fourier transform of a polygonal shape function and its application in electromagnetics the shape function s x y of a planar aperture sigma is defined ashaving value one inside sigma and zero outside the two dimensional fourier transform of s x y when sigma is an n sided polygon is considered and a closed form exact solution is presented applications of these results are described for 1 the calculation of the secondary pattern of a reflector antenna with a noncircular aperture and 2 high frequency transmission through a periodic frequency selective surface

computing highly oscillatory physical optics integral on the polygonal domain by an efficient numerical steepest descent path method in this work the computation of physical optics po type integral with the integrand of quadratic phase and amplitude is studied first we apply the numerical steepest descent path nsdp method to calculate the highly oscillatory po integral on the triangular patch then we rigorously extend the proposed nsdp method to analyze the po integral on polygonal domains furthermore the contributions of critical points on polygonal domains including the stationary phase point resonance and vertex points are comprehensively studied in terms of the nsdp method compared to the traditional high frequency asymptotic hfa method when the wave frequency is not very high but in the high frequency regime the nsdp method has improved the po integral accuracy by one to two digits meanwhile the computational cost by using the proposed nsdp method is independent of the wave frequency

a schwarz generalized eigen oscillation spectral element method gesem for 2 d high frequency electromagnetic scattering in dispersive inhomogeneous media in this paper we propose a parallel schwarz generalized eigen oscillation spectral element method gesem for 2 d complex helmholtz equations in high frequency wave scattering in dispersive inhomogeneous media this method is based on the spectral expansion of complex generalized eigen oscillations for the electromagnetic fields and the schwarz non overlapping domain decomposition iteration method the gesem takes advantages of a special real orthogonality property of the complex eigen oscillations and a new radiation interface condition for the system of equations for the spectral expansion coefficients numerical results validate the high resolution and the flexibility of the method for various materials

implicit explicit one leg methods for nonlinear stiff neutral equations abstract in this paper by adapting the underlying implicit explicit imex one leg methods cf 1 2 a class of extended imex one leg eieol methods are suggested for solving nonlinear stiff neutral equations snes it is proven under some suitable conditions that eieol methods are d convergent of order 2 and stable for nonlinear snes several numerical examples are given to testify the obtained theoretical results and the computational effectiveness of eieol methods moreover a comparison with the fully implicit one leg methods is presented which shows that eieol methods have the higher computational efficiency

errors of linear multistep methods and runge kutta methods for singular perturbation problems with delays abstract this paper is concerned with the error analysis of linear multistep methods and runge kutta methods applied to some classes of one parameter stiff singularly perturbed problems with delays we derive the global error estimates of a stable linear multistep methods and algebraically and diagonally stable runge kutta methods with lagrange interpolation procedure numerical experiments confirm our theoretical analysis

the speed of propagation for kpp type problems ii general domains this paper is devoted to nonlinear propagation phenomena in general unbounded domains of rn for reaction diffusion equations with kolmogorov petrovsky piskunov kpp type nonlinearities this article is the second in a series of two and it is the follow up of the paper 7 which dealt which the case of periodic domains this paper is concerned with general domains and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data we study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition some a priori bounds are proved for large classes of domains the case of exterior domains is also discussed in detail lastly some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited the construction is based on some new heat kernel estimates in such domains

one step methods of any order for neutral functional differential equations a new class of one step methods for the numerical solution of neutral functional differential equations is considered these methods can be taken to have arbitrarily high order this should be compared with other one step methods such as those of tavernini for which the obvious modification to neutral functional differential equations leads only to methods of order one

hα stability of modified runge kutta methods for nonlinear neutral pantograph equations abstract in this paper we investigate h stability of algebraically stable runge kutta methods with a variable stepsize for nonlinear neutral pantograph equations as a result the radau ia radau iia lobatto iiic method the odd stage gauss legendre methods and the one leg method with 1 2 1 are h stable for nonlinear neutral pantograph equations some experiments are given

the two dimensional numerical steepest descent path method for calculating the physical optics scattered fields from different quadratic patches in this article the two dimensional numerical steepest descent path td nsdp method is proposed to calculate the physical optics po scattered fields from the quadratic patches compared with the nsdp method the td nsdp method avoids adopting the complementary error function and hence the td nsdp method could significantly improve the accuracy of the po scattered fields by around four to five digits moreover the computational workload is also frequency independent to demonstrate the efficiencies of the td nsdp method numerical examples for both quadratic convex and concave surface patches are conducted finally the radar cross section rcs simulation from the realistic and complex scatterer is conducted to verify the practicability of this method in electromagnetic em scattering application

existence and uniqueness of the solution of a space time periodic reaction diffusion equation this paper is concerned with the study of the periodic solutions and the entire solutions of the equation 1 tu a t x u q t x u f t x u where the diffusion matrix a the advection term q and the reaction term f are periodic in t and x we prove that the sign of the periodic principal eigenvalue associated with the linearized problem determines the existence and the uniqueness of the periodic solution introducing another eigenvalue we are able to state uniqueness conditions for the entire solution and to derive the asymptotic behavior of the solutions of the associated cauchy problem

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

nonlinear stability and asymptotic stability of implicit euler method for stiff volterra functional differential equations in banach spaces this paper is concerned with the numerical solution of volterra functional differential equations we focus on the stability and asymptotic stability properties of implicit euler method when applied to nonlinear stiff volterra functional differential equations in banach spaces a series of new stability and asymptotic stability results are obtained numerical example is given to confirm the theoretical results

propagation and blocking in periodically hostile environments we study the persistence and propagation or blocking phenomena for a species in periodically hostile environments the problem is described by a reaction diffusion equation with the zero dirichlet boundary condition we first derive the existence of a minimal nonnegative nontrivial stationary solution and study the large time behavior of the solution of the initial boundary value problem in addition to the main goal we then study a sequence of approximated problems in the whole space with reaction terms with very negative growth rates which are outside the domain under investigation finally for a given unit vector by using the information of the minimal speeds of approximated problems we provide a simple geometric condition for the blocking of propagation and we derive the asymptotic behavior of the approximated pulsating travelling fronts moreover for the case of the constant diffusion matrix we provide two conditions for which the limit of approximated minimal speeds is positive

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions

entire solutions of reaction advection diffusion equations with bistable nonlinearity in cylinders this paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction advection diffusion equation with infinite cylinders assume that the equation admits three equilibria two stable equilibria 0 and 1 and an unstable equilibrium it is well known that there are different wave fronts connecting any two of those three equilibria by considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions we establish three different types of entire solutions finally we analyze a model for shear flows in cylinders to illustrate our main results

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

travelling fronts and entire solutions of the fisher kpp equation in ℝn this paper is devoted to time global solutions of the fisher kpp equation in n where f is a c 2 concave function on 0 1 such that f 0 f 1 0 and f 0 on 0 1 it is well known that this equation admits a finite dimensional manifold of planar travelling fronts solutions by considering the mixing of any density of travelling fronts we prove the existence of an infinite dimensional manifold of solutions in particular there are infinite dimensional manifolds of nonplanar travelling fronts and radial solutions furthermore up to an additional assumption a given solution u can be represented in terms of such a mixing of travelling fronts

a numerical study on the stability of a class of helmholtz problems this paper concerns the stability of a class of helmholtz problems in rectangular domains a well known application is the electromagnetic scattering from a rectangular cavity embedded in an infinite ground plane error analysis of numerical methods for cavity problems relies heavily on the stability estimates however it is extremely difficult to derive an optimal stability bound with the explicit dependency on wave numbers in this paper a high order finite element approximation is proposed for calculating the stability bound numerical experiments show that the stability depends strongly on wave numbers in extreme case and it is almost independent on the wave numbers in an average sense our numerical results also help to understand the stability of the multi frequency inverse problems

stability analysis of general linear methods for nonlinear neutral delay differential equations abstract the present paper is concerned with the numerical solution to initial value problems of neutral delay differential equations nddes the sufficient conditions for the stability and asymptotic stability of the general linear methods for a class r of nonlinear systems of nddes are derived numerical test is given that confirm the theoretical results in the end

stability of travelling fronts in a model for flame propagation part i linear analysis we investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number this model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder with neumann boundary conditions in this paper we prove that every travelling wave solution of this model is linearly stable our tools are exponential decay estimates for solutions of elliptic equations in a cylinder and the maximum principle for parabolic equations

nonlinear stability and d convergence of runge kutta methods for delay differential equations abstract this paper deals with the stability and convergence of runge kutta methods with the lagrangian interpolation rklms for nonlinear delay differential equations ddes some new concepts such as strong algebraic stability gdn stability and d convergence are introduced we show that strong algebraic stability of a rkm for odes implies gdn stability of the corresponding rklm for ddes and that a strongly algebraically stable and diagonally stable rkm with order p together with a lagrangian interpolation of order q leads a d convergent rklm of order min p q 1

analytic solutions to a class of nonlinear infinite delay differential equations abstract the infinite delay differential equations iddes are studied and the analytic solution of a class of nonlinear iddes is presented based on the characteristics of the reproducing kernel space w 2 0 besides the exact solution is represented in the form of series it is proved that the n term approximation u n x converges to the exact solution u x of the iddes moreover the approximate error of u n x is monotone decreasing the results of experiments showed that the proposed method in this paper is computationally efficient

stability and error analysis of one leg methods for nonlinear delay differential equations this paper is concerned with the numerical solution of delay dierential equations ddes we focus on the stability behaviour and error analysis of one leg methods with respect to nonlinear ddes the new concepts of gr stability gar stability and weak gar stability are introduced it is proved that a strongly a stable one leg method with linear interpolation is gar stable and that an a stable one leg method with linear interpolation is gr stable weakly gar stable and d convergent of order s if it is consistent of order s in the classical sense c 1999 elsevier science b v all rights reserved ams classication 65l06 65l20

transition semi wave solutions of reaction diffusion equations with free boundaries abstract in this paper we define the transition semi wave solution cf definition 1 1 of the following reaction diffusion equation with free boundaries 1 u t u x x f t x u t r x h t u t h t 0 t r h t u x t h t t r in the homogeneous case i e f t x u f u under the hypothesis f u c 1 0 1 f 0 f 1 0 f 1 0 f u 0 for u 1 we prove that the semi wave connecting 1 and 0 of 1 is unique provided it exists furthermore we prove that any bounded transition semi wave connecting 1 and 0 is exactly the semi wave in the cases where f is kpp fisher type and almost periodic in time space i e f t x u u c t u resp u a x u with c t resp a x being almost periodic applying totally different method we also prove any bounded transition semi wave connecting the unique almost periodic positive solution of u t u c t u resp u x x u a x u 0 and 0 is exactly the unique almost periodic semi wave of 1 finally we provide an example of the heterogeneous equation to show the existence of the transition semi wave without any global mean speeds

stability analysis of numerical methods for linear neutral volterra delay integro differential system this paper is concerned with the stability of numerical methods for linear neutral volterra delay integro differential system a sufficient condition such that the system is asymptotically stable is derived furthermore it is proved that every linear q method with email protected 1 2 1 and a stable bdf method preserve the delay independent stability of its exact solutions finally the numerical experiments are given to demonstrate the conclusions

eulerian gaussian beams for high frequency wave propagation we design an eulerian gaussian beam summation method for solving helmholtz equations in the high frequency regime the traditional gaussian beam summation method is based on lagrangian ray tracing and local ray centered coordinates we propose a new eulerian formulation of gaussian beam theory which adopts global cartesian coordinates level sets and liouville equations yielding uniformly distributed eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources the time harmonic wavefield can be constructed by summing up gaussian beams with ingredients provided by the new eulerian formulation the conventional gaussian beam summation method can be derived from the proposed method there are three advantages of the new method 1 we have uniform resolution of ray distribution 2 we can obtain wavefields excited at different sources by varying only source locations in the summation formula 3 we can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula numerical experiments indicate that the gaussian beam summation method yields accurate asymptotic wavefields even at caustics the new method may be used for seismic modeling and migration

nonlinear stability of linear multistep methods for stiff delay differential equations in banach spaces this paper is concerned with the nonlinear stability properties of a class of linear multistep methods when applied to initial value problems of stiff delay differential equations in banach spaces for the test problem classes d a l b and d a l d b in banach spaces a series of new stability results of the linear multistep methods are obtained

generalized transition fronts for one dimensional almost periodic fisher kpp equations this paper investigates the existence of generalized transition fronts for fisher kpp equations in one dimensional almost periodic media assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction we show that such fronts exist if and only if their average speed is above an explicit threshold this hypothesis is satisfied in particular when the reaction term does not depend on x or in some cases is small enough moreover except for the threshold case the fronts we construct and their speeds are almost periodic in a sense when our hypothesis is no longer satisfied such generalized transition fronts still exist for an interval of average speeds with explicit bounds our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues

existence and asymptotics of travelling waves in a thermo diffusive model in half cylinders part ii oblique boundary conditions abstract in this paper we investigate a system of reaction diffusion equations set in half cylinders and involving oblique boundary conditions this type of system arises in the modelization of burner flames and approach in a suitable sense conical shaped fronts in the whole space we prove the existence of solutions and some properties of the speed of propagation of the travelling fronts in some asymptotic limit

pulsating fronts and front like entire solutions for a reaction advection diffusion competition model in a periodic habitat abstract this paper is devoted to the study of pulsating fronts and pulsating front like entire solutions for a reaction advection diffusion model of two competing species in a periodic habitat under certain assumptions the competition system admits a leftward and a rightward pulsating fronts in the bistable case in this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x axis which turn out to be unique and liapunov stable 2 dimensional manifolds of solutions furthermore the leftward and rightward pulsating fronts are on the boundary of these 2 dimensional manifolds the others behave as the two pulsating fronts propagating from one side of the x axis the faster one then invades the slower one as t these kinds of pulsating front like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat

stability of bdf methods for nonlinear volterra integral equations with delay abstract this paper deals with the stability analysis of the backward differential formulas or bdf methods for nonlinear volterra integral equations with delay vides the presented approach is based on a nonclassical lipschitz condition in particular the criteria on the global and the asymptotic stability of the methods are given

singular limit and homogenization for flame propagation in periodic excitable media this paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media the model involves two parameters the period of the medium l and a singular parameter related to the activation energy the existence of pulsating travelling fronts for fixed and l was proved by berestycki hamel in bh in the present paper we investigate the behaviour of such solutions when more precisely we establish that pulsating travelling fronts behave like travelling waves when the period l is small and we also study the convergence of the solution as goes to zero and l is fixed toward a solution of a free boundary problem

stability of travelling fronts in a model for flame propagation part ii nonlinear stability this paper is concerned with the nonlinear stability of the travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number the model consists in a semilinear parabolic equation in an infinite cylinder with neumann boundary conditions we prove that any solution which is initially close to a travelling wave will converge to a translate of that wave

fronts in reactive convection bounds stability and instability this paper examines a simplified active combustion model in which the reaction influences the flow we consider front propagation in a reactive boussinesq system in an infinite vertical strip nonlinear stability of planar fronts is established for narrow domains when the rayleigh number is not too large planar fronts are shown to be linearly unstable with respect to long wavelength perturbations if the rayleigh number is sufficiently large we also prove uniform bounds on the bulk burning rate and the nusselt number in the kpp reaction case 2003 wiley periodicals inc

transition fronts in time heterogeneous and random media of ignition type abstract the current paper is devoted to the investigation of wave propagation phenomenon in reaction diffusion equations with ignition type nonlinearity in time heterogeneous and random media it is proven that such equations in time heterogeneous media admit transition fronts with time dependent profiles and that such equations in time random media admit transition fronts with random profiles important properties of transition fronts including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts are also obtained

a semi linear elliptic equation in a strip arising in a two dimensional flame propagation model on montre l existence d une solution stationnaire pour un probleme a 2 dimensions pour l equation elliptique semilineaire de la forme u v y u x g u dans la bande s x y r 2 x r 0 0

stability of strong travelling waves for a non local time delayed reaction diffusion equation the paper is concerned with a non local time delayed reaction diffusion equation we prove the time asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength i e the so called strong wavefront by means of the technical weighted energy method when the initial perturbation around the wave is small the exponential convergent rate is also given selection of a suitable weight plays a key role in the proof

existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media we prove the existence of multidimensional traveling wave solutions to the scalar equation for the transport of solutes contaminants with nonlinear adsorption and spatially periodic convection diffusion adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the lax and oleinik entropy conditions in the nondegenerate case we also prove the uniqueness of the traveling waves these traveling waves are analogues of viscous shock profiles they propagate with effective speeds that depend on the periodic porous media only up to their mean states and are given by an averaged rankine hugoniot relation this is a direct consequence of the fact that the transport equation is in conservation form we use the sliding domain method the continuation method spectral theory maximum principles and a priori estimates in the degenerate case the traveling waves are weak solutions of a degenerate parabolic equation and are only holder continuous we obtain them by taking suitable limits on the non degenerate traveling waves the uniqueness of the degenerate traveling waves is open

propagation of reactions in inhomogeneous media consider reaction diffusion equation ut u f x u with x d and general inhomogeneous ignition reaction 0 vanishing at u 0 1 typical solutions 0 u 1 transition from 0 to 1 as time progresses and we study them in the region where this transition occurs under fairly general qualitative hypotheses on f we show that in dimensions d 3 the hausdorff distance of the superlevel sets u e and u 1 e remains uniformly bounded in time for each e 0 1 thus u remains uniformly in time close to the characteristic function of u in the sense of hausdorff distance of superlevel sets we also show that each u e expands with average speed over any long enough time interval between the two spreading speeds corresponding to any x independent lower and upper bounds on f on the other hand these results turn out to be false in dimensions d 4 at least without further quantitative hypotheses on f the proof for d 3 is based on showing that as the solution propagates small values of u cannot escape far ahead of values close to 1 the proof for d 4 is via construction of a counterexample for which this fails such results were before known for d 1 but are new for general non periodic media in dimensions d 2 some are also new for homogeneous and periodic media they extend in a somewhat weaker sense to monostable bistable and mixed reaction types as well as to transitions between general equilibria u u of the pde and to solutions not necessarily satisfying u u u 2016 wiley periodicals inc

pulsating type entire solutions of monostable reaction advection diffusion equations in periodic excitable media abstract we establish the existence of pulsating type entire solutions of reaction advection diffusion equations with monostable nonlinearities in a periodic framework here the nonlinearities include the classic kpp case the pulsating type entire solutions are defined in the whole space and for all time t r by studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions and approaching each other the key techniques are to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions

collocation methods for volterra functional integral equations with non vanishing delays abstract in this paper the existence uniqueness regularity properties and in particular the local representation of solutions for general volterra functional integral equations with non vanishing delays are investigated based on the solution representation we detailedly analyze the attainable global and local convergence order of iterated collocation solutions on invariant meshes it turns out that collocation at the m gauss legendre points neither leads to the optimal global convergence order m 1 nor yields the local convergence order 2m on the whole interval which is in sharp contrast to the case of the classical volterra delay integral equations however if the collocation is based on the m radau ii points the local superconvergence order 2 m 1 will exhibit at all mesh points finally some numerical experiments are performed to confirm our theoretical findings

stability of fronts for a kpp system ii the critical case abstract we study a system of n 1 coupled semilinear parabolic equations on the real line which depends on a small parameter and reduces to the scalar kolmogorov petrovsky piscounov kpp equation when 0 under appropriate scaling the system has a family of traveling fronts parametrized by their speed when 2 as in the scalar kpp case the case of critical speed 2 say is investigated and it is shown that the system inherits some crucial properties of the kpp equation when is small in particular the asymptotic stability of the front in a local and semiglobal sense first we describe the properties of the front and then apply functional arguments to prove its local stability in an adequate weighted sobolev space moreover the decay rate of the perturbations is shown to be polynomial in time finally we show a semiglobal stability property of the front which also is inherited from the scalar kpp equation

reaction diffusion fronts in periodically layered media we compute the effective wavefront speeds of reaction diffusion equations in periodically layered media with coefficients that have small amplitude oscillations around a uniform mean state we compare them with the corresponding wavefront speeds in the uniform state we analyze a one dimensional model where wave propagation is along the layering direction of the medium and a two dimensional shear flow model where wave propagation is othogonal to the layering direction we find that the effective wave speed is smaller in the one dimensional model and is larger in the two dimensional model for both bistable cubic and quadratic nonlinearities of the kolmogorov petrovskii piskunov form we derive approximate expressions for the wave speeds in the bistable case

a multi dimensional bistable nonlinear diffusion equation in a periodic medium in this work we study the existence of wave solutions for a scalar reaction diffusion equation of bistable type posed in a multi dimensional periodic medium roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term here the term wave is used to describe either pulsating travelling wave or standing transition solution as a special case we study a two dimensional heterogeneous allen cahn equation in both cases of slowly varying medium and rapidly oscillating medium we prove that bistability occurs in these two situations and we conclude to the existence of waves connecting u 0 and u 1 moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction

frequency independent approach to calculate physical optics radiations with the quadratic concave phase variations in this work we develop the numerical steepest descent path nsdp method to calculate the physical optics po radiations with the quadratic concave phase variations with the surface integral equation method the physical optics po scattered fields are formulated and further reduced to the surface integrals the high frequency physical critical points contributions including the stationary phase points the boundary resonance points and the vertex points are comprehensively studied via the proposed nsdp method the key contributions of this work are twofold one is that together with the po integrals taking the quadratic parabolic and hyperbolic phase terms this work makes the nsdp theory be complete for treating the po integrals with quadratic phase variations another is that in order to illustrate the transition effect of the high frequency physical critical points in this work we consider and further extend the nsdp method to calculate the po integrals with the coalescence of the high frequency critical points numerical results for the highly oscillatory po integral with the coalescence of the critical points are given to verify the efficiency of the proposed nsdp method the nsdp method could achieve the frequency independent computational workload and error controllable accuracy in all the numerical experiments especially for the case of the coalescence of the high frequency critical points

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

existence of planar flame fronts in convective diffusive periodic media we prove the existence of planar travelling wave solutions in a reaction diffusion convection equation with combustion nonlinearity and self adjoint linear part in rn n 1 the linear part involves diffusion convection terms and periodic coefficients these travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media we use the method of continuation spectral theory and the maximum principle uniqueness and monotonicity properties of solutions follow from a previous paper these properties are essential to overcoming the lack of compactness and the degeneracy in the problem

stability analysis of numerical methods for delay differential equations this paper deals with the stability analysis of step by step methods for the numerical solution of delay differential equations we focus on the behaviour of such methods when they are applied to the linear testproblemu t u t u t with 0 and complex a general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods

generalized traveling waves in disordered media existence uniqueness and stability we prove existence uniqueness and stability of transition fronts generalized traveling waves for reaction diffusion equations in cylindrical domains with general inhomogeneous ignition reactions we also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front in the case of stationary ergodic reactions the fronts are proved to propagate with a deterministic positive speed our results extend to reaction advection diffusion equations with periodic advection and diffusion

transition fronts for the fisher kpp equation this paper is concerned with transition fronts for reaction diffusion equations of the fisher kpp type basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich in the paper we describe the class of transition fronts and we study their qualitative dynamical properties in particular we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate we also classify the transition fronts in the class of measurable superpositions of standard traveling fronts

stability analysis of volterra delay integro differential equations and their backward differentiation time discretization we study the stability of the analytical solutions of initial value problems of a general class of systems of volterra delay integro differential equations numerical methods based on backward differentiation formulae and repeated quadrature formulae are suggested nonlinear and linear stability conditions for the presented methods are derived

existence of pulsating waves of advection reaction diffusion equations of ignition type by a new method abstract the goal of this paper is to study the advection reaction diffusion equation of ignition type u t u q x y u f u xin j x xin existence of planar flame fronts in convective diffusive periodic media arch ration mech anal 121 1992 205 233 berestycki and hamel h berestycki f hamel front propagation in periodic excitable media comm pure appl math 55 2002 949 1032 proved the existence of pulsating waves using the theory of degenerate elliptic equations our goal is to give an alternative and rather natural proof of the same result first we transform the problem so that the existence of pulsating waves is equivalent to the existence of 1 periodic in time solutions of a nonlinear parabolic equation with 1 periodic in time coefficients next we prove the existence of such periodic solutions using a continuation method based on the implicit function theorem

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

entire solutions in monostable reaction advection diffusion equations in infinite cylinders abstract this paper is concerned with entire solutions of a monostable reaction advection diffusion equation in infinite cylinders without the condition f u f 0 by constructing a quasi invariant manifold we prove that there exist two classes of entire solutions furthermore we show that one class of such entire solutions is unique up to space and time translation

quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

stability of numerical methods for delay differential equations abstract consider the following delay differential equation dde y t y t y t t t t 0 with the initial condition y t for t t 0 where and are such that 0 1 0 2 has a unique solution y t the author gives sufficient conditions for the asymptotic stability of the equation 0 1 for which he introduces new definitions of numerical stability the approach is reminiscent of that from the nonlinear stiff ordinary differential equation ode field he investigates stability properties of the class of one point collocation rules in particular the backward euler method turns out to be stable with respect to all the given definitions

optimal superconvergence results for volterra functional integral equations with proportional vanishing delays abstract in this paper we develop a new technique to study the optimal convergence orders of collocation methods for volterra functional integral equations with vanishing delays on quasi geometric meshes basing on a perturbation analysis we show that for m collocation points the global convergence order of the collocation solution is only m however the collocation solution may exhibit superconvergence with order m 1 at the collocation points in particular the local convergence order may attain 2 m 1 at the nodes provided that the collocation is based on the m radau ii points finally some numerical examples are performed to verify our theoretical results

speed up of combustion fronts in shear flows this paper is concerned with the analysis of speed up of reaction diffusion advection traveling fronts in infinite cylinders with periodic boundary conditions the advection is a shear flow with a large amplitude and the reaction is nonnegative with either positive or zero ignition temperature the unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity solving an open problem from berestycki nonlinear pdes in condensed matter and reactive flows kluwer doordrecht 2003 the asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional hormander type condition on the flow

nonlinear stability of runge kutta methods for neutral delay integro differential equations the present paper is concerned with the numerical solution to initial value problems of neutral delay integro differential equations ndides the sufficient conditions for the stability and asymptotic stability of runge kutta methods for a class r a b 1 b 2 c of nonlinear systems of ndides are derived a numerical test that confirms the theoretical results is given in the end

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

the test problem class of volterra functional differential equations in banach space abstract in this paper we introduce the test problem classes which are entitled d 1 2 and d 1 2 respectively with respect to the initial value problems of nonlinear volterra functional differential equations in banach spaces a series of stability results of the analytic solution are obtained and a condition estimate for the class d 0 1 2 which based on logarithmic matrix norm is also obtained the above results extend the existing results for ordinary differential equations

d convergence of general linear methods for stiff delay differential equations abstract this paper is concerned with the numerical solution of delay differential equations ddes we focus on the error behaviour of general linear methods for stiff ddes a restricted type of interpolation procedure is considered for general linear methods d convergence properties of general linear methods with this interpolation procedure are investigated some extant results are unified

existence and nonexistence of traveling waves and reaction diffusion front propagation in periodic media we review the existence results of traveling wave solutions to the reaction diffusion equations with periodic diffusion convection coefficients and combustion bistable nonlinearities we prove that whenever traveling waves exist the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions in the case of bistable nonlinearity and one space dimension we give an example of nonexistence of traveling waves which causes quenching localization of wavefront propagation quenching localization only occurs when the variations of the media from their constant mean values are large enough our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example finally we comment on the role of the effective wave speeds in determining the effective wavefront equation hamilton jacobi equation of the reactiondiffusion equations under the small diffusion fast reaction limit with a formal geometric optics expansion

how travelling waves attract the solutions of kpp type equations we consider in this paper a general reaction diffusion equation of the kpp kolmogorov petrovskii piskunov type posed on an infinite cylinder such a model will have a family of pulsating waves of constant speed larger than a critical speed c the family of all supercritical waves attract a large class of initial data and we try to understand how we describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity the solution will converge to a whole set of translates of the same wave and we identify the convergence dynamics as that of an effective drift around which an effective diffusion process occurs in several nontrivial particular cases we are able to describe the dynamics by an effective equation

a composite preconditioner for the electromagnetic scattering from a large cavity in this paper we propose a composite preconditioner for the electromagnetic scattering from a large cavity the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity the preconditioner proposed here combines the optimal sine transform based approximation with a layered medium model using fast fourier transforms the computational cost of every iteration is o n 2logn on an nxn uniform partition of the unit square numerical results for a model problem show that the new preconditioner is more efficient than those recently considered in the literature for the cavity with a small portion of non layered media we propose a sparse preconditioned conjugate orthogonal conjugate gradient solver combined with the new preconditioner numerical results for a model problem are reported to demonstrate the efficiency of the sparse solver

spatial decay and stability of traveling fronts for degenerate fisher type equations in cylinder abstract this paper is concerned with the spatial decay and asymptotic stability of multidimensional traveling fronts for the degenerate fisher type equation in an unbounded cylinder firstly by applying the moving plane argument and the generalized center manifold theorem we obtain the uniqueness and exponential decay of the traveling front with the critical speed and the non exponential decay of traveling fronts with non critical speeds especially for the p degree fisher equation we get the precise algebraic decaying rates and the higher order expansion of traveling fronts with non critical speeds secondly by applying the spectral analysis and sub super solution method we prove the nonlinear exponential stability of all traveling fronts in some exponentially weighted spaces and the lyapunov stability of traveling fronts with non critical speeds in some polynomially weighted spaces finally by combining the sub super solution method with the nonlinear stability results we get the asymptotic behavior and the asymptotic spreading speeds of the solution for more general initial data which are proved to be determined by the spatial decay of the initial data at one end

an alternative treatment of saddle stationary phase points in physical optics for smooth surfaces in a recent paper on computing the physical optics po integral with a saddle stationary phase point spp by numerical steepest descend method nsdm f vico bondia a new fast physical optics for smooth surfaces by means of a numerical theory of diffraction ieee trans antennas propag vol 58 no 3 pp 773 789 mar 2010 a one dimensional integral was obtained on the path of which there existed higher order up to third poles to avoid tackling the singular integral the abel s summation technique was used to solve the problem ingeniously however it has been shown in some literatures that the integral would be divergent if there were even order poles on the path of integral superficially the integral in f vico bondia s paper does not obey this law and the main aim of the communication is to clear this contradiction we show rigorously that higher order poles have no contributions to the final result to acquire a general law we further consider arbitrary polynomials of higher degrees for the po integral in which the orders of poles can be arbitrary number in such a general case we still show that all higher order poles have no contributions and thus solve the contradiction satisfactorily numerical examples are presented to validate the new derivations and illustrate the accuracy and efficiency of nsdm

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena this paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first order coefficients roughly speaking one of the main results says that the principal eigenvalue with dirichlet boundary conditions is bounded as the amplitude of the coefficients of the first order derivatives goes to infinity if and only if the associated dynamical system has a first integral and the limiting eigenvalue is then determined through the minimization of the dirichlet functional over all first integrals a parabolic version of these results as well as other results for more general equations are given some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts

pulsating fronts for bistable on average reaction diffusion equations in a time periodic environment abstract this paper is devoted to reaction diffusion equations with bistable nonlinearities depending periodically on time these equations admit two linearly stable states however the reaction terms may not be bistable at every time these may well be a periodic combination of standard bistable and monostable nonlinearities we are interested in a particular class of solutions namely pulsating fronts we prove the existence of such solutions in the case of small time periods of the nonlinearity and in the case of small perturbations of a nonlinearity for which we know there exist pulsating fronts we also study uniqueness monotonicity and stability of pulsating fronts

on the numerical stability of volterra integral equations with delay argument abstract stability properties of numerical methods for volterra integral equations with delay arguments are considered some suitable definitions for the stability of the numerical methods are included and also plots of the stability regions for particular cases are included

convergence of one leg methods for neutral delay integro differential equations the error analysis of one leg methods for a class of nonlinear neutral delay integro differential equations ndides is given it is proved that an a stable one leg method with an appropriate quadrature rule applied to ndides is convergent of order at least min p q 12 if the one leg method is consistent of order p2 in the classical sense for odes and the error order of the quadrature rule is o hq 1 numerical examples further confirm the theoretical results

the numerical steepest descent path method for calculating physical optics integrals on smooth conducting quadratic surfaces in this paper we use the numerical steepest descent path nsdp method to analyze the highly oscillatory physical optics po integral on smooth conducting parabolic surfaces including both monostatic and bistatic cases quadratic variations of the amplitude and phase functions are used to approximate the integrand of po integral then the surface po integral is reduced into several highly oscillatory line integrals by invoking the nsdp method these highly oscillatory po line integrals are defined on the corresponding nsdps furthermore the critical point contributions for the po integral are exactly extracted and represented based on the nsdps the proposed nsdp method for calculating the po integral on the smooth conducting surfaces is frequency independent and error controllable compared with the traditional asymptotic expansion approach the nsdp method significantly improves the po integral accuracy by around two digits when the working wave frequencies are not extremely large numerical results are given to validate the nsdp method

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

traveling waves for a boundary reaction diffusion equation abstract we prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature a key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction diffusion equation the so called high energy activation energy limit this explicit formula which is interesting in itself also allows us to get an estimate on the decay at infinity of the traveling wave which turns out to be faster than the usual exponential decay

convergence of runge kutta methods for neutral delay integro differential equations the present paper is concerned with the convergence of numerical methods to solve initial value problems of neutral delay integro differential equations ndides the error estimation of runge kutta methods for a class of nonlinear systems of ndides are derived some numerical experiments are given which confirm the theoretical results

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon