weighted norm inequalities for maximally modulated singular integral operators we present a framework that yields a variety of weighted and vector valued estimates for maximally modulated calderon zygmund singular and maximal singular integrals from a single a priori weak type unweighted estimate for the maximal modulations of such operators we discuss two approaches one based on the good method of coifman and fefferman cf and an alternative method employing the sharp maximal operator as an application we obtain new weighted and vector valued inequalities for the carleson operator proving that it is controlled by a natural maximal function associated with the orlicz space l log l log log log l this control is in the sense of a good inequality and yields strong and weak type estimates as well as vector valued and weighted estimates for the operator in question

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

the sharp weighted bound for general calderon zygmund operators for a general calderon zygmund operator t on r n it is shown that ktfkl2 w c t sup q a q w q w 1 a k fkl2 w for all muckenhoupt weights w 2 a2 this optimal estimate was known as the a2 conjecture a recent result of perez treil volberg reduced the problem to a testing condition on indicator functions which is verified in this paper the proof consists of the following elements i a variant of the nazarov treil volberg method of random dyadic systems with just one random system and completely without bad parts ii a resulting representation of a general calderon zygmund operator as an average of dyadic shifts and iii improvements of the lacey petermichl reguera estimates for these dyadic shifts which allow summing up the series in the obtained representation

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

an alternative local collocation strategy for high convergence meshless pde solutions using radial basis functions this work proposes an alternative decomposition for local scalable meshless rbf collocation the proposed method operates on a dataset of scattered nodes that are placed within the solution domain and on the solution boundary forming a small rbf collocation system around each internal node unlike other meshless local rbf formulations that are based on a generalised finite difference rbf fd principle in the proposed finite collocation method the solution of the pde is driven entirely by collocation of pde governing and boundary operators within the local systems a sparse global collocation system is obtained not by enforcing the pde governing operator but by assembling the value of the field variable in terms of the field value at neighbouring nodes in analogy to full domain rbf collocation systems communication between stencils occurs only over the stencil periphery allowing the pde governing operator to be collocated in an uninterrupted manner within the stencil interior the local collocation of the pde governing operator allows the method to operate on centred stencils in the presence of strong convective fields the reconstruction weights assigned to nodes in the stencils being automatically adjusted to represent the flow of information as dictated by the problem physics this implicit upwinding effect mitigates the need for ad hoc upwinding stencils in convective dominant problems boundary conditions are also enforced within the local collocation systems allowing arbitrary boundary operators to be imposed naturally within the solution construction the performance of the method is assessed using a large number of numerical examples with two steady pdes the convection diffusion equation and the lame navier equations for linear elasticity the method exhibits high order convergence in each case tested greater than sixth order and the use of centred stencils is demonstrated for convective dominant problems in the case of linear elasticity the stress fields are reproduced to the same degree of accuracy as the displacement field and exhibit the same order of convergence the method is also highly stable towards variations in basis function flatness demonstrating significantly improved stability in comparison to finite difference type rbf collocation methods

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

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

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

a numerical method for two dimensional schrodinger equation using collocation and radial basis functions in this paper we propose a numerical scheme to solve the two dimensional 2d time dependent schrodinger equation using collocation points and approximating the solution using multiquadrics mq and the thin plate splines tps radial basis function rbf the scheme works in a similar fashion as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

radial basis function simulation of slow release permanganate for groundwater remediation via oxidation an emerging strategy for remediation of contaminated groundwater is the use of permanganate cylinders for contaminant oxidation the cylinders which are only a few inches in diameter can be placed in wells or pushed directly into the subsurface this work focuses on the modeling and simulation of the reactive process to better understand the design of a group of cylinders for large scale contaminated sites the underlying model is a coupled system of nonlinear partial differential equations accounting for advection dispersion and reactive transport for a contaminant and the permanganate in two spatial dimensions radial basis functions collocation method is used to simulate different spatial arrangements of the cylinders to understand the behavior of the system and gain insight into designing a remediation strategy for a large scale contaminated region since the radial basis function collocation method is a meshless method the locations of the cylinders are not tied to a numerical grid making it an attractive choice for determining optimal placement our focus is to 1 identify a domain of influence measuring the effectiveness of the injected cylinders 2 understand the placement for multiple cylinders required to effectively clean up a given domain and 3 determine a protocol for injecting multiple cylinders over time we provide numerical results showing that domain of influence is a way to measure the effectiveness of installed cylinders domain of influence of one through three sources are simulated placement of two cylinders for an area of 13 f t by 3 f t and three sources for an area of 26 f t by 6 f t are sufficient to clean the contaminant within a reasonable time period the average concentrations of oxidant and contaminant are simulated for the cases of a third cylinder is installed at different time and locations

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

a scalable and implicit meshless rbf method for the 3d unsteady nonlinear richards equation with single and multi zone domains the local hermitian interpolation lhi method is a strong form meshless numerical technique which uses radial basis function rbf interpolants to satisfy linear and nonlinear governing equations and boundary operators recent developments have shown that for linear transport problems applying the pde governing equation directly to the basis functions can greatly improve the accuracy and stability of the resulting solutions in this work the lhi formulation with local pde interpolation is extended to the nonlinear gravity driven richards equation in order to solve unsteady problems involving flow in unsaturated porous media the application of the linearised pde operator to the basis functions incorporates information such as the effective velocity field directly into the solution construction this results in a form of analytical upwinding which helps to stabilise the solution in addition the local interpolation itself satisfies the linearised governing equation allowing for more accurate reconstruction of partial derivatives and hence a more accurate solution the procedure is tested using a 3d infiltration problem with a known analytical solution the performance of the lhi method both with and without local pde interpolation is compared to the finite element method via the femwater software the lhi formulation with pde data centres shows consistent improvement over the femwater solutions reducing errors by several orders of magnitude in addition a procedure is introduced to model strongly heterogeneous and layered soils the physically correct matching conditions are applied over layer interfaces i e continuity of pressure and mass flux the double collocation property of the hermitian rbf method is exploited to enforce both matching conditions at the same set of locations on the layer interface this procedure allows the accurate capture of solutions across such interfaces replicating the required discontinuities in the first derivatives of the pressure profile the multi layer formulation is validated using a transient two layer infiltration problem with the analytical solution replicated to a high precision in a variety of configurations copyright 2010 john wiley sons ltd

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

an efficient method for the numerical solution of hammerstein mixed vf integral equations on 2d irregular domains abstract in this work we adapt the implicity linear collocation method for solving nonlinear two dimensional mixed volterra fredholm integral equations of hammerstein type by using imq rbfs on a non rectangular domain the proposed method is mesh free and it is independent of the geometry of domain convergence analysis of the proposed method together with some benchmark examples is provided which support its reliability and numerical stability

some observations on unsymmetric radial basis function collocation methods for convection diffusion problems in this paper we re investigate the unsymmetric radial basis function rbf collocation method for solving convection diffusion problems with high peclet number as in power and barraco computers and mathematics with applications 2002 43 551 by testing different rbfs and different numbers of nodes we found that the unsymmetric method can still solve high peclet number problems reasonably well by using more nodes and domain decomposition techniques compared to solving one large problem the domain decomposition method is shown to be very efficient and can improve the accuracy especially when the peclet number is not that high from our tests it seems that the rbfs r4 ln r and r8 ln r are not very stable while r6 ln r r5 and r7 perform very well copyright 2003 john wiley sons ltd

a double boundary collocation hermitian approach for the solution of steady state convection diffusion problems in this article a double boundary collocation approach based on the meshless radial basis function hermitian method symmetric method is proposed and compared with the conventional single collocation in the double boundary collocation approach at the boundary collocation points the boundary condition and the governing partial differential equation are required to be satisfied simultaneously instead of only the boundary condition as required in the single collocation we are able to carry out this type of algorithm due to the robustness of the proposed hermite interpolation scheme in which the resulting matrix will be non singular as long as the partial differential operators applied to each point are linearly independent even if in a single node we impose two different differential conditions the results obtained with this new method are characterized by a higher precision especially for the prediction of the fluxes at the boundaries this is due to the higher order of continuity of the approximation at the boundary points imposed by the double collocation

meshless schemes for unsteady navier stokes equations in vorticity formulation using radial basis functions in this paper we provide a new scheme for unsteady incompressible flows in vorticity stream function formulation combined with the radial basis functions method it is an efficient meshless method optimal accuracy can be achieved using this method the efficiency and accuracy are demonstrated by numerical examples

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

efficient shortest path simplex algorithms we consider the specialization of the primal simplex algorithm to the problem of finding a tree of directed shortest paths from a given node to all other nodes in a network of n nodes or finding a directed cycle of negative length two efficient variants of this shortest path simplex algorithm are analyzed and shown to require at most n 1 n 2 2 pivots and o n3 time

weighted weak type 1 1 estimates via rubio de francia extrapolation the classical rubio de francia extrapolation result asserts that if an operator t lp0 u lp0 u is bounded for some p0 1 and every u ap0 then for every 1 1 and every u in this class then for every u a1 1 t is of restricted weak type 1 1 2 for every e 0 t l log l e u lloc1 u moreover for a big class of operators including calderon zygmund maximal operators g functions and the intrinsic square function we obtain a weak type 1 1 estimate with respect to every u a1

multiparameter operators and sharp weighted inequalities this article is concerned with the operators from harmonic analysis which are naturally associated to a multiple parameter family of dilations we are especially interested here in dealing with questions from the theory of such operators whose answers cannot be obtained by a reduction to the case of product operators we also introduce a new tool in order to carry out this multiparameter analysis which comes from the classical theory this is the concept of sharp weighted inequalities where for operators such as the hilbert transform or classical square function one asks for hilbert space inequalities which imply the correct estimates of the norm on lp of these operators as p becomes large 0 introduction in this article our main goal is to study classes of oper ators associated with multiparameter groups of dilations on rn in some sense the consideration of these operators is a natural next step or simplest case after those of the classical calderon zygmund theory and the product space theory it turns out that in order to obtain estimates for the multiparameter operators we shall be led to the following type of extremely classical question how does one obtain l2 estimates on r1 for the hilbert transform or other singular integrals which yield sharp estimates on the lp rl operator norm of h as p approaches 1 or oc these sharp weighted inequalities are of some independent interest and their study will be seen to be essentially equivalent to the deep work of chang wilson and wolff 3 on the exponential square integrability of functions whose area integral is bounded they also have other applications for example they provide an immediate extension of the chang wilson wolff estimates to product spaces as given by pipher in 12 at this point let us be more specific about the exact nature of the multipa rameter operators which we shall consider consider in r3 the dilations given by ps t x y z sx ty9 stz for s91 0 then there is a maximal operator first considered by zygmund ml and singular integrals tl introduced by ricci and stein 13 naturally associated to these dilations the maximal operator af3 is defined by

fractional diffusion equations by the kansa method this study makes the first attempt to apply the kansa method in the solution of the time fractional diffusion equations in which the multiquadrics and thin plate spline serve as the radial basis function in the discretization formulation the finite difference scheme and the kansa method are respectively used to discretize time fractional derivative and spatial derivative terms the numerical solutions of one and two dimensional cases are presented and discussed which agree well with the corresponding analytical solution

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

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 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

on the pointwise convergence of the sequence of partial fourier sums along lacunary subsequences abstract in his 2006 icm invited address konyagin mentioned the following conjecture if s n f stands for the n th partial fourier sum of f and n j j n is a lacunary sequence then s n j f is a e pointwise convergent for any f l log log l in this paper we will show that sup j s n j f 1 c f 1 log log 10 f f 1 as a direct consequence we obtain that s n j f f a e for f l log log l log log log l the discrete walsh model version of this last fact was proved by do and lacey but their methods do not re cover the continuous fourier setting the key ingredient for our proof is a tile decomposition of the operator sup j s n j f which depends on both the function f and on the lacunary structure of the frequencies this tile decomposition called f lacunary is directly adapted to the context of our problem and combined with a canonical mass decomposition of the tiles provides the natural environment to which the methods developed by the author in on the boundedness of the carleson operator near l 1 apply

sharp bounds for general commutators on weighted lebesgue spaces we show that if an operator t is bounded on weighted lebesgue space l 2 w and obeys a linear bound with respect to the a2 constant of the weight then its commutator b t with a function b in bmo will obey a quadratic bound with respect to the a2 constant of the weight we also prove that the kth order commutator t k b b t k 1 b will obey a bound that is a power k 1 of the a2 constant of the weight sharp extrapolation provides corresponding l p w estimates the results are sharp in terms of the growth of the operator norm with respect to the ap constant of the weight for all 1 p 1 all k and all dimensions as examples involving the riesz transforms power functions and power weights show

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

regularized meshless method for antiplane shear problems with multiple inclusions in this paper we employ the regularized meshless method to solve antiplane shear problems with multiple inclusions the solution is represented by a distribution of double layer potentials the singularities of kernels are regularized by using a subtracting and adding back technique therefore the troublesome singularity in the method of fundamental solutions mfs is avoided and the diagonal terms of influence matrices are determined an inclusion problem is decomposed into two parts one is the exterior problem for a matrix with holes subjected to remote shear the other is the interior problem for each inclusion the two boundary densities essential and natural data along the interface between the inclusion and matrix satisfy the continuity and equilibrium conditions a linear algebraic system is obtained by matching boundary conditions and interface conditions finally numerical results demonstrate the accuracy of the present solution good agreements are obtained and compare well with analytical solutions and gong s results

a meshfree method for the numerical solution of the rlw equation in this paper we present a meshfree technique for the numerical solution of the regularized long wave rlw equation this approach is based on a global collocation method using the radial basis functions rbfs different kinds of rbfs are used for this purpose accuracy of the new method is tested in terms of l 2 and l error norms in case of non availability of the exact solution performance of the new method is compared with existing methods stability analysis of the method is established propagation of single and double solitary waves wave undulation and conservation properties of mass energy and momentum of the rlw equation are discussed

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

selection of shape parameter in radial basis functions for solution of time fractional black scholes models abstract the current work aims to exploit two techniques namely residual power series method rpsm and collocation based meshfree method for the solution of time fractional black scholes models with constant and variable coefficients firstly using rpsm we obtain exact solutions of the considered models and then numerical solution by meshfree method computer simulations are performed for three test problems of european options pricing the simulations features excellent agreement with exact solutions accuracy and efficiency of the proposed numerical method is assessed via e2 e and erms error norms convergence of the proposed methods is also analyzed

compactly supported radial basis functions based collocation method for level set evolution in image segmentation the partial differential equation driving level set evolution in segmentation is usually solved using finite differences schemes in this paper we propose an alternative scheme based on radial basis functions rbfs collocation this approach provides a continuous representation of both the implicit function and its zero level set we show that compactly supported rbfs csrbfs are particularly well suited to collocation in the framework of segmentation in addition csrbfs allow us to reduce the computation cost using a kd tree based strategy for neighborhood representation moreover we show that the usual reinitialization step of the level set may be avoided by simply constraining the l1 norm of the csrbf parameters as a consequence the final solution is topologically more flexible and may develop new contours i e new zero level components which are difficult to obtain using reinitialization the behavior of this approach is evaluated from numerical simulations and from medical data of various kinds such as 3 d ct bone images and echocardiographic ultrasound images

a meshless galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions abstract this paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations these types of integral equations arise from boundary value problems of laplace s equations with nonlinear robin boundary conditions the discrete galerkin method together with the inverse multiquadric radial basis functions established on scattered points is utilized to approximate the solution the discrete galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method the proposed scheme uses a special accurate quadrature formula via the nonuniform gauss legendre integration rule to compute logarithm like singular integrals appeared in the scheme since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain it is meshless and does not depend to the domain form we also investigate the error analysis of the proposed method illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates

endpoint estimates for commutators of singular integral operators abstract we prove endpoint estimates for commutatotrs of singular integrals with bmo functions we first show that they satisfy l log l type inequalities and then that there is a hardy space type estimate where the usual atomic hardy space is replaced by an appropriate subspace

extrapolation of weights revisited new proofs and sharp bounds abstract we use an appropriate factorization of the a p weights to give another proof of the extrapolation theorem of rubio de francia it provides sharp bounds in terms of the a p constant of the weights then we extend the result to more general settings including off diagonal and partial range extrapolation among the applications we prove by iteration a multivariable extrapolation theorem and give a sharp bound for calderon zygmund operators on l p w for weights in a q q p

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

a meshfree method for inverse wave propagation using collocation and radial basis functions abstract a stable and highly accurate meshfree scheme based on strong form collocation associated with radial basis functions and explicit time integration is proposed to solve ill posed inverse wave propagation problems tikhonov regularization technique and the l curve criterion are employed to deal with the noise measurement data before solving the governing equation to get a stable solution appropriate weights are first derived for imposing boundary conditions and known conditions from the measurement data to yield the optimal convergence for inverse problems compared to the required iterative procedures in common methods no iteration is needed in the proposed method which leads to the time saving computation and avoids the instability resulting from the iterative methods dispersion analysis describes that this method has much less dispersion error than the classical finite element method fem and popular weak form meshfree methods and stability analysis is performed to predict the critical time step for the explicit time integration another outstanding advantage of this method is that it is effortless to extend the solution scheme of one dimensional inverse problem to high dimensional inverse problems furthermore the influences of the high noise into measurement data are evaluated which demonstrate that even handling with very high noise in the input data the proposed algorithm can achieve stable and high accuracy solutions

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

solving partial differential equations by bkm combined with ddm the boundary knot method bkm has recently been developed as an inherently meshless integration free boundary type collocation technique for the numerical discretization of general partial differential equation systems unlike the method of fundamental solutions the use of non singular radial basis functions in the bkm avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain but the condition number of the coefficient matrix of the linear algebraic equations which is derived from the homogeneous boundary value problem becomes very high as the total number of boundary knots increase which in turn affects the accuracy of the computation result combined with domain decomposition method we can avoid this problem easily in this paper we will solve homogeneous and non homogeneous partial differential equations using bkm combined with overlapped ddm from the numerical experiments we can find that it can solve pdes accurately

on the k shortest path trees problem abstract we address the problem of finding the k best path trees connecting a source node with any other non source node in a directed network with arbitrary lengths the main result in this paper is the proof that the k th shortest path tree is adjacent to at least one of the previous k 1 shortest path trees consequently we design an o f n m c max km time and o k m space algorithm to determine the k shortest path trees in a directed network with n nodes m arcs and maximum absolute length c max where o f n m c max is the best time needed to solve the shortest simple paths connecting a source node with any other non source node

extrapolation on lp μ we solve the extrapolation problem concerning bounded operators on lp that is we give end point estimates for sublinear operators t such that t lp lp is bounded with constant less than or equal to 1 p 1 m applications to the hardy littlewood maximal operator the hilbert transform and composition of operators are also given

the numerical solution of nonlinear two dimensional volterra fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis in this paper we present a numerical method for solving two dimensional nonlinear volterra fredholm integral equations of the second kind the method approximates the solution by the discrete collocation method based on radial basis functions rbfs constructed on a set of disordered data the proposed method is meshless since it does not require any background mesh or domain elements error analysis of this method is also investigated numerical examples which compare the proposed method with 2d tfs method 4 approve its supremacy in terms of accuracy and computational cost using various rbfs we have concluded that mq rbf is the best choice for the proposed method

numerical solution of the nonlinear klein gordon equation using radial basis functions the nonlinear klein gordon equation is used to model many nonlinear phenomena in this paper we propose a numerical scheme to solve the one dimensional nonlinear klein gordon equation with quadratic and cubic nonlinearity our scheme uses the collocation points and approximates the solution using thin plate splines tps radial basis functions rbf the implementation of the method is simple as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

error estimate in fractional differential equations using multiquadratic radial basis functions the purpose of this paper is to investigate the collocation method based on multiquadric mq radial basis functions rbfs for fractional differential equations fdes in the process of doing this a scattered data rbf interpolation is considered a posterior error bound for the smooth solutions of fdes in the proposed method is established in the l 2 norm using an alternate characterization of the native space numerical results are presented which confirm the theoretical prediction of the convergence behavior of the proposed method

boundary knot method for 2d and 3d helmholtz and convection diffusion problems under complicated geometry the boundary knot method bkm of very recent origin is an inherently meshless integration free boundary type radial basis function collocation technique for the numerical discretization of general partial differential equation systems unlike the method of fundamental solutions the use of non singular general solution in the bkm avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain the purpose of this paper is to extend the bkm to solve 2d helmholtz and convection diffusion problems under rather complicated irregular geometry the method is also first applied to 3d problems numerical experiments validate that the bkm can produce highly accurate solutions using a relatively small number of knots for inhomogeneous cases some inner knots are found necessary to guarantee accuracy and stability the stability and convergence of the bkm are numerically illustrated and the completeness issue is also discussed

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

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

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

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

a radial basis functions based finite differences method for wave equation with an integral condition the hyperbolic partial differential equation which contains integral condition in place of classical boundary condition arises in many application of modern physics and technologies in this article we propose a numerical method to solve the hyperbolic equation with nonlocal boundary condition using radial basis function based finite difference method several numerical experiments are presented and compared with some existing method to demonstrate the efficiency of the proposed method

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

a new pivot selection rule for the network simplex algorithm we present a new network simplex pivot selection rule which we call the minimum ratio pivot rule and analyze the worst case complexity of the resulting network simplex algorithm we consider networks with n nodes m arcs integral arc capacities and integral supplies demands of nodes we define a 0 1 valued penalty for each arc of the network the minimum ratio pivot rule is to select that eligible arc as the entering arc whose addition to the basis creates a cycle with the minimum cost to penalty ratio we show that the so defined primal network simplex algorithm solves minimum cost flow problem within o n pivots and in o m n log n time where is any upper bound on the sum of all arc flows in every feasible flow for assignment and shortest path problems our algorithm runs in o n 2 pivots and o nm n 2 log n time

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

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

a meshless integration free and boundary only rbf technique based on the radial basis function rbf nonsingular general solution and dual reciprocity method drm this paper presents an inherently meshless integration free boundary only rbf collocation technique for numerical solution of various partial differential equation systems the basic ideas behind this methodology are very mathematically simple in this study the rbfs are employed to approximate the inhomogeneous terms via the drm while nonsingular general solution leads to a boundary only rbf formulation for homogenous solution the present scheme is named as the boundary knot method bkm to differentiate it from the other numerical techniques in particular due to the use of nonsingular general solutions rather than singular fundamental solutions the bkm is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions the efficiency and utility of this new technique are validated through a number of typical numerical examples completeness concern of the bkm due to the sole use of the nonsingular part of complete fundamental solution is also discussed

new extrapolation estimates given a sublinear operator t satisfying that t a lp cp 1 a lp for every measurable set a and every 1 0 1 r tf y dy1 log r m f x 1 log f x d x this estimate allows us to improve yano s extrapolation theorem and also to obtain that for every f l log l r 1 r tf y dylogr f 1 other types of extrapolation results are also given

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

the scaling network simplex algorithm in this paper we present a new primal simplex pivot rule and analyze the worst case complexity of the resulting simplex algorithm for the minimum cost flow the assignment and the shortest path problems we consider networks with n nodes m arcs integral arc capacities bounded by an integer number u and integral arc costs whose magnitudes are bounded by an integer number c our pivot rule may be regarded as a scaling version of dantzig s pivot rule our pivot rule defines a threshold value which is initially at most 2c and the rule permits any arc with a violation of at least 2 to be the editing variable we select the leaving arc so that strong feasibility of the basis is maintained when there is no arc satisfying this rule then we replace by 2 and repeat the process the algorithm terminates when 1 we show that the simplex algorithm using this rule performs o nmu log c pivots and can be implemented to run in o m2u log c time specializing these results for the assignment and short

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

sharp reverse holder property for a weights on spaces of homogeneous type abstract in this article we present a new proof of a sharp reverse holder inequality for a weights then we derive two applications a precise open property of muckenhoupt classes and as a consequence of this last result we obtain a simple proof of a sharp weighted bound for the hardy littlewood maximal function involving a constants m l p w c 1 p 1 w a p a 1 p where 1 p w 1 1 p and c is a dimensional constant our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak reverse holder inequality which is still sufficient to prove the open property for a p classes and the l p boundedness of the maximal function in this latter case the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure and the geometric constant of the quasimetric

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

unconditionally stable mfltd method for the full wave electromagnetic simulation a new unconditionally stable numerical method for the time domain simulation of electromagnetic problems using a combination of the laguerre polynomials in the time domain and radial basis functions rbf in the spatial domain named mesh free laguerre time domain mfltd method is described unconditionally stable laguerre based method may be computationally much more efficient than the fdtd methods and using rbfs allows computing the problems with complex shapes in a non uniform point s distribution a numerical example at the final part of the communication shows the efficiency of the presented method in problems with sparse and coarse regions using the mfltd method for this example we can achieve a 96 reduction in the computation time and get an acceptable degree of accuracy in comparison to the conventional mesh free time domain mftd method

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

an alternative approach for numerical solutions of the navier stokes equations conventional approaches for solving the navier stokes equations of incompressible fluid dynamics are the primitive variable approach and the vorticity velocity approach in this paper an alternative approach is presented in this approach pressure and one of the velocity components are eliminated from the governing equations the result is one higher order partial differential equation with one unknown for two dimensional problems or two higher order partial differential equations with two unknowns for three dimensional problems a meshless collocation method based on radial basis functions for solving the navier stokes equations using this approach is presented the proposed method is used to solve a two and a three dimensional test problem of which exact solutions are known it is found that with appropriate values of the method parameters solutions of satisfactory accuracy can be obtained copyright 2006 john wiley sons ltd

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

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

the fuzzy shortest path problem and its most vital arcs abstract the shortest path problem is to find the shortest distance between two specified nodes in a network an arc is called a single most vital arc in the network if its removal from the network results in the greatest increase in the shortest distance the availability of arcs plays a key role in network problems the most vital arcs problems provide a means by which the importance of arc s availability can be measured in the traditional shortest path problem the arc lengths are assumed to be crisp numbers in this paper we consider the case that the arc lengths are fuzzy numbers in particular we derive the membership function of the shortest distance by using a fuzzy linear programming approach based on this result we then propose an algorithm for finding the single most vital arc in a network

a compact five point stencil based on integrated rbfs for 2d second order differential problems in this paper a compact 5 point stencil for the discretisation of second order partial differential equations pdes in two space dimensions is proposed we employ integrated radial basis functions in one dimension 1d irbfs to construct the approximations for the dependent variable and its derivatives over the three nodes in each direction of the stencil certain nodal values of the second order derivatives are incorporated into the approximations with the help of the integration constants in the case of elliptic pdes one algebraic equation is formed at each interior node and the obtained final system of which each row has 5 non zero entries is solved iteratively using a picard scheme in the case of parabolic pdes discretised with a crank nicolson procedure a set of three simultaneous algebraic equations is established at each interior node and the three equations are then combined to form two tridiagonal equations through the implicit elimination approach linear and non linear test problems including lid driven cavity flow and natural convection between the outer square and the inner cylinder are considered to verify the proposed stencil

a meshless method for solving nonlinear two dimensional integral equations of the second kind on non rectangular domains using radial basis functions with error analysis in this paper we present a numerical method for solving two dimensional nonlinear fredholm integral equations of the second kind on a non rectangular domain the method utilizes radial basis functions rbfs constructed on scattered points as a basis in the discrete collocation method the proposed scheme is meshless since 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 nonlinear integral equation to the solution of a nonlinear system of algebraic equations error analysis is presented for this method finally numerical examples are included to show the validity and efficiency of the new technique

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

numerical investigation on convergence of boundary knot method in the analysis of homogeneous helmholtz modified helmholtz and convection diffusion problems abstract this paper concerns a numerical study of convergence properties of the boundary knot method bkm applied to the solution of 2d and 3d homogeneous helmholtz modified helmholtz and convection diffusion problems the bkm is a new boundary type meshfree radial function basis collocation technique the method differentiates from the method of fundamental solutions mfs in that it does not need the controversial artificial boundary outside physical domain due to the use of non singular general solutions instead of the singular fundamental solutions the bkm is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method drm therefore when applied to inhomogeneous problems the error of the drm confounds the bkm accuracy in approximation of homogeneous solution while the latter essentially distinguishes the bkm mfs and boundary element method in order to avoid the interference of the drm this study focuses on the investigation of the convergence property of the bkm for homogeneous problems the given numerical experiments reveal rapid convergence high accuracy and efficiency mathematical simplicity of the bkm