what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

a decomposition theorem for bounded solutions and the existence of periodic solutions of periodic differential equations abstract we prove a decomposition theorem for bounded uniformly continuous mild solutions to periodic evolution equations of the form dx dt a t x f t with in general unbounded periodic a periodic f and compact monodromy operator by this theorem every bounded uniformly continuous mild solution to is a sum of a periodic solution to and a quasi periodic solution to its homogeneous equation an analog of this for bounded solutions has been proved for abstract functional differential equations dx dt ax f t x t f t with finite delay where a generates a compact semigroup as an immediate consequence the existence of such a solution implies the existence of a periodic solution to the inhomogeneous equation as well as a formula for its fourier coefficients this even for the classical case of equations improves considerably the previous results on the subject

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

error estimates of a spectral petrov galerkin method for two sided fractional reaction diffusion equations abstract we study regularity and the spectral method for two sided fractional diffusion equations with a reaction term we show that the regularity of the solution in weighted sobolev spaces can be greatly improved compared to that in standard sobolev spaces with this regularity we prove an optimal error estimate for the spectral petrov galerkin method numerical results are presented to verify our theoretical convergence orders

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

a generalized spectral collocation method with tunable accuracy for variable order fractional differential equations we generalize existing jacobi gauss lobatto collocation methods for variable order fractional differential equations using a singular approximation basis in terms of weighted jacobi polynomials of the form 1 pm x mu p j a b x where mu 1 in order to derive the differentiation matrices of the variable order fractional derivatives we develop a three term recurrence relation for both integrals and derivatives of these weighted jacobi polynomials hence extending the three term recurrence relationship of jacobi polynomials the new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter mu even for cases where we do not know explicitly the form of singularity in the solution at the boundaries

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

periodic solutions for p laplacian neutral rayleigh equation with a deviating argument abstract by using topological degree theory and some analytical skill some criteria to guarantee the existence of periodic solutions are derived for p laplacian neutral rayleigh equation with a deviating argument of the following form p x t c x t f x t g x t t e t

evolution semigroups and sums of commuting operators a new approach to the admissibility theory of function spaces this paper is concerned with conditions for the admissibility of a translation invariant function space m with respect to a well posed linear evolution equation du dt au f t t r we propose a new approach to this problem by considering the sum of two commuting operators d dt dm and the operator of multiplication by a on m on the one hand the closure of this operator is the infinitesimal generator of the so called evolution semigroup associated with on the other hand the generator g of this semigroup relates a mild solution u of to the forcing term f by the rule gu f consequently various spectral criteria of the type dm a for the admissibility of the function space m with respect to can be proved in an elegant manner moreover they can be naturally extended to general classes of differential equations including higher order and abstract functional differential equations applications and examples are provided to illustrate the obtained results

stepanov like pseudo almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay abstract in this paper we prove the invariance of stepanov like pseudo almost periodic functions under bounded linear operators furthermore we obtain existence and uniqueness theorems of pseudo almost periodic mild solutions to evolution equations u t a t u t h t and u t a t u t f t b u t t c t s u s d s f t on r assuming that a t satisfy acquistapace terreni conditions that the evolution family generated by a t has exponential dichotomy that r 0 a is almost periodic that b c t s t s are bounded linear operators that f is lipschitz with respect to the second argument uniformly in the first argument and that h f f are stepanov like pseudo almost periodic for p 1 and continuous to illustrate our abstract result a concrete example is given

differentiability of solutions with respect to parameters in neutral differential equations with state dependent delays in this paper we consider a class of nonlinear neutral differential equations with state dependent delays in both the neutral and the retarded terms we study well posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite dimensional parameters in a pointwise sense and also in the c norm

two periodic solutions of neutral difference systems depending on two parameters in this paper we establish existence multiplicity and nonexistence of periodic solutions for a class of first order neutral difference systems our approach is based on a fixed point theorem in cones as well as some analysis techniques

periodic solutions for a kind of second order neutral differential systems with deviating arguments by means of the generalized continuation theorem of coincidence degree theory a kind of second order neutral differential systems with deviating arguments as followsd 2dt 2 x t cx t r ddtgradf x t gradg x email protected t p t is studied a new result on the existence of periodic solutions is obtained which relates to the deviating arguments t t and r

positive periodic solutions of periodic neutral lotka volterra system with state dependent delays by using a fixed point theorem of strict set contraction some new criteria are established for the existence of positive periodic solutions of the following periodic neutral lotka volterra system with state dependent delays dxi t dt xi t ri t j 1naij t xj t j 1nbij t xj t ij t x1 t xn t j 1ncij t xj t ij t x1 t xn t i 1 2 n where ri aij bij cij c r r i j 1 2 n are periodic functions and ij ij c rn 1 r i 1 2 n are periodic functions with respect to their first arguments respectively

periodic solution of neutral lotka volterra system with periodic delays abstract a nonautonomous n species lotka volterra system with neutral delays is investigated a set of verifiable sufficient conditions is derived for the existence of at least one strictly positive periodic solution of this lotka volterra system by applying an existence theorem and some analysis techniques where the assumptions of the existence theorem are different from that of gaines and mawhin s continuation theorem r e gaines j l mawhin coincidence degree and nonlinear differential equations springer verlag berlin 1977 and that of abstract continuation theory for k set contraction w petryshyn z yu existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems nonlinear anal 6 1982 943 969 moreover a problem proposed by freedman and wu h i freedman j wu periodic solution of single species models with periodic delay siam j math anal 23 1992 689 701 is answered

periodic solutions of linear and quasilinear neutral functional differential equations abstract in this paper we will use the fourier expansions to generalize hale s alternative results for the existence of periodic solutions of linear retarded functional differential equations to neutral equations using some a priori estimation and the leray schauder degree theory we will obtain some existence theorems of periodic solutions for a class of quasilinear neutral equations the results also generalize the works of hatvani and krisztin on retarded equations

a composite coincidence degree with applications to boundary value problems of neutral equations we present a topological degree theory for the nonlinear problem l i b x g x with applications to a class of boundary value problems of neutral equations where l is an unbounded fredholm operator of index zero b is condensing and g is l compact

periodic and almost periodic solutions of volterra integral differential equations with infinite memory abstract conditions are given which guarantee that if t 0 is sufficiently small then x t 0 de s x t s f t has a unique t periodic solution x for each continuous t periodic function f the vectors x and f are n dimensional the matrix function e s is n n with bounded total variation the proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non zero fourier frequencies are bounded away from zero the results are applied to study certain perturbations of the above equation

a fast finite volume method for conservative space time fractional diffusion equations discretized on space time locally refined meshes abstract a fast finite volume method was developed for conservative space time fractional diffusion partial differential equations in two space dimensions in the method a locally refined composite spatial mesh is used to resolve the boundary layers of the solutions and a graded temporal mesh is used to resolve the singularity of the solutions near the initial time a fast numerical method was developed by fully exploring the structure of the stiffness matrices numerical results show the utility of the method

periodic solutions for generalized lienard neutral equation with variable parameter abstract by means of mawhin s continuation theorem existence criteria is established for the periodic solutions to a neutral equation with variable parameter it is worth stating that the coefficient c is not a constant which is different from the corresponding ones of past work

on positive periodic solutions of a nonautonomous neutral delay n species competitive system abstract the purpose of this paper is to investigate a general neutral delay n species competitive nonautonomous system different from gaines and mawhin s continuation theorem r e gaines j l mawhin coincidence degree and nonlinear differential equations springer verlag berlin 1977 and abstract continuation theory for k set contraction w petryshyn z yu existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems nonlinear anal 6 1982 943 969 r e gaines j l mawhin lecture notes in mathematics vol 586 springer verlag berlin 1977 a new existence theorem is introduced with the help of the theorem and some techniques a set of sufficient conditions are derived for the existence of at least one strictly positive componentwise periodic solution as an application we also examine some special cases which have been studied extensively in the literature some known results are improved and generalized

positive periodic solutions of higher dimensional functional difference equations with a parameter by using krasnoselskii s fixed point theorem and upper and lower solutions method we find some sets of positive values determining that there exist positive t periodic solutions to the higher dimensional functional difference equations of the form x n 1 a n x n h n fxn n n z where a n diag a1 n a2 n am n h n diag h1 n h2 n hm n aj hj z r z z are t periodic j 1 2 m t 1 0 x z rm f r m r m where r m x1 xm t rm xj 0 j 1 2 m r x r x 0

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

massera problem for non autonomous retarded differential equations abstract in this paper under acquistapace terreni conditions we study the existence of periodic solutions for some non autonomous semilinear partial functional differential equation with delay we assume that the linear part is non densely defined the delayed part is assumed to be periodic with respect to the first argument using a fixed point theorem for multivalued mapping some sufficient conditions are given to prove the existence of periodic solutions an example is shown to illustrate our results

periodic solutions to functional differential equations on considere un systeme d equations differentielles fonctionnelles x g t x t ou g r b r n est t periodiquement et b est un espace de phase de fonctions continues qui appliquent 0 dans r n on introduit les concepts de limitation b uniforme et de limitation ultime b uniforme on donne des conditions suffisantes a l existence d une solution t periodique

a variation of constants formula for abstract functional differential equations in the phase space abstract for linear functional differential equations with infinite delay in a banach space a variation of constants formula is established in the phase space as an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line

lie and conditional symmetries of the three component diffusive lotka volterra system lie and q conditional symmetries of the classical three component diffusive lotka volterra system in the case of one space variable are studied the group classification problems for finding lie symmetries and q conditional symmetries of the first type are completely solved notably non lie symmetries q conditional symmetry operators for a multi component nonlinear reaction diffusion system are constructed for the first time the results are compared with those derived for the two component diffusive lotka volterra system the conditional symmetry obtained for the non lie reduction of the three component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found particularly the exact solution describing different scenarios of competition between three species is constructed

spectral approximation of a variable coefficient fractional diffusion equation in one space dimension abstract in this article we consider the approximation of a variable coefficient two sided fractional diffusion equation fde having unknown u by introducing an intermediate unknown q the variable coefficient fde is rewritten as a lower order constant coefficient fde a spectral approximation scheme using jacobi polynomials is presented for the approximation of q q n the approximate solution to u u n is obtained by post processing q n an a priori error analysis is given for q q n and u u n two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates

crank nicolson galerkin spectral method for solving two dimensional time space distributed order weakly singular integro partial differential equation abstract the fractional pdes based upon the distributed order fractional derivative have several applications in physics the two dimensional time space distributed order weakly singular integro partial differential model is investigated by a combination of finite difference and galerkin spectral methods a second order finite difference formula is employed to approximate the temporal variable in this stage the stability and convergence of the semi discrete scheme are proved then the galerkin spectral method based on the modified jacobi polynomials is applied to discrete the space variable also in this step the error estimate of the full discrete scheme is studied finally two test problems have been presented to confirm the theoretical results

stepanov like pseudo almost automorphic mild solutions to nonautonomous evolution equations abstract we obtain new existence and uniqueness theorems of pseudo almost automorphic mild solutions to several nonautonomous evolution equations with stepanov like pseudo almost automorphic terms two explanation examples are given

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

a remarkable class of non classical symmetries of the steady two dimensional boundary layer equations we study the steady two dimensional boundary layer equations in the flat and axisymmetric case to show that some similarity reductions already found in the literature with ad hoc new methods of reduction are indeed invariant solutions under the action of non classical symmetries as introduced by bluman and cole moreover we show that the celebrated von mises transformation that reduces the two dimensional flat boundary layer equations to a second order evolution equation is indeed a backlund transformation related to a non classical symmetry

complete guiding sets for a class of almost periodic differential equations abstract a functional differential equation admitting a complete guiding set always has an a priori bounded solution the introduction of this concept and the idea of reducibility allow the authors to apply a well known method for the autonomous case to a non autonomous problem a previous result of ortega and tineo is extended to a non hyperbolic semilinear functional differential equation with almost periodic coefficients in its lineal part the non hyperbolicity condition is established in terms of the sacker sell spectrum of the associated linear system

two periodic solutions of second order neutral functional differential equations in this paper we consider a type of second order neutral functional differential equations we obtain some existence results of multiplicity and nonexistence of positive periodic solutions our approach is based on a fixed point theorem in cones

periodic solutions of abstract neutral functional differential equations abstract we characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a umd space

periodic solutions to a second order p laplacian neutral functional differential system abstract by analyzing some properties of the linear difference operator a a x t x t c x t first and then using an extension of mawhin s continuation theorem a second order p laplacian neutral functional differential system as follows d d t p x t c x t f t x t x t t x t is studied some new results on the existence of periodic solutions is obtained the result is related to the deviating arguments and meanwhile the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature

on the existence of periodic solutions to rayleigh differential equation of neutral type in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t c x t f x t g t x t t e t in the critical case c 1 by analyzing some properties of the linear difference operator a a x t x t c x t and using mawhin s continuation theorem some new results are obtained

existence of periodic solutions for a p laplacian neutral functional differential equation abstract in this paper the author study the existence of periodic solutions for a p laplacian neutral functional differential equation as follows p u t j 1 n c j u t r j f u t u t t g u t j 1 n j t g u t j t p t by analyzing some properties of operator a c t c t a x t x t i 1 n c i x t r i and then using continuation theorem of coincidence degree theory developed by mawhin some new results on the existence of periodic solutions are obtained the example shows that the results of this paper are more general and easily applicable and improve and generalize some corresponding ones of the known literature

the existence uniqueness and global attractivity of periodic solution for a type of neutral functional differential system with delays in this article by applying the coincidence degree theory and constructing a suitable lyapunov functional the authors study a kind of neutral functional differential system with delays some sufficient conditions which guarantee the existence uniqueness and global attractivity of periodic solution for this system are obtained

a petrov galerkin finite element method for fractional convection diffusion equations in this work we develop variational formulations of petrov galerkin type for one dimensional fractional boundary value problems involving either a riemann liouville or caputo derivative of order alpha in 3 2 2 in the leading term and both convection and potential terms they arise in the mathematical modeling of asymmetric superdiffusion processes in heterogeneous media the well posedness of the formulations and sharp regularity pickup of the variational solutions are established a novel finite element method fem is developed which employs continuous piecewise linear finite elements and shifted fractional powers for the trial and test space respectively the new approach has a number of distinct features it allows the derivation of optimal error estimates in both the l 2 d and h 1 d norms and on a uniform mesh the stiffness matrix of the leading term is diagonal and the resulting linear system is well conditioned further in the riemann liouville case an enriched fem is propose

symmetries and heir equations for the laminar boundary layer model iterations of the nonclassical symmetries method applied to both the prandtl equation and prandtl system give rise to new nonlinear equations which inherit the lie symmetry algebra of the original prandtl equation system invariant solutions of these heir equations supply new solutions of the original equation system

adaptive finite element method for fractional differential equations using hierarchical matrices abstract a robust and fast solver for the fractional differential equation fdes involving the riesz fractional derivative is developed using an adaptive finite element method it is based on the utilization of hierarchical matrices h matrices for the representation of the stiffness matrix resulting from the finite element discretization of the fdes we employ a geometric multigrid method for the solution of the algebraic system of equations we combine it with an adaptive algorithm based on a posteriori error estimation a posteriori error estimation based adaptive algorithm is used to deal with general type singularities arising in the solution of the fdes through various test examples we demonstrate the efficiency of the method and the high accuracy of the numerical solution even in the presence of singularities the proposed technique has been verified effectively through fundamental examples including riesz left right riemann liouville fractional derivative and furthermore it can be readily extended to more general fractional differential equations with different boundary conditions and low order terms

existence and asymptotic stability of periodic solution for evolution equations with delays abstract in this paper we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a hilbert space h u t a u t f t u t u t 1 u t n t r where a d a h h is a positive definite selfadjoint operator f r h n 1 h is a nonlinear mapping which is periodic in t and 1 2 n are positive constants we present essential conditions on the nonlinearity f to guarantee that the equation has periodic solutions or an asymptotically stable periodic solution the discussion is based on analytic semigroups theory and an integral inequality with delays

existence of periodic solutions for a p laplacian neutral functional differential equation with multiple variable parameters abstract by means of mawhin s continuation theorem a kind of p laplacian neutral functional differential equation with multiple variable parameters as follows p u t i 1 n c i t u t r i f u t u t t g u t j 1 n j t g u t j t e t is studied the interesting thing is that the coefficient c is not a constant by analyzing some properties of operator a c t c t a x t x t i 1 n c i t x t r i and then using mawhin s continuation theorem some new results on the existence of periodic solutions are obtained

a generalized spectral collocation method with tunable accuracy for fractional differential equations with end point singularities we develop spectral collocation methods for fractional differential equations with variable order with two end point singularities specifically we derive three term recurrence relations for both integrals and derivatives of the weighted jacobi polynomials of the form 1 x mu 1 1 x mu 2 p j a b x a b mu 1 mu 2 1 which leads to the desired differentiation matrices we apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end point singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters mu 1 and mu 2 even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method

periodic solutions of neutral differential equation with multiple deviating arguments in this paper by means of an abstract continuous theorem of k set contractive operator and some analysis techniques the authors study the existence of periodic solution of the nonlinear neutral differential equation with multiple deviating argumentsddt x t cx email protected email protected i 1 email protected i t f x email protected i t email protected j 1 email protected j t g x email protected j t p t a new result is obtained

schaefer type theorem and periodic solutions of evolution equations abstract two fixed point theorems for the sum of contractive and compact operators are obtained in this paper which generalize and improve the corresponding results in h schaefer uber die methode der a priori schranken math ann 129 1955 415 416 t a burton integral equations implicit functions and fixed points proc amer math soc 124 1996 2383 2390 v i istr tescu fixed point theory an introduction reidel dordrecht 1981 t a burton k colleen a fixed point theorem of krasnoselskii schaefer type math nachr 189 1998 23 31 d r smart fixed point theorems cambridge univ press cambridge 1980 as the applications for the results we obtain the existence of periodic solutions for some evolution equations with delay which extend the corresponding results in t a burton b zhang periodic solutions of abstract differential equations with infinite delay j differential equations 90 1991 357 396

periodic solutions for some nonlinear differential equations of neutral type where f ir x ir ir is a caratheodory function f x is 2n periodic for every x ci e r and r e io 2n this problem has been studied by sadovskii in 151 by hale and mawhin in 6 in the case ial 1 to the previous one our interest is particularly motivated by the fact that represents a simple and meaningful model for a general phenomenon which often arises in the study of functional differential equations in fact as is well known in some cases a delayed differential equation can be seen as a perturbation of an ordinary differential equation in particular this occurs when the delayed term in the equation can be considered smaller than the ordinary one or dominated by it when such domination is available many good properties can be proved and several results such as those of 6 and 16 take advantage of this fact on the contrary when no domination is present the situation is much more complicated and in general very little is known in problem the delayed term can be considered dominated by the ordinary one whenever ial 1 as a matter of fact in this case one can prove the invertibility of the differential operator l x x a r ux the compactness of its inverse and even some a priori estimates for the solutions consequently one can reduce to a fixed point problem and solve it by means of some topological degree argument however when 11 1 so that the weight of the two terms x t and x t t is the same the above properties fail and moreover for some delays the kernel of the differential operator lx x a r is infinite dimensional therefore one must look for different methods problem is thus a simple but nevertheless significant example of the considerations illustrated by hale in 5 about the difficulty of undertaking a unified approach in the study of functional differential equations at the same time it provides an analogy between neutral equations and the wave equation in the line suggested by hale in 5

a crank nicolson adi galerkin legendre spectral method for the two dimensional riesz space distributed order advection diffusion equation abstract in the paper a crank nicolson alternating direction implicit adi galerkin legendre spectral scheme is presented for the two dimensional riesz space distributed order advection diffusion equation the gauss quadrature has a higher computational accuracy than the mid point quadrature rule which is proposed to approximate the distributed order riesz space derivative so that the considered equation is transformed into a multi term fractional equation moreover the transformed equation is solved by discretizing in space by the adi galerkin legendre spectral scheme and in time using the crank nicolson difference method stability and convergence analysis are verified for the numerical approximation a lot of numerical results are demonstrated to justify the theoretical analysis

linearized stability for a class of neutral functional differential equations with state dependent delays abstract in this paper we formulate a stability theorem by means of linearization around a trivial solution in the case of autonomous neutral functional differential equations with state dependent delays we prove that if the trivial solution of the linearized equation is exponentially stable then the trivial solution of the nonlinear equation is exponentially stable as well as an application of the main result explicit stability conditions are given

boundedness and almost periodicity of solutions of partial functional differential equations abstract we study necessary and sufficient conditions for the abstract functional differential equation x ax fx t f t to have almost periodic quasi periodic solutions with the same structure of spectrum as f the main conditions are stated in terms of the imaginary solutions of the associated characteristic equations and the spectrum of the forcing term f the obtained results extend recent results to abstract functional differential equations

a spectral method of exponential convergence for singular solutions of the diffusion equation with general two sided fractional derivative an open problem in the numerical solution of fractional partial differential equations fpdes is how to obtain high order accuracy for singular solutions even for smooth right hand sides solutions of fpdes are singular here we consider the one dimensional diffusion equation with general two sided fractional derivative characterized by a parameter p in 0 1 for p 1 2 we recover the riesz fractional derivative while for p 1 0 we obtain the one sided fractional derivative we employ a petrov galerkin projection in a properly weighted sobolev space with two sided jacobi polyfracnomials as basis and test functions in particular we derive these two sided jacobi polyfractonomials as eigenfunctions of a sturm liouville problem with weights uniquely determined by the parameter p we provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term i e for smooth data corresponding to singular solutions we obtain exponential convergence wh

periodic solutions to neutral differential equation with deviating arguments by employing the continuation theorem of coincidence degree theory and some analysis techniques we study a kind of neutral differential equation with deviating arguments as follows x t cx t r f x t g x t t p t some new results on the existence of periodic solutions are obtained our result generalizes main work of wang app math lett 12 1999 41 44

existence and global attractivity of positive periodic solutions for delay lotka volterra competition patch systems with stocking abstract by means of mawhin s continuation theorem of coincidence degree and lyapunov functional a set of easily verifiable criteria are established for the existence and global attractivity of positive periodic solutions for delay lotka volterra competition patch system with stocking x 1 t h 1 t x 1 t a 1 t b 1 t x 1 t c 1 t y t d 1 t x 2 t 1 x 1 t s 1 t x 2 t h 2 t x 2 t a 2 t b 2 t x 2 t d 2 t x 1 t 2 x 2 t s 2 t y t h 3 t y t a 3 t b 3 t y t t 0 k s y t s ds c 3 t x 1 t s 3 t some known results are generalized and improved

on the existence of periodic solutions for neutral functional differential equation abstract by means of mawhin s continuation theorem existence criteria are established for the periodic solutions of neutral functional differential equation

periodic solutions to a kind of neutral functional differential equation in the critical case in this paper we consider a kind of neutral functional differential equation as follows x t cx t gt xt t e t in the critical case c 1 by employing the topological degree theory and some analysis techniques we obtain some new results on the existence of periodic solutions

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

periodicity on a class of neutral impulsive delay system we present some sufficient conditions for the existence of positive periodic solutions of a class of neutral impulsive delay differential system when our results applied to the differential system without impulse a known result in the literature is corrected moreover one example is given to illustrate the results obtained

efficient spectral galerkin methods for fractional partial differential equations with variable coefficients efficient spectral galerkin algorithms are developed to solve multi dimensional fractional elliptic equations with variable coefficients in conserved form as well as non conserved form these algorithms are extensions of the spectral galerkin algorithms for usual elliptic pdes developed in 24 more precisely for separable fpdes we construct a direct method by using a matrix diagonalization approach while for non separable fpdes we employ a preconditioned bicgstab method with a suitable separable fpde with constant coefficients as preconditioner the cost of these algorithms is of o n d 1 flops where d is the space dimension we derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries we also present ample numerical results to validate the algorithms and error estimates

periodic solutions for functional differential equations with infinite lead and delay abstract this paper discusses the existence of periodic solutions for functional differential equations with infinite lead and delay on the basis of fixed point theorems topological degree theories comparison principles and differential inequality methods some general results are proved

two periodic solutions of n dimensional neutral functional difference systems in this paper we establish existence multiplicity and nonexistence of periodic solutions for a class of first order n dimensional neutral functional difference systems our approach requires the krasnoselskii fixed point theorem in cones

positive periodic solutions for a class of delay differential equations abstract in this paper we employ fixed point theorem and functional equation theory to study the existence of positive periodic solutions of the delay differential equation x t t x t t x 2 t t x t t x t moreover new explicit conditions for the related population growth models are also given

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations abstract in this paper we use the coincidence degree theory to establish new results on the existence and uniqueness of t periodic solutions for the first order neutral functional differential equation of the form x t b x t g 1 t x t g 2 t x t p t

positive periodic solutions of infinite delay functional differential equations depending on a parameter in this paper we employ the upper and lower solutions method to study the existence of positive periodic solutions to a system of infinite delay equations with a email protected t a t x t email protected t x t

permanence and existence of a positive periodic solution to a periodic stage structured system with infinite delay in this paper we consider a periodic non autonomous competitive stage structured system with infinite delay for the interaction between n species the adult members of which are in competition for each of the n species the model incorporates a time delay which represents the time from birth to maturity of that species infinite delay is introduced which denotes the influential effect of the entire past history of the system on the current competition interactions we first prove by using the comparison principle that if the growth rates are sufficiently large then the solutions are uniformly permanent then by using horn s fixed point theorem we show that the system with finite delay has a positive periodic solution as a consequence of this result we prove that even the system with infinite delay admits a positive periodic solution

existence of periodic solutions for a pp laplacian neutral functional differential equation in this paper the author study the existence of periodic solutions for a pp laplacian neutral functional differential equation as follows p u t j 1ncju t rj f u t u t t g u t j 1n j t g u t j t p t by analyzing some properties of operator a ct ct ax t x t i 1ncix t ri and then using continuation theorem of coincidence degree theory developed by mawhin some new results on the existence of periodic solutions are obtained the example shows that the results of this paper are more general and easily applicable and improve and generalize some corresponding ones of the known literature

optimal convergence rates for semidiscrete finite element approximations of linear space fractional partial differential equations under minimal regularity assumptions abstract we consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by jin et al 2015 and ervin et al 2018 the error estimates are proved by using two approaches one approach is to apply the duality argument in johnson 1987 for the heat equation to consider the error estimates for the linear space fractional partial differential equations this argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution another approach is to use the approximate solution operators of the corresponding fractional elliptic problems this argument can be extended to consider more general linear space fractional partial differential equations numerical examples are given to show that the numerical results are consistent with the theoretical results

sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions this paper provides a rigorous and delicate analysis for exponential decay of jacobi polynomial expansions of analytic functions associated with the bernstein ellipse using an argument that can recover the best estimate for the chebyshev expansion we derive various new and sharp bounds of the expansion coefficients which are featured with explicit dependence of all related parameters and valid for degree n ge 1 we demonstrate the sharpness of the estimates by comparing with existing ones in particular the very recent results in siam j numer anal 50 2012 pp 1240 1263 we also extend this argument to estimate the gegenbauer gauss quadrature remainder of analytic functions which leads to some new tight bounds for quadrature errors

a fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation abstract we develop a fast solver for the spectral element method sem applied to the two sided fractional diffusion equation on uniform geometric and graded meshes by approximating the singular kernel with a degenerate kernel we construct a hierarchical matrix h matrix to represent the stiffness matrix of the sem and provide error estimates which we verify numerically we can solve efficiently the h matrix approximation problem using a hierarchical lu h lu decomposition method which reduces the computational cost to o r 2 n d log 2 n o r 3 n d log n where r is the rank of the h lu decomposition n d is the total number of degrees of freedom and n is the number of elements however we lose the high accuracy of the sem to prevent this we solve the corresponding preconditioned system by using the h matrix approximation problem as a preconditioner hence recovering the high order accuracy of the sem numerical results show that the condition number of the preconditioned system is independent of the polynomial degree p and grows with the number of elements but at modest values of the rank r here r is the rank of submatrices of the h matrix approximation it is below order 10 in our experiments which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system this reduction is higher in the two sided fractional derivative compared to one sided fractional derivative the corresponding cost is o r 2 n d log 2 n o r 3 n d log n o n d 2 moreover by using a structured mesh uniform or geometric mesh we can further reduce the computational cost to o r 2 n d log 2 n o r 3 n d log n o p 2 n log n for the preconditioned system furthermore we compare the present method with the adaptive cross approximation aca which is kernel independent and relies on the coefficient matrix the comparison shows that the accuracy using the present algorithm is better than the one using aca for the h matrix approximation problem and furthermore we have observed that the aca method may fail to converge in some cases