on the concept of general solution for impulsive differential equations of fractional order q 1 2 in this paper we provide the formulas of general solution for some impulsive differential equations of fractional order q 1 2

on solving systems of multi pantograph equations via spectral tau method abstract the current manuscript focuses on solving systems of multi pantograph equations the spectral tau method is applied for solving systems of multi pantograph equations with shifted jacobi polynomials as basis functions the convergence analysis of the proposed technique is also investigated we introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods

multiple positive solutions for a class of integral inclusions this paper deals with sufficient conditions for the existence of at least two positive solutions for a class of integral inclusions arising in the traffic theory to show our main results we apply a norm type expansion and compression fixed point theorem for multivalued map due to agarwal and o regan a note on the existence of multiple fixed points for multivalued maps with applications j differential equation 160 2000 389 403

splitting methods for the nonlocal fowler equation we consider a nonlocal scalar conservation law proposed by andrew c fowler to describe the dynamics of dunes and we develop a numerical procedure based on splitting methods to approximate its solutions we begin by proving the convergence of the well known lie formula which is an approximation of the exact solution of order one in time we next use the split step fourier method to approximate the continuous problem using the fast fourier transform and the finite difference method our numerical experiments confirm the theoretical results

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

a structure theorem for actions of semisimple lie groups we consider a connected semisimple lie group g with finite center an admissible probability measure on g and an ergodic g u space x v we first note lemma 0 1 that x v has a unique maximal projective factor of the form g q vo where q is a parabolic subgroup of g and then prove 1 theorem 1 if every noncompact simple factor of g has real rank at least two then the maximal projective factor is nontrivial unless v is a g

generic stationary measures and actions let g be a countably infinite group and let be a generating probability measure on g we study the space of stationary borel probability measures on a topological g space and in particular on z g where z is any perfect polish space we also study the space of stationary measurable g actions on a standard nonatomic probability space equip the space of stationary measures with the weak topology when has finite entropy we show that a generic measure is an essentially free extension of the poisson boundary of g when z is compact this implies that the simplex of stationary measures on z g is a poulsen simplex we show that this is also the case for the simplex of stationary measures on 0 1 g we furthermore show that if the action of g on its poisson boundary is essentially free then a generic measure is isomorphic to the poisson boundary next we consider the space of stationary actions equipped with a standard topology known as the weak topology here we show that when g has property t the ergodic actions are meager we also construct a group g without property t such that the ergodic actions are not dense for some finally for a weaker topology on the set of actions which we call the very weak topology we show that a dynamical property e g ergodicity is topologically generic if and only if it is generic in the space of measures there we also show a glasner king type 0 1 law stating that every dynamical property is either meager or residual

attractivity for functional volterra integral equations of convolution type in this paper we investigate the existence of attractive and uniformly locally attractive solutions for a functional nonlinear integral equation with a general kernel we use methods and techniques of fixed point theorems and properties of measure of noncompactness we extend and generalize results obtained by other authors in the context of fractional functional differential equations

solving groundwater flow problems by conjugate gradient methods and the strongly implicit procedure the performance of the preconditioned conjugate gradient method with three preconditioners is compared with the strongly implicit procedure sip using a scalar computer the preconditioners considered are the incomplete cholesky iccg and the modified incomplete cholesky miccg which require the same computer storage as sip as programmed for a problem with a symmetric matrix and a polynomial preconditioner polcg which requires less computer storage than sip although polcg is usually used on vector computers it is included here because of its small storage requirements in this paper published comparisons of the solvers are evaluated all four solvers are compared for the first time and new test cases are presented to provide a more complete basis by which the solvers can be judged for typical groundwater flow problems based on nine test cases the following conclusions are reached 1 sip is actually as efficient as iccg for some of the published linear two dimensional test cases that were reportedly solved much more efficiently by iccg 2 sip is more efficient than other published comparisons would indicate when common convergence criteria are used and 3 for problems that are three dimensional nonlinear or both and for which common convergence criteria are used sip is often more efficient than iccg and is sometimes more efficient than miccg

on the attractivity of solutions for a class of multi term fractional functional differential equations in this paper we present some alternative results concerning with the existence and attractivity dependence of solutions for a class of nonlinear fractional functional differential equations in our consideration we apply the well known schauder fixed point theorem in conjunction with the technique of measure of noncompactness moreover we provide some examples to illustrate the effectiveness of the obtained results

weakly singular gronwall inequalities and applications to fractional differential equations abstract we obtain some new gronwall type inequalities which are applicable to some weakly singular volterra integral equations similar to the ones first studied by d henry the main interest is that we consider cases with a double singularity and we obtain explicit l bounds rather than l 1 bounds furthermore our bounds involve the exponential function and not the mittag leffler function as in some previous works we give applications to some volterra integral equations with a doubly singular kernel that arise from caputo fractional differential equations where as opposed to previous papers we have a singularity in the nonlinearity

on integral equations of urysohn volterra type the existence of solutions to urysohn volterra equation in a locally convex topological hausdorff space has been established the schauder tychonoff theorem is the main tool in our analysis

study on existence of solutions for some nonlinear functional integral equations abstract using the technique of measures of noncompactness in banach algebra we employ abstract fixed point theorems such as darbo s theorem to study the existence solution in banach algebra c 0 a for some functional integral equations which include many key integral and functional equations that arise in nonlinear analysis

generalization of a retarded gronwall like inequality and its applications abstract this paper generalizes a lipovan s result of gronwall like inequalities j math anal appl 252 2000 389 to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals from our result bihari s result and pinto s result can be deduced as some special cases our result can be applied to give the global existence and estimate solutions we also apply it to improve an estimation for almost periodicity of an invariant manifold

on the solution of a nonlinear integral equation on the basis of a fixed point technique and cubic b spline scaling functions in this paper we apply the fixed point method to solve some nonlinear functional volterra integral equations which appear in many physical chemical and biological problems in each iteration of this method cubic semi orthogonal compactly supported b spline wavelets are used as basis functions to approximate the solution also the convergence of this numerical method is investigated and some examples are presented to show the accuracy and convergence of the method

on local attractivity and asymptotic stability of solutions of a quadratic volterra integral equation the existence of solutions of a nonlinear quadratic volterra integral equation is studied in our considerations we apply the technique of measures of noncompactness in conjunction with the classical schauder fixed point principle such an approach allows us to obtain a result on the existence of solutions of an equation in question which are uniformly locally attractive or asymptotically stable

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

a class of nonlinear non instantaneous impulsive differential equations involving parameters and fractional order abstract in this article we study asymptotic and smooth properties of solutions to nonlinear non instantaneous impulsive differential equations involving parameters of integer order and fractional order we introduce the concept of continuous dependence and differentiability of solutions and establish sufficient conditions to guarantee the solution depends continuously and is differentiable on the initial condition impulsive parameters and junction parameters finally two models of non instantaneous impulsive logistic equations are given to illustrate our results

existence of solutions for integral inclusions this paper presents sufficient conditions for the existence of solutions to a nonlinear integral inclusion being a volterra counterpart of an integral inclusion arising in traffic theory our results are obtained via a martelli type fixed point theorem and lead to new existence principles

on quadratic integral equations in orlicz spaces abstract in this paper we study the quadratic integral equation of the form x t g t x t a b k t s f s x s d s several existence theorems for a e monotonic solutions in orlicz spaces are proved for strongly nonlinear functions f the presented method of the proof can be easily extended to different classes of solutions

numerical method for volterra equation with a power type nonlinearity abstract in this work we prove that a family of explicit numerical methods is convergent when applied to a nonlinear volterra equation with a power type nonlinearity in that case the kernel is not of lipschitz type therefore the classical analysis cannot be utilized we indicate several difficulties that arise in the proofs and show how they can be remedied the tools that we use consist of variations on discreet gronwall s lemmas and comparison theorems additionally we give an upper bound on the convergence order we conclude the paper with a construction of a convergent method and apply it for solving some examples

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

a fixed point theorem for condensing operators and applications to hammerstein integral equations in banach spaces abstract we prove a fixed point theorem of furi pera type for condensing operators this will then be used to establish the existence of solutions to certain integral equations of hammerstein type in banach spaces

existence results for fractional order semilinear functional differential equations with nondense domain abstract in this paper we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the riemann liouville derivative our approach is based on integrated semigroup theory the banach contraction principle and the nonlinear alternative of leray schauder type

unstructured additive schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients this paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes the effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a a1 ad and then considering certain positive coefficient sequences a m in which some of these values approach 0 or infty as m rightarrow infty we consider the performance of additive schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid we show that the preconditioned conjugate gradient method converges in both the energy and the euclidean norms with a number of iterations which grows only logarithmically in the size of the maximum jump cal j m max a k m a l m k l 1 ldots d as m rightarrow infty the result is obtained by a careful analysis of the preconditioned matrix it cannot be obtained by the usual procedure of estimating condition numbers a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in cal j m recent results of chan smith and zou have shown that by using this preconditioner together with the conjugate gradient method the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh our results now show that this method is also highly resilient to discontinuous coefficients even if no attention is paid to the coefficient discontinuity in the construction of the solver

approximate controllability for fractional semilinear parabolic equations abstract in this paper we deal with the control systems described by a large class of fractional semilinear parabolic equations firstly we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate banach space secondly we introduce a suitable concept on a mild solution for this kind of fractional parabolic equations and present the existence and uniqueness of mild solution by utilizing the theory of semigroup of linear operator nonlinear analysis method and fixed point theorem then the approximate controllability of the fractional semilinear parabolic equations is formulated and proved at the end of the paper an example is given to illustrate our main results

fixed point theorems in the frechet space c r and functional integral equations on an unbounded interval abstract the purpose of this paper is to establish fixed point theorems for operators in the frechet space of continuous functions on the real half axis in our considerations we apply the technique of measures of noncompactness in conjunction with the tikchonov fixed point theorem the obtained results are applied in the proof of the solvability of a nonlinear functional integral equation with the initial value moreover we show that solutions of that equation are uniformly globally asymptotically attractive the results presented in the paper allow to improve existence theorems for integral and functional equations obtained earlier in many research papers

fixed point method for solving nonlinear quadratic volterra integral equations we consider a paper of banas and sadarangani 2008 11 which deals with monotonicity properties of the superposition operator and their applications an application of the monotonicity properties is to study the solvability of a quadratic volterra integral equation in this paper we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic volterra integral equation then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

on a measure of noncompactness in the holder space ck γ ω and its application abstract in this article based on the concept of relative compactness in the holder space c k where k 0 0 1 and is a compact subset of r n we present a new measure of noncompactness on c k moreover as an application we study the existence of solution of a nonlinear integral equation in the holder space by applying darbo fixed point theorem

exploring delayed mittag leffler type matrix functions to study finite time stability of fractional delay differential equations abstract in this paper we introduce a concept of delayed two parameters mittag leffler type matrix function which is an extension of the classical mittag leffler matrix function with the help of the delayed two parameters mittag leffler type matrix function we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations via the variation of constants method in addition we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations thereafter we present finite time stability results of nonlinear fractional delay differential equations under mild conditions on nonlinear term finally an example is presented to illustrate the validity of the main theorems

on the concept of general solution for impulsive differential equations of fractional order q 0 1 in this paper for impulsive differential equations with fractional order q 0 1 we show that the formula of solutions in cited papers are incorrect secondly we find out a formula of the general solution for impulsive cauchy problem with caputo fractional derivative q 0 1 further for a kind of impulsive fractional differential equations system with special initial value we come to an existence result for it by applying fixed point methods

a new approach for solving integro differential equations of variable order abstract in this work we consider a class of nonlinear integro differential equations of variable order existence uniqueness and stability results are discussed for solving the considered equations operational matrices based on the shifted legendre polynomials are used first we approximate the unknown function and its derivatives in terms of the shifted legendre polynomials then by substituting these approximations into the equation and using the properties of the shifted legendre polynomials together with the collocation points the main problem is reduced to a system of nonlinear algebraic equations an error bound is proved for the approximate solution obtained by the proposed method finally some illustrative examples are included to show the efficiency and accuracy of the proposed scheme

feedback control for fractional impulsive evolution systems this paper is mainly concerned with feedback control systems governed by fractional impulsive evolution equations involving riemann liouville derivatives in reflexive banach spaces we firstly give an existence and uniqueness result of mild solutions for the equations by applying the banach s fixed point theorem next by using the filippove theorem and the cesari property a new set of sufficient assumptions are formulated to guarantee the existence result of feasible pairs we also present an existence result of optimal control pairs for the lagrange problem

a darbo fixed point theory approach towards the existence of a functional integral equation in a banach algebra abstract this paper investigates the nonlinear functional integral equation comprising of hadamard fractional operator using the concept of measure of non compactness existence of solutions in banach algebra has been studied under certain relevant assumptions in conjunction with fixed point theory finally an example has been considered to substantiate the validity of the result

on some measures of noncompactness in the frechet spaces of continuous functions abstract in this paper the frechet spaces of continuous functions defined on a bounded or an unbounded interval with values in the space of all real sequences are considered for those frechet spaces new regular measures of noncompactness are constructed and several properties of these measures are established the results obtained are further applied to infinite systems of functional integral equations

investigation on the existence of solutions for some nonlinear functional integral equations abstract we prove an existence theorem for a nonlinear integral equation which is a volterra counterpart of an integral equation arising in traffic theory using the technique of the measure of noncompactness in banach algebra we prove an existence theorem for a nonlinear functional integral equation basic fixed point theorems such as darbo s theorem are employed to obtain the aforementioned aims in banach algebra

on existence and asymptotic stability of solutions of a nonlinear integral equation we prove an existence theorem for a nonlinear integral equation being a volterra counterpart of an integral equation arising in the traffic theory the method used in the proof allows us to obtain additional characterization in terms of asymptotic stability of solutions of an equation in question

on the oscillatory behavior of solutions of nonlinear fractional differential equations the study of oscillation theory for fractional differential equations has been initiated by grace et al 14 in this paper we establish some new criteria for the oscillation of fractional differential equations with the caputo derivative of the form c d a x t e t f t x t a 1 1 2 we also present the conditions under which all solutions of this equation are asymptotic to at b as t for some real numbers a b we shall employ a different technique rather than that in 14

nonexistence of periodic solutions and s asymptotically periodic solutions in fractional difference equations nonexistence of periodic solution is proved for periodic right hand side existence of s asymptotically periodic solutions is discussed existence and uniqueness results are obtained for s asymptotically periodic solution nontrivial estimations of a ratio of gamma functions are used in this paper it is shown that a fractional difference equation with a periodic right hand side can not possess a periodic solution but it can have an s asymptotically periodic solution sufficient conditions are proved for the existence of a unique s asymptotically periodic solution an example is also given illustrating the obtained results

qualitative analysis for solutions of a certain more generalized two dimensional fractional differential system with hadamard derivative based on an elementary inequality and two gronwall bellman type integral inequality systems this paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain fractional systems with hadamard derivative established in a recent paper by the authors the major generalizations come from linearization method and de singular approach after overcoming some difficulties in the hadamard type singular kernel

on existence and local attractivity of solutions of a quadratic volterra integral equation of fractional order in the paper we study the existence of solutions of a nonlinear quadratic volterra integral equation of fractional order this equation is considered in the banach space of real functions defined continuous and bounded on an unbounded interval moreover we show that solutions of this integral equation are locally attractive

the numerical solution of nonlinear two dimensional volterra fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis in this paper we present a numerical method for solving two dimensional nonlinear volterra fredholm integral equations of the second kind the method approximates the solution by the discrete collocation method based on radial basis functions rbfs constructed on a set of disordered data the proposed method is meshless since it does not require any background mesh or domain elements error analysis of this method is also investigated numerical examples which compare the proposed method with 2d tfs method 4 approve its supremacy in terms of accuracy and computational cost using various rbfs we have concluded that mq rbf is the best choice for the proposed method

integral inequalities of systems and the estimate for solutions of certain nonlinear two dimensional fractional differential systems this paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain two dimensional fractional differential systems established in a recent paper of the authors the main generalizations come from an elementary inequality and by means of the modification of medved s de singular approach

an inverse time dependent source problem for a time space fractional diffusion equation abstract this paper is devoted to identify a time dependent source term in a time space fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point the existence and uniqueness of a weak solution for the corresponding direct problem with homogeneous dirichlet boundary condition are proved we provide the uniqueness and a stability estimate for the inverse time dependent source problem based on the separation of variables we transform the inverse source problem into a first kind volterra integral equation with the source term as the unknown function and then show the ill posedness of the problem further we use a boundary element method combined with a generalized tikhonov regularization to solve the volterra integral equation of the fist kind the generalized cross validation rule for the choice of regularization parameter is applied to obtain a stable numerical approximation to the time dependent source term numerical experiments for six examples in one dimensional and two dimensional cases show that our proposed method is effective and stable

a boundary condition capturing multigrid approach to irregular boundary problems we propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions in this method we adopt a matrix free approach i e we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices as many other robust multigrid methods do the main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function numerical results are given to compare our multigrid method with black box and algebraic multigrid methods

domain decomposition for multiscale pdes we consider additive schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic pdes with highly variable coefficients in contrast to standard analyses we do not assume that the coefficients can be resolved by a coarse mesh this situation arises often in practice for example in the computation of flows in heterogeneous porous media in both the deterministic and monte carlo simulated stochastic cases we consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid we perform a new analysis of the preconditioned matrix which shows rather explicitly how its condition number depends on the variable coefficient in the pde as well as on the coarse mesh and overlap parameters the classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements by contrast our new results show that with a good choice of subdomains and coarse space basis functions the preconditioner can still be robust even for large coefficient variation inside domains when the classical method fails to be robust in particular our estimates prove very precisely the previously made empirical observation that the use of low energy coarse spaces can lead to robust preconditioners we go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary i e two scale media model problems moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case we show also how the analysis extends in a straightforward way to multiplicative versions of the schwarz method

the poisson formula for groups with hyperbolic properties the poisson boundary of a group g with a probability measure is the space of ergodic components of the time shift in the path space of the associated random walk via a generalization of the classical poisson formula it gives an integral representation of bounded harmonic functions on g in this paper we develop a new method of identifying the poisson boundary based on entropy estimates for conditional random walks it leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the poisson boundary with natural topological boundaries for several classes of groups word hyperbolic groups and discontinuous groups of isometries of gromov hyperbolic spaces groups with inflnitely many ends cocompact lattices in cartan hadamard manifolds discrete subgroups of semisimple lie groups

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

approximate controllability of fractional evolution systems with riemann liouville fractional derivatives in this paper we deal with the control systems governed by fractional evolution differential equations involving riemann liouville fractional derivatives in banach spaces our main purpose in this article is to establish suitable assumptions to guarantee the existence and uniqueness results of mild solutions under these conditions the approximate controllability of the associated fractional evolution systems involving riemann liouville fractional derivatives is formulated and proved

convergence and stability of compact finite difference method for nonlinear time fractional reaction diffusion equations with delay abstract this paper is concerned with numerical solutions of nonlinear time fractional reaction diffusion equations with time delay a linearized compact finite difference scheme is proposed to solve the equations in terms of a new developed fractional gronwall type inequality convergence and stability of the proposed scheme are obtained numerical experiments are given to illustrate the theoretical results

existence results for nonlinear integral equations the schauder tychonoff theorem in locally convex topological hausdorff spaces and a nonlinear alternative of leray schauder type are used to establish some existence results for volterra and hammerstein integral equations

existence theory for nonlinear fredholm and volterra integral equations on half open intervals in this chapter we present existence theory for the nonlinear fredholm integral equation y t h t smallint 0 t k t s g s y s ds 4 1 1 and the nonlinear volterra integral equation y t h t smallint 0 t k t s g s y s ds 4 1 2 when both are defined on the compact interval 0 t naturally we first concern ourselves with existence principles for both equations

fractional order integral equations of two independent variables in this paper we present some results concerning the existence the uniqueness and the attractivity of solutions for some functional integral equations of riemann liouville fractional order by using some fixed point theorems

hybrid fixed points of multivalued operators in metric spaces with applications abstract this paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space our claim is also illustrated with applications to a class of integral inclusions for proving the existence results

application of simulation function and measure of noncompactness for solvability of nonlinear functional integral equations and introduction to an iteration algorithm to find solution abstract the aim of this article is to investigate the existence of solution for the nonlinear functional integral equations with the help of measure of noncompactness simulation function and generalized darbo fixed point theorem also we provided example for the applicability of obtained results to the theory of integral equations also we introduce an iteration algorithm by modified homotopy perturbation and adomian decomposition method to find solution of the above problem with high accuracy finally we have a discussion about convergence and upper bound of error

some new generalized gronwall bellman type discrete fractional inequalities some new gronwall bellman type discrete fractional inequalities are established explicit bounds for unknown functions in fractional inequalities are derived properties of solutions of one certain fractional difference equation are researched in this paper we present some new generalized gronwall bellman type discrete fractional inequalities including fractional sum inequalities and fractional difference inequalities these inequalities are of new forms compared with the existing results so far in the literature and can be used to provide explicit bounds for unknown functions concerned for illustrating the effectiveness of the presented results we apply them to research qualitative and quantitative properties for the solutions of an ivp with respect to a fractional difference equation

an efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs a mathematical model for the prediction of fluid pressures is given by a time dependent diffusion equation application of the finite element method leads to a system of linear equations a complication is that the underground consists of layers with very large differences in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the cg method has been observed moreover a classical termination criterion is not valid in this problem after diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability for the ic preconditioner the same behavior is observed to annihilate the effect of the extreme eigenvalues a deflated cg method is used the convergence rate improves considerably and the termination criterion becomes again reliable finally a cheap approximation of the eigenvectors is proposed

parallel computation of flow in heterogeneous media modelled by mixed finite elements abstract in this paper we describe a fast parallel method for solving highly ill conditioned saddle point systems arising from mixed finite element simulations of stochastic partial differential equations pdes modelling flow in heterogeneous media each realisation of these stochastic pdes requires the solution of the linear first order velocity pressure system comprising darcy s law coupled with an incompressibility constraint the chief difficulty is that the permeability may be highly variable especially when the statistical model has a large variance and a small correlation length for reasonable accuracy the discretisation has to be extremely fine we solve these problems by first reducing the saddle point formulation to a symmetric positive definite spd problem using a suitable basis for the space of divergence free velocities the reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive schwarz domain decomposition preconditioner the result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field numerical experiments exhibit almost optimal levels of parallel efficiency the domain decomposition solver doug http www maths bath ac uk parsoft used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures

the global attractivity and asymptotic stability of solution of a nonlinear integral equation abstract in this paper we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation some new results are given

analysis of feti methods for multiscale pdes in this paper we study a variant of the finite element tearing and interconnecting feti method which is suitable for elliptic pdes with highly heterogeneous multiscale coefficients x in particular coefficients with strong variation within subdomains and or jumps that are not aligned with the subdomain interfaces using energy minimisation and cut off arguments we can show rigorously that for an arbitrary positive coefficient function alpha in l infty omega the condition number of the preconditioned feti system can be bounded by c 1 log h h 2 where h is the subdomain diameter and h is the mesh size and where the function c depends only on the coefficient variation in the vicinity of subdomain interfaces in particular if alpha omega i varies only mildly in a layer i of width near the boundary of each of the subdomains i then c alpha mathcal o h eta 2 independent of the variation of in the remainder i i of each subdomain and independent of any jumps of across subdomain interfaces the quadratic dependence of c on h can be relaxed to a linear dependence under stronger assumptions on the behaviour of in the interior of the subdomains our theoretical findings are confirmed in numerical tests

finite difference methods with non uniform meshes for nonlinear fractional differential equations in this article finite difference methods with non uniform meshes for solving nonlinear fractional differential equations are presented where the non equidistant stepsize is non decreasing the rectangle formula and trapezoid formula are proposed based on the non uniform meshes combining the above two methods we then establish the predictor corrector scheme the error and stability analysis are carefully investigated at last numerical examples are carried out to verify the theoretical analysis besides the comparisons between non uniform and uniform meshes are given where the non uniform meshes show the better performance when dealing with the less smooth problems

overlapping schwarz methods on unstructured meshes using non matching coarse grids we consider two level overlapping schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions standard finite element interpolation from the coarse to the fine grid may be used our theory requires no assumption on the substructures that constitute the whole domain so the substructures can be of arbitrary shape and of different size the global coarse mesh is allowed to be non nested to the fine grid on which the discrete problem is to be solved and neither the coarse mesh nor the fine mesh need be quasi uniform in addition the domains defined by the fine and coarse grid need not be identical the one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which neumann boundary conditions are given in this general setting our algorithms have the same optimal convergence rate as the usual two level overlapping domain decomposition methods on structured meshes the condition number of the preconditioned system depends only on the possibly small overlap of the substructures and the size of the coarse grid but is independent of the sizes of the subdomains

analysis of feti methods for multiscale pdes part ii interface variation in this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting feti method and a variant thereof all floating feti we consider a scalar elliptic equation in a two or three dimensional domain with a highly heterogeneous multiscale diffusion coefficient this coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains in other words the subdomain partitioning does not need to resolve any jumps in the coefficient under suitable assumptions we derive bounds for the condition numbers of one level and all floating feti that are robust with respect to strong variations in the contrast in the coefficient and that are explicit in some geometric parameters associated with the coefficient variation in particular robustness holds for face edge and vertex islands in high contrast media as a central tool we prove and use new weighted poincare and discrete sobolev type inequalities that are explicit in the weight our theoretical findings are confirmed in a series of numerical experiments

existence of nondecreasing solutions of a quadratic integral equation of volterra type nondecreasing solutions of the quadratic integral equations is an interesting subject in the area of mathematics sciences and engineering using the theory of measures of noncompactness and applying a new method we prove the existence of nondecreasing solutions of a quadratic integral equation of volterra type in c i

existence of solutions to singular integral equations abstract nonnegative solutions are established for singular integral equations of the form y t h t t 0 k t s f s y s ds for t 0 t here f may be singular at y 0

random walks on random coset spaces with applications to furstenberg entropy we determine the range of furstenberg entropy for stationary ergodic actions of nonabelian free groups by an explicit construction involving random walks on random coset spaces

some extensions of the lanczos ortiz theory of canonical polynomials in the tau method lanczos and ortiz placed the canonical polynomials c p s in a central position in the tau method in addition ortiz devised a recursive process for determining c p s consisting of a generating formula and a complementary algorithm coupled to the formula in this paper a we extend the theory so as to include in the formalism also the ordinary linear differential operators with polynomial coefficients d with negative height h max n n m n n 0 where m n denotes the degree of dx n b we establish a basic classification of the c p s q m x and their orders m m as primary or derived depending respectively on whether 3n n m n m or such n does not exist and we state a classification of the indices n n as generic m n n h singular m n n h and indefinite dx n 0 then a formula which gives the set of primary orders is proved c in the rather frequent case in which all c p s are primary we establish for differential operators d with any height h a recurrency formula which generates bases of the polynomial space and their multiple c p s arising from distinct x n n n so that no complementary algorithmic construction is needed the primary c p s so produced are classified as generic or singular depending on the index n d we establish the general properties of the multiplicity relations of the primary c p s and of their associated indices it becomes clear that ortiz s formula generates for h 0 the generic c p s in terms of the singular and derived c p s while singular and derived c p s and the multiples of distinct indices are constructed by the algorithm

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations abstract some new weakly singular integral inequalities of gronwall bellman type are established which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations integral equations and evolution equations some applications to fractional differential and integral equations are also indicated

multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions multilevel schwarz methods are developed for a conforming finite element approximation of second order elliptic problems we focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions we establish a condition number estimate for the iterative operator which is independent of the coefficients and grows at most as the square of the number of levels we also characterize a class of distributions of the coefficients called quasi monotone for which the weighted l 2 projection is stable and for which we can use the standard piecewise linear functions as a coarse space in this case we obtain optimal methods i e bounds which are independent of the number of levels and subregions we also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas we consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods

several integral inequalities and their applications in nonlinear differential systems in this paper several new integral inequalities are presented which are effective in dealing with the integro differential inequalities whose variable exponents are greater than one compared with existing integral inequalities those proposed here can be applied to more complicated differential equations the notions of uniform lipschitz stability are generalized and the relations between these notions are analyzed several sufficient conditions about uniform lipschitz asymptotic stability of nonlinear systems are established by the proposed integral inequalities these sufficiently conditions can be similarly generalized to linearly perturbed differential systems that appear in the literature finally an example of uniform lipschitz asymptotic stability of nonlinear differential systems is shown

row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients linear systems with large differences between the coefficients called discontinuous coefficients often arise when physical phenomena in heterogeneous media are modeled by partial differential equations pdes such problems are usually solved by domain decomposition techniques but these can be difficult to implement when subdomain boundaries are complicated or the grid is unstructured it is known that for such systems diagonal scaling can sometimes improve the eigenvalue distribution and the convergence properties of some algorithm preconditioner combinations however there seems to be no study outlining both the usefulness and limitations of this approach it is shown that l 2 scaling of the equations is a generally useful preconditioner for such problems when the system matrices are nonsymmetric but only when the off diagonal elements are small to moderate tests were carried out on several nonsymmetric linear systems with discontinuous coefficients derived from convection diffusion elliptic pdes with small to moderate convection terms it is shown that l 2 scaling improved the eigenvalue distribution of the system matrix by reducing their concentration around the origin very significantly furthermore such scaling improved the convergence properties of restarted gmres and bi cgstab with and without the ilu 0 preconditioner since ilu 0 is theoretically oblivious to diagonal scaling these results indicate that l 2 scaling also improves the runtime numerical stability

new criteria on finite time stability of fractional order hopfield neural networks with time delays in this article the finite time stability fts of fractional order hopfield neural networks with time delays fhnntds is studied a widely used inequality in investigating the stability of the fractional order neural networks is fractional order gronwall inequality related to the mittag leffler function which cannot be directly used to study the stability of the factional order neural networks with time delays in the existing works related to fractional order gronwall inequality with time delays the order 0 was divided into two cases 0 0 5 and 0 5 in this article a new fractional order gronwall integral inequality with time delay and the unified form for all the fractional order 0 is developed which can be widely applied to investigate fts of various fractional order systems with time delays based on this new inequality a new criterion for the fts of fhnntds is derived compared with the existing criteria in which fractional order 0 1 was divided into two cases 0 0 5 and 0 5 1 the obtained results in this article are presented in the unified form of fractional order 0 1 and convenient to verify more importantly the criteria in this article are less conservative than some existing ones finally two numerical examples are given to demonstrate the validity of the proposed results

existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay we deal in this paper with the mild solution for fractional semilinear differential equations with infinite delay d ax t ax t f t x t bx t email protected 0 t x t email protected t email protected 0 with t 0 and 0 0 on a complex banach space x by means of classical fixed points methods

tau approximation method for the weakly singular volterra hammerstein integral equations in this paper we propose a useful method based on the tau method with arbitrary bases to find the numerical solution of weakly singular voletrra hammerstein integral equations in this scheme an operational approach using the orthogonal polynomial bases is presented for converting the problem under consideration to its matrix vector representation error analysis of this method is also presented some numerical examples are provided to illustrate the accuracy and computational efficiency of the method

the construction of projection vectors for a deflated iccg method applied to problems with extreme contrasts in the coefficients abstract to predict the presence of oil and natural gas in a reservoir it is important to know the fluid pressure in the rock formations a mathematical model for the prediction of the fluid pressure history is given by a time dependent diffusion equation application of the finite element method leads to systems of linear equations a complication is that the underground consists of layers with very large contrasts in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the iccg method has been observed and a classical termination criterion is not valid in this problem in 19 we have shown that the number of small eigenvalues of the diagonally scaled matrix is equal to the number of high permeability domains which are not connected to a dirichlet boundary in this paper the proof is extended to an incomplete cholesky decomposition to annihilate the bad effect of these small eigenvalues on the convergence the deflated iccg method is used in 19 we have shown how to construct a deflation subspace for the case of a set of more or less parallel layers that subspace proved to be a good approximation of the span of the small eigenvectors as a result of this the convergence of diccg is independent of the contrasts in the permeabilities in this paper it is shown how to construct deflation vectors even in the case of very irregular shaped layers and layers with so called inclusions a theoretical investigation and numerical experiments show that the diccg method is not sensitive to small perturbations of the deflation vectors the efficiency of the diccg method is illustrated by numerical experiments