improved and more feasible numerical methods for riesz space fractional partial differential equations abstract in this article the numerical solutions of fractional order partial differential equations with riesz space fractional derivatives on a finite domain have been considered two kinds of equations have been considered the riesz fractional diffusion equation and the riesz fractional advection dispersion equation the riesz fractional diffusion equation is obtained from the standard diffusion equation by replacing the second order space derivative with the riesz fractional derivative of order where 1 2 the riesz fractional advection dispersion equation is obtained from the standard advection dispersion equation by replacing the first and second order space derivatives with the riesz fractional derivatives of order 1 1 and 1 2 respectively to obtain the solutions firstly the system of odes is obtained using the improved matrix transform method with respect to the space variable after this the 3 1 pade approximation can be used to construct the exponential matrix in the analytic solution of the ode and we obtain two difference schemes it has also been shown that the two difference schemes are unconditionally stable and feasible using the matrix analysis method finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

on some classes of almost automorphic functions and applications to fractional differential equations abstract we study in this paper the properties of c n almost automorphic and asymptotically c n almost automorphic functions a new concept with values in a banach space we then give a new result related to the existence and uniqueness of an asymptotically almost automorphic solution of a semilinear fractional differential equation of the form d x t a x t f t x t b x t with t r 0 1 where a generates a family of resolvent family s t and f satisfies some lipschitz conditions

chaos synchronization of the fractional lu system abstract chaos synchronization of the fractional lu system is theoretically and numerically studied by two methods the suitable conditions for achieving synchronization of this fractional differential system are derived by using the laplace transform theory numerical simulations coincide with the theoretical analysis

linear approximation of transfer function with a pole of fractional power two methods for the linear approximation of a transfer function with a pole of fractional power are presented analog circuit models are developed and their frequency response curves and step response curves are compared it was found that the pade method gives a better approximation than wang and hsia s method within the frequency limit as specified

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

short communication modification of the application of a contraction mapping method on a class of fractional differential equation this letter discusses an application of a modified contraction mapping method to the proof of a global existence and uniqueness theorem of a cauchy problem for some version of differential equation of fractional order in banach space an essential condition in such applications which have previously been considered in many literatures will be dispensed with

synchronization in fractional order differential systems an symmetrically coupled system consisting of identical fractional order differential systems including chaotic and non chaotic systems is investigated in this paper such a coupled system has in its synchronous state a mode decomposition by which the linearized equation can be decomposed into motions transverse to and parallel to the synchronous manifold furthermore the decomposition can induce a sufficient condition on synchronization of the overall system which guarantees if satisfied that a group synchronization is achieved two typical numerical examples fractional brusselators and the fractional rossler system are used to verify the theoretical prediction the theoretical analysis and numerical results show that the lower the order of the fractional system the longer the time for achieving synchronization at a fixed coupling strength

bounded solutions to a class of semilinear integro differential equations in banach spaces abstract let a be the generator of an immediately norm continuous c 0 semigroup defined on a banach space x we study the existence and uniqueness of bounded solutions for the semilinear integro differential equation with infinite delay u t a u t t e t s a u s d s f t u t t r r for each f r x x satisfying diverse lipschitz type conditions sufficient conditions are established for the existence and uniqueness of an almost periodic almost automorphic and asymptotically almost periodic solution among other types of distinguished solutions these results have significance in viscoelasticity theory finally an example is presented to illustrate the feasibility and effectiveness of the results

conformable derivative approach to anomalous diffusion abstract by using a new derivative with fractional order referred to conformable derivative an alternative representation of the diffusion equation is proposed to improve the modeling of anomalous diffusion the analytical solutions of the conformable derivative model in terms of gauss kernel and error function are presented the power law of the mean square displacement for the conformable diffusion model is studied invoking the time dependent gauss kernel the parameters related to the conformable derivative model are determined by levenberg marquardt method on the basis of the experimental data of chloride ions transportation in reinforced concrete the data fitting results showed that the conformable derivative model agrees better with the experimental data than the normal diffusion equation furthermore the potential application of the proposed conformable derivative model of water flow in low permeability media is discussed

controllability of fractional integro differential evolution equations with nonlocal conditions this paper concerns the controllability for a class of fractional integro differential evolution equations with nonlocal initial conditions by using the fractional calculus measure of noncompactness and the monch fixed point theorem we obtain a controllability result for the nonlocal cauchy problem of the fractional integro differential evolution equations involving noncompact semigroups and the nonlocal functions without lipschitz continuity an example is given to illustrate the effectiveness of the abstract results

on discontinuous galerkin methods discontinuous galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present several alternatives for treating the diffusion and advective fluxes have been introduced this report summarizes some of the methods that have been proposed several numerical examples are included in the paper these present discontinuous galerkin solutions of one dimensional problems with a scalar variable results are presented for diffusion reaction problems and advection diffusion problems we discuss the performance of various formulations with respect to accuracy as well as stability of the method copyright 2003 john wiley sons ltd

a newly developed qp relaxation method for element connectivity parameterization to achieve stress based topology optimization for geometrically nonlinear structures the aim of this work is to present a novel computational approach to employ the stress based topology optimization method stom to minimize the volume subject to the locally defined stress constraints of a geometrically nonlinear structure in the framework of the element connectivity parameterization ecp method considering the locally defined stress constraints in topology optimization to is a classic and challenging engineering problem and successful optimization procedures have recently been developed using the density based to method for linear elastic structures however no study has yet considered the static failure constraint when using to for a geometrically nonlinear structure therefore the present study develops a novel computational approach for the stom for a geometrically nonlinear structure to successfully optimize a geometrically nonlinear structure the unstable element issue must be properly addressed in addition to the stress singularity issue the existence of a large number of constraints and the highly nonlinear behavior of the local stress constraints to effectively resolve these issues this research adopts the ecp method to interpolate and optimize the connectivities among solid finite elements furthermore we find that a stress singularity issue linked to the local optima issue arises in the ecp method that is different from that of the density based to by investigating the singularity behavior in detail we develop a new qp relaxation method that is suitable for the ecp method to demonstrate the improved capability of the proposed ecp method with the modified qp relaxation several two dimensional to problems are solved

asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions abstract of concern is a class of abstract semilinear integrodifferential equations with nonlocal initial conditions under some suitable hypotheses we establish some new theorems about the existence of asymptotically almost automorphic solutions to the integrodifferential equations moreover an example is given to illustrate our results

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

bounded mild solutions of perturbed volterra equations with infinite delay abstract we study existence and regularity of bounded mild solutions on the real line to perturbed integral equations with infinite delay in the space of almost periodic functions in the bohr sense the space of compact almost automorphic functions the space of almost automorphic functions and the space of asymptotically almost automorphic functions

isometries of csl algebras we show that every jordan isomorphism of csl algebras whose restriction to the diagonal of the algebra is a selfadjoint map is the sum of an isomorphism and an anti isomorphism it follows that every surjective linear isometry of csl algebras is the sum of an isomorphism and an anti isomorphism followed by a unitary multiplication

optimal layout design of three dimensional geometrically non linear structures using the element connectivity parameterization method the topology design optimization of three dimensional geometrically non linear continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization to overcome these difficulties the element connectivity parameterization ecp method with two implementation formulations is proposed in ecp structural layouts are represented by inter element connectivity which is controlled by the stiffness of element connecting zero length links depending on the link location ecp may be classified as an external ecp e ecp or an internal ecp i ecp in this paper i ecp is newly developed to substantially enhance computational efficiency the main idea in i ecp is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level as for ecp implementation with commercial software e ecp developed earlier for two dimensional problems is easier to use even for three dimensional problems because it requires only numerical analysis results for design sensitivity calculation the characteristics of the i ecp and e ecp methods are compared and these methods are validated with numerical examples copyright 2006 john wiley sons ltd

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

coprime factorizations and stability of fractional differential systems we give a frequency domain approach to stabilization for a large class of systems with transfer functions involving fractional powers of s a necessary and sufficient criterion for bibo stability is given and it is shown how to construct coprime factorizations and associated bezout factors in order to parametrize all stabilizing controllers of these systems

controllability of fractional evolution nonlocal impulsive quasilinear delay integro differential systems in this work the controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro differential systems in a banach space has been established by using the theory of fractional calculus fixed point technique and also we introduced a new concept called a u resolvent family as an application that illustrates the abstract results an example is given

isometries of real hilbert c modules abstract let t v w be a surjective real linear isometry between full real hilbert c modules over real c algebras a and b respectively we show that the following conditions are equivalent a t is a 2 isometry b t is a complete isometry c t preserves ternary products d t preserves inner products e t is a module map when a and b are commutative we give a full description of the structure of t

generalized fractional controller for singular systems of differential equations abstract in this article we consider a class of singular linear systems of first order and introduce a generalized fractional order feedback controller of caputo type the closed loop system in question is a singular system of differential equations having both first and fractional order derivatives we provide a comprehensive theory for the existence and uniqueness of solutions as well as for the stability of the system with inclusion of the fractional order controller an example of a singular system with a fractional order proportional integral controller as well as an example on a 3 bus power system with inclusion of a fractional order damping controller is given to illustrate our theory

almost automorphic mild solutions to fractional differential equations abstract we introduce the concept of resolvent families to prove the existence of almost automorphic mild solutions to the differential equation d t u t a u t t n f t 1 2 n z considered in a banach space x where f r x is almost automorphic we also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t 1 2 assuming f t x is almost automorphic in t for each x x satisfies a global lipschitz condition and takes values on x finally we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t u t 1 2 under analogous conditions as in the previous case

on defining the distributions δ k and δ k by fractional derivatives how to define products and powers of distributions is a difficult and not completely understood problem and has been investigated from several points of views since schwartz established the theory of distributions around 1950 many fields such as differential equations or quantum mechanics require such operations in this paper we use caputo fractional derivatives and the following generalized taylor s formula for 0 1 t i 0 m c d 0 t i 0 i 1 t i c d 0 t m 1 m 1 1 t m 1 to give meaning to the distributions k x and k x for all k r these can be regarded as powers of dirac delta functions and have applications to quantum theory at the end of this paper the distributions log t and t 2 are given by the sequence and the neutrix limit

non probabilistic reliability based topology optimization of geometrically nonlinear structures using convex models this paper describes a non probabilistic reliability based topology optimization method for the design of continuum structures undergoing large deformations the variation of the structural system is treated with the multi ellipsoid convex model which is a realistic description of the parameters being inherently uncertain but bounded or lacking sufficient probabilistic data the formulation of the optimal design is established as a volume minimization problem with non probabilistic reliability constraints on the geometrically nonlinear structural behaviour in order to circumvent numerical difficulties in solving the nested double loop optimization problem a performance measure based approach is employed to transform the constraint on the reliability index into one on the concerned performance in conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints the optimization problem is solved by gradient based mathematical programming methods three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method the obtained optimal solutions show the importance of incorporating various uncertainties in the design problem moreover it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation

regularity of mild solutions for a class of fractional order differential equations in this article we show sufficient conditions ensuring the existence and uniqueness of a mild solution to the equation d au t au t d a 1f t u t 1 a 2 email protected r in the same space where f belongs here a is a sectorial operator defined in a banach space x d a is the fractional derivative in the riemann liouville sense and f x email protected x for each email protected x where w x is a vector valued subspace of the space of continuous and bounded functions the subspaces w x that we will consider in this article are the space of periodic almost periodic almost automorphic and compact almost automorphic vector valued functions among others in particular we extend and unify recent results established for the equation in the papers agarwal et al 2010 cuevas et al 2010 and cuevas and lizama 2008

isometries of non self adjoint operator algebras abstract we study isometries of certain non self adjoint operator algebras by means of the structure of the complete holomorphic vector fields on their unit balls and the associated partial jordan triple products we show that isometries of nest sub algebras of b h are of the form t utw or t ujt jw where u w are suitable unitary operators and j a fixed involution of h

design sensitivity analysis and topology optimization of displacement loaded non linear structures abstract a continuum based design sensitivity analysis dsa method for geometrically nonlinear systems with nonhomogeneous boundary conditions is developed to topologically optimize the displacement loaded nonlinear structures in the adjoint variable method the solution space requires just homogeneous boundary conditions even if the original system has nonhomogeneous ones a design sensitivity expression for the instantaneous rigidity functional is derived for the displacement loaded nonlinear topology optimization the tangent stiffness is obtained at the end of the equilibrium iterations in the nonlinear analysis of the original system this stiffness is used in the dsa so that no iteration would be necessary to evaluate the design sensitivity expressions in force loaded systems the solution dose not converge easily because the material distributes sparsely sometimes during optimization however when the displacement loaded system is used there is no convergence difficulty

on the morita equivalence of tensor algebras our objective is two fold first we want to develop a notion of morita equivalence for c correspondences that guarantees that if two c correspondences e and f are morita equivalent then the tensor algebras of e and f taoeu and taofu are strongly morita equivalent in the sense of 8 the toeplitz algebras toeu and tofu are strongly morita equivalent in the sense of rieffel 32 see 33 also and the cuntz pimsner algebras 28 ooeu and oofu are strongly morita equivalent in the same sense we were inspired by earlier work of curto williams and the first author in 15 and combes 14 these papers investigate conditions implying that the crossed product of two c dynamical systems are strongly morita equivalent given that the coefficient algebras are strongly morita equivalent abadie eilers and exel 1 were similarly inspired and developed a theory of morita equivalence for special c correspondences our analysis extends theirs but our arguments are different see example 2 5 and remarks 3 3 and 3 6 for a comparison of their results and ours second we want to investigate the converse implication that is we study the following question if taoeu and taofu are strongly morita equivalent in the sense of 8 when are e and f morita equivalent in the sense we define for this converse direction we focus on the tensor algebras rather than the toeplitz and cuntz pimsner algebras because very little can be said in the self adjoint setting here we were inspired by arveson s paper 2 in which it is proved that two ergodic measure preserving transformations are conjugate if and only if certain non self adjoint crossed products associated with the transformations are algebraically isomorphic see 3 also his analysis was further extended by k saito and the first author to cover non self adjoint crossed products built from automorphisms of general von neumann algebras they showed that non self adjoint crossed products built from properly outer automorphisms of von neumann algebras are isomorphic if and only if the automorphisms are conjugate see 22 and 23 our principal result in the converse direction theorem 7 2 asserts that if e and f are two aperiodic correspondences and if taoeu is strongly morita equivalent to taofu in the sense of 8 then e and f are morita equivalent as we shall see in 5 the notion of aperiodicity is an exact analogue for correspondences of the concept of free action in ergodic theory free action was a central hypothesis in arveson s

chaos synchronization of the chua system with a fractional order chaos synchronization of two identical chua systems with the same fractional order is studied by utilizing the pecora carroll pc method the active passive decomposition pad method the one way coupling method and the bidirectional coupling one the sufficient conditions for achieving synchronization between these two systems are derived via the laplace transformation theory numerical simulations show the effectiveness of the theoretical analyses

topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique abstract this paper presents a simple but effective additive hyperelasticity technique to circumvent numerical difficulties in solving the material density based topology optimization of elastic structures undergoing large displacements by adding a special hyperelastic material to the design domain excessive distortion and numerical instability occurred in the low density or intermediate density elements are thus effectively alleviated during the optimization process the properties of the additional hyperelastic material are established based on a new interpolation scheme which allows the nonlinear mechanical behaviour of the remodelled structure to achieve an acceptable approximation to the original structure in conjunction with the adjoint variable scheme for sensitivity analysis the topology optimization problem is solved by a gradient based mathematical programming algorithm numerical examples are given to demonstrate the effectiveness of the proposed method

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

nonlinear dynamic response topology optimization using the equivalent static loads method abstract a novel method for nonlinear dynamic response topology optimization is proposed using the equivalent static loads esls method the esls are the loads that generate the same response field of linear static analysis as that of nonlinear dynamic analysis at each time step in the proposed procedure nonlinear dynamic analysis is performed esls are made and linear static topology optimization is carried out with the esls the process cyclically proceeds until the convergence criterion which is specifically defined for this problem is satisfied since the density method for topology optimization is utilized the low density finite elements can cause mesh distortion in nonlinear dynamic analysis transformation variables are introduced for a new update method for the incorporating process of the topology results into nonlinear dynamic analysis also a new objective function is proposed to minimize the peaks of the time dependent transient responses a couple of standard problems and a practical problem are solved to validate the proposed method

cp h extendable maps between hilbert modules and cph semigroups abstract one may ask which maps between hilbert modules allow for a completely positive extension to a map acting block wise between the associated extended linking algebras in these notes we investigate in particular those cp extendable maps where the 22 corner of the extension can be chosen to be a homomorphism the cp h extendable maps we show that they coincide with the maps considered by asadi 4 by bhat ramesh and sumesh 9 and by skeide 28 we also give an intrinsic characterization that generalizes the characterization by abbaspour tabadkan and skeide 1 of homomorphically extendable maps as those which are ternary homomorphisms for general strictly cp extendable maps we give a factorization theorem that generalizes those of asadi of bhat ramesh and sumesh and of skeide for cp h extendable maps this theorem may be viewed as a unification of the representation theory of the algebra of adjointable operators and the ksgns construction then we examine semigroups of cp h extendable maps so called cph semigroups as an application we illustrate their relation with a new sort of generalized dilation of cp semigroups cph dilations

theoretical aspects of the internal element connectivity parameterization approach for topology optimization the internal element connectivity parameterization i ecp method is an alternative approach to overcome numerical instabilities associated with low stiffness element states in non linear problems in i ecp elements are connected by zero length links while their link stiffness values are varied therefore it is important to interpolate link stiffness properly to obtain stably converging results the main objective of this work is two fold 1 the investigation of the relationship between the link stiffness and the stiffness of a domain discretizing patch by using a discrete model and a homogenized model and 2 the suggestion of link stiffness interpolation functions the effects of link stiffness penalization on solution convergence are then tested with several numerical examples the developed homogenized i ecp model can also be used to physically interpret an intermediate design variable state