on a fractional integral equation of periodic functions involving weyl riesz operator in banach algebras abstract in this paper we study the existence of periodic solutions for a nonlinear integral equation of periodic functions involving weyl riesz fractional integral operator under the mixed generalized lipschitz caratheodory and monotonicity conditions the fixed point theorems due to dhage are the main tool in carrying out our proofs

approximation orders for natural splines in arbitrary dimensions based on variational properties we generalize the approximation properties of the univariate natural cubic spline to splines in arbitrary dimensions

the use of dual reciprocity boundary element method in coupled thermoviscoelasticity a boundary element formulation is presented in a unified form for the analysis of thermoviscoelasticity problems the formulation contains the thermoelastic material as a special case the boundary only nature of boundary element method is retained through the use of particular integral method where the particular solutions are evaluated with the aid of dual reciprocity approximation the proposed formulation can be used in both coupled and uncoupled thermoviscoelasticity analyses and it permits performing the analysis in terms of fundamental solutions of viscoelastodynamics and diffusion thermal equation and eliminates the need for using the complicated fundamental solutions of coupled thermoviscoelasticity the formulation is performed in fourier space where any viscoelastic model can be simulated via the correspondence principle the determination of the response in time space requires the inversion which can be carried out conveniently by using the fast fourier transform algorithm for assessment some sample problems both uncoupled and coupled are considered and whenever possible comparisons are given with the exact data it is found that the formulation developed in the study even with the simplest base function proposed in literature may be used reliably in thermoviscoelasticity analysis at least for the problems with finite solution domains

the constant variation formulae for singular fractional differential systems with delay abstract this paper considers the caputo singular fractional differential systems with delay and the riemann liouville singular fractional differential systems with delay a new function is defined by the d inverse matrix and function two fundamental solutions are given the constant variation formulae for singular fractional differential systems with delay are obtained

approximation using scattered shifts of a multivariate function the approximation of a general d variate function f by the shifts r d of a fixed functionoccurs in many applications such as data fit ting neural networks and learning theory when hz d is a dilate of the integer lattice there is a rather complete understanding of the approximation problem 6 18 using fourier techniques however in most applications the center set is either given or can be chosen with complete freedom in both of these cases the shift invariant setting is too restrictive this paper studies the approximation problem in the caseis arbitrary it establishes approximation theorems whose error bounds reflect the local density of the points in two different settings are analyzed the first is when the setis prescribed in advance in this case the theorems of this paper show that in analogy with the classical univariate spline ap proximation improved approximation occurs in regions where the density is high the second setting corresponds to the problem of non linear approximation in that setting the setcan be chosen using information about the target function f we discuss how to best make these choices and give estimates for the approximation error

a mixed method for dirichlet problems with radial basis functions we present a simple discretization by radial basis functions for the poisson equation with dirichlet boundary condition a lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition this simplifies previous attempts to use radial basis functions in the interior domain to approximate the solution and on the boundary to approximate the multiplier which technically requires that the mesh norm in the interior domain is significantly smaller than that on the boundary numerical experiments confirm the theoretical results

a semi discrete scheme for solving nonlinear hyperbolic type partial integro differential equations using radial basis functions in this paper we propose an effective numerical method for solving nonlinear volterra partial integro differential equations these equations include the partial differentiations of an unknown function and the integral term containing the unknown function as the memory of system radial basis functions and finite difference method as the main techniques play the important role to reduce a nonlinear partial integro differential equation to a linear system of equations some examples are demonstrated to describe the method numerical results confirm the validity and efficiency of the presented method

eigenvalue and stability of singular differential delay systems the eigenvalue and the stability of singular differential systems with delay are considered firstly we investigate some properties of the eigenvalue then give the exact exponential estimation for the fundamental solution and finally discuss the necessary and sufficient condition of uniform asymptotic stability

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions multivariate interpolation of smooth data using smooth radial basis functions is considered the behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically explicit criteria for different types of limits are given using the results for the limits the dependence of the error on the shape parameter of the radial basis function is investigated the mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error

solutions of fractional multi order integral and differential equations using a poisson type transform abstract we consider a wide class of integral and ordinary differential equations of fractional multi orders 1 1 1 2 1 m depending on arbitrary parameters i 0 i r i 1 m denoting the differentiation operators by d d i i and by l l i i the corresponding integrations operators right inverse to d we first observe that d and l can be considered as operators of the generalized fractional calculus respectively as generalized fractional derivatives and integrals a solution of the homogeneous ode of this kind d y z y z 0 0 is the recently introduced multi index mittag leffler function e 1 i i z we find a poisson type integral transformation p generalizing the classical poisson integral formula that maps the cosm function into the multi index mittag leffler function and also transforms the simpler differentiation and integration operators of integer order m 1 dm d dz m and lm the m fold integration into the operators d and l thus from the known solution of the volterra type integral equation with the m fold integration lm via p as a transformation transmutation operator we find the corresponding solution of the integral equation y z l z f z then a solution of the fractional multi order differential equation d y z y z f z comes out in an explicit form as a series of integrals involving fox s h functions for each particularly chosen r h s function f z such a solution can be evaluated as an h function special cases of the equations considered here lead to solutions in terms of the mittag leffler bessel struve lommel and hyper bessel functions and some other known generalized hypergeometric functions

a meshless galerkin method for dirichlet problems using radial basis functions in this paper a numerical method is given for partial differential equations which combines the use of lagrange multipliers with radial basis functions it is a new method to deal with difficulties that arise in the galerkin radial basis function approximation applied to dirichlet also mixed boundary value problems convergence analysis results are given several examples show the efficiency of the method using tps or sobolev splines

robust stability analysis of fractional order uncertain singular nonlinear system with external disturbance this paper investigates robust stability for fractional order fo singular nonlinear systems the fo system is disturbed by external uncertainty and disturbance a central analysis technique is enabled by proposing a fundamental boundedness lemma for the first time this lemma is used for robust stability analysis of fo systems especially for mittag leffler stability analysis of fo nonlinear systems more importantly how to obtain a more accurate bound is given to reduce conservative an fo proportional derivative pd controller is proposed to normalize the fo singular system furthermore a criterion for stability of the normalized fo nonlinear systems is provided by linear matrix inequalities lmis finally two illustrative simulation examples are presented to illustrate effectiveness of the proposed stability notion

on the existence and uniqueness of solutions of a class of fractional differential equations abstract in this paper we investigate the existence and uniqueness of solutions for the following class of multi order fractional differential equations d 1 1 1 d n n n u t i 1 n d i i i u t d i n i i u t f t u t t 0 1 u 0 0 i 1 n i 1 i 0 i 0 1 i n where d i n i i denotes the generalized erdelyi kober operator of fractional derivative of order i moreover some properties concerning the positive maximal minimal and continuation of solutions are obtained

existence and uniqueness of holomorphic solutions for fractional cauchy problem by employing majorant functions the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations the cauchy problems are introduced furthermore the analytic continuation of solutions is studied

a fast procedure for model search in multidimensional contingency tables summary a procedure to select the simplest acceptable models for a multidimensional contingency table is proposed it is based on two rules first that if a model is accepted then all models that include it are considered to be accepted and secondly that if a model is rejected then all its submodels are considered to be rejected two versions are described one for the class of graphical models and the other for the class of hierarchical log linear models application of both versions to a six dimensional table is illustrated the procedure can be regarded as an alternative to fitting all possible models made computationally feasible by application of the two rules it is a generalization of the procedure proposed by havr anek 1984 but is in many cases considerably faster

the l 2 approximation order of surface spline interpolation we show that if the open bounded domain c r d has a sufficiently smooth boundary and if the data function f is sufficiently smooth then the l p norm of the error between f and its surface spline interpolant is o p 1 2 1 p where p min m m d 2 d p and m is an integer parameter specifying the surface spline in case p 2 this lower bound on the approximation order agrees with a previously obtained upper bound and so we conclude that the l 2 approximation order of surface spline interpolation is m 1 2

the controllability of fractional control systems with control delay this paper discusses the controllability of fractional control systems with control delay we firstly give the solution expression for fractional control systems with control delay then give the necessary and sufficient conditions for the controllability of fractional control systems with control delay

meshless galerkin method using radial basis functions based on domain decomposition in this present paper we will give an overview about the combination of rbf meshless method with domain decomposition especially galerkin method based on the analysis for this method before we will consider the combination of projection domain decomposition method with radial basis function both of galerkin approximation and collocation method are discussed some examples are given to show the efficiency of this method

operators on radial functions a general theory is provided that allows to write multi variate fourier transforms or convolutions of radial functions as very simple univariate operations as a byproduct an interesting group of operators i x r with i i i i i is defined it contains the classical derivatives as i1 d dr and is intimately connected to the fourier transform applications to the construction of new positive definite radial and to new identities for special functions are included

set valued system of fractional differential equations with hysteresis in this paper we investigate the global solvability in l 1 0 1 of a set valued system of nonlinear fractional differential equations with hysteresis some existence theorems for both single and multivalued systems are proved

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

a note on the meshless method using radial basis functions in this paper we prove some inverse inequalities for the finite dimensional subspace interpolated by radial basis functions the properties of these inequalities are worse than those in finite element spaces these bad properties result in a huge condition number for both the collocation method and galerkin method using radial basis functions as applications theoretical estimates of the condition numbers of a meshless method using radial basis functions are given

bem for transient 2d elastodynamics using multiquadric functions a new boundary element model for transient dynamic analysis of 2d structures is presented the dual reciprocity method drm is reformulated for the 2d elastodynamics by using the multiquadric radial basis functions mq the required kernels for displacement and traction particular solutions are derived some terms of these kernels are found to be singular therefore a new smoothing technique is proposed to solve this problem hence the limiting values of relevant kernels are computed the validity and strength of the proposed formulation are demonstrated throughout several numerical applications it is proven from the results that the present formulation is more stable than the traditional drm which uses the conical 1 r function especially in predicting results in the far time zone

solving nonlinear differential equations of mechanics with the boundary element method and radial basis functions the boundary element method is a very effective method for solving linear differential equations to use it also in the consideration of non linear problems some different procedures were developed among them the dual reciprocity method and the particular integral method both procedures use interpolation conditions for the approximation with radial basis functions in this paper a method is presented which avoids problems connected with interpolation by means of quasi interpolation it is possible to solve differential equations of the kind mu p u approximately the application to two non linear problems of plate theory yield good results hints to a theoretical examination of the method including sufficient conditions for feasibility and convergence are given 1997 by john wiley sons ltd

an improved order of approximation for thin plate spline interpolation in the unit disc we show that the l p norm of the error in thin plate spline interpolation in the unit disc decays like o h gamma p 1 2 where gamma p min 2 1 2 p under the assumptions that the function to be approximated is c infty and that the interpolation points contain the finite grid hj j in mathbb z 2 vert hj vert 1 h

error and stability estimates for surface divergence free rbf interpolants on the sphere recently a new class of surface divergence free radial basis function interpolants has been developed for surfaces in 3 in this paper several approximation results for this class of interpolants will be derived in the case of the sphere s 2 in particular sobolev type error estimates are obtained as well as optimal stability estimates for the associated interpolation matrices in addition a bernstein estimate and an inverse theorem are also derived numerical validation of the theoretical results is also given

the uniform convergence of thin plate spline interpolation in two dimensions let f be a function from cal r 2 to cal r that has square integrable second derivatives and let s be the thin plate spline interpolant to f at the points underline v i i 1 2 ldots n in cal r 2 we seek bounds on the error f underline x s underline x when underline x is in the convex hull of the interpolation points or when underline x is close to at least one of the interpolation points but need not be in the convex hull we find for example that if underline x is inside a triangle whose vertices are any three of the interpolation points then f underline x s underline x is bounded above by a multiple of h where h is the length of the longest side of the triangle and where the multiplier is independent of the interpolation points further if cal d is any bounded set in cal r 2 that is not a subset of a single straight line then we prove that a sequence of thin plate spline interpolants converges to f uniformly on cal d specifically we require h rightarrow 0 where h is now the least upper bound on the numbers d underline x cal v underline x in cal d and where d underline x cal v underline x in cal r 2 is the least euclidean distance from underline x to an interpolation point our method of analysis applies integration by parts and the cauchy schwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation

controlled rounding and cell perturbation statistical disclosure limitation methods for tabular data rounding methods are common techniques in many statistical offices to protect sensitive information when publishing data in tabular form classical versions of these methods do not consider protection levels while searching patterns with minimum information loss and therefore typically the so called auditing phase is required to check the protection of the proposed patterns this paper presents a mathematical model for the whole problem of finding a protected pattern with minimum loss of information and proposes a branch and cut algorithm to solve it it also describes a new methodology closely related to the classical controlled rounding methods but with several advantages the new methodology is named cell perturbation and leads to a different optimization problem which is simpler to solve than the previous problem this paper presents a cutting plane algorithm for finding an exact solution of the new problem which is a pattern guaranteeing the same protection level requirements but with smaller loss of information when compared with the classical controlled rounding optimal patterns the auditing phase is unnecessary on the solutions generated by the two algorithms the paper concludes with computational results on real world instances and discusses a modification in the objective function to guarantee statistical properties in the solutions

obtaining cell counts for contingency tables from rounded conditional frequencies abstract we present an integer linear programming formulation and solution procedure for determining the tightest bounds on cell counts in a multi way contingency table given knowledge of a corresponding derived two way table of rounded conditional probabilities and the sample size the problem has application in statistical disclosure limitation which is concerned with releasing useful data to the public and researchers while also preserving privacy and confidentiality previous work on this problem invoked the simplifying assumption that the conditionals were released as fractions in lowest terms rather than the more realistic and complicated setting of rounded decimal values that is treated here the proposed procedure finds all possible counts for each cell and runs fast enough to handle moderately sized tables

a comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations abstract in this article we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations a series of test examples was solved with these two schemes different problems with different type of governing equations and boundary conditions particular emphasis was paid to the ability of these schemes to solve the steady state convection diffusion equation at high values of the peclet number from the examples tested in this work it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better however the unsymmetric method has the advantage of being simpler to implement two main features about the results obtained in this work are worthy of special attention first with the symmetric method it was possible to solve convection diffusion problems at a very high peclet number without the need of any artificial damping term and second with these two approaches symmetric and unsymmetric it is possible to impose free boundary conditions for problems in unbounded domains

perturbed kernel approximation on homogeneous manifolds current methods for interpolation and approximation within a native space rely heavily on the strict positive definiteness of the underlying kernels if the domains of approximation are the unit spheres in euclidean spaces then zonal kernels kernels that are invariant under the orthogonal group action are strongly favoured in the implementation of these methods to handle real world problems however some or all of the symmetries and positive definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution perturbation analysis is therefore needed to address the stability problem encountered in this paper we study two kinds of perturbations of positive definite kernels small random perturbations and perturbations by dunkl s intertwining operators c dunkl y xu orthogonal polynomials of several variables encyclopedia of mathematics and its applications vol 81 cambridge university press cambridge 2001 we show that with some reasonable assumptions a small random perturbation of a strictly positive definite kernel can still provide vehicles for interpolation and enjoy the same error estimates we examine the actions of the dunkl intertwining operators on zonal strictly positive definite kernels on spheres we show that the resulted kernels are strictly positive definite on spheres of lower dimensions

inverse and saturation theorems for radial basis function interpolation while direct theorems for interpolation with radial basis functions are intensively investigated little is known about inverse theorems so far this paper deals with both inverse and saturation theorems for an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use in case of thin plate spline interpolation we also give certain saturation theorems

improved accuracy of lp approximation to derivatives by radial basis function interpolation the present paper considers the approximation to a function and its derivatives by radial basis function interpolation and its derivatives respectively on the sobolev space it is known that due to edge effects we lose some accuracy near the boundary thus the goal of this paper is to show that the convergence rate of the approximation error can be at least doubled when a certain boundary condition is met

scattered data interpolation on embedded submanifolds with restricted positive definite kernels sobolev error estimates in this paper we present error estimates for kernel interpolation at scattered sites on manifolds the kernels we consider will be obtained by the restriction of positive definite kernels on math

some exact solutions to stefan problems with fractional differential equations some exact solutions to the first second and extended stefan problems with fractional time derivative described in the caputo sense are given by means of fractional green s function and wright function in this paper by the aid of simple calculations many results of differential equations of integer order can be obtained as special cases of the results given by this paper

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

the drm md integral equation method an efficient approach for the numerical solution of domain dominant problems this work presents a multi domain decomposition integral equation method for the numerical solution of domain dominant problems for which it is known that the standard boundary element method bem is in disadvantage in comparison with classical domain schemes such as finite difference fdm and finite element fem methods as in the recently developed green element method gem in the present approach the original domain is divided into several subdomains in each of them the corresponding green s integral representational formula is applied and on the interfaces of the adjacent subregions the full matching conditions are imposed in contrast with the gem where in each subregion the domain integrals are computed by the use of cell integration here those integrals are transformed into surface integrals at the contour of each subregion via the dual reciprocity method drm using some of the most efficient radial basis functions known in the literature on mathematical interpolation in the numerical examples presented in the paper the contour elements are defined in terms of isoparametric linear elements for which the analytical integrations of the kernels of the integral representation formula are known as in the fem and gem the obtained global matrix system possesses a banded structure however in contrast with these two methods gem and non hermitian fem here one is able to solve the system for the complete internal nodal variables i e the field variables and their derivatives without any additional interpolation finally some examples showing the accuracy the efficiency and the flexibility of the method for the solution of the linear and non linear convection diffusion equation are presented copyright 1999 john wiley sons ltd

a unified mathematical programming framework for different statistical disclosure limitation methods this paper concerns statistical disclosure control methods to minimize information loss while keeping small the disclosure risk from different data snoopers this issue is of primary importance in practice for statistical agencies when publishing data it is assumed that the sensitive data have been identified by practitioners in the statistical offices and the paper addresses the secondary problem of protecting these data with different methods all defined in a unified mathematical framework a common definition of protection is used in four different methodologies two integer linear programming models described in the literature for the cell suppression methodology are extended to work also for the controlled rounding methodology in addition two relaxed variants are presented using two associated linear programming models called partial cell suppression and partial controlled rounding respectively a final discussion shows how to combine the four methods and how to implement a cutting plane approach for the exact and heuristic resolution of the combinatorial problems in practice this approach was implemented in argus a software package of disclosure limitation tools

a unified theory of radial basis functions native hilbert spaces for radial basis functions ii abstract this contribution provides a new formulation of the theory of radial basis functions in the context of integral operators instead of fourier transforms the most important tools now are expansions into eigenfunctions this unifies the theory of radial basis functions in r d with the theory of zonal functions on the sphere sd 1 and the theory of kernel functions on riemannian manifolds new characterizations of native spaces and positive definite functions are included the paper is a self contained continuation of an earlier survey r schaback international series of numerical mathematics vol 132 birkhauser basel 1999 pp 255 282 over the native spaces associated to not necessarily radial basis functions

an application of the ltdrm to transient diffusion problems with nonlinear material properties and nonlinear boundary conditions in this paper we shall present an application of the laplace transform dual reciprocity method ltdrm to the solution of transient diffusion problems with nonlinear material properties and nonlinear boundary conditions two methods with and without utilizing the kirchhoff transformation will be discussed the nonlinear terms in both governing equation and boundary conditions are linearized first followed by the removal of the time variable from the linearized differential system with the laplace transformation and then the problem is solved by the dual reciprocity method as a result a solution at any specific time can be obtained without step by step calculations in time domain and computations of domain integrals to show the efficiency and accuracy of the present method several problems are analyzed and their solutions are compared with those obtained either analytically or numerically

some results of the degenerate fractional differential system with delay an analytic study on linear systems of degenerate fractional differential equations with constant coefficients is presented we discuss the existence and uniqueness of solutions for the initial value problem of linear degenerate fractional differential systems the exponential estimation of the degenerate fractional differential system with delay and sufficient conditions for the finite time stability for the system are obtained finally an example is provided to illustrate the effectiveness of the presented analytical approaches

controlled rounding of tabular data controlled rounding is a procedure that perturbs tabular data collected from respondents in such a way as to preserve the anonymity of the respondents while maintaining the integrity of the data the three dimensional controlled rounding problem is np complete this paper describes a binary tree search algorithm for solving three dimensional problems based on linear programming techniques computational results obtained from tests with over 31 000 randomly generated tables and 292 real life tables have shown that this algorithm can effectively find controlled roundings when they exist or determine that no solution exists the computational results also demonstrate that the running time of the search algorithm can be cut in half by using a heuristic for initializing the simplex basis for the linear programming problems

dynamical analysis of fractional order modified logistic model in this paper we study a fractional differential equation model of the single species multiplicative allee effect first we study the stability of equilibrium points further we give some sufficient conditions ensuring the existence and uniqueness of integral solution in the last section we perform several numerical simulations to validate our analytical findings

drm multidomain mass conservative interpolation approach for the bem solution of the two dimensional navier stokes equations abstract the dual reciprocity method drm is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals in this approach the nonlinear terms are usually approximated by an interpolation applied to the convective terms of the navier stokes equations in this paper we introduce a radial basis function interpolation scheme for the velocity field that satisfies the continuity equation mass conservative the proposed method performs better than the classical interpolation used in the drm approach to represent such a field the new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the nonlinear terms in the navier stokes equations

all well posed problems have uniformly stable and convergent discretizations this paper considers a large class of linear operator equations including linear boundary value problems for partial differential equations and treats them as linear recovery problems for functions from their data well posedness of the problem means that this recovery is continuous discretization recovers restricted trial functions from restricted test data and it is well posed or stable if this restricted recovery is continuous after defining a general framework for these notions this paper proves that all well posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well posedness of the problem and independent of the computational discretization provided that sufficiently many test data are used the solutions of discretized problems converge when enlarging the trial spaces and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions this allows new and very simple proofs of convergence rates for generalized finite elements symmetric and unsymmetric kansa type collocation and other meshfree methods like meshless local petrov galerkin techniques it is also shown that for a fixed trial space weak formulations have a slightly better convergence rate than strong formulations but at the expense of numerical integration since convergence rates are reduced to those coming from approximation theory and since trial spaces are arbitrary this also covers various spectral and pseudospectral methods all of this is illustrated by examples

regularization of linear descriptor systems with variable coefficients we study linear descriptor control systems with rectangular variable coefficient matrices we introduce condensed forms for such systems under equivalence transformations and use these forms to detect whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback we show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system plus some solution components that are constrained to be zero

improved error bounds for scattered data interpolation by radial basis functions if additional smoothness requirements and boundary conditions are met the well known approximation orders of scattered data interpolants by radial functions can roughly be doubled

high order numerical derivatives for one dimensional scattered noisy data based on the smoothing spline approximation in this paper we propose a regularization method for computing high order numerical derivatives from one dimensional noisy data the convergence rates under two different choices of the regularization parameter are obtained numerical examples show that the proposed method is effective and stable

on systems of linear fractional differential equations with constant coefficients abstract this paper deals with the study of linear systems of fractional differential equations such as the following system 1 y a x y b x where y is the riemann liouville or the caputo fractional derivative of order 0 2 a x a 11 x a 1 n x a n 1 x a nn x b x b 1 x b n x are matrices of known real functions in a way analogous to the usual case we show how a generalized matrix exponential function and certain fractional green function in connection with the mittag leffler type functions would allow us to obtain an explicit representation of the general solution to the system 1 when a is a constant matrix

l p error estimates for shifted surface spline interpolation on sobolev space the accuracy of interpolation by a radial basis function is usually very satisfactory provided that the approximant f is reasonably smooth however for functions which have smoothness below a certain order associated with the basis function no approximation power has yet been established hence the purpose of this study is to discuss the lp approximation order 1 p of interpolation to functions in the sobolev space wpk with k max 0 d 2 d p we are particularly interested in using the shifted surface spline which actually includes the cases of the multiquadric and the surface spline moreover we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met

fractional order nonlinear systems with delay in iterative learning control in this paper we discuss a p type iterative learning control ilc scheme for a class of fractional order nonlinear systems with delay by introducing the norm and using gronwall inequality the sufficient condition for the robust convergence of the tracking errors is obtained based on this convergence the p type ilc updating laws can be determined

local mass conservative hermite interpolation for the solution of flow problems by a multi domain boundary element approach abstract this paper presents a local hermite radial basis function interpolation scheme for the velocity and pressure fields the interpolation for velocity satisfies the continuity equation mass conservative interpolation while the pressure interpolation obeys the pressure equation additionally the dual reciprocity boundary element method drbem is applied to obtain an integral representation of the navier stokes equations then the proposed local interpolation is used to obtain the values of the field variables and their partial derivatives at the boundary of the sub domains this interpolation allows one to obtain the boundary values needed for the integral formulas for velocity and pressure at some nodes within the sub domains in the proposed approach the boundary elements are merely used to parameterize the geometry but not for the evaluation of the integrals as it is usually done the presented multi domain approach is different from the traditional ones in boundary elements because the resulting integral equations are non singular and the boundary data needed for the boundary integrals are approximated using a local interpolation some accurate results for simple stokes problems and for the navier stokes equations at low reynolds numbers up to re 400 were obtained

meshless method with ridge basis functions abstract meshless collocation methods for the numerical solutions of pdes are increasingly adopted due to their advantages including efficiency and flexibility and radial basis functions are popularly employed to represent the solutions of pdes motivated by the advantages of ridge basis function representation of a given function such as the connection to neural network fast convergence as the number of terms is increased better approximation effects and various applications in engineering problems a meshless method is developed based on the collocation method and ridge basis function interpolation this method is a truly meshless technique without mesh discretization it neither needs the computation of integrals nor requires a partition of the region and its boundary moreover the method is applied to elliptic equations to examine its appropriateness numerical results are compared to that obtained from other meshless methods and influence factors of accuracy for numerical solutions are analyzed

free mesh radial basis function collocation approach for the numerical solution of system of multi ion electrolytes the main objective of this work is to show the versatility of an innovative mesh free numerical scheme which allows the solution of the coupled transport equations governing the evolution of a system of multi ion electrolytes by mean of the use of radial basis functions interpolation and a nesting approach the nesting approach is used in order to capture the detail behaviour of the solution near a solid wall i e at the diffusion reaction layer as a numerical example the problem of silver deposition on a rotating disk is analysed the versatility of the proposed numerical scheme is proved by solving the proposed electrochemical phenomena imposing three different kinds of boundary conditions at the solid wall dirichlet neumann and robin the obtained numerical results are compared with simplified analytical approximation of the corresponding problems the effect of the ligand concentration on the others species is considered and a comparison between the obtained concentration profiles for the complex ags2o and ag s2o3 when the concentration of the ligand s2o takes different values is shown copyright 2005 john wiley sons ltd

radial basis function interpolation in the quantum trajectory method optimization of the multi quadric shape parameter this paper investigates multi quadric radial basis function rbf interpolation and its application in the quantum trajectory method qtm for wave packet propagation in the multi quadric r r2 2 1 2 r is the radial distance from the observation point to the origin of the basis function and is known as the shape parameter due to its affect on the functional form of the basis function the quality of any rbf interpolation scheme is dictated by the choice of this parameter many recent studies have investigated a suitable means for obtaining an optimized time independent parameter the purpose of this study however is to not only to find this optimized shape parameter but also to analyze its time dependence in four different dynamical models the anisotropic harmonic oscillator the downhill ramp the uphill ramp and a harmonic oscillator coupled with a downhill ramp to obtain the optimized shape parameter at each time step an algorithm similar to the leave one out method of cross validation is utilized the results for each of the four models are presented and the feasibility and necessity of employing a shape parameter optimization algorithm for each of the models is discussed

on the optimal shape parameters of radial basis functions used for 2 d meshless methods abstract a radial point interpolation meshless or radial pim method was proposed by authors to overcome the possible singularity associated with only polynomial basis the radial pim used multiquadric mq or gaussian as basis functions these two radial basis functions all included shape parameters although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed how these shape parameters affect the accuracy of the radial pim has not been studied yet this paper studied the effect of shape parameters on the numerical accuracy of radial pim a range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix error of energy and irregularity of node distribution it is observed that the widely used shape parameters for mq and reciprocal mq basis are not even close to their optimums the optimal shape parameters are found in this paper to be simply q 1 03 and r 1 42 for mq basis and c 0 003 0 03 for gaussian basis

on condition number of meshless collocation method using radial basis functions in this paper we will give an estimate for the condition number of meshless collocation method using radial basis functions