numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

some practical experience with the time integration of dissipative equations different methods for the numerical integration of evolution dissipative partial differential equations are tested with the kuramoto sivashinsky equation discretizations in space include galerkin and nonlinear galerkin methods for integration in time three different codes are used including standard stiff ode methods numerical tests show that standard codes for stiff ode render a gain of computing time of several orders of magnitude with respect problem tailored methods

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

staggered time integrators for wave equations we consider variations of the adams bashforth backward differentiation and runge kutta families of time integrators to solve systems of linear wave equations on uniform time staggered grids these methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders we investigate the accuracy and stability of these methods analytically experimentally and through the use of a novel root portrait technique

on the super fixed point property in product spaces abstract we prove that if f is a finite dimensional banach space and x has the super fixed point property for nonexpansive mappings then f x has the super fixed point property with respect to a large class of norms including all l p norms 1 p this provides a solution to the super version of the problem of khamsi 1989

the dunford pettis and the kadec klee properties on tensor products of jb triples we recall that a banach spacex satisfies the dunford pettis property dpp if every weakly compact operator t from x to another banach space is completely continuous i e t maps weakly cauchy sequences into norm convergent sequences the question if the projective tensor product of two banach spaces having the dpp also has this property has focused the attention of several researcher in 1983 m talagrand found a banach space x such that x has the schur property and l1 0 1 e does not satisfy the dpp see 34 four years latter r ryan showed that x y satisfies the dpp and contains no copies of 1 whenever x and y have both properties 33 f bombal and i villanueva prove in 7 see also 6 20 that whenever k1 and k2 are two infinite compact hausdorff spaces then the projective tensor product of c k1 and c k2 has the dpp if and only if k1 and k2 are scattered that is c k1 and c k2 contains no copies of 1 the interesting ideas developed by bombal fernandez and villanueva in 7 6 are the inspiration and the starting point for the present paper in the

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

uniform smoothness implies super normal structure property we prove that a uniformly smooth banach space x has supernormal structure property more precisely we prove that if the modulus of smoothness of x satisfies lim 0 x 1 2 then x and x are superreflexives and have super normal structure property

a method to compute statistics of large noise induced perturbations of nonlinear schrodinger solitons we describe in detail the application of importance sampling to numerical simulations of large noise induced perturbations in soliton based optical transmission systems governed by the nonlinear schrodinger equation the method allows one to concentrate the samples in monte carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors and it yields computational speedups of several orders of magnitude over standard monte carlo simulations we demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude frequency timing and phase fluctuations in a prototypical soliton based communication system

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

relatively weakly open sets in closed balls of banach spaces and real jb triples of finite rank we prove that given a real jb triple x there exists a nonempty relatively weakly open subset of the closed unit ball of x with diameter less than 2 if and only if the banach space of x is isomorphic to a hilbert space moreover we give the structure of real jb triples whose banach spaces are isomorphic to hilbert spaces such real jb triples are also characterized in two different purely algebraic ways

error analysis and applications of the fourier galerkin runge kutta schemes for high order stiff pdes an integrating factor mixed with runge kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations in this paper the novel hybrid fourier galerkin runge kutta scheme with the aid of an integrating factor is proposed to solve nonlinear high order stiff pdes error analysis and properties of the scheme are provided application to the approximate solution of the nonlinear stiff korteweg de vries the 3rd order pde dispersive equation kuramoto sivashinsky the 4th order pde dissipative equation and kawahara the 5th order pde equations are presented comparisons are made between this proposed scheme and the competing method given by kassam and trefethen it is found that for kdv ks and kawahara equations the proposed method is the best

the daugavet property of c algebras jb triples and of their isometric preduals a banach space x is said to have the daugavet property if every rank one operator t x x satisfies id t 1 t we give geometric characterizations of this property in the settings of c algebras jb triples and their isometric preduals we also show that in these settings the daugavet property passes to ultrapowers and thus it is equivalent to an stronger property called the uniform daugavet property

a composite runge kutta method for the spectral solution of semilinear pdes a new composite runge kutta rk method is proposed for semilinear partial differential equations such as korteweg de vries nonlinear schrodinger kadomtsev petviashvili kp kuramoto sivashinsky ks cahn hilliard and others having high order derivatives in the linear term the method uses fourier collocation and the classical fourth order rk method except for the stiff linear modes which are treated with a linearly implicit rk method the composite rk method is simple to implement indifferent to the distinction between dispersive and dissipative problems and as efficient on test problems for ks and kp as any other generally applicable method

time discretization and stability regions for dissipative dispersive kuramoto sivashinsky equation arising in turbulent gas flow over laminar liquid abstract in this study spatio temporal discretization for semilinear dissipative partial differential equations type is introduced analyzed and implemented the model studied here is the dispersively kuramoto sivashinsky equation with an additional term representing the dispersive term arising in turbulent gas flow over laminar liquid tseluiko and kalliadasis 2011 this additional term is multiplied by a parameter that represents the influence of the turbulent gas flow our objective is to examine the effect of this additional term on the dynamics of the kuramoto sivashinsky equation characterized by its chaotic behavior this is achieved by combining the exponential time differencing crank nicolson etd cn scheme derived by kleefed et al 2012 and fourier pseudospectral schemes for temporal and spatial stepping respectively the method is known to be stable and second order convergent in addition a theoretical study and a plot of stability regions of etd cn were performed showing its effectiveness for the stiff problem studied here

separate weak continuity of the triple product in dual real jb triples we prove that if e is a real jb triple having a predual e then e is the unique predual of e and the triple product on e is separately sigma e e continuous

adaptive multiquadric collocation for boundary layer problems an adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two point boundary value problems using a multiquadric integral formulation the second derivative of the solution is approximated by multiquadric radial basis functions this approach is combined with a coordinate stretching technique the required variable transformation is accomplished by a conformal mapping an iterated sine transformation a new error indicator function accurately captures the regions of the interval with insufficient resolution this indicator is used to adaptively add data centres and collocation points the method resolves extremely thin layers accurately with fairly few basis functions the proposed adaptive scheme is very robust and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally the effectiveness of our new method is demonstrated on two examples with boundary layers and one example featuring an interior layer it is shown in detail how the adaptive method refines the resolution

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

on super fixed point property and super weak compactness of convex subsets in banach spaces abstract for a nonempty convex set c of a banach space x a self mapping on c is said to a linear respectively affine isometry if it is the restriction of a linear respectively affine isometry defined on the whole space x by means of super weakly compact set theory established in the recent years in this paper we first show that a nonempty closed bounded convex set of a banach space has super fixed point property for affine or equivalently linear isometries if and only if it is super weakly compact and the super fixed point property and the super weak compactness coincide on every closed bounded convex subset of a banach space under equivalent renorming sense with the application of fabian montesinos zizler s renorming theorem we finally show that every strongly super weakly compact generated banach space can be renormed so that every weakly compact convex set has super fixed point property

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

a split step approach for the 3 d maxwell s equations split step procedures have previously been used successfully in a number of situations e g for hamiltonian systems such as certain nonlinear wave equations in this study we note that one particular way to write the 3 d maxwell s equations separates these into two parts requiring only the solution of six uncoupled 1 d wave equations the approach allows arbitrary orders of accuracy in both time and space and features in many cases unconditional stability

wave operator bounds for one dimensional schrodinger operators with singular potentials and applications boundedness of wave operators for schrodinger operators in one space dimension for a class of singular potentials admitting finitely many dirac delta distributions is proved applications are presented to for example dispersive estimates and commutator bounds

uniformly nonsquare banach spaces have the fixed point property for nonexpansive mappings abstract it is shown that if the modulus x of nearly uniform smoothness of a reflexive banach space satisfies x 0 1 then every bounded closed convex subset of x has the fixed point property for nonexpansive mappings in particular uniformly nonsquare banach spaces have this property since they are properly included in this class of spaces this answers a long standing question in the theory

non monte carlo formulations and computational techniques for the stochastic non linear schrodinger equation stochastic ordinary and partial differential equations sopdes in various forms arise and are successfully utilized in the modeling of a variety of physical and engineered systems such as telecommunication systems electronic circuits cosmological systems financial systems meteorological and climate systems while the theory of stochastic partial and especially ordinary differential equations is more or less well understood there has been much less work on practical formulations and computational approaches to solving these equations in this paper we concentrate on the stochastic non linear schrodinger equation snlse that arises in the analysis of wave propagation phenomena mainly motivated by its predominant role as a modeling tool in the design of optically amplified long distance fiber telecommunication systems we present novel formulations and computational methods for the stochastic characterization of the solution of the snlse our formulations and techniques are not aimed at computing individual realizations i e sample paths for the solution of the snlse a la monte carlo instead starting with the snlse we derive new systems of differential equations and develop associated computational techniques the numerical solutions of these new equations directly produce the ensemble averaged stochastic characterization desired for the solution of the snlse in a non monte carlo manner without having to compute many realizations needed for ensemble averaging

fourier methods with extended stability intervals for the korteweg de vries equation a full leap frog fourier method for integrating the korteweg de vries kdv equation u t uu x varepsilon u xxx 0 results in an o n 3 stability constraint on the time step where n is the number of fourier modes used this stability limit is much more restrictive than the accuracy limit for many applications in this paper we propose a method for which the stability limit is extended by treating the linear dispersive u xxx term implicitly thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit the implicit method is implemented without solving linear systems by integrating in time in the fourier space and discretizing the nonlinear uu x term by leap frog a second method we propose uses basis functions which solve the linear part of the kdv equation and leap frog for time integration a linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that

a pseudospectral procedure for the solution of nonlinear wave equations with examples from free surface flows an algorithm for the solution of general isotropic nonlinear wave equations is presented the algorithm is based on a symmetric factorization of the linear part of the wave operator followed by its exact integration through an integrating factor in spectral space the remaining nonlinear and forcing terms can be handled with any standard pseudospectral procedure solving the linear part of the wave operator exactly effectively eliminates the stiffness of the original problem characterized by a wide range of temporal scales the algorithm is tested and applied to several problems of three dimensional long surface waves solitary wave propagation interaction diffraction and the generation of waves by flow over slowly varying bottom topography other potential applications include waves in rotating and stratified flows and wave interaction with more pronounced topographic features

banach space properties sufficient for normal structure in this paper we establish lower bounds for the weakly convergent sequence coefficient wcs x of a banach space x in terms of some well known moduli and coefficients by mean of these bounds we identify several properties of geometrical nature which imply normal structure we show that these properties are strictly more general than other previously known sufficient conditions for normal structure

sufficient conditions for uniform normal structure of banach spaces and their duals in this paper we present some sufficient conditions for the uniform normal structure of banach spaces and their dual spaces more precisely many known conditions implying the uniform normal structure of banach spaces are shown to imply uniform normal structure of their dual spaces as well

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained

the fixed point property in jb triples and preduals of jbw triples abstract we prove that for every member x in the class of real or complex jb triples or preduals of jbw triples the following assertions are equivalent 1 x has the fixed point property 2 x has the super fixed point property 3 x has normal structure 4 x has uniform normal structure 5 the banach space of x is reflexive as a consequence a real or complex c algebra or the predual of a real or complex w algebra having the fixed point property must be finite dimensional

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined