nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

nonlinear evolution type equations and their exact solutions using inverse variational methods we present the role of invariants in obtaining exact solutions of differential equations firstly conserved vectors of a partial differential equation p d e allow us to obtain reduced forms of the p d e for which some of the lie point symmetries in vector field form are easily concluded and therefore provide a mechanism for further reduction secondly invariants of reduced forms of a p d e are obtainable from a variational principle even though the p d e itself does not admit a lagrangian in this latter case the reductions carry all the usual advantages regarding noether symmetries and double reductions the examples we consider are nonlinear evolution type equations such as the korteweg devries equation but a detailed analysis is made on the fisher equation which describes reaction diffusion waves in biology inter alia other diffusion type equations lend themselves well to the method we describe e g the fitzhugh nagumo equation which is briefly discussed some aspects of painleve properties are also suggested

stability of moving mesh systems of partial differential equations moving mesh methods based on the equidistribution principle ep are studied from the viewpoint of stability of the moving mesh system of differential equations for fine spatial grids the moving mesh system inherits the stability of the original discretized partial differential equation pde unfortunately for some pdes the moving mesh methods require so many spatial grid points that they no longer appear to be practical failures and successes of the moving mesh method applied to three reaction diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh pde

a comparison of differential quadrature methods for the solution of partial differential equations abstract discrete ordinate methods for the solution of partial differential equations are examined and compared a novel approach for the discrete representation of differential operators based on split range polynomial expansions is introduced the utility of the method is demonstrated for the case of differentiation of functions involving steep gradients the solution of burgers equation is presented to illustrate the effectiveness of the technique for the solution of nonlinear partial differential equations exhibiting nearly discontinuous solutions comparisons are made with other differential quadrature methods

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

the unexplained accuracy of the lagrange mesh method the lagrange mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a gauss quadrature in order to analyze its accuracy four different lagrange mesh calculations based on the zeros of laguerre polynomials are compared with exact variational calculations based on the corresponding laguerre basis the comparison is performed for three solvable radial potentials the morse harmonic oscillator and coulomb potentials the results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy even in the presence of the centrifugal singularity the same property holds for the approximate wave functions this striking accuracy remains unexplained

universality class for domain growth in random magnets a 2d ising model with random ferromagnetic bonds is studied by monte carlo simulation following a quench from t infinity to t tc the domain size grows as l t approximately in t to x at late times the data are consistent with the theoretical prediction chi 4 the exponent defined by si 0 si t approximately l t and the scaling functions for the spatial correlations are very close to those of the pure system suggesting that pure and random systems belong to the same universality class

nonlinear dynamic response and active vibration control for piezoelectric functionally graded plate the nonlinear dynamic response and active vibration control of the piezoelectric functionally graded plate are analyzed in this paper based on higher order shear plate theory and elastic piezoelectric theory the nonlinear geometric and constitutive relations of the piezoelectric functionally graded plate are established and then the nonlinear motion equations of the piezoelectric functionally graded plate are obtained through hamilton s variational principle the nonlinear active vibration control of the structure is carried out with adoption of the negative velocity feedback control algorithm by applying finite difference method the whole problem is solved by using iterative method synthetically in numerical examples the effects of mechanical load electric load the volume fraction and the geometric parameters on the dynamic response and vibration control of the piezoelectric fgm plate are investigated

diffusive approximation of fisher s equation abstract we show that fisher s equation is the relaxation limit of a discrete kinetic model with two speeds in which the collision term contains a source the analysis can be directly applicable to obtain a numerical approximation to fisher s equation in terms of the relaxing system

dynamic stability of curved panels with cutouts the parametric instability behaviour of curved panels with cutouts subjected to in plane static and periodic compressive edge loadings are studied using finite element analysis the first order shear deformation theory is used to model the curved panels considering the effects of transverse shear deformation and rotary inertia the theory used is the extension of dynamic shear deformable theory according to sanders first approximation for doubly curved shells which can be reduced to love s and donnell s theories by means of tracers the effects of static and dynamic load factors geometry boundary conditions and the cutout parameters on the principal instability regions of curved panels with cutouts are studied in detail using bolotin s method quantitative results are presented to show the effects of shell geometry and load parameters on the stability boundaries results for plates are also presented as special cases and are compared with those available in the literature

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

three dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels abstract the free vibration problem of a homogeneous isotropic thick cylindrical shell or panel subjected to a certain type of simply supported edge boundary conditions is considered for this problem the governing equations of three dimensional linear elasticity are employed and solved by using a new iterative approach which in practice leads to the prediction of the exact frequencies of vibration in the particular case of a flat plate or a complete shell excellent agreement is achieved between results based on the present approach and corresponding results based on other exact analyses available in the literature for certain cylindrical panel geometries comparisons are also made between corresponding numerical results based on the present analysis and certain two dimensional shell theories

complex traveling wave solutions to the fisher equation abstract there is the widespread existence of wave phenomena in physics chemistry and biology this clearly necessitates a study of traveling waves in depth and of the modeling and analysis involved in the present paper we study the fisher equation by means of the first integral method which is based on the ring theory of commutative algebra a traveling wave solution is obtained which indicates that the analytical solutions in the literature are particular cases of our result

dynamic stability of simply supported composite cylindrical shells under partial axial loading abstract the parametric vibration of a simply supported composite circular cylindrical shell under periodic partial edge loadings is discussed in this article donnell s nonlinear shallow shell theory considering first order shear deformation theory is used to model the shell the applied partial edge loading is represented in terms of a fourier series and stress distributions within the cylindrical shell are determined by prebuckling analysis the governing equations of the dynamic instability of shells are derived in terms of displacements u v w and rotations x employing the galerkin and bolotin methods the dynamic instability regions are computed using the expression for the stress function derived in this paper the pre buckling stresses in the cylindrical shell due to partial loading can be calculated explicitly numerical results are presented to show the influence of radius to thickness ratio different partial edge loading distributions and shear deformation on the dynamic instability regions the linear and nonlinear responses in the stable and unstable regions are presented to bring out the characteristic features of the dynamic instability regions such as the existence of beats its dependence on forcing frequency and effect of nonlinearity on the response the effect of dynamic load amplitude on the nonlinear response is also studied it is found that for higher values of dynamic loading the shell exhibits chaotic behavior

the errors in calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points abstract we discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points we propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices

non algebraic domain growth for phase ordering dynamics in a random field discusses the three stage numerical exposition of the effects of quenched disorder on phase ordering dynamics the authors study the effects of random fields on domain growth the numerical results indicate that the domain growth law is logarithmic for both the cases with non conserve and conserved order parameter this is compatible with theoretical expectations they also study the dynamical scaling of the structure factor for both the non conserved and conserved cases

numerical study of fisher s reaction diffusion equation by the sinc collocation method fisher s equation which describes a balance between linear diffusion and nonlinear reaction or multiplication is studied numerically by the sinc collocation method the derivatives and integrals are replaced by the necessary matrices and a system of algebraic equations is obtained to approximate solution of the problem the error in the approximation of the solution is shown to converge at an exponential rate numerical examples are given to illustrate the accuracy and the implementation of the method the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large both the limiting wave fronts and the limiting speed are independent of the initial values

a unified approach for the solution of the fokker planck equation this paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the fokker planck equation the unified features of the discrete singular convolution algorithm are discussed it is demonstrated that different implementations of the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point three benchmark stochastic systems the repulsive wong process the black scholes equation and a genuine nonlinear model are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the fokker planck equation via a time dependent method an additional example the incompressible euler equation is used to further validate the present approach for more difficult problems numerical results indicate that the present unified approach is robust and accurate for solving the fokker planck equation

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

a new numerical scheme for the fisher equation in the context of the one dimensional fisher equation 1937 the authors study a new numerical scheme for reaction diffusion equations proposed by oono and puri 1987 they find that this scheme enables a reasonable simulation of the fisher equation at high values of the time increment where conventional schemes are not applicable for lower values of the time increment the new scheme compares favourably with conventional schemes

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

crossover in growth law and violation of superuniversality in the random field ising model we study the nonconserved phase ordering dynamics of the d 2 3 random field ising model quenched to below the critical temperature motivated by the puzzling results of previous work in two and three dimensions reporting a crossover from power law to logarithmic growth together with superuniversal behavior of the correlation function we have undertaken a careful investigation of both the domain growth law and the autocorrelation function our main results are as follows we confirm the crossover to asymptotic logarithmic behavior in the growth law but at variance with previous findings we find the exponent in the preasymptotic power law to be disorder dependent rather than being that of the pure system furthermore we find that the autocorrelation function does not display superuniversal behavior this restores consistency with previous results for the d 1 system and fits nicely into the unifying scaling scheme we have recently proposed in the study of the random bond ising model

non algebraic domain growth in binary alloys with quenched disorder the authors present detailed numerical results from a computationally efficient cell dynamical system model of domain growth in binary alloys with quenched disorder their numerical results suggest that the domain growth law for the disordered case is compatible with r t approximately 1nt x where x has a weak dependence on the disorder amplitude however it is possible that their simulations do not access the true asymptotic regime they also find that the scaled structure factor for the disordered case is independent of the amplitude of disorder and is the same as that for the pure system

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

non algebraic domain growth in random magnets a cell dynamical approach the authors develop a novel numerical approach based on a computationally efficient cell dynamical system cds model for studying the kinetics of ordering in systems described by a non conserved order parameter with quenched disorder evolving from unstable initial states they use this model to study the kinetics of domain growth in a coarse grained version of the random exchange ising model their numerical data strongly indicate quantitative agreement with the theoretically predicted asymptotic growth law over a limited range of disorder amplitudes they also compare their observations with those in laboratory experiments and make important predictions regarding dynamical scaling in these systems

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

the wave equation on hyperbolic spaces abstract we study the dispersive properties of the wave equation associated with the shifted laplace beltrami operator on real hyperbolic spaces and deduce new strichartz estimates for a large family of admissible pairs as an application we obtain local well posedness results for the nonlinear wave equation

a multidomain spectral method for supersonic reactive flows this paper has a dual purpose it presents a multidomain chebyshev method for the solution of the two dimensional reactive compressible navier stokes equations and it reports the results of the application of this code to the numerical simulations of high mach number reactive flows in recessed cavity the computational method utilizes a newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations the results of the simulations are relevant to recessed cavity flame holders

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

chebyshev pseudospectral multi domain technique for viscous flow calculation abstract a novel chebyshev pseudospectral multi domain technique is introduced for the numerical solution of the navier stokes equations in the primitive variable formulation careful consideration is given to the proper interface condition in the multi domain method which decomposes the solution domain into subdomains by overlapping one grid point a finite difference approximation is used for the time discretization which transforms the system to a set of coupled equations which is then solved by an efficient computational boundary cb method in the cb method the pressure neumann boundary conditions are simply obtained by re arranging the discrete algebraic equations resulting from chebyshev pseudospectral multi domain approximation for the space variables to demonstrate the effectiveness of this technique we present some accurate computational results for calculating the driven cavity flows with re up to 10 000 and for the flow past a circular cylinder with re up to 100 the results for both applications are in good agreement with those of previous researchers

computational solution of two dimensional unsteady pdes using moving mesh methods numerical experiments are described which illustrate some important features of the performance of moving mesh methods for solving two dimensional partial differential equations pdes here we are concerned with algorithms based on moving mesh methods proposed by w huang and r d russell siam j sci comput 20 998 1999 we show that the accuracy of the computations is strongly influenced by the choice of monitor function and we present a monitor function which yields a higher rate of convergence than those that are commonly used in an earlier paper g beckett j a mackenzie a ramage and d m sloan j comput phys 167 372 2001 we demonstrated a robust and efficient algorithm for problems in one space dimension in which the mesh equation is decoupled from the physical pde and the time step is controlled automatically the present work extends this algorithm to deal with problems in two space dimensions

a proof that the discrete singular convolution dsc lagrange distributed approximating function ldaf method is inferior to high order finite differences finite differences approximate the mth derivative of a function u x by a series j n nd j m u x j where the x j are the grid points the closely related discrete singular convolution dsc and lagrange distributed approximating function ldaf methods treated here as a single algorithm approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a by means of fourier analysis and error theorems we show that the dsc is worse than the standard finite differences in differentiating exp ikx for all k when a a f d where a f d 1 n 1 with n as the stencil width is the value of the dsc parameter that makes its weights most closely resemble those of finite differences for a

novel domain growth law in random magnets abstract we investigate the kinetics of domain growth in ising magnets where a fraction 1 p of the magnetic atoms or ions spins are randomly substituted by non magnetic impurities we argue that close to the percolation threshold p c the statistical self similarity of the underlying structure gives rise to a novel crossover in the growth law we propose a method to detect any evidence of this new prediction from the kinetics of domain growth in the dilute ising model dim during intermediate time scales by carrying out monte carlo simulations not at p p c but at slightly higher spin concentrations we analyze the results of our extensive monte carlo simulation of the strongly diluted two dimensional ising model and find the growth to be consistent with the proposed scenario we also compare our observations with those in the recent experiments on the kinetics of ordering in rb 2 co p mg 1 p f 4

strongly disordered chain of impedances theoretical analysis and numerical results the authors study the input impedance of a chain of random resistors chosen according to a broad distribution i e decaying as a power law for large arguments as a function of frequency the modulus of the input impedance varies as a power law while its phase has a well defined value a simple theoretical analysis based on estimations of sums of random variables yields the value of the exponents which are in perfect agreement with numerical simulations even when logarithmic corrections come into play the analysis can be extended further to investigate the sample to sample fluctuations of the impedance again agreement with simulations is very good

vibration analysis by discrete singular convolution this paper explores the utility of a discrete singular convolution algorithm for vibration analysis a number of different realizations of singular convolution kernels are selected to illustrate the present algorithm vibration analysis of strings rods beams diatomic molecules membranes waveguides and thin plates are utilized to test numerical accuracy and speed of convergence of the present approach numerical experiments indicate that the discrete singular convolution is a simple and reliable algorithm for vibration analysis

scaling in the aging dynamics of the site diluted ising model we study numerically the phase ordering kinetics of the two dimensional site diluted ising model the data can be interpreted in a framework motivated by renormalization group concepts apart from the usual fixed point of the nondiluted system there exist two disorder fixed points characterized by logarithmic and power law growth of the ordered domains this structure gives rise to a rich scaling behavior with an interesting crossover due to the competition between fixed points and violation of superuniversality

the three dimensional dilute ising magnet the dilute ising model with a p 0 8 fraction of the sites occupied by spins is simulated on l3 systems for l up to 300 using a single cluster algorithm of wolff in the range of reduced temperature 0 002 t tc tc 0 03 the susceptibility appears to fit a simple power with an effective exponent eff 1 36

ergodicity of the stochastic fractional reaction diffusion equation abstract a stochastic reaction diffusion equation with fractional dissipation is considered the smaller the exponent of the equation is the weaker the dissipation of the equation is the equation is discussed in detail when the exponent changes the aim is to prove the well posedness existence and uniqueness of an invariant measure as well as strong law of large numbers and convergence to equilibrium without the analytic property of the semigroup some methods are used to overcome the difficulties to get the energy estimates the results in this paper can be applied to the classic reaction diffusion equation with wiener noise

uniformly accurate diffusive relaxation schemes for multiscale transport equations many transport equations such as the neutron transport radiative transfer and transport equations for waves in random media have a diffusive scaling that leads to the diffusion equations in many physical applications the scaling parameter mean free path may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic diffusive regimes within one problem and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter in an earlier work jin pareschi and toscani siam j numer anal 35 1998 pp 2405 2439 we handled this numerical problem for discrete velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws jin and xin comm pure appl math 48 1995 pp 235 277 such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit yet explicitly implementable schemes that work with high resolution uniformly with respect to the relaxation parameter in this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities when one uses the even and odd parities of the transport equation

non equilibrium dynamics of disordered systems understanding the broad continuum of relevant time scales via a strong disorder rg in configuration space we show that an appropriate description of the non equilibrium dynamics of disordered systems is obtained through a strong disorder renormalization procedure in configuration space that we define for any master equation with transitions rates between configurations the idea is to eliminate iteratively the configuration with the highest exit rate to obtain renormalized transition rates between the remaining configurations the multiplicative structure of the new generated transition rates suggests that for a very broad class of disordered systems the distribution of renormalized exit barriers defined as will become broader and broader upon iteration so that the strong disorder renormalization procedure should become asymptotically exact at large time scales we have checked numerically this scenario for the non equilibrium dynamics of a directed polymer in a two dimensional random medium

strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds abstract we prove the global in time strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds we obtain estimates for the full set of wave admissible indices including the endpoint the key points are the properties of the microlocalized spectral measure of laplacian on this setting showed in 18 and a littlewood paley squarefunction estimate as applications we prove the global existence and scattering for a family of nonlinear wave equations on this setting

nonlinear waves on 3d hyperbolic space in this article global in time dispersive estimates and strichartz estimates are explored for the wave equation on three dimensional hyperbolic space due to the negative curvature extra dispersion is noted as compared to the euclidean case and a wider range of strichartz estimates is proved using these small data global existence to semilinear wave equations is shown for a range of powers that is broader than that known for euclidean space

cluster monte carlo algorithms abstract the swendsen wang and wolff monte carlo algorithms are described in some detail using the potts model as an example various generalizations are then reviewed and some applications are discussed two complete fortran programs for the algorithms are provided

solution of evolutionary partial differential equations using adaptive finite differences with pseudospectral post processing a coordinate transformation approach is described that enables pseudospectral methods to be applied efficiently to unsteady differential problems with steep solutions the work is an extension of a method presented by mulholland huang and sloan for the adaptive pseudospectral solution of steady problems a coarse grid is generated by a moving mesh finite difference method that is based on equidistribution and this grid is used to construct a time dependent coordinate transformation a sequence of spatial transformations may be generated at discrete points in time or a single transformation may be generated as a continuous function of space and time the differential problem is transformed by the coordinate transformation and then solved using a method that combines pseudospectral discretisation in space with a suitable integrator in time numerical results are presented for unsteady problems in one space dimension

a pseudospectral method of solution of fisher s equation in this paper we develop an accurate and efficient pseudospectral solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations the collocation method employed is based on chebyshev gauss lobatto quadrature points we compare results for a single domain as well as for a subdivision of the main domain into subintervals instabilities that occur in the numerical solution for a single domain analogous to those found by others are attributed to round off errors arising from numerical features of the discrete second derivative matrix operator however accurate stable solutions of fisher s equation are obtained with a multidomain pseudospectral method a detailed comparison of the present approach with the use of the sine interpolation is also carried out

profile decompositions for wave equations on hyperbolic space with applications the goal for this paper is twofold our first main objective is to develop bahouri gerard type profile decompositions for waves on hyperbolic space recently such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy with an eye towards further applications we develop this theory in a fairly general framework which includes the case of waves on hyperbolic space perturbed by a time independent potential our second objective is to use the profile decomposition to address a specific nonlinear problem namely the question of global well posedness and scattering for the defocusing energy critical semi linear wave equation on three dimensional hyperbolic space possibly perturbed by a repulsive time independent potential using the concentration compactness rigidity method introduced by kenig and merle we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as t rightarrow pm infty this proof serves as a blueprint for the arguments in the work 45 by the authors where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane

parametric instability of doubly curved panels subjected to non uniform harmonic loading the parametric instability characteristics of doubly curved panels subjected to various in plane static and periodic compressive edge loadings including partial and concentrated edge loadings are studied using finite element analysis the first order shear deformation theory is used to model the doubly curved panels considering the effects of transverse shear deformation and rotary inertia the theory used is the extension of dynamic shear deformable theory according to the sander s first approximation for doubly curved shells which can be reduced to love s and donnell s theories by means of tracers the effects of static load factor aspect ratio radius to thickness ratio shallowness ratio boundary conditions and the load parameters on the principal instability regions of doubly curved panels are studied in detail using bolotin s method quantitative results are presented to show the effects of shell geometry and load parameters on the stability boundaries results for plates and cylindrical shells are also presented as special cases and are compared with those available in the literature

spectral differencing with a twist spectral collocation methods have become very useful in providing highly accurate solutions to differential equations a straightforward implementation of these methods involves the use of spectral differentiation matrices to obtain optimal accuracy these matrices must be computed carefully we demonstrate that naive algorithms for computing these matrices suffer from severe loss of accuracy due to roundoff errors several improvements are analyzed and compared a number of numerical examples are provided demonstrating significant differences between the sensitivity of the forward problem and the inverse problem

free vibration analysis of functionally graded curved panels using a higher order finite element formulation free vibration analysis of functionally graded curved panels is carried out using a higher order formulation a c0 finite element formulation is used to carry out the analysis the element consists of nine degrees of freedom per node with higher order terms in the taylor s series expansion which represents the higher order transverse cross sectional deformation modes the formulation includes sanders approximation for doubly curved shells considering the effects of rotary inertia and transverse shear a realistic parabolic distribution of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided material properties are assumed to be temperature independent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents heat conduction between ceramic and metal constituents is neglected the accuracy of the formulation is validated by comparing the results with those available in the literature effects of panel geometry parameters and boundary conditions are studied

numerical solution of fisher s equation using a moving mesh method the paper investigates the viability of using moving mesh methods to simulate travelling wave solutions of fisher s equation results are presented that illustrate the weaknesses in moving mesh methods based on equidistribution of some popular monitor functions it is shown that knowledge of the differential equation and the travelling wave solution may be used to construct a monitor function that yields accurate results with suitably chosen moving mesh methods a comparison is made between a moving mesh partial differential equation and a moving mesh differential algebraic equation for the evolution in time

dynamic instability of composite plates subjected to non uniform in plane loads abstract in this paper the dynamic instability of a shear deformable composite plate subjected to periodic non uniform in plane loading is studied for four sets of boundary conditions the static component and the dynamic component of the applied periodic in plane loading are assumed to vary according to either parabolic or linear distributions initially the plate membrane problem is solved using the ritz method to evaluate the plate in plane stress distributions within the prebuckling range due to the applied non uniform in plane edge loading subsequently using the evaluated stress distribution within the plate the equations governing the plate instability boundaries are formulated via hamilton s variational principle employing galerkin s method these partial differential equations are reduced into a set of ordinary differential equations mathieu type of equations describing the plate dynamic instability behaviour following bolotin s method the instability regions are determined from the boundaries of instability which represents the periodic solution of the differential equations with period t and 2 t to the mathieu equations the instability regions are determined for uniform linear and parabolic dynamic in plane loads using first order and second order approximations numerical results are also presented to bring out the effects of span to thickness ratio shear deformation aspect ratio boundary conditions and static load factor on the instability regions

on the transition from the ginzburg landau equation to the extended fisher kolmogorov equation abstract the ginzburg landau gl equation generically describes the behaviour of small perturbations of a marginally unstable basic state in systems on unbounded domains in this paper we consider the transition from this generic situation to a degenerate co dimension 2 case in which the gl approach is no longer valid instead of studying a general underlying model problem we consider a two dimensional system of coupled reaction diffusion equations in one spatial dimension we show that near the degeneration the behaviour of small perturbations is governed by the extended fisher kolmogorov efk equation at leading order the relation between the gl equation and the efk equation is quite subtle but can be analysed in detail the main goal of this paper is to study this relation which we do asymptotically the asymptotic analysis is compared to numerical simulations of the full reaction diffusion system as one approaches the co dimension 2 point we observe that the stable stationary periodic patterns predicted by the gl equation evolve towards various different families of stable stationary but not necessarily periodic so called multi bump solutions in the literature these multi bump patterns are shown to exist as solutions of the efk equation but there is no proof of the asymptotic stability of these solutions our results suggest that these multi bump patterns can also be asymptotically stable in large classes of model problems

effect of noise on spinodal decomposition the long time behaviour of two dimensional systems undergoing spinodal decomposition is studied numerically with the aid of a cell dynamical approach both without and with noise in both cases the representative length scale of the pattern behaves as l t approximately tphi where the exponent phi crosses over from approximately 0 28 to approximately 0 33 the crossover time increases with an increase in amplitude of the noise

wavelets generated by using discrete singular convolution kernels this paper explores the connection between wavelet methods and an efficient computational algorithm the discrete singular convolution dsc many new dsc kernels are constructed and they are identified as wavelet scaling functions two approaches are proposed to generate wavelets from dsc kernels two well known examples the canny filter and the mexican hat wavelet are found to be special cases of the present dsc kernel generated wavelets a family of wavelet generators proposed in this paper are found to form an infinite dimensional lie group which has an invariant subgroup of translation and dilation if dsc kernels form an orthogonal system they are found to span a wavelet subspace in a multiresolution analysis

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

dynamic stability of cylindrical panels with transverse shear effects abstract the dynamic stability of simply supported isotropic cylindrical panels under combined static and periodic axial forces are investigated an extension of donnells shell theory to a first order shear deformation theory is used and a system of mathieu hill equations are obtained via a normal mode expansion and the parametric resonance response was analyzed using bolotins method results are compared with those obtained using the classical shell theory the effects of the thickness to radius ratio on the instability regions are examined in detail

vibration and stability analyses of cross ply laminated circular cylindrical shells abstract an improved analytical procedure is introduced in the context of free vibration and stability problems of cross ply laminated circular cylindrical shells the donnell shear deformation type theory and donnell s classical theory are used to illustrate the procedure numerical results are presented for shells with a variety of boundary conditions special attention is given to the axisymmetric problems and new results not yet found in the literature for stability and vibration are reported it is recommended that the new technique be used for generating the levy type solutions in vibration and stability problems of laminated shell panels and plates when the number and order of equations are higher than those of the classical theories

comparison of the nodal integral method and nonstandard finite difference schemes for the fisher equation the relationship between the so called nonstandard finite difference schemes and the nodal integral method nim is investigated using the fisher equation as a model problem exact and best finite difference schemes are reviewed first next the nim for the fisher equation is developed it is shown that the nim leads to a nonstandard evaluation of the derivatives moreover the resulting scheme possesses the desirable characteristics of the nonstandard finite difference schemes such as the nonlocal evaluation of the nonlinear terms thus the nim provides a systematic framework to obtain schemes similar to the best finite difference schemes numerical results for a propagating front problem show that the nim can capture the shape and speed of the front very accurately results also show that the best finite difference scheme is stable for large grid sizes but only at the cost of inaccuracy in the front propagation speed additional results are obtained using the nim for symmetric and asymmetric initial conditions these describe the interaction of two fronts of advantageous genes that approach each other and merge it is noted that the traveling fronts evolving from certain asymmetric initial conditions become indistinguishable from those that evolve from symmetric initial conditions

crank nicolson quasi wavelets method for solving fourth order partial integro differential equation with a weakly singular kernel in this paper we study a novel numerical scheme for the fourth order partial integro differential equation with a weakly singular kernel in the time direction a crank nicolson time stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term and the quasi wavelets numerical method for space discretization our interest in the present paper is a continuation of the investigation in yang et al 33 where we study discretization in time by using the forward euler scheme the comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order partial integro differential equation with a weakly singular kernel we also tested the method proposed on several one and two dimensional problems with very promising results besides in order to demonstrate the power of the quasi wavelets method in comparison with standard discretization methods we also consider the high frequency oscillation problems with the integro differential term

on traveling wave solutions of fisher s equation in two spatial dimensions it is shown for the quadratic fisher equation in two spatial dimensions that along with a plane wave there exist several other traveling waves with nontrivial front geometry some of the solutions are found in explicit form others are constructed approximately the dispersion relationship and velocity curvature dependence generated by these solutions are studied

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

high accuracy approximation techniques for analytic functions a generalization of the familiar mesh point technique for numerical approximation of functions is presented high accuracy and very rapid convergence may be obtained by thoughtful choice of the reference function chosen for interpolation between the mesh points in particular derivative operators are represented by highly nonlocal matrices but this is no drawback when one has computing machines to perform the algebraic manipulations some examples are given from familiar quantum mechanical problems

option valuation by using discrete singular convolution this paper explores the utility of a discrete singular convolution dsc algorithm for solving the black scholes equation both european and american style options which include all nontrivial plain option pricing problems are considered to test the accuracy and to examine the efficiency of the present algorithm adaptive meshes are constructed to enhance the performance of the dsc algorithm all the present results are validated either by the analytical solution or by the standard binomial tree method extensive comparisons are carried out with two standard finite difference schemes and two binomial models of high speed convergence numerical experiments reveal that the present approach is accurate efficient and reliable for financial derivative valuations

the near equivalence of five species of spectrally accurate radial basis functions rbfs radial basis function rbf interpolants have become popular in computer graphics neural networks and for solving partial differential equations in many fields of science and engineering in this article we compare five different species of rbfs gaussians hyperbolic secant sech s inverse quadratics multiquadrics and inverse multiquadrics we show that the corresponding cardinal functions for a uniform unbounded grid are all approximated by the same function c x 1 sin x sinh x for some constant which depends on the inverse width parameter shape parameter of the rbf and also on the rbf species the error in this approximation is exponentially small in 1 for sech s and inverse quadratics and exponentially small in 1 2 for gaussians the error is proportional to 4 for multiquadrics and inverse multiquadrics the error in all cases is small even for o 1 these results generalize to higher dimensions the gaussian rbf cardinal functions in any number of dimensions d are without approximation the tensor product of one dimensional gaussian cardinal functions c d x 1 x 2 x d j 1 d c x j for other rbf species we show that the two dimensional cardinal functions are well approximated by the products of one dimensional cardinal functions again the error goes to zero as 0 the near identity of the cardinal functions implies that all five species of rbf interpolants are almost the same despite the great differences in the rbf s themselves

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

a direct spectral collocation poisson solver in polar and cylindrical coordinates in this paper we present a direct spectral collocation method for the solution of the poisson equation in polar and cylindrical coordinates the solver is applied to the poisson equations for several different domains including a part of a disk an annulus a unit disk and a cylinder unlike other poisson solvers for geometries such as unit disks and cylinders no pole condition is involved for the present solver the method is easy to implement fast and gives spectral accuracy we also use the weighted interpolation technique and nonclassical collocation points to improve the convergence

pseudospectral method for fisher equation in a disk abstract in this paper we develop a mixed jacobi fourier pseudospectral method for solving the fisher equation in a disc fisher s equation always plays a large role in many fields such as logistic population growth models tissue engineering nuclear reactions and neurophysiology etc it is very important to study how to solve these equations numerically in this work we employ the generalized jacobi approximation to simulate the singularity of solutions at the regional center some mixed jacobi fourier interpolation approximation results are established which play important roles in numerical simulation of various problems defined in a disc as an application the mixed jacobi fourier pseudospectral scheme is provided for the fisher equation in a disk the generalized stability and convergence of the proposed scheme are proved some numerical results are presented to demonstrate the efficiency of this new algorithm

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

comment on the discrete ordinate method in the kinetic theory of gases abstract the discrete ordinate do method for solving the linear integro differential equations of kinetic theory is investigated and other numerical methods for solving this type of problem are reviewed it is pointed out that the do method can be regarded as a special case of the method of weighted residuals an efficient method for calculating the matrix d ij representing the differential operator is described and it is shown that the method does not necessarily require the use of the nodes of a gaussian quadrature

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

field driven interface dynamics of a random soft spin ising model the kinetics of an interface between spin up and spin down domains in a soft spin ising model with quenched random fields and driving magnetic field h is studied numerically within a discrete time dynamics at zero temperature spins are updated in parallel starting from a flat interface it is found that for fields smaller than a threshold field hc the interface is pinned while above hc the mean velocity of the interface increases proportional to h hc for two and three dimensional systems

thermally induced dynamic instability of laminated composite conical shells abstract thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method the laminated composite conical shells are subjected to static and periodic thermal loads the linear instability approach is adopted in the present study a set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs the formulation begins with three dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium after proper nondimensionalization asymptotic expansion and successive integration we obtain recursive sets of differential equations at various levels the method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible using the method of differential quadrature and bolotin s method and imposing the orthonormality and solvability conditions on the present asymptotic formulation we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner the principal instability regions of cross ply conical shells with simply supported simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory

a soluble model of domain growth in one dimensional disordered systems we introduce a new algorithm for domain growth in disordered systems at low temperatures based on the tendency for such a system to freeze into metastable states we apply the algorithm to the d 1 random bond ising model for a variety of bond distributions the analytical forms we obtain for the two point correlation function and autocorrelation function agree well with numerical simulations of the model these forms are not the same as those of the pure ising model with glauber dynamics

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved

comparison of the discrete singular convolution and three other numerical schemes for solving fisher s equation in this paper a discrete singular convolution dsc algorithm is introduced to solve fisher s equation to which obtaining an accurate and reliable traveling wave solution is a challenging numerical problem two novel numerical treatments a moving frame scheme and a new asymptotic scheme are designed to overcome the subtle difficulties involved in the numerical solution of fisher s equation the moving frame scheme is proposed to provide accurate and efficient spatial resolution of the problem the new asymptotic scheme is introduced to facilitate the correct prediction of the wave speed after long time integrations the resulting dsc algorithm is able to correctly predict long time traveling wave behavior the performance of the present algorithm is demonstrated through extensive numerical experiments as well as through a comparison with three other standard approaches i e methods of the fourier pseudospectral the accurate spatial derivatives and the crank nicolson schemes the spatial and temporal accuracies of all four methods are analyzed and numerically verified