shaw problem with a single source sink the free boundary is described as a function of three parameters whose dependence on time is evaluated through two different yet complementary methods possible mechanisms for nonexistence beyond a critical time are found in the case of both a contracting and expanding fluid region

c semigroups and almost periodic solutions of evolution equations abstract the paper is concerned with the existence of almost periodic mild solutions to evolution equations of the form u t au t f t where a generates a c semigroup and f is almost periodic using the evolution c semigroup associated with the equation under consideration we obtain various sufficient conditions for the existence of almost periodic solutions to which extend previous ones to a more general class of ill posed equations involving c semigroups

fractional calculus based fdtd algorithm for ultrawideband electromagnetic characterization of arbitrary dispersive dielectric materials a novel finite difference time domain algorithm for modeling ultrawideband electromagnetic pulse propagation in arbitrary multirelaxed dispersive media is presented the proposed scheme is based on a general yet computationally efficient series representation of the fractional derivative operators associated with the permittivity functions describing the frequency dispersion properties of a given dielectric material dedicated uniaxial perfectly matched layer boundary conditions are derived and implemented in combination with the basic time marching scheme moreover a total field scattered field formulation is adopted in order to analyze the material response under plane wave excitation compared with alternative numerical methodologies available in the scientific literature the proposed technique features a significantly enhanced accuracy in the solution of complex electromagnetic propagation problems involving higher order dispersive dielectrics such as the ones typically encountered in geoscience and bioengineering applications

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations abstract we investigate spectral criteria for the existence of almost periodic solutions to linear 1 periodic evolution equations of the formdx dt a t x f t with in general unbounded a t and almost periodicf using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set e isp f then the above equation has an almost periodic mild solutionxfwhich is unique if one requiressp xf 2 k k z sp f we emphasize that our method allows us to treat the equations without assumption on the existence of floquet representation this improves recent results on the subject in addition we discuss some particular cases in which the spectrum of monodromy operator does not intersect the unit circle and apply the obtained results to study the asymptotic behavior of solutions finally an application to parabolic equations is considered

stability and almost periodicity of trajectories of periodic processes abstract we prove that if the monodromy operator v of a linear periodic process u t in a banach space e is power bounded has countable peripheral spectrum and if its peripheral point spectrum satisfies a certain natural and simple duality condition which always holds in reflexive spaces then every positive trajectory u u 0 x 0 x e is asymptotically almost periodic if in particular the peripheral point spectrum of v is empty then every positive trajectory is asymptotically stable we also obtain results on almost periodicity of complete bounded trajectories and consider conditions under which nontrivial complete trajectories exist

asymptotic behavior of nonlinear evolution equations in banach spaces abstract in this study we show how yosida s approach for the approximation of the derivative obtains new results for evolution systems using this method we obtain multivalued time dependent perturbation results in addition we consider translation invariant subspaces y of the bounded and uniformly continuous functions to obtain criteria for the existence of solutions u y to the equation u t a t u t u t f t t r or of solutions u which are asymptotically close to y for the inhomogeneous differential equation u t a t u t u t f t t 0 u 0 u 0 in general banach spaces where a t denotes a possibly nonlinear time dependent dissipative operator particular examples of the space y are spaces of functions with various properties of almost periodicity and more general types of asymptotic behavior furthermore an application to functional differential equations is described

on the spectrum complete trajectories and asymptotic stability of linear semi dynamical systems abstract let t t t 0 be a c0 semigroup in a banach space x with generator a we prove that if t t t 0 is bounded and sun reflexive and the sun dual semigroup is not asymptotically stable then there exist bounded complete trajectories under t t t 0 provided i there is t0 0 such that ran t t0 is dense in x or ii a i r questions of almost periodicity of complete trajectories are also discussed and a new proof of our earlier theorem jointly with yu i lyubich on asymptotic stability is given

strongly φ like functions of order α in two dimensional free boundary problems abstract in this paper we apply certain results in the theory of univalent functions to investigate the time evolution of the free boundary of a viscous fluid for a planar flow problem in the hele shaw cell model under injection to this end we prove that the property of strongly likeness of order 0 1 is preserved in time for both inner and outer problems under the assumption of nonzero small surface tension

schaefer type theorem and periodic solutions of evolution equations abstract two fixed point theorems for the sum of contractive and compact operators are obtained in this paper which generalize and improve the corresponding results in h schaefer uber die methode der a priori schranken math ann 129 1955 415 416 t a burton integral equations implicit functions and fixed points proc amer math soc 124 1996 2383 2390 v i istr tescu fixed point theory an introduction reidel dordrecht 1981 t a burton k colleen a fixed point theorem of krasnoselskii schaefer type math nachr 189 1998 23 31 d r smart fixed point theorems cambridge univ press cambridge 1980 as the applications for the results we obtain the existence of periodic solutions for some evolution equations with delay which extend the corresponding results in t a burton b zhang periodic solutions of abstract differential equations with infinite delay j differential equations 90 1991 357 396

asymptotics for periodic systems abstract this paper investigates the asymptotic behaviour of solutions of periodic evolution equations starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications for this class of systems we analyse in detail the spectral properties of the associated monodromy operator showing in particular that it is a so called ritt operator under a natural resonance condition this allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions in particular we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential we illustrate our general results by applying them to concrete problems including the one dimensional wave equation with periodic damping

circular spectrum and bounded solutions of periodic evolution equations abstract we consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u a t u h t u f t where a t is in general an unbounded operator depending 1 periodically on t h is 1 periodic in t is small and f is a bounded and continuous function that is not necessarily uniformly continuous we propose a new approach to the spectral theory of functions via the concept of circular spectrum and then apply it to study the linear equations u a t u f t with general conditions on f for small we show that the perturbed equation inherits some properties of the linear unperturbed one the main results extend recent results in the direction saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f

asymptotically almost periodic solutions of evolution equations in banach spaces abstract tile linear abstract evolution equation u t au t t t r is considered where a d a e e is the generator of a strongly continuous semigroup of operators in the banach space e starting from analogs of kadets and loomis theorems for vector valued almost periodic functions we show that if a i r is countable and r e is asymptotically almost periodic then every bounded and uniformly continuous solution u to is asymptotically almost periodic provided e tu t has uniformly convergent means for all a i r related results on eberlein weakly asymptotically almost periodic periodic asymptotically periodic and c0 solutions of as well as on the discrete case of solutions of difference equations are included

boundedness and almost periodicity of solutions of partial functional differential equations abstract we study necessary and sufficient conditions for the abstract functional differential equation x ax fx t f t to have almost periodic quasi periodic solutions with the same structure of spectrum as f the main conditions are stated in terms of the imaginary solutions of the associated characteristic equations and the spectrum of the forcing term f the obtained results extend recent results to abstract functional differential equations

a characteristic equation for non autonomous partial functional differential equations abstract we characterize the exponential dichotomy of non autonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time periodic case from this we deduce robustness results we further study the almost periodicity of solutions to the inhomogeneous equation our approach is based on the spectral theory of evolution semigroups

fdtd for nth order dispersive media previously a method for applying the finite difference time domain fdtd method to dispersive media with complex permittivity described by a function with a single first order pole was presented this method involved the recursive evaluation of a discrete convolution and was therefore relatively efficient in this work the recursive convolution approach is extended to media with dispersions described by multiple second order poles the significant change from the first order implementation is that the single backstore variable for each second order pole is complex the approach is demonstrated for a pulsed plane wave incident on a medium with a complex permittivity described by two second order poles and excellent agreement is obtained with the exact solution

stability and phase error analysis of fd td in dispersive dielectrics four fd td extensions for the modeling of pulse propagation in debye or lorentz dispersive media are analyzed through studying the stability and phase error properties of the coupled difference equations corresponding to maxwell s equations and to the equations for the dispersion for good overall accuracy the author shows that all schemes should be run at their courant stability limit and that the timestep should finely resolve the medium timescales particularly for debye schemes it should be at least spl delta t 10 sup spl minus 3 spl tau while for lorentz schemes it should be spl delta t 10 sup spl minus 2 spl tau where spl tau is a typical medium relaxation time a numerical experiment with a debye medium confirms this the author has determined that two of the discretizations for debye media are totally equivalent in the lorentz medium case the author establishes that the method that uses the polarization differential equation to model dispersion is stable for all wavenumbers and that the method using the local in time constitutive relation is weakly unstable for modes with wavenumber k such that k spl delta x spl pi 2

frequency dependent fdtd methods using z transforms the frequency dependent finite difference time domain method fd sup 2 td method has been shown to be capable of correctly calculating electromagnetic propagation through media whose dielectric properties are frequency dependent however as researchers search for more elaborate applications the formulation of the fd sup 2 td methods becomes more complex in this work the mathematics of the fd sup 2 td method is developed using z transform theory this has the advantages of presenting a clearer formulation and allowing researchers to draw on the literature of systems analysis and signal processing disciplines

almost periodicity of mild solutions of inhomogeneous periodic cauchy problems abstract we consider a mild solution u of a well posed inhomogeneous cauchy problem u t a t u t f t on a banach space x where a is periodic for a problem on r we show that u is asymptotically almost periodic if f is asymptotically almost periodic u is bounded uniformly continuous and totally ergodic and the spectrum of the monodromy operator v contains only countably many points of the unit circle for a problem on r we show that a bounded uniformly continuous solution u is almost periodic if f is almost periodic and various supplementary conditions are satisfied we also show that there is a unique bounded solution subject to certain spectral assumptions on v f and u

tauberian theorems and stability of solutions of the cauchy problem let f r x be a bounded strongly measurable function with values in a banach space x and let ie be the singular set of the laplace transform f in ir suppose that e is countable and 0 e i uf s u du 0 uniformly for s 0 as 0 for each in e it is shown that t 0 e i uf u du f i 0 as t for each in r e in particular f t 0 if f is uniformly continuous this result is similar to a tauberian theorem of arendt and batty it is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on buc r x and it implies several results concerning stability of solutions of cauchy problems

almost periodic solutions of first and second order cauchy problems abstract almost periodicity of solutions of first and second order cauchy problems on the real line is proved under the assumption that the imaginary resp real spectrum of the underlying operator is countable related results have been obtained by ruess v and basit our proof uses a new idea it is based on a factorisation method which also gives a short proof of the vector valued version of loomis classical theorem saying that a bounded uniformly continuous function from r into a banach spacexwith countable spectrum is almost periodic ifc0 x our method can also be used for solutions on the half line this is done in a separate paper

weak almost periodicity and the strong ergodic limit theorem for periodic evolution systems the first ergodic theorems for semigroups of nonlinear operators date back to work done by j b baillon cl 23 in 1975 1976 where he proved in 2 that i i lim l t j t s i x dt exists for an odd nonlinear contraction semigroup s t a 0 in real hilbert space in subsequent years various authors have extended his results to motions and almost orbits of nonlinear strongly continuous semigroups in uniformly convex banach spaces e g see 4 10 22 24 26 27 30 31 33 351 as well as 4 10 241 for yet further references these results apply to strong and integral solutions of the quasi autonomous nonlinear cauchy problem

propagation in linear dispersive media finite difference time domain methodologies finite difference time domain fdtd methodologies are presented for electromagnetic wave propagation in two different kinds of linear dispersive media an nth order lorentz and an mth order debye medium the temporal discretization is accomplished by invoking the central difference approximation for the temporal derivatives that appear in the first order differential equations from this the final equations are temporally advanced using the classical leapfrog method one dimensional scattering from a dielectric slab is chosen for a test case provided that the maximum operating frequency times the time step is small and that the wave is adequately resolved in space as shown in the error analysis the agreement between the computed and exact solutions will be excellent the attached data which are associated with the four pole lorentz dielectric and the five pole debye medium verify this assertion