approximation orders and shape preserving properties of the multiquadric trigonometric b spline quasi interpolant the purpose of the paper is to derive some properties of the multiquadric trigonometric b spline quasi interpolant firstly the paper captures its approximation orders for high order derivatives based on the error estimate one can choose the shape parameter properly such that the quasi interpolant gives optimal approximations to high order derivatives moreover the approximation orders also show that from the theoretical point of view the considered quasi interpolant can be applied when the approximation of high order derivatives is needed i e numerical solution of some pdes construction of lyapunov function etc secondly the paper derives some shape preserving properties of the quasi interpolant these properties suggest that the quasi interpolant may be used in the geometric modeling cad cam for instance that requires shape preservation finally to illustrate the validity of the results some numerical examples are presented both theoretical and numerical results demonstrate that the quasi interpolant cannot only provide excellent approximations to high order derivatives but also preserve the shapes well

solving time dependent engineering problems with multiquadrics abstract this paper deals with the solution of time dependent problems the multiquadric radial basis function method is formulated with a new approach for transient problems one and two dimensional problems are considered the forward difference and the crank nicolson time marching schemes for parabolic cases are considered the central difference integration method of the newmark family is considered for hyperbolic problems the method proves its accuracy in four numerical examples

a quasi radial basis functions method for american options pricing abstract based on the idea of quasi interpolation and radial basis functions approximation a fast and accurate numerical method is developed for solving the black scholes equation for valuation of american options prices since the method does not require solving a resultant full matrix the ill conditioning problem resulting from using the radial basis functions as a global interpolant can be avoided the method has been shown to be effective in solving problems with free boundary condition as indicated in the numerical computation for the american option pricing an excellent approximation of the solution as well as the free optimal exercise boundary can be obtained

radial spline collocation method for static analysis of beams conventional spline collocation method scm uses equally spaced knots in the computation of approximation however when load distribution geometry material property of structure are discontinuous it will greatly reduce accuracy the goal of this paper is mainly to develop unequally spaced knots to improve the disadvantage of using equally spaced knots applying the concept of radial basis functions rbf the spline function is transformed into radial spline function rsf consequently rsf can conveniently calculate the values for unequally spaced knots so that the complicated beam problems with discontinuous characteristics can be solved accurately and efficiently by the proposed radial spline collocation method rscm in this paper the proposed rscm also can be extended to solve other structural problems such as frame plate shell etc

a multiquadric quasi interpolation with linear reproducing and preserving monotonicity in this paper we develop a multiquadric mq quasi interpolation which has the properties of linear reproducing and preserving monotonicity moreover we give its approximation error by theoretic analysis and illustrate the effect by means of two examples one of the examples is to approach the linear combination of two sine functions with different frequencies another is to approximate a function with discontinuity from the results of the examples we believe that the present mq quasi interpolation is feasible

moments in infinite channel queues we consider a new approach to the problem of evaluating the moments of the steady state number of customers in system b for an infinite channel queue having general input our new approach uses the rate conservation law as a consequence we show that for a variety of g gi queues var b as long as service times have finite first moment in addition we give some ordering relations for the variance of b when comparing two alternative systems with convex ordered inter arrival time distributions

locally distinguishing inputs for systems on lie groups and parallelism this paper is concerned with invariant systems on lie groups a sufficient condition for a piecewise constant input to distinguish between any two distinct initial states contained in some open neighbourhood of the state space is given this condition involves the parallel translation of vectors along curves thus relating observability to parallelism

explicit and implicit meshless methods for linear advection diffusion type partial differential equations simple mesh grid free explicit and implicit numerical schemes for the solution of linear advection diffusion problems is developed and validated herein unlike the mesh or grid based methods these schemes use well distributed quasi random points and approximate the solution using global radial basis functions the schemes can be seen as generalized finite differences with random points instead of a regular grid system this allows the computation of problems with complex shaped boundaries in higher dimensions with no need for complex mesh grid structure and with no extra implementation difficulties

applying multiquadric quasi interpolation to solve burgers equation abstract in this paper we develop a kind of univariate multiquadric mq quasi interpolation and use it to solve burgers equation with viscosity at first we construct the mq quasi interpolation which possesses the properties of linear reproducing and preserving monotonicity next we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable then we verify our method for two examples with distinguishing initial value condition one example is tested for three reynolds number that is r 10 r 100 and r 10 000 from the numerical experiments we see that the presented method in this paper is valid although the accuracy of the technique is not higher than hon and mao s one another example is used to examine the travelling of the shock we can improve the accuracy by selecting an appropriate shape parameter and using a higher accurate mq quasi interpolation the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

solving high order partial differential equations with indirect radial basis function networks this paper reports a new numerical method based on radial basis function networks rbfns for solving high order partial differential equations pdes the variables and their derivatives in the governing equations are represented by integrated rbfns the use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high order derivatives the proposed rbfn method is verified successfully through the solution of thin plate bending and viscous flow problems which are governed by biharmonic equations for thermally driven cavity flows the solutions are obtained up to a high rayleigh number

numerical solution of poisson s equation using radial basis function networks on the polar coordinate this paper introduces a variant of direct and indirect radial basis function networks drbfns and irbfns for the numerical solution of poisson s equation we use transformation from cartesian coordinates to polar ones and use drbfn and irbfn methods on the basis of a multiquadric approximation scheme we have experienced that the result shows better accuracy than previously known ones also our new way of solution does not influence the condition number

the map and blend scattered data interpolant on a sphere abstract the map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere this technique consists of two different mappings of the sphere to planar domains solving two corresponding scattered data interpolation problems on the planar domains and then the interpolant on the sphere is formed by blending these two planar interpolants if the user has software for solving the scattered data interpolation problem on a planar domain only a small programming effort is needed to implement the map and blend interpolant on the sphere although any interpolant to scattered data on a planar domain can be used in our general technique we use the multiquadric radial basis method

on a subclass of certain convex functions with negative coefficients coefficient distortion covering and coefficient inequalities are determined for univalent functions with negative coefficients that is convex of order a extreme points for this class is also determined

a quasi interpolation scheme for periodic data based on multiquadric trigonometric b splines abstract multiquadric mq quasi interpolation has been studied extensively in the literature however the mq quasi interpolant is not well defined for periodic data since its kernel the mq function itself is not periodic note that in many applications the data may arise from a closed curve surface and thus possess some kind of periodicity for example analysis of geodetic and meteorological data constructing active contours estimating the region of attraction of dynamical systems and so on therefore it is meaningful to construct a quasi interpolant whose kernel itself is periodic for periodic data combining the construction techniques of the mq quasi interpolant and the trigonometric b spline quasi interpolant the paper constructs such a quasi interpolant the quasi interpolant couples together the periodicity of the trigonometric b spline quasi interpolant and the smoothness of the mq quasi interpolant moreover both the quasi interpolant and its derivatives are periodic the quasi interpolant covers the trigonometric b spline quasi interpolant as a special case in addition the error estimate shows that a proper shape parameter can be chosen such that the quasi interpolant provides the same approximation order as a trigonometric b spline quasi interpolant for a periodic function furthermore the quasi interpolant gives better approximations to high order derivatives than the trigonometric b spline quasi interpolant as illustrated by numerical examples presented at the end of the paper

multiquadric method for the numerical solution of a biphasic mixture model a computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model the model includes a set of constitutive equations for the fluid flows through the solid phase a set of momentum equations for stress strain equilibrium and a continuity equation for the solid phase and the fluid phase the numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading the numerical results are compared with spilker s penalty finite element method

error estimates of local multiquadric based differential quadrature lmqdq method through numerical experiments summary in this article we present an error estimate of the derivative approximation by the local multiquadricbased differential quadrature lmqdq method radial basis function is different from the polynomial approximation in which taylor series expansion is not applicable so the present analysis is performed through the numerical solution of poisson equation it is known that the approximation error of lmqdq method depends on three factors i e local density of knots h free shape parameter c and number of supporting knots ns by numerical experiments their contribution to the approximation error and correlation were studied and analysed in this paper an error estimate o h c n is thereafter proposed in which n is a positive constant and determined by the number of supporting knots ns copyright 2005 john wiley sons ltd

solving high order ordinary differential equations with radial basis function networks this paper is concerned with the application of radial basis function networks rbfns for numerical solution of high order ordinary differential equations odes two unsymmetric rbf collocation schemes namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process are developed to solve high order odes directly and the latter is found to be considerably superior to the former good accuracy and high rate of convergence are obtained with the proposed indirect method

numerical computation of three dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method abstract in this paper the local multiquadric differential quadrature lmqdq method is applied on three dimensional incompressible flow problems the lmqdq method is among the newly proposed mesh free methods unlike the traditional differential quadrature dq method the weighting coefficients of lmqdq method are determined by using the radial basis functions rbfs as the trial functions instead of high order polynomials the main concern of this paper is to discuss the effectiveness of using lmqdq method to solve 3 d incompressible navier stokes n s equations in the primitive variable form three dimensional lid driven cavity flow problem with reynolds numbers of 100 400 and 1000 was chosen as a test case to validate the lmqdq method the computed velocity profiles along the vertical and horizontal centerlines are given and compared with available data in the literature

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

approximation to the k th derivatives by multiquadric quasi interpolation method quasi interpolation is very useful in the study of approximation theory and its applications since it can yield solutions directly without the need to solve any linear system of equations based on the good performance simple to evaluate of multiquadric functions the advantages approximation order convexity and monotonlcity preserving of multiquadric quasi interpolation have been widely discussed however it is usually only for the approximation properties of the function itself and the good properties for derivatives whereas the high order derivatives have been largely ignored in this paper we go further into the approximation properties to the k th derivatives by using multiquadric quasi interpolation furthermore we develop two kinds of multiquadric quasi interpolation schemes on bounded interval x 0 x n whose derivatives converge to the corresponding derivatives of the approximated functions finally the numerical experiments are presented to confirm the accuracy of the presented scheme both theoretical results and numerical examples show this scheme provides good accuracy even if the data points in x 0 x n are irregularly distributed

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

a new spectral meshless radial point interpolation smrpi method for the two dimensional fredholm integral equations on general domains with error analysis in this article we present a numerical method to solve two dimensional fredholm integral equations of the second kind on general domains the method utilizes meshless and spectral collocation techniques but it is not traditional meshless collocation method the point interpolation method with the help of strictly positive definite radial basis functions is used to construct shape functions as a basis functions in the frame of spectral collocation methods these basis functions shape functions have kronecker delta function property since the proposed method is meshless it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional integral equation to the solution of a linear system of algebraic equations convergence analysis with error estimates are given with full discussion furthermore some numerical examples are presented to show the validity and efficiency of smrpi

dynamically knots setting in meshless method for solving time dependent propagations equation abstract in order to display a function with some finite discrete sampling data efficiently we require more sampling data where the function is more oscillatory and less sampling data where the function is more flat if the function is happen to be a solution of partial differential equation and is solved by finite elements method then we should construct finer element near the singularity however we do not know where the oscillation or even shocks will happen to a function which is a solution of non linear partial differential equation therefore we cannot preset the finer elements near such points a trivial method is taking very dense knots or very fine elements everywhere to keep the accuracy this strategy cost the computation time another idea is to move the position of the sampling points according to the varying of the function with the time we observed that this approach could be achieved by lagrangian formulation for finite difference method however the lagrangian formulation does not possess shape preserving and variation diminishing properties it is difficult to achieve the approach for finite elements methods too because the moving knots will destroy the topology of the mesh recently the meshless method becomes to topic to solve partial differential equation numerically the meshless method does not require any mesh or any structure of the knots sampling points therefore we can move the knots freely to simulate the problem the only restriction is to keep the knots no overlapping this paper is a test of our approach for the burger s equation with radial basis quasi interpolation

a radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory abstract the free vibration analysis of functionally graded plates is performed by the multiquadric radial basis function method and a higher order shear deformation theory the multiquadric method is a truly meshless method allowing a fast and simple domain and boundary discretization we use an homogenization technique for material properties based on the mori tanaka scheme numerical tests show that the method is reliable and produces good results

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

on a method of atkinson for evaluating domain integrals in the boundary element method we have shown how to couple a numerical method due to atkinson for computing particular solutions to a class of elliptic differential equations with a variety of boundary element methods which alleviates the problem of domain discretization for solving inhomogeneous equations when the inhomogeneous term is of the form typically used in the dual reciprocity method atkinson s formula is such that only boundary integrals need to be approximated numerically if this method is coupled with the method of potentials it results in a computational technique which requires neither boundary nor domain discretization some numerical results are given validating our approach

a quasi interpolation method for solving stiff ordinary differential equations based on the idea of quasi interpolation and radial basis functions approximation a numerical method is developed to quasi interpolate the forcing term of differential equations by using radial basis functions a highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions this overcomes the ill conditioning problem resulting from using the radial basis functions as a global interpolant error estimation is given for a particular second order stiff differential equation with boundary layer the result of computations indicates that the method can be applied to solve very stiff problems with the use of multiquadric a special class of radial basis functions it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation copyright 2000 john wiley sons ltd

some extensions of radial basis functions and their applications in artificial intelligence abstract radial basis functions have recently found interesting applications in artificial intelligence and in particular in the problem of learning to perform a particular task from a set of examples however in many practical cases the radial basis functions method cannot be applied in a straight forward manner because it does not take into account some features that are typical of the problem of learning from examples in this paper we show some extensions of the standard theory introduced in order to deal with a large number of examples and with problems in which different variables play very different roles we present some examples and also point out some open problems

on quantum state observability and measurement we consider the problem of determining the state of a quantum system given one or more readings of the expectation value of an observable the system is assumed to be a finite dimensional quantum control system for which we can influence the dynamics by generating all the unitary evolutions in a lie group we investigate to what extent by an appropriate sequence of evolutions and measurements we can obtain information on the initial state of the system we present a system theoretic viewpoint of this problem in that we study the observability of the system in this context we characterize the equivalence classes of indistinguishable states and propose algorithms for state identification

applying multiquadric quasi interpolation for boundary detection in this paper we propose a novel scheme for simulating geometric active contours geometric flow of one kind applying multiquadric mq quasi interpolation we first represent the geometric flow in its parametric form then we obtain the numerical scheme by using the derivatives of the quasi interpolation to approximate the spatial derivative of each dependent variable and a forward difference to approximate the temporal derivative of each dependent variable the resulting scheme is simple efficient and easy to implement also images with complex boundaries can be more easily proposed on the basis of the good properties of the mq quasi interpolation several biomedical and astronomical examples of applications are shown in the paper comparisons with other methods are included to illustrate the validity of the method

a numerical solution of the nonlinear controlled duffing oscillator by radial basis functions in this research a new numerical method is applied to investigate the nonlinear controlled duffing oscillator this method is based on the radial basis functions rbfs to approximate the solution of the optimal control problem by using the collocation method we apply legendre gauss lobatto points for rbfs center nodes in order to use the numerical integration method more easily then the method of lagrange multipliers is used to obtain the optimum of the problems for this purpose different applications of rbfs are used the differential and integral expressions which arise in the dynamic systems the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients illustrative examples are included to demonstrate the validity and applicability of the technique

numerical solution of burgers fisher equation by cubic b spline quasi interpolation abstract in this paper numerical solution of the burgers equation is presented based on the cubic b spline quasi interpolation at first the cubic b spline quasi interpolation is introduced moreover the numerical scheme is presented by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable the accuracy of the proposed method is demonstrated by some test problems the numerical results are found in good agreement with exact solutions the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

starlike convex and close to convex functions of complex order the main purpose of this paper is to introduce two classes p b l b and r b l b b a complex number with re b 0 of functions which are analytic in the unit disc coefficient estimates for functions in these classes and radii of close to convexity starlikeness and convexity are obtained further applications in the fractional calculus are given for functions in p b l b and r b l b

the asymptotic cardinal function of the multiquadratic ϕ r r2 c2 12as c a radial basis function approximation has the form s x j rd yj x xj 2 x rd where 0 r is some given function yj j zd are real coefficients and the centres xj j zd are points in rd it is known that radial basis function approximations using the multiquadric r r2 c2 12 possess many useful and interesting properties when the centres form an infinite regular lattice we analyse the limiting case as c and identify a class of functions that arise as uniform limits of the multiquadric interpolants in the univariate case we observe that the cardinal function for the multiquadratic becomes the sinc function as c the limit of the multivariate cardinal function is also identified

on a new family of radial basis functions mathematical analysis and applications to option pricing abstract in this paper we introduce a new family of infinitely smooth and nearly locally supported radial basis functions rbfs derived from the general solution of a heat equation arising from the american option pricing problem these basis functions are expressed in terms of the repeated integrals of the complementary error function and provide highly efficient tools to solve the free boundary partial differential equation resulting from the related option pricing model we introduce an integral operator with a function dependent lower limit which is employed as a basic tool to prove the radial positive definiteness of the proposed basis functions and could be of independent interest in the rbf theory we then show that using the introduced functions as expansion bases in the context of an rbf based meshless collocation scheme we could exactly impose the transparent boundary condition accompanying the heat equation we prove that the condition numbers of the resulting collocation matrices are orders of magnitude less than those arising from other popular rbf families used in current literature some other properties of these bases such as their fourier transforms as well as some useful representations in terms of positive borel measures will also be discussed

on univalent functions with negative coefficients sharp results concerning coefficients distortion covering theorems for univalent functions with negative coefficients are investigated also closure theorem and extreme points for this class are also determined furthermore integral operators and applications in the fractional calculus for functions in the class are studied here these results extent some known theorems

on local observability for invariant systems on lie groups and coset spaces the purpose of this short paper is to provide a proof of a local observability criterion for invariant systems on lie groups and coset spaces the criterion was originally presented with an ad hoc proof based on lie theoretic arguments the alternative proof presented here shows that the criterion essentially comes from the classical observability results involving the observability codistribution and its properties in this way it unifies the two approaches and extends the application field of the classical results

multi symplectic quasi interpolation method for hamiltonian partial differential equations abstract in this paper we propose a multi symplectic quasi interpolation method for solving multi symplectic hamiltonian partial differential equations based on the method of lines we first discretize the multi symplectic pdes using quasi interpolation method and then employ appropriate time integrators to obtain the full discrete system the local conservation properties including multi symplectic conservation laws energy conservation laws and momentum conservation laws are discussed in detail for illustration we provide two concrete examples the nonlinear wave equation and the nonlinear schrodinger equation the salient feature of our multi symplectic quasi interpolation method is that it is valid both on uniform grids and nonuniform grids the numerical results show the good accuracy and excellent conservation properties of the proposed method

a collocation meshless method based on local optimal point interpolation this paper deals with the use of the local optimal point interpolating lopi formula in solving partial differential equations pdes with a collocation method lopi is an interpolating formula constructed by localization of optimal point interpolation formulas that reproduces polynomials and verifies the delta kronecker property this scheme results in a truly meshless method that produces high quality output and accurate solutions

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

solution and control of pde via global optimization methods abstract based on the concept of equivalent solutions not to be confused with approximations to the exact solution a new consideration is given to ill posed 1 2 and overdetermined pde problems and to problems with nonexistent solutions 3 then a new method based on full global optimization techniques is developed for solution and control of processes described by partial differential equations the ideas are illustrated by examples and a case study is presented in comparison with the quasi reversibility method 4

an inverse problem of identifying the control function in two and three dimensional parabolic equations through the spectral meshless radial point interpolation abstract this study proposes a kind of spectral meshless radial point interpolation smrpi for solving two and three dimensional parabolic inverse problems on regular and irregular domains the smrpi is developed for identifying the control parameter which satisfies the semilinear time dependent two and three dimensional diffusion equation with both integral overspecialization and overspecialization at a point in the spatial domain this method is based on a combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions are used to construct shape functions which act as basis functions in the frame of smrpi it is proved that the scheme is stable with respect to the time variable in h1 and convergent by the order of convergence o t the results of numerical experiments are compared to analytical solutions to confirm the accuracy and efficiency of the presented scheme

multicloud multigrid convergence with a meshless operator the primary objective of this work is to develop and test a new convergence acceleration technique we call multicloud multicloud is well founded in the mathematical basis of multigrid but relies on a meshless operator on coarse levels the meshless operator enables extremely simple and automatic coarsening procedures for arbitrary meshes using arbitrary fine level discretization schemes the performance of multicloud is compared with established multigrid techniques for structured and unstructured meshes for the euler equations on two dimensional test cases results indicate comparable convergence rates per unit work for multicloud and multigrid however because of its mesh and scheme transparency multicloud may be applied to a wide array of problems with no modification of fine level schemes as is often required with agglomeration techniques the implication is that multicloud can be implemented in a completely modular fashion allowing researchers to develop fine level algorithms independent of the convergence accelerator for complex three dimensional problems

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

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

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

improved accuracy of multiquadric interpolation using variable shape parameters given n scattered data points we examined the problem of finding n variable multiquadric mq shape parameters or r2 values because the problem of finding the optimal r2 values is a nonlinear one we optimized these parameters numerically by minimizing the root mean square rms errors the resulting r2 values varied over many orders of magnitude we have tested this approach on a number of univariate and bivariate franke s problems and found that the rms error reduction was substantial

local radial point interpolation mlrpi method for solving time fractional diffusion wave equation with damping the purpose of the current investigation is to determine numerical solution of time fractional diffusion wave equation with damping for caputo s fractional derivative of order 1 2 a meshless local radial point interpolation mlrpi scheme based on galerkin weak form is analyzed the reason of choosing mlrpi approach is that it does not require any background integrations cells instead integrations are implemented over local quadrature domains which are further simplified for reducing the complication of computation using regular and simple shape the unconditional stability and convergence with order o 6 2 are proved where is time stepping also several numerical experiments are illustrated to verify theoretical analysis

observability and observers for nonlinear systems this paper deals with observability and observers for general nonlinear systems in the non control affine case we characterize systems that are observable independently of the inputs an exponential observer for these systems is also exhibited

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

quasi interpolation for linear functional data quasi interpolation has been studied extensively in the literature however most studies of quasi interpolation are usually only for discrete function values or a finite linear combination of discrete function values note that in practical applications more commonly we can sample the linear functional data the discrete values of the right hand side of some differential equations rather than the discrete function values e g remote sensing seismic data etc therefore it is more meaningful to study quasi interpolation for the linear functional data the main result of this paper is to propose such a quasi interpolation scheme error estimate of the scheme is also given in the paper based on the error estimate one can find a quasi interpolant that provides an optimal approximation order with respect to the smoothness of the right hand side of the differential equation the scheme can be applied in many situations such as the numerical solution of the differential equation construction of the lyapunov function and so on respective examples are presented in the end of this paper

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

edged based smoothed point interpolation method for acoustic radiation with perfectly matched layer abstract it is known that for exterior acoustics the domain based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the sommerfeld radiation condition in this work by using the polynomial and radial bases with specific node selection schemes the point interpolation method with the generalized gradient smoothing technique is employed to improve the accuracy for acoustic computations the appropriately softened acoustical stiffness of the discrete model is constructed based on the edge based smoothing domains via the gradient smoothing technique in addition the perfectly matched layer technique enables the smoothed point interpolation method to investigate exterior acoustics the complex coordinate transformation is utilized to determine the proper parameters for the perfectly matched layer through theoretical and numerical analyses several typical acoustic problems are analyzed revealing that better accuracy is obtained by the present method compared with the finite element method especially for high wavenumbers

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

observability of systems on lie groups and coset spaces the purpose of this paper is to study the observability of a class of systems for which the state space is a lie group and the output space is a cosec space the study of this type of system was initiated by brockett siam j control 10 1972 pp 265 284 in this paper brockett s observability results are generalized and necessary and sufficient conditions for observability are obtained effective algorithms are established to verify such conditions finally as an application some disturbance decoupling problems are considered

a kind of multiquadric quasi interpolation operator satisfying any degree polynomial reproduction property to scattered data in this paper by virtue of using the linear combinations of the shifts of f x to approximate the derivatives of f x and waldron s superposition idea 2009 we modify a multiquadric quasi interpolation with the property of linear reproducing to scattered data on one dimensional space such that a kind of quasi interpolation operator l r 1f has the property of r 1 email protected z r 0 degree polynomial reproducing and converges up to a rate of r 2 there is no demand for the derivatives of f in the proposed quasi interpolation l r 1f so it does not increase the orders of smoothness of f finally some numerical experiments are shown to compare the approximation capacity of our quasi interpolation operators with that of wu schaback s quasi interpolation scheme and feng li s quasi interpolation scheme

motion control of drift free left invariant systems on lie groups in this paper we address the constructive controllability problem for drift free left invariant systems on finite dimensional lie groups with fewer controls than state dimension we consider small spl epsiv amplitude low frequency periodically time varying controls and derive average solutions for system behavior we show how the pth order average formula can be used to construct open loop controls for point to point maneuvering of systems which require up to p 1 iterations of lie brackets to satisfy the lie algebra controllability rank condition in the cases p 2 3 we give algorithms for constructing these controls as a function of structure constants that define the control authority i e the actuator capability of the system the algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with o spl epsiv sup p accuracy in general exactly if the lie algebra is nilpotent the methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs

a kind of improved univariate multiquadric quasi interpolation operators in this paper a kind of improved univariate multiquadric quasi interpolation operators is proposed by using hermite interpolating polynomials error analysis shows that the convergence rate of the operators depends heavily on the shape parameter c which indicates that our operators could provide the desired smoothness and precision by choosing a suitable value of c numerical examples show that the operators provide a high degree of accuracy moreover operators are applied to the fitting of discrete solutions of initial value problems

solving partial differential equation by using multiquadric quasi interpolation abstract in this paper we use a kind of univariate multiquadric mq quasi interpolation to solve partial differential equation pde we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement our numerical experiment includes two examples one is solving viscid burgers equation for initial trapezoidal conditions another is simulating the interaction of two waves travelling in opposite direction from the numerical experiment we can see that the present scheme is valid