the pseudospectral method for solving differential eigenvalue problems abstract a simple pseudospectral method is presented for the numerical solution of linear differential eigenvalue problems the method does not produce the spurious eigenvalues which generally occur when such problems are solved by the spectral tau method the pseudospectral method is presented using two model problems and the presentation contains a useful algorithm for the computation of the spectral differentiation matrices at general collocation points numerical results are offered for a set of benchmark problems including the orr sommerfeld equation for stability of plane poiseuille flow between parallel plates the results indicate that the new pseudospectral method is comparable in accuracy to the tau method

regularization and error estimate for an initial inverse nonlocal diffusion problem abstract we study the backward problem of determining the initial condition for a system of parabolic diffusion equations which is severely ill posed in the sense of hadamard to stabilize the solution we develop the quasi reversibility qr and fourier truncation methods to construct the regularized solutions we also investigate the error estimates and convergence rates between the regularized solutions and the true solution in l 2 and h 1 norm a numerical scheme is presented

pullback attractors in v for non autonomous 2d navier stokes equations and their tempered behaviour abstract in this paper the asymptotic behaviour of the solutions to a non autonomous 2d navier stokes model is analyzed when the initial datum belongs to v for two frameworks the universe of fixed bounded sets and also for another universe given by a tempered condition the existence of pullback attractors in these different universes is established and thanks to regularity properties the relation between these several families of attractors and the corresponding in h is successfully studied finally two results about the tempered behaviour in v and h 2 2 of the pullback attractors when time goes to are obtained

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

asymptotic behaviour of a phase field model with three coupled equations without uniqueness abstract we prove the existence of weak solutions for a phase field model with three coupled equations with unknown uniqueness and state several dynamical systems depending on the regularity of the initial data then the existence of families of global attractors level set depending for the corresponding multi valued semiflows is established applying an energy method finally using the regularizing effect of the problem we prove that these attractors are in fact the same

time stepping methods for the simulation of the self assembly of nano crystals in matlab on a gpu partial differential equations describing the patterning of thin crystalline films are typically of fourth or sixth order they are quasi or semilinear and they are mostly defined on simple geometries such as rectangular domains for the numerical simulation of these kinds of problems spectral methods are an efficient approach we apply several implicit explicit schemes to one recently derived pde that we express in terms of coefficients of trigonometric interpolants while the simplest imex scheme turns out to have the mildest step size restriction higher order sbdf schemes tend to be more unstable and exponential time integrators are fastest for the calculation of very accurate solutions we implemented a reduced model in the expint package syntax 3 and compared various exponential schemes a convexity splitting approach was employed to stabilize the sbdf1 scheme we show that accuracy control is crucial when using this idea therefore we present a time adaptive sbdf1 sbdf1 2 step method that yields convincing results reflecting the change in timescales during topological changes of the nanostructures the implementation of all presented methods is carried out in matlab we used the open source gpumat package to gain up to 5 fold runtime benefits by carrying out calculations on a low cost gpu without having to prescribe any knowledge in low level programming or cuda implementations and found comparable speedups as with matlab s pct or with gpumat run on octave

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

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

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

the fidelity of exponential and imex integrators for wave turbulence introduction of a new near minimax integrating factor scheme abstract exponential and imex integrators are considered for wave turbulence problems which are stiff systems that often require long time simulations applying a multi scale asymptotic analysis to the difference equations for a simple wave turbulence problem reveals that exponential runge kutta methods incorrectly model interactions between resonant waves unless they are specifically designed otherwise and implicit explicit imex integrators introduce dispersive and dissipative errors that severely degrade the accuracy of slow energy exchange between resonant waves the same analysis shows integrating factor if methods to be the most suitable numerical experiments performed on the resonant triad model and the majda mclaughlin tabak mmt model verify the discrete asymptotic analysis in the mmt problem exponential time differencing methods require a time step more than two orders of magnitude smaller than if methods to achieve comparable accuracy and imex methods require more than six orders of magnitude smaller time steps a novel near minimax rational approximation of the matrix exponential is proposed to be used for integrating factor methods and is demonstrated to be as accurate as the if method but without the need to compute a matrix exponential

generalized integrating factor methods for stiff pdes the integrating factor if method for numerical integration of stiff nonlinear pdes has the disadvantage of producing large error coefficients when the linear term has large norm we propose a generalization of the if method and in particular construct multistep type methods with several orders of magnitude improved accuracy we also consider exponential time differencing etd methods and point out connections with a particular application of the commutator free lie group methods we present a new fourth order etdrk method with improved accuracy the methods considered are compared in several numerical examples

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

rooted tree analysis of runge kutta methods with exact treatment of linear terms we investigate a class of time discretization schemes called etd runge kutta methods where the linear terms of an ordinary differential equation are treated exactly while the other terms are numerically integrated by a one step method these schemes proposed by previous authors can be regarded as modified runge kutta methods whose coefficients are matrices instead of scalars from this viewpoint we reexamine the notion of consistency convergence and order to provide a mathematical foundation for new methods applying the rooted tree analysis expansion theorems of both the strict and numerical solutions are proved and two types of order conditions are defined several classes of formulas with up to four stages that satisfy the conditions are constructed and it is shown that the power series of matrices employed as their coefficients can be determined using the order conditions

on the relation between two different concepts of pullback attractors for non autonomous dynamical systems abstract for an abstract dynamical system we establish under minimal assumptions the existence of d attractor i e a pullback attractor for a given class d of families of time varying subsets of the phase space we relate this concept with the usual attractor of fixed bounded sets pointing out its usefulness in order to ensure the existence of this last attractor in particular situations moreover we prove that under a simple assumption these two notions of attractors generate in fact the same object this is then applied to a navier stokes model improving some previous results on attractor theory

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

beyond conventional runge kutta methods in numerical integration of odes and daes by use of structures and local models there are two parts in this paper in the first part we consider an overdetermined system of differential algebraic equations daes we are particularly concerned with hamiltonian and lagrangian systems with holonomic constraints the main motivation is in finding methods based on gauss coefficients preserving not only the constraints symmetry symplecticness and variational nature of trajectories of holonomically constrained hamiltonian and lagrangian systems but also having optimal order of convergence the new class of s s gauss lobatto specialized partitioned additive runge kutta spark methods uses greatly the structure of the daes and possesses all desired properties in the second part we propose a unified approach for the solution of ordinary differential equations odes mixing analytical solutions and numerical approximations the basic idea is to consider local models which can be solved efficiently for example analytically and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction in order to preserve also symmetry we define the new class of symmetrized runge kutta methods with local model srklm

asymptotic behaviour of nonlocal p laplacian reaction diffusion problems abstract in this paper we focus on studying the existence of attractors in the phase spaces l 2 and l p among others for time dependent p laplacian equations with nonlocal diffusion and nonlinearities of reaction diffusion type firstly we prove the existence of weak solutions making use of a change of variable which allows us to get rid of the nonlocal operator in the diffusion term thereupon the regularising effect of the equation is shown applying an argument of a posteriori regularity since under the assumptions made we cannot guarantee the uniqueness of weak solutions in addition this argument allows to ensure the existence of an absorbing family in w 0 1 p this leads to the existence of the minimal pullback attractors in l 2 l p and some other spaces as l p relationships between these families are also established

an implementation of the exponential time differencing scheme to the magnetohydrodynamic equations in a spherical shell over the last decade there has been renewed interest in applying exponential time differencing etd time stepping schemes to the solution of stiff systems in this paper we present an implementation of such a scheme to the fully spectral solution of the incompressible magnetohydrodynamic equations in a spherical shell one problem associated with etd schemes is the accurate calculation of the necessary matrices we implement and discuss in detail a variety of different methods including direct computation contour integration spectral expansions and recurrence relations we compare the accuracy of six different second order methods in determining the evolution of a three dimensional magnetic field under the action of a prescribed time dependent flow of electrically conducting fluid and find that for the timestep restriction imposed by the nonlinear terms etd methods are no more accurate than linearly implicit methods which have the significant advantage of being easier to implement however etd methods are more readily extendable than those which are linearly implicit and will become much more advantageous at higher order

order conditions for commutator free lie group methods we derive order conditions for commutator free lie group integrators for certain problems these schemes can be good alternatives to the runge kutta munthe kaas schemes especially when applied to stiff problems or to homogeneous manifolds with large isotropy groups the order conditions correspond to certain subsets of the set of ordered rooted trees we discuss ways to select these subsets and their combinatorial properties we also suggest how the reuse of flow calculations can be included in order to reduce the computational cost in the case that at most two flow calculations are admitted in each stage the order conditions simplify substantially we derive families of fourth order schemes which effectively use only five flow calculations per step

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

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

exponential integrator methods for systems of non linear space fractional models with super diffusion processes in pattern formation abstract nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in science and engineering however solving these type of models is computationally demanding in this paper we propose an exponential integrator method for space fractional models as an attractive and easy to code alternative for other existing second order exponential integrator methods this scheme is based on using a real distinct poles discretization for the underlying matrix exponentials one of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency the scheme is established to be second order convergent and proven to be robust for nonlinear space fractional reaction diffusion problems involving non smooth initial data our approach is exhibited by solving a system of two dimensional problems which exhibits pattern formation and has applications in cell division empirically super diffusion processes are displayed by investigating the effect of the fractional power of the underlying laplacian operator on the pattern formation found in these models furthermore the superiority of our method over competing second order etd schemes bdf2 scheme and imex schemes is demonstrated

efficient l stable method for parabolic problems with application to pricing american options under stochastic volatility efficient l stable numerical method for semilinear parabolic problems with nonsmooth initial data is proposed and implemented to solve heston s stochastic volatility model based pde for pricing american options under stochastic volatility the proposed new method is also used to solve two asset american options pricing problem cox and matthews s m cox p c matthews exponential time differencing for stiff systems journal of computational physics 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic problems kassam and trefethen a k kassam l n trefethen fourth order time stepping for stiff pdes siam journal on scientific computing 26 4 2005 1214 1233 showed that while computing certain functions involved in the cox matthews schemes severe cancelation errors can occur which affect the accuracy and stability of the schemes kassam and trefethen proposed complex contour integration technique to implement these schemes in a way that avoids these cancelation errors but this approach creates new difficulties in choosing and evaluating the contour integrals for larger problems we modify the etdrk schemes using positivity preserving pade approximations of the matrix exponential functions and construct computationally efficient parallel version using splitting technique as a result of this approach it is required only to solve several backward euler linear problems in serial or parallel

weak poincare inequalities and l2 convergence rates of markov semigroups abstract in order to describe l 2 convergence rates slower than exponential the weak poincare inequality is introduced it is shown that the convergence rate of a markov semigroup and the corresponding weak poincare inequality can be determined by each other conditions for the weak poincare inequality to hold are presented which are easy to check and which hold in many applications the weak poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes some typical examples are given to illustrate the general results in particular our results are applied to the stochastic quantization of field theory in finite volume moreover a sharp criterion of weak poincare inequalities is presented for poisson measures on configuration spaces

on a backward problem for the kirchhoff s model of parabolic type abstract we study for the first time the backward problem for nonlocal nonlinear boundary value problem of kirchhoff s model of parabolic type first we show that the problem is severely ill posed in the sense of hadamard we propose two methods the fourier truncation method for stabilizing the problem with homogeneous source and the quasi reversibility method for regularizing the problem with nonlinear source under some priori assumptions on the exact solution we establish some stability estimates in the h 0 1 norm

functional inequalities and spectrum estimates the infinite measure case abstract as a continuation of 13 where a poincare type inequality was introduced to study the essential spectrum on the l 2 space of a probability measure this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite high order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality criteria of the inequality and estimates of the inequality constants are presented finally some concrete examples are considered to illustrate the main results in particular estimates of high order eigenvalues obtained in this paper are sharp as checked by two examples on the euclidean space

a new class of split exponential propagation iterative methods of runge kutta type sepirk for semilinear systems of odes abstract in this paper the framework of the exponential propagation iterative methods of runge kutta type epirk is extended to construct split epirk sepirk integrators for semilinear systems of odes the structure of the sepirk methods possesses the flexibility and generality that allows construction of very efficient schemes we demonstrate how bicolored trees based b series can be used to derive the order conditions for the new integrators an algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed the numerical results illustrate the advantages offered by the new class of integrators the experiments also address the comparative performance of split vs non split epirk methods and the question of improving efficiency by optimizing coefficients of the sepirk schemes it is shown that specific schemes can be custom built to improve computational efficiency for a particular problem

on group fourier analysis and symmetry preserving discretizations of pdes in this paper we review some group theoretic techniques applied to discretizations of pdes inspired by the recent years active research in lie group and exponential time integrators for differential equations we will in the first part of the paper present algorithms for computing matrix exponentials based on fourier transforms on finite groups as an example we consider spherically symmetric pdes where the discretization preserves the 120 symmetries of the icosahedral group this motivates the study of spectral element discretizations based on triangular subdivisions in the second part of the paper we introduce novel applications of multivariate non separable chebyshev polynomials in the construction of spectral element bases on triangular and simplicial sub domains these generalized chebyshev polynomials are intimately connected to the theory of root systems and weyl groups used in the classification of semi simple lie algebras and these polynomials share most of the remarkable properties of the classical chebyshev polynomials such as near optimal lebesgue constants for the interpolation error the existence of fft based algorithms for computing interpolants and pseudo spectral differentiation and existence of gaussian integration rules the two parts of the paper can be read independently

functional inequalities for empty essential spectrum abstract in terms of the equivalence of poincare inequality and the existence of spectral gap the super poincare inequality is suggested in the paper for the study of essential spectrum it is proved for symmetric diffusions that such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator this inequality recovers known sobolev and nash type ones it is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below some results are also proved for a more general setting including symmetric jump processes moreover estimates of inequality constants are also presented which lead to a proof of a result on ultracontractivity suggested recently by d stroock finally concentration of reference measures for super poincare inequalities is studied the resulting estimates extend previous ones for poincare and log sobolev inequalities

a new method for fast transforms in parity mixed pdes part ii application to confined rotating convection to demonstrate the utility of the parity filtering methods described by vasil et al g vasil n brummell k julien a new method for fast transforms in parity mixed pdes part i numerical techniques and analysis j comput phys 2008 we introduce a numerical code designed to solve for rayleigh benard convection in a confined rotating box using the new methods we have formulated that is using a straightforward pseudospectral framework we incorporate techniques for efficiently computing parity mixed coriolis accelerations in a time dependent numerical solver the goals of the presented numerical code are to provide a tool to investigate aspects of confined rotating convection experiments with a simple model and to illustrate the application of parity filtering in our numerical tests we find that a correct accounting for parity leads to clear and interesting behavior that has been observed in laboratory experiments but that has not been observed in previous numerical simulations in periodic domains

variable step implementation of etd methods for semilinear problems abstract spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations exponential integrators solve exactly the linear part which is assumed to be responsible of the stiffness of the problem while the nonlinear terms are approximated numerically in this paper a variable step implementation of the exponential time differencing methods of multistep type is considered a local error estimate of exponential type whose computation is for free is introduced and some implementation issues for diagonalizable matrices are addressed numerical results showing the behaviour of the proposed variable step implementation are provided too

long time behavior of a non autonomous parabolic equation with nonlocal diffusion and sublinear terms abstract this paper is devoted to study the asymptotic behavior of a time dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth namely we extend some previous results from the literature obtaining existence uniqueness and continuity results analyzing the stationary problem and decay of the solutions of the evolutionary problem and finally under more general assumptions ensuring the existence of pullback attractors for the associated dynamical system in both l 2 and h 1 norms relationships among these objects are established using regularizing properties of the equation

exponential rosenbrock methods of order five construction analysis and numerical comparisons the aim of this paper is to construct high order exponential rosenbrock methods and to analyze their convergence properties for the time discretization of large scale systems of stiff differential equations we present a new and simple approach for deriving the stiff order conditions these conditions allow us to construct new pairs of embedded methods of high order as an example we present a fifth order method with five stages for particular problems the order conditions can be simplified it is then even possible to construct a method of order 5 with three stages only the error analysis is performed in an abstract framework of strongly continuous semigroups that allows us to treat semilinear evolution equations in banach spaces convergence results are proved for variable step size implementations to demonstrate the efficiency of the new integrators we give some numerical experiments in matlab in particular numerical comparisons for semilinear parabolic pdes in one and two space dimensions are presented

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations

functional inequalities on abstract hilbert spaces and applications we study the essential spectrum and the semigroup property for self adjoint operators on abstract hilbert spaces by using functional inequalities some known results obtained on the l 2 space w r t a measure space are generalized the functional inequality is also used to study non symmetric semigroups

asymptotic behaviour of nonlocal reaction diffusion equations abstract the existence of a global attractor in l 2 is established for a reaction diffusion equation on a bounded domain in r d with dirichlet boundary conditions where the reaction term contains an operator f l 2 l 2 which is nonlocal and possibly nonlinear existence of weak solutions is established but uniqueness is not required compactness of the multivalued flow is obtained via estimates obtained from limits of galerkin approximations in contrast with the usual situation these limits apply for all and not just for almost all time instants

existence and upper semi continuity of pullback attractors of a p laplacian equation with delay in this paper we study the dynamics of a non autonomous p laplacian equation with delay we first prove the existence and uniqueness of solutions for this equation by using the galerkin approximations then applying the sobolev compactness theorem ascoli azerla theorem and energy method we show the existence of pullback attractors for the associated process in space c h 0 lp as well as in space c h 0 l2 the upper semi continuity result of the pullback attractors is also established here