the overlapped radial basis function finite difference rbf fd method a generalization of rbf fd abstract we present a generalization of the rbf fd method that computes rbf fd weights in finite sized neighborhoods around the centers of rbf fd stencils by introducing an overlap parameter 0 1 such that 1 recovers the standard rbf fd method and 0 results in a full decoupling of stencils we provide experimental evidence to support this generalization and develop an automatic stabilization procedure based on local lebesgue functions for the stable selection of stencil weights over a wide range of values we provide an a priori estimate for the speedup of our method over rbf fd that serves as a good predictor for the true speedup we apply our method to parabolic partial differential equations with time dependent inhomogeneous boundary conditions neumann in 2d and dirichlet in 3d our results show that our method can achieve as high as a 60 speedup in 3d over existing rbf fd methods in the task of forming differentiation matrices

a galerkin radial basis function method for the schrodinger equation in this article we consider the discretization of the time dependent schrodinger equa tion using radial basis functions rbfs we formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions since we can show that time stability of the discretization is not guaranteed for an rbf collocation method we propose to employ a galerkin ansatz instead for gaussians it is shown that exponential convergence is obtained up to a point where a systematic error from the domain where no basis functions are centered takes over the choice of the shape parameter and of the resolved region is studied numerically compared to the fourier method with periodic boundary conditions the basis functions can be centered in a smaller domain which gives increased accuracy with the same number of points

image reconstruction from limited range projections using orthogonal moments a set of orthonormal polynomials is proposed for image reconstruction from projection data the relationship between the projection moments and image moments is discussed in detail and some interesting properties are demonstrated simulation results are provided to validate the method and to compare its performance with previous works

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

approximation on non tensor domains including squircles part iii single domain spectral methods have been largely restricted to tensor product bases on a tensor product grid to break the tensor barrier we study approximation in a domain bounded by a squircle the zero isoline of b x y x 2 y 2 1 the boundary varies smoothly from a circle 1 to the square polynomial least squares hyperinterpolation converges geometrically as long as the number of points p is at least double the number of basis functions n the polynomial grid was made denser near the boundaries chebyshev like by depositing grid points along wisely chosen contours of b gaussian radial basis functions rbfs were more robust in the sense that they too converged geometrically but hyperinterpolation p n and a chebyshevized grid were unnecessary a uniform grid truncated to include only those points within the squircle was satisfactory even without interpolation points on the boundary although boundary points are a cost effective improvement for a given number of points p however rbf interpolation was only slightly more accurate than polynomial hyperinterpolation and needed twice as many basis functions interpolation costs can be greatly reduced by exploiting the invariance of the squircle bounded domain to the eight elements d 4 dihedral group

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

error bounds for gmls derivatives approximations of sobolev functions this paper provides the error estimates for generalized moving least squares gmls derivatives approximations of a sobolev function in l p norms and extends them for local weak forms of dmlpg methods sometimes they are called diffuse or uncertain derivatives but precisely they are direct approximants of exact derivatives which possess the optimal rates of convergence gmls derivatives approximations are different from the standard derivatives of mls approximation while they are much easier to evaluate at considerably lower cost in this paper the same orders of convergence with comparison to the standard derivatives are obtained for them

quadratic serendipity finite elements over convex polyhedra summary the conventional approach to construct quadratic elements for an n sided polygon will yield n n 1 2 shape functions which increases the computational effort it is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion floater and lai proposed a systematic way to construct higher order approximations over arbitrary polygons using the generalized barycentric coordinates and triangular coordinates this approach ensures 2n shape functions with nodes only on the boundary of the polygon in this paper we extend the polygonal splines approach to three dimensions and construct serendipity shape functions over hexahedra and convex polyhedra this is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates the quadratic shape functions possess kronecker delta property and satisfy constant linear and quadratic precision the accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests the numerical results show that the quadratic serendipity elements pass the patch test yield optimal convergence rate and can tolerate extreme mesh distortion this article is protected by copyright all rights reserved

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

maximal attractor and inertial set for eguchi oki matsumura equation abstract we consider the eguchi oki matsumura equation describing phase transition in binary alloys we show that the corresponding semigroup possesses a maximal attractor and prove the existence of an inertial set

development and validation of a 3d rbf spectral model for coastal wave simulation abstract with the objective of simulating wave propagation in the nearshore zone for engineering scale applications a two dimensional 2dv model based on the euler zakharov equations 73 54 is extended to three dimensions 3d to maintain the flexibility of the approach with the goal of applying the model to irregularly shaped domains the horizontal plane is discretized with scattered nodes the horizontal derivatives are then estimated using the radial basis function finite difference rbf fd method while a spectral approach is used in the vertical dimension a sensitivity analysis examined the robustness of the rbf fd approach as a function of rbf parameters when estimating the derivatives of a representative function for a targeted stencil size between 20 and 30 nodes piecewise smooth ps polyharmonic spline phs functions are recommended avoiding the use of infinitely smooth is rbfs which are less appropriate for the desired applications because of their dependence on a shape parameter comparisons of simulation results to observations from two wave basin experiments show that nonlinear effects induced by complex bottom bathymetries are reproduced well by the model with the recommended rbf approach validating the use of this method for 3d simulations of wave propagation

fractional time quantum dynamics application of the fractional calculus to quantum processes is presented in particular the quantum dynamics is considered in the framework of the fractional time schr odinger equation se which differs from the standard se by the fractional time derivative ensuremath partial ensuremath partial t ensuremath rightarrow ensuremath partial ensuremath alpha ensuremath partial t ensuremath alpha it is shown that for ensuremath alpha 1 2 the fractional se is isospectral to a comb model an analytical expression for the green s functions of the systems are obtained the semiclassical limit is discussed

evolution equations for phase separation and ordering in binary alloys we explore two phenomenological approaches leading to systems of coupled cahn hilliard and cahn allen equations for describing the dynamics of systems which can undergo first order phase separation and order disorder transitions simultaneously starting from the same discrete lattice free energy function in the first approach a quasicontinuum limit is taken for this discrete energy and the evolution of the system is then assumed to be given by gradient flow in the second approach a discrete set of gradient flow evolution equations is derived for the lattice dynamics and a quasicontinuum limit is then taken we demonstrate in the context of bcc fe al binary alloys that it is important that variables be chosen that accommodate the variations in the average concentration as well as the underlying ordered structure of the possible coexistent phases only then will the two approaches lead to roughly the same continuum descriptions we conjecture that in general the number of variables necessary to describe the dynamics of such systems is equal ton 1 n 2 1 wheren 1 is given by the dimension of the span of the bases of the irreducible representations needed to describe the symmetry groups of the possible equilibrium phases andn 2 is the number of chemical components n 1 of these variables are nonconserved and the remaining are conserved and represent the average concentrations for the fe al alloys this implies a description of one conserved order parameter and one nonconserved order parameter the resultant description is given by a cahn hilliard equation coupled to a cahn allen equation via the lower order nonlinear terms the rough equivalence of the two phenomenological methods adds credibility to the validity of the resulting evolution equations a similar description should also be valid for alloy systems in which the structure of the competing phases is more complicated

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

time dependent schrodinger like equation with nonlocal term we investigate a time dependent schrodinger like equation in presence of a nonlocal term by using the method of variable separation and the green function approach we analyze the green function for different forms of the memory kernel and the nonlocal term results for delta potential energy function are presented distributed order memory kernels are also considered and the asymptotic behaviors of the green function are derived by using tauberian theorem the obtained results for the green function for the considered schrodinger like equation may be transformed to those for the probability distribution function of a diffusion like equation with memory kernel and can be used to explain various anomalous diffusive behaviors

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

the fractional finite hankel transform and its applications in fractal space in the present work a generalized finite hankel transform is derived which is useful in solving equations in fractal dimension df and involving a fractal diffusion coefficient d0r the corresponding inversion formula is established and some properties are given then the transform is successfully used to solve a class of time fractional diffusion equations in fractional spatial dimension with an absorbent term and schrodinger equation in fractional dimensional space green s functions and exact wave function of the above problems are found

time dependent solutions for a fractional schrodinger equation with delta potentials we investigate for an arbitrary initial condition the time dependent solutions for a fractional schrodinger equation in the presence of delta potentials by using the green function approach the solutions obtained show an anomalous spreading asymptotically characterized by a power law behavior which is governed by the order of the fractional spatial operator present in the schrodinger equation

a radial basis functions method for fractional diffusion equations one of the ongoing issues with fractional diffusion models is the design of an efficient high order numerical discretization this is one of the reasons why fractional diffusion models are not yet more widely used to describe complex systems in this paper we derive a radial basis functions rbf discretization of the one dimensional space fractional diffusion equation in order to remove the ill conditioning that often impairs the convergence rate of standard rbf methods we use the rbf qr method 1 33 by using this algorithm we can analytically remove the ill conditioning that appears when the number of nodes increases or when basis functions are made increasingly flat the resulting rbf qr based method exhibits an exponential rate of convergence for infinitely smooth solutions that is comparable to the one achieved with pseudo spectral methods we illustrate the flexibility of the algorithm by comparing the standard rbf and rbf qr methods for two numerical examples our results suggest that the global character of the rbfs makes them well suited to fractional diffusion equations they naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from a second order to a fractional order diffusion model as such they should be considered as one of the methods of choice to discretize fractional diffusion models of complex systems

moment computation for objects with spline curve boundary a new approach is proposed for computation of area and geometric moments for a plane object with a spline curve boundary the explicit formulae are obtained for area and low order moment calculation the complexity of calculation depends on the moment order spline degree and the number of control points used in spline representation the formulae proposed use the advantage that the sequence of spline control points is cyclic it allowed us to reduce substantially the number of summands in them the formulae might be useful in different applications where it is necessary to perform measurements for shapes with a smooth boundary

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

some solutions to the space fractional schrodinger equation using momentum representation method the space fractional schrodinger equation with linear potential delta function potential and coulomb potential is studied under momentum representation using fourier transformation by use of mellin transform and its inverse transform we obtain the energy levels and wave functions expressed in h function for a particle in linear potential field the wave function expressed also by the h function and the unique energy level of the bound state for the particle of even parity state in delta function potential well which is proved to have no action on the particle of odd parity state is also obtained the integral form of the wave functions for a particle in coulomb potential field is shown and the corresponding energy levels which have been discussed in laskin s paper phys rev e 66 056108 2002 are proved to satisfy an equality of infinite limit of the h function all of these results contain the ones of the standard quantum mechanics as their special cases

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

stable integration rules with scattered integration points a general method for near best approximations to functionals on r d using scattered data information is applied for producing stable multidimensional integration rules the rules are constructed to be exact for polynomials of degree 6m and for a quasi uniform distribution of the integration points it is shown that the approximation order is o h m 1 where h is an average distance between the data points c 1999 elsevier science b v all rights reserved

fractional schrodinger equation with noninteger dimensions the spatial and time dependent solutions of the schrodinger equation incorporating the fractional time derivative of distributed order and extending the spatial operator to noninteger dimensions are investigated they are obtained by using the green function approach in two situations the free case and in the presence of a harmonic potential the results obtained show an anomalous spreading of the wave packet which may be related to an anomalous diffusion process

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

on the role of polynomials in rbf fd approximations iii behavior near domain boundaries abstract radial basis function generated finite difference rbf fd approximations generalize grid based regular finite differences to scattered node sets these become particularly effective when they are based on polyharmonic splines phs augmented with multi variate polynomials phs poly one key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill conditioning the strengths of this approach were previously shown to be especially striking for approximations near domain boundaries where the stencils become highly one sided due to the polynomial runge phenomenon regular fd approximations of high accuracy will in such cases have very large weights well into the domain the inclusion of phs type rbfs in the process of generating weights makes it possible to avoid this adverse effect with that as motivation this study aims at gaining a better understanding of the behavior of phs poly generated rbf fd approximations near boundaries illustrating it in 1 d 2 d and 3 d

time fractional schrodinger equation fox s h functions and the effective potential after introducing the formalism of the general space and time fractional schrodinger equation we concentrate on the time fractional schrodinger equation and present new results via the elegant language of fox s h functions we show that the general time dependent part of the wave function for the separable solutions of the time fractional schrodinger equation is the mittag leffler function with an imaginary argument by two different methods after separating the mittag leffler function into its real and imaginary parts in contrast to existing works we show that the total probability is 1 and decays with time introducing the effective potential approach we also write the mittag leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts in the light of these we reconsider the simple box problem

exact solutions of fractional schrodinger like equation with a nonlocal term we study the time space fractional schrodinger equation with a nonlocal potential by the method of fourier transform and laplace transform the green function and hence the wave function is expressed in terms of h functions graphical analysis demonstrates that the influence of both the space fractal parameter and the nonlocal parameter on the fractional quantum system is strong indeed the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative

an algebraic cubature formula on curvilinear polygons abstract we implement in matlab a gauss like cubature formula on bivariate domains whose boundary is a piecewise smooth jordan curve curvilinear polygons the key tools are green s integral formula together with the recent software package chebfun to approximate the boundary curve close to machine precision by piecewise chebyshev interpolation several tests are presented including some comparisons of this new routine chebfungauss with the recent splinegauss that approximates the boundary by splines

parameter estimation for the fractional schrodinger equation using bayesian method in this paper the fractional schrodinger equation is studied the bayesian method is put forward to estimate some relevant parameters of the equation results show that the estimated values can fit well with the exact solution the varying initial values and maximum iterations have little effect on the estimated results it indicates that the bayesian method is efficient for the multi parameter estimation for the fractional schrodinger equation this method can also be used to estimate parameters for the fractional schrodinger equation in other potential field

the time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems abstract in this paper a time fractional fourier law is obtained from fractional calculus according to the fractional fourier law a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built the fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases in addition we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized h function using integral transformation methods the fractional heat conduction equation in the case 0 1 interpolates the standard heat conduction equation 1 and the localized heat conduction equation 0 finally numerical results are presented graphically for various values of order of fractional derivative

cad and mesh repair with radial basis functions in this paper we present a process that includes both model mesh repair and mesh generation the repair algorithm is based on an initial mesh that may be either an initial mesh of a dirty cad model or stl triangulation with many errors such as gaps overlaps and t junctions this initial mesh is then remeshed by computing a discrete parametrization with radial basis functions rbf s we showed in 1 that a discrete parametrization can be computed by solving partial differential equations pde s on an initial correct mesh using finite elements paradoxically the meshless character of the rbf s makes it an attractive numerical method for solving the pde s for the parametrization in the case where the initial mesh contains errors or holes in this work we implement the orthogonal gradients method to be described in 2 as a rbf solution method for solving pde s on arbitrary surfaces different examples show that the presented method is able to deal with errors such as gaps overlaps t junctions and that the resulting meshes are of high quality moreover the presented algorithm can be used as a hole filling algorithm to repair meshes with undesirable holes the overall procedure is implemented in the open source mesh generator gmsh 3

fast computation of legendre moments of polyhedra abstract the three dimensional 3d orthogonal moments are an efficient tool for object reconstruction and 3d image analysis however until now 3d orthogonal moments have not been analysed in detail from the point of view of reducing the computational complexity in this paper we present a recursive algorithm for fast computation of legendre moments of polyhedra first a gaussian theorem is employed to transform the volume integral into a surface one the double integral can then be deduced from the simple integral by a green s theorem finally the recursive relationship is investigated as one can see the proposed method decreases the computational complexity tremendously

image analysis by modified legendre moments in the paper a new set of orthogonal moments based on the modified legendre polynomials is introduced three properties of the modified legendre polynomials which are orthogonality orthogonal invariance and the characteristic that an interval on the center of 1 1 covers more zeros than do that on the edge of 1 1 are discussed detailedly the orthogonality of the proposed moments ensures minimal information redundancy in a moment set the orthogonal invariance of the proposed polynomials makes the proposed moments have the property of translation invariance and the third property ensures that the modified legendre moments have superior feature representation capabilities over legendre moments in analyzing small images for small images the description by the modified legendre moments is better than that by the legendre moments and the chebyshev moments in terms of image reconstruction errors theoretical and experimental measures of performance are carried out to investigate the image representation capabilities of the proposed moments for images and noisy images the computational aspects of the moments using recurrence integral symmetry and translation invariance are also discussed experimental results are shown

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

meshless cubature over the disk using thin plate splines in some recent papers the construction of meshless interpolatory cubature formulas using radial basis functions has been studied in particular thin plate splines allow us to conveniently use green s formula and give good results on scattered samples of small moderate size over polygons here we discuss the extension to meshless cubature over the disk and its practical implementation

comparison of adaptive sampling methods for generation of surrogate aerodynamic models a surrogate modeling strategy using effective interpolation and sampling methods facilitates a reduction in the number of computational fluid dynamics simulations required to construct an aerodynamic model to a specified accuracy in this paper two adaptive sampling strategies are compared for generating surrogate models based on kriging and radial basis function interpolation respectively the relationships between the two model formulations are discussed and three test cases are considered including analytic functions and recovery of aerodynamic coefficients for two example applications longitudinal flight mechanics analysis for the dlr f12 aircraft and structural loads analysis of an rae2822 airfoil for the airfoil example models of cl cd and cm were constructed with the two sampling strategies using euler boundary layer coupled computational fluid dynamics and a three dimensional flight envelope of incidence mach and reynolds number the two sampling approaches direct some samples toward

polynomials and potential theory for gaussian radial basis function interpolation we explore a connection between gaussian radial basis functions and polynomials using standard tools of potential theory we find that these radial functions are susceptible to the runge phenomenon not only in the limit of increasingly flat functions but also in the finite shape parameter case we show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices while not imposing restrictions on the shape parameter or number of points

an insight into rbf fd approximations augmented with polynomials abstract radial basis function generated finite differences rbf fd based on the combination of polyharmonic splines phs with high degree polynomials have recently emerged as a powerful and robust numerical approach for the local interpolation and derivative approximation of functions over scattered node layouts among the key features i high orders of accuracy can be achieved without the need of selecting a shape parameter or the issues related to numerical ill conditioning and ii the harmful edge effects associated to the use of high order polynomials better known as runge s phenomenon can be overcome by simply increasing the stencil size for a fixed polynomial degree the present study complements our previous results providing an analytical insight into rbf fd approximations augmented with polynomials it is based on a closed form expression for the interpolant which reveals the mechanisms underlying these features including the role of polynomials and rbfs in the interpolant the approximation error and the behavior of the cardinal functions near boundaries numerical examples are included for illustration

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

generalized time dependent schrodinger equation in two dimensions under constraints we investigate a generalized two dimensional time dependent schrodinger equation on a comb with a memory kernel a dirac delta term is introduced in the schrodinger equation so that the quantum motion along the x direction is constrained at y 0 the wave function is analyzed by using green s function approach for several forms of the memory kernel which are of particular interest closed form solutions for the cases of dirac delta and power law memory kernels in terms of fox h function as well as for a distributed order memory kernel are obtained further a nonlocal term is also introduced and investigated analytically it is shown that the solution for such a case can be represented in terms of infinite series in fox h functions green s functions for each of the considered cases are analyzed and plotted for the most representative ones anomalous diffusion signatures are evident from the presence of the power law tails the normalized green s functions obtained in this work are of broader interest

existence of solution for eguchi oki matsumura equation describing phase separation and order disorder transition in binary alloys abstract in this paper we consider the eguchi oki matsumura equation which consists of the fourth and second order coupled equations of parabolic type it is shown that this system admits the unique global solution

on the role of polynomials in rbf fd approximations radial basis function generated finite difference rbf fd approximations generalize classical grid based finite differences fd from lattice based to scattered node layouts this greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas many different types of radial functions have been considered in this rbf fd context in this study we find that i polyharmonic splines phs in conjunction with supplementary polynomials provide a very simple way to defeat stagnation also known as saturation error and ii give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter in follow up studies we will focus on how to best use these hybrid rbf polynomial bases for fd approximations in the contexts of solving elliptic and hyperbolic type pdes

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

evaluation of convergence behavior of metamodeling techniques for bridging scales in multi scale multimaterial simulation the effectiveness of several metamodeling techniques viz the polynomial stochastic collocation method adaptive stochastic collocation method a radial basis function neural network a kriging method and a dynamic kriging method is evaluated this is done with the express purpose of using metamodels to bridge scales between micro and macro scale models in a multi scale multimaterial simulation the rate of convergence of the error when used to reconstruct hypersurfaces of known functions is studied for sufficiently large number of training points stochastic collocation methods generally converge faster than the other metamodeling techniques while the dkg method converges faster when the number of input points is less than 100 in a two dimensional parameter space because the input points correspond to computationally expensive micro meso scale computations the dkg is favored for bridging scales in a multi scale solver

error saturation in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation one restriction is error saturation which occurs for many types of rbfs including gaussian rbfs of the form f x a h exp email protected 2 x h 2 in the limit h 0 for fixed a the error does not converge to zero but rather to e s a previous studies have theoretically determined the saturation error for gaussian rbf on an infinite uniform interval and for the same with a single point omitted the gap enormously increases e s a we show experimentally that the saturation error on the unit interval email protected 1 1 is about 0 06exp 0 47 a 2 email protected huge compared to the o email protected a 2 exp email protected 2 email protected 2 saturation error for a grid with one point omitted we show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points the saturation error can be avoided by choosing email protected 1 the flat limit but the condition number of the interpolation matrix explodes as o exp p 2 email protected 2 the best strategy is to choose the largest a which yields an acceptably small saturation error if the user chooses an error tolerance d then a o p t i m u m d 1 2log d 0 06

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

computing eigenmodes ofelliptic operators using radial basis functions radial basis function rbf approximations have been successfully used to solve boundary value problems numerically we show that rbfs can also be used to compute eigenmodes of elliptic operators particular attention is given to the laplacian operator in two dimensions including techniques to avoid degradation of the solution near the boundaries for regions with corner singularities special functions must be added to the basis to maintain good convergence numerical results compare favorably to basic finite element methods

evaluation of kriging based surrogate models constructed from mesoscale computations of shock interaction with particles abstract macro scale computations of shocked particulate flows require closure laws that model the exchange of momentum energy between the fluid and particle phases closure laws are constructed in this work in the form of surrogate models derived from highly resolved mesoscale computations of shock particle interactions the mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of mach number and particle volume fraction two kriging based methods viz the dynamic kriging method dkg and the modified bayesian kriging method mbkg are evaluated for their ability to construct surrogate models with sparse data i e using the least number of mesoscale simulations it is shown that if the input data is noise free the dkg method converges monotonically convergence is less robust in the presence of noise the mbkg method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments this work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows

polynomial approximation and cubature at approximate fekete and leja points of the cylinder the paper deals with polynomial interpolation least square approximation and cubature of functions dened on the rectangular cylinder k d 1 1 with d the unit disk the nodes used for these processes are the approximate fekete points afp and the discrete leja points dlp extracted from suitable weakly admissible meshes wams of the cylinder from the analysis of the growth of the lebesgue constants approximation and cubature errors we show that the afp and the dlp extracted from wam are good points for polynomial approximation and numerical integration of functions dened on the cylinder

solution of a modified fractional diffusion equation abstract recently a modified fractional diffusion equation has been proposed i sokolov j klafter from diffusion to anomalous diffusion a century after einstein s brownian motion chaos 15 2005 026103 a v chechkin r gorenflo i m sokolov v yu gonchar distributed order time fractional diffusion equation frac calc appl anal 6 3 2003 259279 i m sokolov a v checkin j klafter distributed order fractional kinetics acta phys pol b 35 2004 1323 for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term in this letter we give the solution of the modified equation on an infinite domain in contrast to the solution of the traditional fractional diffusion equation the solution of the modified equation requires an infinite series of fox functions instead of a single fox function

solutions for a schrodinger equation with a nonlocal term we obtain time dependent solutions for a schrondiger equation in the presence of a nonlocal term by using the green function approach these solutions are compared with recent results obtained for the fractional schrodinger equation as well as for the usual one the nonlocal term incorporated in the schrodinger equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution

orthogonal polynomials in matlab a suite of matlab programs has been developed as part of the book orthogonal polynomials computation and approximation oxford university press oxford 2004 by gautschi the package contains routines for generating orthogonal polynomials as well as routines dealing with applications in this paper a brief review of the first part of the package is given dealing with procedures for generating the three term recurrence relation for orthogonal polynomials and more general recurrence relations for sobolev orthogonal polynomials moment based methods and discretization methods and their implementation in matlab are among the principal topics discussed

an exact method for computing the area moments of wavelet and spline curves we present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis using green s theorem we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product the multidimensional filter coefficients are precomputed exactly as the solution of a two scale relation to demonstrate the performance improvement of the new method we compare it with existing methods such as pixel based approaches and approximation of the region by a polygon we also propose an alternate scheme when the scaling function is sinc x

bivariate lagrange interpolation at the padua points computational aspects the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order of growth o log 2 n here we show four families of padua points for interpolation at any even or odd degree n and we present a stable and efficient implementation of the corresponding lagrange interpolation formula based on the representation in a suitable orthogonal basis we also discuss extension of non polynomial padua like interpolation to other domains such as triangles and ellipses we give complexity and error estimates and several numerical tests

pricing european and american options by radial basis point interpolation a rbpi method for pricing european and american options is proposed rbpi has not yet been used for option pricing the overall efficiency is improved by coupling rbpi with other approaches three different algorithms for american options are tested and compared numerical results demonstrate the effectiveness of the techniques employed we propose the use of the meshfree radial basis point interpolation rbpi to solve the black scholes model for european and american options the rbpi meshfree method offers several advantages over the more conventional radial basis function approximation nevertheless it has never been applied to option pricing at least to the very best of our knowledge in this paper the rbpi is combined with several numerical techniques namely an exponential change of variables which allows us to approximate the option prices on their whole spatial domain a mesh refinement algorithm which turns out to be very suitable for dealing with the non smooth options payoff and an implicit euler richardson extrapolated scheme which provides a satisfactory level of time accuracy finally in order to solve the free boundary problem that arises in the case of american options three different approaches are used and compared the projected successive overrelaxation method psor the bermudan approximation and the penalty approach numerical experiments are presented which demonstrate the computational efficiency of the rbpi and the effectiveness of the various techniques employed

radial basis function generated finite differences for option pricing problems abstract in this paper we present a numerical method to price options based on radial basis function generated finite differences rbf fd in space and the backward differentiation formula of order 2 bdf 2 in time we use gaussian rbfs that depend on a shape parameter e the choice of this parameter is crucial for the performance of the method we chose e as const h 1 and we derive suitable values of the constant for different stencil sizes in 1d and 2d this constant is independent of the problem parameters such as the volatilities of the underlying assets and the interest rate in the market in the literature on option pricing with rbf fd a constant value of the shape parameter is used we show that this always leads to ill conditioning for decreasing h whereas our proposed method avoids such ill conditioning we present numerical results for problems in 1d 2d and 3d demonstrating the useful features of our method such as discretization sparsity flexibility in node placement and easy dimensional extendability which provide high computational efficiency and accuracy

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

stable finite difference scheme for a model equation of phase separation a stable conservative and practical finite difference scheme is presented in order to solve numerically a model equation of phase separation in some binary alloys the model system is introduced by eguchi oki and matsumura in the attempt of theoretically describing such phenomena and extends the well known cahn hilliard equation even numerical solutions are hard to obtain for this dynamics our scheme utilizes a lyapunov functional that is the free energy of the system whose effectiveness is demonstrated by numerical implementations

generalized fractional schrodinger equation with space time fractional derivatives in this paper the generalized fractional schrodinger equation with space and time fractional derivatives is constructed the equation is solved for free particle and for a square potential well by the method of integral transforms fourier transform and laplace transform and the solution can be expressed in terms of mittag leffler function the green function for free particle is also presented in this paper finally we discuss the relationship between the cases of the generalized fractional schrodinger equation and the ones in standard quantum

local radial basis function based differential quadrature method and its application to solve two dimensional incompressible navier stokes equations local radial basis function based differential quadrature method is presented in detail in this paper the method is a natural mesh free approach like the conventional differential quadrature dq method it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots which may be distributed randomly however different from the conventional dq method the weighting coefficients in present method are determined by taking the radial basis functions rbfs instead of high order polynomials as the test functions the method works in a similar fashion as conventional finite difference schemes but with truly mesh free property in this paper we mainly concentrate on the multiquadric rbfs since they have exponential convergence the effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied and how the value of optimal c varies with the number of local support knots is also numerically demonstrated the proposed method is validated by its application to the simulation of natural convection in a square cavity excellent numerical results are obtained on an irregular knot distribution

enhancing finite differences with radial basis functions polynomials are used together with polyharmonic spline phs radial basis functions rbfs to create local rbf finite difference rbf fd weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences fd the presented method replicates the convergence properties of fd but for arbitrary node layouts it is tested on the 2d compressible navier stokes equations at low mach number relevant to atmospheric flows test cases are taken from the numerical weather prediction community and solved on bounded domains thus attention is given on how to handle boundaries with the rbf fd method as well as a novel implementation for hyperviscosity comparisons are done on cartesian hexagonal and quasi uniform node layouts consideration and guidelines are given on phs order polynomial degree and stencil size the main advantages of the present method are 1 capturing the basic physics of the problem surprisingly well even at very coarse resolutions 2 high order accuracy without the need of tuning a shape parameter and 3 the inclusion of polynomials eliminates stagnation saturation errors a matlab code is given to calculate the differentiation weights for this novel approach

decay of a potential vortex in a generalized oldroyd b fluid the velocity field and the adequate shear stress corresponding to the decay of a potential vortex in a generalized oldroyd b fluid are determined by means of hankel and laplace transforms the exact solutions written in terms of the generalized g and r functions are presented as a sum of the newtonian solutions and the adequate non newtonian contributions these solutions can be easy specialized to give the solutions for generalized maxwell fluids and ordinary oldroyd b or maxwell fluids performing the same motion the influence of the fractional parameters as well as that of the material parameters on the decay of the vortex is emphasized by graphical illustrations

numerical comparison of least square based finite difference lsfd and radial basis function based finite difference rbffd methods in this paper two mesh free methods i e least square based finite difference lsfd and radial basis function based finite difference rbffd are compared numerically in terms of their accuracy and efficiency these two mesh free methods are based on different approximation schemes that is the least square approximation and radial basis function rbf approximation the two mesh free methods exhibit very different behaviors in many ways in this study we examine the performance of the two methods by applying them to two example problems poisson equation and two dimensional incompressible viscous lid driven cavity flow and some interesting findings are observed

meshless cubature by green s formula abstract by coupling the flexibility of thin plate splines interpolation with green s integral formula we obtain a meshless cubature method from scattered samples of small moderate size on convex as well as nonconvex and even multiply connected polygonal domains

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

implementation of material interface conditions in the radial point interpolation meshless method we propose a systematic approach to accurate imposition of material interface conditions for the meshless radial point interpolation method rpim a new set of equations for updating fields at interfaces is derived a piecewise polynomial and a modified radial basis function are proposed and applied to account for field discontinuities near the interfaces in contrast to the previous work the proposed approach incorporates material properties and reduces geometrical dependence of the interpolation function numerical results show that the proposed approach significantly improves the simulation accuracy of the rpim at a cost of a small overhead in computational time