approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

three step iterations for variational inequalities and nonexpansive mappings in this paper we introduce a new three step iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality using the technique of updating the solution we show that the sequence converges strongly to a common element of two sets under some control conditions results proved in this paper may be viewed as an improvement and refinement of the recent results of noor and huang m aslam noor z huang three step methods for nonexpansive mappings and variational inequalities appl math comput in press and yao and yao y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput in press

strong convergence of a general iterative method for variational inequality problems and fixed point problems in hilbert spaces abstract in this paper we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods our results improve and extend the recent ones announced by chen et al j m chen l j zhang t g fan viscosity approximation methods for nonexpansive mappings and monotone mappings doi 10 1016 j jmaa 2006 12 088 iiduka and tahakshi h iiduka w takahashi strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings nonlinear anal 61 2005 341 350 yao and yao y h yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput doi 10 1016 j amc 2006 08 062 and many others

two extragradient methods for generalized mixed equilibrium problems nonexpansive mappings and monotone mappings in this paper we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone lipschitz continuous mapping we obtain some weak convergence theorems for the sequences generated by these processes in hilbert spaces the results in this paper generalize extend and unify some well known weak convergence theorems in the literature

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

the high order compact numerical algorithms for the two dimensional fractional sub diffusion equation in this paper performing the average operators on the space variables a numerical scheme with third order temporal convergence for the two dimensional fractional sub diffusion equation is considered for which the unconditional stability and convergence in l1 l norm are strictly analyzed for 0 0 9569347 by using the discrete energy method therewith adding small perturbation terms we construct a compact alternating direction implicit difference scheme for the two dimensional case finally some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all 0 1

a relaxed extragradient like method for a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem abstract very recently takahashi and takahashi s takahashi w takahashi strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a hilbert space nonlinear anal 69 2008 1025 1033 suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a hilbert space in this paper based on takahashi takahashi s iterative method and well known extragradient method we introduce a relaxed extragradient like method for finding a common solution of a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a hilbert space and then obtain a strong convergence theorem utilizing this theorem we establish some new strong convergence results in fixed point problems variational inequalities mixed equilibrium problems and systems of generalized equilibria

towards a statistical theory of texture evolution in polycrystals most technologically useful materials possess polycrystalline microstructures composed of a large number of small monocrystalline grains separated by grain boundaries the energetics and connectivity of the grain boundary network play a crucial role in determining the properties of a material across a wide range of scales a central problem in materials science is to develop technologies capable of producing an arrangement of grains a texture that provides for a desired set of material properties one of the most challenging aspects of this problem is to understand the role of topological reconfigurations during coarsening here we propose an upscaling procedure suitable for large complex systems the procedure is based on numerical experimentation combined with stochastic tools and consists of large scale numerical simulations of a system at a microscopic level statistical analysis of the microscopic data and formulation of the model based on stochastic characteristics predicted by the statistical analysis this approach promises to be valuable in establishing the effective model of microstructural evolution in realistic two and three dimensional systems to test ideas we use our upscaling procedure to study the mesoscopic behavior of a reduced one dimensional network of grain boundaries despite the simplicity of its formulation this model exhibits highly nontrivial behavior characterized by growth and disappearance of grain boundaries and develops probability distributions similar to those observed in higher dimensional simulations here we focus on the grain deletion events which are common to all coarsening systems

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

solving linear and non linear space time fractional reaction diffusion equations by the adomian decomposition method in this paper we consider linear and non linear space time fractional reaction diffusion equations stfrde on a finite domain the equations are obtained from standard reaction diffusion equations by replacing a second order space derivative by a fractional derivative of order 1 2 and a first order time derivative by a fractional derivative of order 0 1 we use the adomian decomposition method to construct explicit solutions of the linear and non linear stfrde finally some examples are given

strong convergence theorems for strongly nonexpansive sequences the aim of this paper is to prove a strong convergence theorem for a pair of sequences of nonexpansive mappings in a hilbert space where one of them is a strongly nonexpansive sequence and provide some applications of the theorem

numerical methods for solving a two dimensional variable order anomalous subdiffusion equation australian research council dp0559807 dp0986766 national natural science foundation of china 10271098 natural science foundation of fujian province 2009j01014

strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in banach spaces in this paper we introduce a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of a general variational inequality and the set of common fixed points of an infinite family of k i strict pseudocontractions in a banach space we obtain some strong convergence theorems under suitable conditions the results obtained in this paper improve and extend the recent ones announced by many others

algorithms for approximating minimization problems in hilbert spaces in this paper we study the following minimization problem minx fix s w m2 12 x 2 h x where b is a bounded linear operator m 0 is some constant h is a potential function for c f fix t is the set of fixed points of nonexpansive mapping s and w is the solution set of an equilibrium problem this paper introduces two new algorithms one implicit and one explicit that can be used to find the solution of the above minimization problem

hybrid noor iteration method for nonexpansive mappings in hilbert spaces in this paper hybrid noor iteration method was introduced for nonexpansive mappings in hilbert spaces the sufficient and necessary conditions that the iteration sequences converge strongly to a fixed point of a nonexpansive mapping are obtained in hilbert spaces

a hybrid algorithm for pseudo contractive mappings abstract in this paper we suggest and analyze a hybrid algorithm for pseudo contractive mappings in hilbert spaces further we prove the strong convergence of the proposed iterative algorithm for lipschitz pseudo contractive mappings in hilbert spaces

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

variational inequalities with generalized monotone operators we investigate the variational inequality with pseudomonotone operators in the sense of karamardian in banach spaces new existence results which extend many known results in infinite dimensional spaces are derived under rather weak assumptions new uniqueness results which also seem to be new even in finite dimensional spaces are also derived in particular new existence and uniqueness results for the complementarity problem with pseudomonotone operators in banach spaces are obtained also existence and uniqueness results for minimization problems with pseudoconvex functions in banach spaces are obtained

approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method in this paper we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real hilbert space furthermore we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters

a computational wavelet method for variable order fractional model of dual phase lag bioheat equation abstract in this study we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue several models with different levels of simplifications have been proposed one of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time is the 2d transient dual phase lag dpl variable order fractional energy equation for finding the numerical solution of this general case we propose an efficient semi discrete method based on the two dimensional legendre wavelets 2d lws precisely the variable order fractional derivatives of the model are discretized in the first stage and then the response of the model is expanded by the 2d lws consequently the main problem is transformed into an equivalent system of algebraic equations which can be simply tackled the stability of the proposed method is examined theoretically and experimentally also the procedure is described for one example to examine the computational efficiency of method the experimental results show the stability and spectral accuracy of the proposed method according to the achieved results increasing the fractional order from 0 1 to 1 0 leads to increment of maximum tissue temperatures by about 29 near the center of the targeted region

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

an analysis of the rayleigh stokes problem for a generalized second grade fluid we study the rayleigh stokes problem for a generalized second grade fluid which involves a riemann liouville fractional derivative in time and present an analysis of the problem in the continuous space semidiscrete and fully discrete formulations we establish the sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v v including v in l 2 omega v l2 a space semidiscrete galerkin scheme using continuous piecewise linear finite elements is developed and optimal with respect to initial data regularity error estimates for the finite element approximations are derived further two fully discrete schemes based on the backward euler method and second order backward difference method and the related convolution quadrature are developed and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data numerical results for one and two dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method and to verify the convergence theory

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

a galerkin finite element method for the modified distributed order anomalous sub diffusion equation abstract in this paper we consider a modified time distributed order anomalous sub diffusion equation for the description of processes becoming less anomalous over the course of time first we discretize the integral term using a mid point quadrature rule and transform the time distributed order fractional diffusion equation into a multi term equation then applying the backward difference method in time and galerkin finite element method in space we present a fully discrete scheme to solve the time distributed order fractional diffusion equation moreover the stability and convergence of the proposed method are also discussed numerical examples are given to verify the accuracy and stability of our scheme

viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems abstract in this paper we first introduce a general iterative scheme by halpern approximation method for finding a common element of the set of solutions of an equilibrium problem the set of solutions of the variational inequality for an infinite family of i inverse strongly monotone mappings and the set of common fixed points for a family of infinitely nonexpansive mappings in a hilbert space then we extend the halpern iterative scheme to a general viscosity approximation scheme by a simple and different method as applications we utilize our results to investigate some convergence problems for strictly pseudocontractive mappings and maximal monotone operators the results obtained in this paper extend and improve the recent ones announced by many others

a discontinuous petrov galerkin method for time fractional diffusion equations we propose and analyze a time stepping discontinuous petrov galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time fractional subdiffusion problems we prove the existence uniqueness and stability of approximate solutions and derive error estimates to achieve high order convergence rates from the time discretizations the time mesh is graded appropriately near t 0 to compensate for the singular temporal behavior of the exact solution near t 0 caused by the weakly singular kernel but the spatial mesh is quasi uniform in the l infty 0 t l 2 omega norm 0 t is the time domain and omega is the spatial domain for sufficiently graded time meshes a global convergence of order k m alpha 2 h r 1 is shown where 0 alpha 1 is the fractional exponent k is the maximum time step h is the maximum diameter of the elements of the spatial mesh and m and r are the degrees of approximate solutions in time an

convergence of the grunwald letnikov scheme for time fractional diffusion using bivariate generating functions we prove convergence of the grunwald letnikov difference scheme for the fractional diffusion equation in one space dimension with and without central linear drift in the fourier laplace domain as the space and time steps tend to zero in a well scaled way this implies convergence in distribution weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle the difference schemes allow also interpretation as discrete random walks for fractional diffusion with central linear drift we show that in the fourier laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

general pade approximation method for time space fractional diffusion equation in recent years fractional differential equations have attracted much attention due to their wide application in this paper we present a novel numerical method for the space time riesz caputo fractional diffusion equation which discrete the riesz derivative by a fourth order fractional compact difference scheme then the above space time riesz caputo fractional diffusion equation change into a fractional ordinary differential equation fode system again using laplace and inverse laplace transforms one can get the analytical solution of the fode system furthermore we approximate the mittag leffler function by the global pade approximation and obtain the numerical method for the space time riesz caputo fractional diffusion equation finally two numerical examples are presented to show that the numerical results are in good line with our theoretical analysis

a fourier method and an extrapolation technique for stokes first problem for a heated generalized second grade fluid with fractional derivative in this article we consider stokes first problem for a heated generalized second grade fluid with fractional derivative sfp hgsgf implicit and explicit numerical approximation schemes for the sfp hgsgf are presented the stability and convergence of the numerical schemes are discussed using a fourier method in addition the solvability of the implicit numerical approximation scheme is also analyzed a richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed finally a numerical test is given the numerical results demonstrate the good performance of our theoretical analysis

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems abstract the purpose of this work is to introduce a hybrid projection method for finding a common element of the set of a generalized equilibrium problem the set of solutions to a variational inequality and the set of fixed points of a strict pseudo contraction in a real hilbert space

on modified iterative method for nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above two sets under some parameters controlling conditions

numerical methods with fourth order spatial accuracy for variable order nonlinear stokes first problem for a heated generalized second grade fluid stokes first problem has in recent years received much attention in this paper we focus on the variable order nonlinear stokes first problem for a heated generalized second grade fluid a numerical scheme with fourth order spatial accuracy is developed to solve the problem the stability solvability and convergence of the numerical scheme are discussed via fourier analysis an improved numerical scheme is also developed in addition a numerical example is given and the numerical results support the effectiveness of our theoretical analysis results

on mann implicit iterations for strongly accretive and strongly pseudo contractive mappings abstract for a lipschitz strongly accretive map considered by chidume in c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 it is known that a classical picard type iteration process converges strongly to a zero of the operator he also proved that the rate of convergence in this case is at least as fast as a geometric progression in this paper we study the mann implicit iteration sequence for strongly accretive and strongly pseudo contractive mappings we showed that this implicit scheme gives better convergence rate estimate presented results improve the corresponding results of c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 l liu approximation of fixed points of a strictly pseudo contractive mapping proc am math soc 125 2 1997 1363 1366 w r sastry g v r babu approximation of fixed points of strictly pseudo contractive mappings on arbitrary closed convex sets in a banach space proc amer math soc 128 2000 2907 2909 y song r chen viscosity approximative methods to cesaro means for non expansive mappings appl math comput 186 2 2007 1120 1128

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

quadratic spline collocation method for the time fractional subdiffusion equation in this paper exploiting the quadratic spline collocation qsc method we numerically solve the time fractional subdiffusion equation with dirichelt boundary value conditions the coefficient matrix of the discretized linear system is investigated in detail theoretical analyses and numerical examples demonstrate the proposed technique can enjoy the global error bound with o 3 h 3 under the l norm provided that the solution v x t has four order continual derivative with respects to x and t and it can achieve the accuracy of o 4 h 4 at collocation points where h are the step sizes in time and space respectively

convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications in this paper we consider the problem of convergence of an iterative algorithm for a system of generalized variational inequalities and a nonexpansive mapping strong convergence theorems are established in the framework of real banach spaces

a new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings in this paper we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for a inverse strongly monotone mapping in hilbert spaces we prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main result improve and extend plubtieng and punpaeng s corresponding result s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings applied mathematics and computation 197 2008 548 558 using this theorem we obtain three corollaries

weak and strong convergence theorems for countable lipschitzian mappings and its applications abstract we use mann s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of lipschitzian mappings finally we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

iterative algorithms for a general system of generalized nonlinear mixed composite type equilibria in this paper we consider and study a general system of generalized nonlinear mixed composite type equilibria in hilbert spaces first we prove the existence and uniqueness of the solution for this system of generalized nonlinear mixed composite type equilibria second the mann iterative method with errors is extended to develop some new iterative algorithms for finding approximate solutions for this system of generalized nonlinear mixed composite type equilibria we also derive the strong convergence of the sequences generated by these iterative algorithms in hilbert spaces

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

stability and convergence of finite difference schemes for a class of time fractional sub diffusion equations based on certain superconvergence this paper is devoted to the construction and analysis of finite difference methods for solving a class of time fractional subdiffusion equations based on the certain superconvergence at some particular points of the fractional derivative by the traditional first order grunwald letnikov formula some effective finite difference schemes are derived the obtained schemes can achieve the global second order numerical accuracy in time which is independent of the values of anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme respectively are established for the one dimensional problem along with the strict analysis on the unconditional stability and convergence of these schemes by the discrete energy method furthermore the extension to the two dimensional case is also considered numerical experiments support the correctness of the theoretical analysis and effectiveness of the new developed difference schemes the superconvergent points of the traditional first order grunwald letnikov formula to approximate the riemann liouville fractional derivatives are used to construct the several finite difference schemes for solving a class of time fractional sub diffusion equations the obtained schemes can achieve the global second order accuracy in time which is independent of the anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme are presented respectively the strict stability and convergence analysis on the resulting difference schemes are given by the discrete energy method the extension to the two dimensional case is also discussed the compact difference scheme with the convergence of second order in time and fourth order in space is derived along with the strict stability and convergence analysis several numerical experiments are implemented to illustrate the computational efficiency numerical accuracy and robustness of the proposed schemes

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

discrete random walk models for symmetric levy feller diffusion processes we propose a variety of models of random walk discrete in space and time suitable for simulating stable random variables of arbitrary index 0 2 in the symmetric case we show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous markovian stochastic processes which we refer to as levy feller diffusion processes

mixed spline function method for reaction subdiffusion equations in this paper we develop two classes of finite difference schemes for the reaction subdiffusion equations by using a mixed spline function in space direction forward and backward difference in time direction respectively it has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters by fourier method we prove that one class of difference scheme is unconditionally stable and convergent the other is conditionally stable and convergent finally some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes

entire solutions of higher order abstract cauchy problems abstract we consider the abstract cauchy problem for the higher order differential equation u n t i 0 n 1 a i u i t 0 t 0 u j 0 u j 0 j n 1 wherea0 an 1are closed linear operators in a complex banach spacee it is shown that if there exist 0 2 r 0 and a positive integerhsuch that nr and hakr 0 k n 1 are bounded in c arg e i 2 r wherer n n 1i 0 iai 1 then there exists a dense subsetgofensuch that for every initial value u0 un 1 g has a unique entire solution that is a solution which can be extended analytically to the whole complex plane as a corollary we get that the cauchy problems for the parabolic higher order linear differential equations have a unique entire solution for a dense set of initial data

a compact adi method and its extrapolation for time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions a compact alternating direction implicit adi finite difference method is proposed for two dimensional time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions the unconditional stability and convergence of the method is proved the error estimates in the weighted l 2 l 2 and l l norms are obtained the proposed method has the fourth order spatial accuracy and the temporal accuracy of order min 2 1 min 2 1 where 0 1 0 1 is the order of the fractional derivative in order to further improve the temporal accuracy two richardson extrapolation algorithms are presented numerical results demonstrate the accuracy of the compact adi method and the high efficiency of the extrapolation algorithms

a fractional alternating direction implicit method for a multi term time space fractional bloch torrey equations in three dimensions abstract the time space fractional bloch torrey equation ts fbte has been proposed to simulate anomalous diffusion in the human brain however the development of numerical methods and theoretical analysis for multi term time space fractional bloch torrey equations in three dimensions is still limited in this paper we consider a class of 3 d multi term time space fractional bloch torrey equations 3d mtts fbtes firstly we adopt a fractional centred difference scheme to discretize the riesz fractional derivative and propose a fractional alternating direction implicit method fadim for the model secondly the solvability stability and convergence of the method are investigated finally numerical results are presented to support our theoretical analysis in addition to demonstrate the applicability of our method we consider a coupled 3 d fbte as an example to solve and exhibit the effects on the behaviour of the transverse magnetization

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

a meshless method for solving the time fractional advection diffusion equation with variable coefficients abstract in this paper an efficient and accurate meshless method is proposed for solving the time fractional advection diffusion equation with variable coefficients which is based on the moving least square mls approximation in the proposed method firstly the time fractional derivative is approximated by a finite difference scheme of order o t 2 0 1 and then the mls approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the caputo sense also the validity of the proposed method is investigated in error analysis discussion the main aim is to show that the meshless method based on the mls shape functions is highly appropriate for solving fractional partial differential equations fpdes with variable coefficients the efficiency and accuracy of the proposed method are verified by solving several examples

a cubic trigonometric b spline collocation approach for the fractional sub diffusion equations an algorithm for fractional sub diffusion model using cubic trigonometric b spline approach is presented the proposed scheme is unconditionally stable using von neumann approach error norms confirm the accuracy and efficiency of presented scheme the proposed method provides more accurate results as compare to cubic b spline method a cubic trigonometric b spline collocation approach for the numerical solution of fractional sub diffusion equation is presented in this paper the approach is based on the usual finite difference scheme to discretize the time derivative while the approximation of the second order derivative with respect to space is obtained by the cubic trigonometric b spline functions with the help of grunwald letnikov discretization of the riemann liouville derivative the scheme is shown to be stable using the fourier method and the accuracy of the scheme is tested by application to a test problem the results of the numerical test verify the accuracy and efficiency of the proposed algorithm

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main theorem extends a recent result of yao and yao y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings applied mathematics and computation 186 2 2007 1551 1558

algorithms with strong convergence for a system of nonlinear variational inequalities in banach spaces abstract in this paper a general system of nonlinear variational inequality problem in banach spaces was considered which includes some existing problems as special cases for solving this nonlinear variational inequality problem we construct two methods which were inspired and motivated by korpelevich s extragradient method furthermore we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality

weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations abstract let h be a real hilbert space k a nonempty closed convex subset of h and t k k a lipschitz pseudocontraction with a nonempty fixed point set weak and strong convergence theorems for the iterative approximation of fixed points of t are proved furthermore if t h h is a lipschitz monotone operator and f r t where r t denotes the range of t weak and strong convergence theorems for the iterative approximation of solutions of the operator equation tx f are proved

a finite element splitting extrapolation for second order hyperbolic equations splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel this article discusses a finite element splitting extrapolation for second order hyperbolic equations with time dependent coefficients this method possesses a higher degree of parallelism less computational complexity and more flexibility than richardson extrapolation while achieving the same accuracy by means of domain decomposition and isoparametric mapping some grid parameters are chosen according to the problem the multiparameter asymptotic expansion of the d quadratic finite element error is also established the splitting extrapolation formulas are developed from this expansion an approximation with higher accuracy on a globally fine grid can be computed by solving a set of smaller discrete subproblems on different coarser grids in parallel some a posteriori error estimates are also provided numerical examples show that this method is efficient for solving discontinuous problems and nonlinear hyperbolic equations

continuous time random walks on moving fluids the scheme of the continuous time random walk ctrw is generalized to include the possibility of a moving background it is shown that this generalization reproduces in the macroscopic limit the usual diffusion advection equation and the properties of standard diffusion in a shear flow the new formalism is then used to derive the corresponding macroscopic equation for ctrw s with infinite mean squared step length and with infinite mean waiting time in a moving fluid for these two ctrw s we finally include an analysis of the dispersion in three different two dimensional linear shear flows

integrated semigroups and higher order abstract equations abstract let e be a banach space b 0 b n 1 closed linear operators on e and f 0 e this paper is devoted to the study of well posedness of the cauchy problem for higher order abstract differential equations formula using the theory of integrated semigroups we obtain some concise and useful criteria

numerical algorithm for two dimensional fractional stokes first problem for a heated generalized second grade fluid with smooth and non smooth solution abstract in this paper we consider the numerical solution of the two dimensional fractional stokes first problem for a heated generalized second grade fluid the proposed method is based on the l1 finite difference scheme for the temporal direction while the legendre spectral method for the spatial direction numerical stability and convergence of the method are rigorously established at the mean time the case of non smooth solution is considered by adding some correction terms numerical experiments are included which verify the theoretical predictions

discrete maximal regularity of time stepping schemes for fractional evolution equations in this work we establish the maximal ell p regularity for several time stepping schemes for a fractional evolution model which involves a fractional derivative of order alpha in 0 2 alpha ne 1 in time these schemes include convolution quadratures generated by backward euler method and second order backward difference formula the l1 scheme explicit euler method and a fractional variant of the crank nicolson method the main tools for the analysis include operator valued fourier multiplier theorem due to weis math ann 319 735 758 2001 doi 10 1007 pl00004457 and its discrete analogue due to blunck stud math 146 157 176 2001 doi 10 4064 sm146 2 3 these results generalize the corresponding results for parabolic problems

on variational inclusion and common fixed point problems in hilbert spaces with applications abstract the purpose of this paper is to introduce a general iterative method for finding a common element of the solution set of quasi variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings in the framework hilbert spaces strong convergence of the sequences generated by the purposed iterative scheme is obtained

a characterization of norm continuity of propagators for second order abstract differential equations abstract in this paper we obtain a concise characterization of norm continuity for t 0 of propagators for the complete second order abstract differential equation on a banach space e u t bu t au t 0 t 0 where b l e as a consequence we discover that a strongly continuous cosine operator function or operator group is norm continuous for t 0 if and only if its generator is bounded

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

a new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization abstract in this paper we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for inverse strongly monotone mappings in a hilbert space under suitable conditions some strong convergence theorems for approximating a common element of the above three sets are obtained as applications at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings the results presented in the paper extend and improve some recent results of yao and yao y y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput 186 2 2007 1551 1558 plubtieng and punpaeng s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings appl math comput 2007 doi 10 1016 j amc 2007 07 075 s takahashi and w takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2006 506 515 su shang and qin y f su m j shang x l qin an iterative method of solution for equilibrium and optimization problems nonlinear anal 2007 doi 10 1016 j na 2007 08 045 and chang cho and kim s s chang y j cho j k kim approximation methods of solutions for equilibrium problem in hilbert spaces dynam systems appl in print