a two stage filter for smoothing multivariate noisy data on unstructured grids abstract experimental data as well as numerical simulations are very often affected by noise random fluctuations that distort the final output or the intermediate products of a numerical process in this paper a new method for smoothing data given on nonstructured grids is proposed it takes advantage of the smoothing properties of kernel weighted averaging and least squares techniques the weighted averaging is performed following a shepard like procedure where gaussian kernels are employed while least squares fitting is reduced to the use of very few basis functions so as to improve smoothness though at the price of interpolation accuracy once we have defined an n point grid and want to make a smoothed fit at a given grid point this method reduces to consideration of all m point stencils for a given value of m where m n that include that point and to make a least squares fitting for that particular point in each of those stencils finally the various results thus obtained are weight averaged the weights being inversely proportional to the distance of the point to the middle of the stencil then this process is repeated for all the grid points so as to obtain the smoothing of the input function though this method is generalized for the multivariate case one two and five dimensional test cases are shown as examples of the performance of this method

the maximal operator on generalized orlicz spaces abstract in this note i present a sufficient condition for the boundedness of the maximal operator on generalized orlicz spaces the result includes as special cases the optimal condition for orlicz spaces as well as the essentially optimal conditions for variable exponent lebesgue spaces and the double phase functional

variational nonlinear elliptic equations in nonreflexive musielak spaces abstract we investigate the existence and uniqueness of variational solutions for the dirichlet problem associated with variational nonlinear elliptic equations in musielak spaces we assume neither 2 nor 2 condition on the functions but suppose that they satisfy some structural assumptions that allow to define the variational framework and to obtain poincare type inequalities within the approach we use the galerkin method is coupled with an abstract existence result by f e browder 1966

a pocket guide to nonlinear differential equations in musielak orlicz spaces abstract the musielak orlicz setting unifies variable exponent orlicz weighted sobolev and double phase spaces they inherit technical difficulties resulting from general growth and inhomogeneity in this survey we present an overview of developments of the theory of existence of pdes in the setting including reflexive and non reflexive cases as well as isotropic and anisotropic ones particular attention is paid to problems with data below natural duality in absence of lavrentiev s phenomenon

green s function and comparison principles for first order periodic differential equations with piecewise constant arguments in this paper we obtain the expression of the green s function related with a first order periodic differential equation with piecewise constant argument we derive comparison results for the treated linear operator by studying the sign of the obtained green s function

steady flow of non newtonian fluids monotonicity methods in generalized orlicz spaces abstract our purpose is to show the existence of weak solutions to steady flow of non newtonian incompressible fluids with nonstandard growth conditions of the cauchy stress tensor we are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening since l p framework is not sufficient to capture the described model we describe the growth conditions with the help of a general x dependent convex function and formulate our problem in generalized orlicz spaces

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

generalized orlicz spaces and related pde abstract we prove the boundedness of the maximal operator in generalized orlicz spaces defined on subsets of r n the proof is based on an extension result for functions we study generalized sobolev orlicz spaces and establish density of smooth functions and the poincare inequality as applications we establish the existence of solutions of the laplace equation with zero and non zero right hand side further we systematize assumptions for functions and prove several basic tools needed for the study of differential equations of generalized orlicz growth

on the application of kam theory to discontinuous dynamical systems abstract so far the application of kolmogorov arnold moser kam theory has been restricted to smooth dynamical systems since there are many situations which can be modeled only by differential equations containing discontinuous terms such as state dependent jumps e g in control theory or nonlinear oscillators it is shown by a series of transformations how kam theory can be used to analyze the dynamical behaviour of such discontinuous systems as well the analysis is carried out for the example x x a sgn x p t with p c 6 being periodic it is known that all solutions are unbounded for small a 0 we prove that all solutions are bounded for a 0 sufficiently large and that there are infinitely many periodic and quasiperiodic solutions in this case

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

on the integral manifold for a system of differential equations with piecewise constant argument the existence of an integral manifold for a system of differential equations with piecewise constant argument is studied

green s function for second order differential equations with piecewise constant arguments abstract in this paper the second order nonautonomous differential equations with piecewise constant arguments x t t t x t t t t x t c t x t d t x t t t 0 t x 0 x t x 0 x t is considered we obtain the green s function to express the unique solution for the system

on the reduction principle for differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a new type of differential equations with piecewise constant argument epcag more general than epca k l cooke j wiener retarded differential equations with piecewise constant delays j math anal appl 99 1984 265 297 j wiener generalized solutions of functional differential equations world scientific singapore 1993 the reduction principle v a pliss the reduction principle in the theory of the stability of motion izv akad nauk sssr ser mat 27 1964 1297 1324 in russian v a pliss integral sets of periodic systems of differential equations nauka moskow 1977 in russian is proved for epcag the structure of the set of solutions is specified we establish also the existence of global integral manifolds of quasilinear epcag in the so called critical case and investigate the stability of the zero solution

new approach for the existence of pseudo almost periodic solutions for some second order differential equation with piecewise constant argument abstract the following equation d 2 d t 2 x t px t 1 qx 2 t 1 2 f t is considered and necessary and sufficient conditions are given in order to ensure the existence and uniqueness of pseudo almost periodic solutions

on favard s theorems abstract in this paper we improve and extend the classical favard s theorems i e favard s theorem of the module containment favard s theorem of linear differential equations we study favard s theory of linear differential equations with piecewise constant argument an example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument the equivalence between almost automorphic functions and n almost periodic ones is studied

pseudo monotone mappings in sobolev orlicz spaces and nonlinear boundary value problems on unbounded domains where r is a domain not necessarily bounded in r in the case where the a have at most polynomial growth in u and its derivatives this problem can be studied in the context of the mappings of monotone type from a sobolev space into its conjugate when the functions a even the top order terms do not obey any polynomial growth it is customary to use a sobolev orlicz space rather than a sobolev space although at the cost of some degree of complexity in the arguments caused mainly by the fact that orlicz spaces are not reflexive in general we shall be dealing here with the latter case the concept of a pseudo monotone operator was introduced to take into account the difference of the properties of the highest order functions a in their top order variables from the lower order derivatives when the domain r is bounded and a polynomial growth condition is satisfied it is a classical result to derive the pseudo monotonicity of the mapping t induced by the operator a in a natural manner from a relatively simple set of hypotheses leray lions conditions on the functions a 3 131 an analogous conclusion for the case of sobolev orlicz spaces was also shown to be true 161 for unbounded domains r other methods were developed to obtain some

on the almost automorphy of bounded solutions of differential equations with piecewise constant argument abstract in this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x t a x t f t t r where a is a bounded linear operator in x and f is an x valued almost automorphic function

on periodic solutions of differential equations with piecewise constant argument the periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type m u akhmet integral manifolds of differential equations with piecewise constant argument of generalized type nonlinear anal tma 66 2007 367 383 m u akhmet on the reduction principle for differential equations with piecewise argument of generalized type j math anal appl 336 2007 646 663 is addressed we consider the critical case when associated linear homogeneous system admits nontrivial periodic solutions criteria of existence of periodic solutions of such equations are obtained one of the main auxiliary results of our paper is an analogue of gronwall bellman lemma for functions with piecewise constant and retarded advanced type arguments dependence of solutions on the parameter is investigated appropriate examples are given to show our results

algorithms for interpolation and localization in irregular 2d meshes abstract in numerical simulations of power systems discrete models and mesh free models are very often used simultaneously for example in particle in cell pic codes the aim of this paper is to introduce new techniques for interpolation as well as for localization in irregular fourpoint meshes these algorithms extend the existing methods for regular grids to grids consisting of non equidistant convex four point meshes they were developed in order to obtain short cpu times and possible vectorization

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

first order differential equations with piecewise constant arguments and nonlinear boundary value conditions this paper is devoted to the study of differential equations with piecewise constant arguments coupled with nonlinear boundary value conditions under suitable assumptions on the data of the equation by means of the method of weakly coupled lower and upper solutions we derive the existence of extremal solutions and extremal quasi solutions moreover some results are given concerning the uniqueness of solutions furthermore we deduce some maximum principles related to the linear equation which allow us to develop the monotone iterative method some illustrative examples are also presented

variational inequalities in orlicz sobolev spaces on etudie l existence de solutions d inegalites variationnelles associees a des operateurs differentiels elliptiques

linearization near the integral manifold for a system of differential equations with piecewise constant argument abstract in this paper we prove a linearization result via an integral manifold for a system of differential equations with piecewise constant argument

on the second order differential equation with piecewise constant argument and almost periodic coefficients in this paper we study the existence of almost and quasi periodic solutions to two classes of second order differential equations as a corollary it is shown that periodic and unbounded solutions can coexist for the equation x t 2x t bx t f t which is different from the case b 0 this phenomena is due to the piecewise constant argument and illustrates a crucial difference between ordinary differential equations and differential equations with piecewise constant argument the results are extended to nonlinear equations

oscillations in systems of differential equations with piecewise constant argument abstract a linear system of differential equations with the argument t 1 2 is studied where denotes the greatest integer function of special interest are oscillatory and periodic properties of the solutions which depend on the eigenvalues of a certain matrix associated with the system therefore the results can be extended to more general dynamical systems

local rectangular refinement with application to nonreacting and reacting fluid flow problems a new solution adaptive gridding method has been developed for the solution of discretized systems of coupled nonlinear elliptic partial differential equations on rectangular domains such a method is required for the numerical solution of realistic combustion problems in which physical quantities may vary by orders of magnitude over one tenth of a millimeter at atmospheric pressure or over micrometers at higher pressures the local rectangular refinement lrr method maintains orthogonality at grid line intersections but lifts the tensor product restriction common to traditional grids producing unstructured grids governing equations are discretized throughout the domain using newly derived forms and newton s method is used to solve the resulting system on a simple test case with a known solution the lrr method and its new discretizations are found to be more accurate than gridding methods representative of those appearing previously in the literature for the more realistic problem of nonreacting driven square cavity flow the lrr solution agrees very well with previously published data when the lrr method is applied to a practical reacting flow a rich axisymmetric laminar bunsen flame with complex chemistry multicomponent transport and an optically thin radiation submodel grid spacing highly influences the inner flame s position which stabilizes only with adequate refinement the vorticity velocity formulation of the governing equations is shown to produce valid results when used in conjunction with the lrr gridding technique furthermore each lrr grid is used to form a nonuniform equivalent tensor product etp grid and also in most cases an equispaced fully refined fr grid these additional grids are supersets of the lrr grids and thus contain refinement in exactly the same regions performance comparisons between the lrr etp and fr grids indicate that the lrr method provides substantial savings in execution time and computer memory requirements without compromising solution accuracy

method of lyapunov functions for differential equations with piecewise constant delay we address differential equations with piecewise constant argument of generalized type 5 8 and investigate their stability with the second lyapunov method despite the fact that these equations include delay stability conditions are merely given in terms of lyapunov functions that is no functionals are used several examples one of which considers the logistic equation are discussed to illustrate the development of the theory some of the results were announced at the 14th international congress on computational and applied mathematics iccam2009 antalya turkey in 2009

regularity results for a class of obstacle problems with nonstandard growth abstract we consider the obstacle problem related to the following energy with nonstandard growth d u p x log e d u d x we investigate the regularity properties of solutions to the obstacle problems along with a suitable assumptions on the variable exponent p and the obstacle

on quasi periodic solutions of differential equations with piecewise constant argument we study the existence of quasi periodic solutions to differential equations with piecewise constant argument epca for short it is shown that epca with periodic perturbations possess a quasi periodic solution and no periodic solution the appearance of quasi periodic rather than periodic solutions is due to the piecewise constant argument this new phenomenon illustrates a crucial difference between ode and epca the results are extended to nonlinear equations

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

regularity results for a new class of functionals with non standard growth conditions we prove that local minimizers of the functional f u du p x log e du dx are of class c0 for every 0 1 has logarithmic modulus of continuity moreover in case the exponent p x 1 is a holder continuous function we establish that minimizers of f u are of class c1 for some 0 1

shepard s interpolation for solution adaptive methods in numerical simulations of fluid dynamics problems solution adaptive methods have proven to be very powerful the implementation of the modified shepard s interpolation to the structured grids used in cfd is suggested and described in this paper which takes advantage of the logical grid structure this technique which is demonstrated to be robust efficient smoother and a more accurate alternative to linear interpolation is used in the remapping step during the solution adaption procedure applications to the solutions of the incompressible navier stokes equations both 2d and 3d are included

integral manifolds of differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a general type of differential equations with piecewise constant argument epcag the existence of global integral manifolds of the quasilinear epcag is established when the associated linear homogeneous system has an exponential dichotomy the smoothness of the manifolds is investigated the existence of bounded and periodic solutions is considered a new technique of investigation of equations with piecewise argument based on an integral representation formula is proposed appropriate illustrating examples are given

stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type this paper is concerned with the numerical properties of q methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments using two q methods namely the one leg q method and the linear q method the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained and the conditions of oscillations for the q methods are also obtained it is proved that oscillations of the analytic solution are preserved by the q methods furthermore the relationships between stability and oscillations are revealed some numerical experiments are presented to illustrate our results

pseudo almost periodic solutions for equation with piecewise constant argument abstract by using the roughness theory of exponential dichotomies and the contraction mapping some sufficient conditions are obtained for the existence and uniqueness of pseudo almost periodic solution of the above differential equation with piecewise constant argument d x d t a t x t j 0 r a j t x t j g t x t x t x t r