the use of pde centres in the local rbf hermitian method for 3d convective diffusion problems in this work an extension is proposed to the local hermitian interpolation lhi method a meshless numerical method based on interpolation with small and heavily overlapping radial basis function rbf systems this extension to the lhi method uses interpolation functions which themselves satisfy the partial differential equation pde to be solved in this way a much improved reconstruction of partial derivatives can be obtained resulting in significantly improved accuracy in many cases the implementation algorithm is described and is validated via three convection diffusion reaction problems for steady and transient situations a crank nicolson implicit time stepping technique is used for the time dependent problems in the proposed approach a form of analytical upwinding is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function which includes the desired information about the convective velocity field the implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one dimensional travelling front problem at peclet numbers of 500 1000 2000 5000 and infinity which corresponds to a shock front in the case of infinity in addition the accuracy of the numerical scheme is validated against a one dimensional steady state solution exhibiting strong boundary layer effects and also against a steady and a transient three dimensional convection diffusion problem on irregular datasets all the test cases are validated against the corresponding analytical solutions finally the effect of various interpolation stencil configurations is investigated and some important limitations on local data centre distribution are identified

implementation of a generalized exponential basis functions method for linear and non linear problems this is the accepted version of the following article mossaiby f ghaderian m rossi r implementation of a generalized exponential basis functions method for linear and non linear problems international journal for numerical methods in engineering on line jul 2015 which has been published in final form at http dx doi org 10 1002 nme 4985

a green s discrete transformation meshfree method for simulating transient diffusion problems this manuscript presents the formulation and application of the green s discrete transformation method gdtm for the meshfree simulation of transient diffusion problems including those with moving boundaries the gdtm implements a linear combination of time dependent green s basis functions defined on a set of source points to approximate the field in the form of a solution series a discrete transformation is implemented to evaluate unknown coefficients of this series which eliminates the need to use time integration schemes we will study the optimal number and location of the gdtm source points that yield the highest level of accuracy while maintaining a manageable condition number for the resulting linear system of equations the optimal values of these parameters which are inherently independent of the domain geometry are determined such that the basis functions have appropriate features for approximating the field a comprehensive convergence study is presented to show the precision and convergence rate of the gdtm for modeling various diffusion problems we also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies heat transfer in a turbine blade thermal response of a porous material and localized pitting corrosion in stainless steel copyright c 2015 john wiley sons ltd

local radial basis function based gridfree scheme for unsteady incompressible viscous flows a local radial basic function rbf based gridfree scheme has been developed to solve unsteady incompressible navier stokes equations in primitive variables the velocity pressure decoupling is obtained by making use of a fractional step algorithm the scheme is validated over a variety of benchmark problems and found a very good agreement with the existing results comparisons with the benchmark solutions show that the developed local rbf gridfree scheme is stable and produces accurate results on domains discretized even with non uniform distribution of nodal points

a novel meshless method for fully nonlinear advection diffusion reaction problems to model transfer in anisotropic media abstract this article presents the new version of the backward substitution method bsm for simulating transfer in anisotropic and inhomogeneous media governed by linear and fully nonlinear advection diffusion reaction equations adres the key idea of the method is to formulate a general analytical expression of the solution in the form of the series over a basis system which satisfies the boundary conditions with any choice of the free parameters the radial basis functions rbfs of the different types are used to generate the basis system for expressing the solution then the expression is substituted into the adre under consideration and the free parameters are determined by the collocation inside the solution domain as a result we separate the approximation of the boundary conditions and the approximation of the pde inside the solution domain this approach leads to an important improvement of the accuracy of the approximate solution and can be easily extended onto irregular domain problems furthermore the proposed method is extended to general fully nonlinear adres in combination with the quasilinearization technique some numerical results and comparisons are provided to justify the advantages of the proposed method

multi level meshless methods based on direct multi elliptic interpolation a short overview on the direct multi elliptic interpolation and the related meshless methods for solving partial differential equations is given a new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain an error estimation is also derived to implement the method quadtree based multi level methods are used the approach avoids the use of large dense and ill conditioned matrices and significantly reduces the computational cost

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

rbf fd formulas and convergence properties the local rbf is becoming increasingly popular as an alternative to the global version that suffers from ill conditioning in this paper we study analytically the convergence behavior of the local rbf method as a function of the number of nodes employed in the scheme the nodal distance and the shape parameter we derive exact formulas for the first and second derivatives in one dimension and for the laplacian in two dimensions using these formulas we compute taylor expansions for the error from this analysis we find that there is an optimal value of the shape parameter for which the error is minimum this optimal parameter is independent of the nodal distance our theoretical results are corroborated by numerical experiments

comparison of finite element techniques for solidification problems the accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed grid finite element techniques using the analytical solution for this problem as a reference in the finite element formulation of the problem different time stepping schemes are compared the implicit euler backward algorithm combined with an iterative scheme and two three time level methods the lees algorithm and a dupont algorithm which are both applied as non iterative schemes furthermore different methods for handling the evolution of latent heat are examined an approximation method suggested by lemmon and one suggested by del giudice both using the enthalpy formulation as well as a fictitious heat flow method presented by rolph and bathe results of calculations performed with the consistent heat capacity matrix are compared with those performed with a lumped heat capacity matrix

convergence of unsymmetric kernel based meshless collocation methods this paper proves convergence of variations of the unsymmetric kernel based collocation method introduced by kansa in 1986 since then this method has been very successfully used in many applications though it may theoretically fail in special situations and though it had no error bound or convergence proof up to now thus it is necessary to add assumptions or to make modifications our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates these rates improve with the smoothness of the solution the domain and the kernel providing the trial spaces but they are currently not yet optimal and deserve refinement they are based on rates of approximation to the residuals by nonstationary meshless kernel based trial spaces and they are independent of the type of differential operator the results are applicable to large classes of linear problems in strong form provided that there is a smooth solution and the test and trial discretizations are chosen with some care our analysis does not require assumptions like ellipticity and it can be extended to ill posed problems

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

on unsymmetric collocation by radial basis functions solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by kansa in 1990 it rewrites the problem as a generalized interpolation problem and the solution is obtained by solving a possibly large linear system the method has been used successfully in a variety of applications but a proof of nonsingularity of the linear system was still missing this paper shows that a general proof of this fact is impossible however numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort

preconditioned conjugate gradients radial basis functions and toeplitz matrices abstract radial basis functions provide highly useful and flexible interpolants to multivariate functions further they are beginning to be used in the numerical solution of partial differential equations unfortunately their construction requires the solution of a dense linear system therefore much attention has been given to iterative methods in this paper we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data thus our method applies to the corresponding toeplitz matrices the number of iterations required to achieve a given tolerance is independent of the number of variables

reconstruct multiscale functions using different rbfs in different subdomains in practice we usually encounter many multiscale functions for example in the fields of condensed phase physics computational chemistry and nano structure it would be very difficult to reconstruct these functions in this paper we will provide an efficient meshless method we would reconstruct these functions by two or more rbfs the efficiency would be demonstrated by numerical experiments

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

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

local method of approximate particular solutions for two dimensional unsteady burgers equations the local method of approximate particular solutions lmaps is first proposed to solve burgers equations in avoiding the ill conditioning problem the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low order linear systems within the local supporting domain then the global solutions can be obtained by reformulating the local matrix in the global and sparse matrix the obtained large sparse linear systems are directly solved instead of using more complicated iterative method the numerical experiments have shown that the proposed method is suitable for solving the two dimensional unsteady burgers equations with high accuracy and efficiency

application and improvement of a recent approximate analytical solution of richards equation a method for developing wetting front profiles for the one dimensional richards equation is given the method is obtained by simplifying and extending a recent approximate solution and brings together features from several different studies difficulties associated with profile development are discussed by applying the method to several examples providing a deeper understanding of the solutions of richards equation for simplicity we illustrate the improvements for soil water diffusivities and conductivities which have a power law dependence on the water content although they should apply in general and the appropriate generalization is given in the conclusion

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

exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption several authors have now used the backlund transformation to linearize richard s equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities after transformation a convection diffusion equation is obtained to date all the analytical results making use of the backlund transform have been based on a specified constant flux that is assumed to exist at the surface boundary we generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux the solution contains a function that is specified by a volterra integral equation in practice the volterra equation must be evaluated numerically this task is accomplished using a relatively straightforward algorithm the analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping our new analytical results therefore also apply to this class of solute transport processes the general form of the adsorption isotherm for which exact results can be obtained is derived this isotherm appears capable of representing a relatively large family of physically relevant cases although this topic remains open for future investigation

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

optimal variable shape parameter for multiquadric based rbf fd method in this follow up paper to our previous study in bayona et al 2011 2 we present a new technique to compute the solution of pdes with the multiquadric based rbf finite difference method rbf fd using an optimal node dependent variable value of the shape parameter this optimal value is chosen so that to leading order the local approximation error of the rbf fd formulas is zero in our previous paper bayona et al 2011 2 we considered the case of an optimal constant value of the shape parameter for all the nodes our new results show that if one allows the shape parameter to be different at each grid point of the domain one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique we analyze the same examples studied in bayona et al 2011 2 both with structured and unstructured grids and compare our new results with those obtained previously we also find that if there are a significant number of nodes for which no optimal value of the shape parameter exists then the improvement in accuracy deteriorates significantly in those cases we use generalized multiquadrics as rbfs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter

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

clean two and three dimensional analytical solutions of richards equation for testing numerical solvers 1 this technical note derives clean analytical solutions of richards equation for three dimensional unsaturated groundwater flow clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional fourier series terms two dimensional versions of these solutions are also given the primary purpose for the solutions is to test linear and nonlinear solvers in finite difference volume element computer programs for accuracy and scalability using architectures ranging from pcs to parallel high performance computers this derivation starts from the quasi linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content this allows a transformation to be used to create a linear partial differential equation separation of variables and fourier series are then used to obtain the final solution physically reasonable material properties are also used a total of four solutions are given in this technical note steady state and transient solutions for two different boundary conditions of the sample problem

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

chaotic dynamics in nonlinear duopoly game with heterogeneous players in this study we investigate the dynamics of a nonlinear discrete time duopoly game where the players have heterogeneous expectations two players with different expectations are considered one is boundedly rational and the other thinks with adaptive expectations the stability conditions of the equilibria are discussed we show how the dynamics of the game depend on the model parameters we demonstrate that as some parameters of the game are varied the stability of nash equilibrium is lost through period doubling bifurcation the chaotic features are justified numerically via computing lyapunov exponents sensitive dependence on initial conditions and the fractal dimension

meshless methods for one dimensional oscillatory fredholm integral equations abstract in this paper efficient and simple algorithms based on levin s quadrature theory and our earlier work involving local radial basis function rbf and chebyshev differentiation matrices are adopted for numerical solution of one dimensional highly oscillatory fredholm integral equations this work is focused on the comparative performance of local rbf meshless and pseudospectral procedures we have tested the proposed methods on phase functions with and without stationary phase point s both on uniform and chebyshev grid points the proposed procedures are shown accurate and efficient and therefore provide a reliable platform for the numerical solution of integral equations from the numerical results we draw some conclusions about accuracy efficiency and robustness of the proposed approaches

generalized finite differences using fundamental solutions summary it is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions the fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a dirac delta distribution fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary only global approximations in the domain of interest in this work the same characteristic of the fundamental solutions that of solving the homogeneous equation exactly is used but not to build a global approximation on the contrary local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid copyright q 2009 john wiley sons ltd

multilevel compact radial functions based computational schemes for some elliptic problems abstract compactly supported radial basis functions cs rbfs have been recently introduced in the context of the dual reciprocity method as a possible cure of dense matrices and ill conditioning problems when using the classical radial basis functions however the support scaling factor and slow convergence rate of the cs rbfs have also raised issues on the effectiveness of the cs rbfs in this paper two multilevel schemes have been proposed to alleviate these problems

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

exponential dichotomy roughness on banach spaces in the present paper we extend existing results on exponential dichotomy roughness for linear ode systems to infinite dimensional banach space we give new conditions for the existence of exponential dichotomy roughness in infinite dimensional space and in the finite interval case we also improve previous results by indicating the exact values of the dichotomic constants of the perturbed equation

two algorithms for fast 2d node generation application to rbf meshless discretization of diffusion problems and image halftoning abstract mesh generation techniques for traditional mesh based numerical approaches such as fem and fvm have now reached a good degree of maturity there is no such an acknowledged background when dealing with node generation techniques for meshless numerical approaches despite their theoretical simplicity and efficiency furthermore node generation can be put in connection with some well known image approximation techniques two node generation algorithms are here proposed and employed in the numerical solution of 2d steady state diffusion problems by means of a local radial basis function rbf meshless method finally such algorithms are also tested for grayscale image approximation through stippling

a boundary meshless method using chebyshev interpolation and trigonometric basis function for solving heat conduction problems a boundary meshless method has been developed to solve the heat conduction equations through the use of a newly established two stage approximation scheme and a trigonometric series expansion scheme to approximate the particular solution and fundamental solution respectively as a result no fundamental solution is required and the closed form of approximate particular solution is easy to obtain the effectiveness of the proposed computational scheme is demonstrated by several examples in 2d and 3d we also compare our proposed method with the finite difference method and the other meshless method showed in sarler and vertnik comput math appl 2006 51 1269 1282 excellent numerical results have been observed copyright 2007 john wiley sons ltd

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

the quasilinear approximation for unsaturated porous media flow this paper gives a review of the quasilinear approximation an exponential relationship between hydraulic conductivity and moisture potential and the uses this has been put to since the time of its creation one of the major attractions of this approximation is that under steady regimes the highly nonlinear richards equation governing unsaturated flow can be linearized this simplification has enabled analytic and semianalytic solutions to be found for many unsaturated flow problems these are reviewed and the current research areas in which the quasilinear approximation is being used are discussed

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

the generalized finite difference method for long time dynamic modeling of three dimensional coupled thermoelasticity problems abstract in this study a new framework for the efficient and accurate solutions of three dimensional 3d dynamic coupled thermoelasticity problems is presented in our computations the krylov deferred correction kdc method a pseudo spectral type collocation technique is introduced to perform the large scale and long time temporal simulations the generalized finite difference method gfdm a relatively new meshless method is then adopted to solve the resulting boundary value problems the gfdm uses the taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables the method thus is truly meshless that can be applied for solving problems merely defined over irregular clouds of points for problem with complicated geometries this paper also examines a new distance criterion for adaptive selection of nodes in the gfdm simulations preliminary numerical experiments show that the kdc accelerated gfdm methods are very promising for accurate and efficient long time and large scale dynamic simulations

non linear heat conduction in composite bodies a boundary element formulation the present paper discusses the numerical solution of steady state non linear heat conduction problems in composite bodies by using the boundary element method two kinds of non linearities are considered the temperature dependence of the thermal conductivity and boundary conditions of the radiative type by introducing the integral of conductivity as a new variable the governing equation of the problem becomes linear in the transform space transformed boundary conditions of the dirichlet and neumann types are also linear but convective boundary conditions become non linear also discontinuities arise in the value of the integral of conductivity across the interface between materials with different properties since continuity of temperature is imposed the problem is numerically solved by discretizing the external and interface boundaries of the region under consideration with constant boundary elements and applying an iterative scheme of the newton raphson type

an adaptive rbf finite collocation approach to track transport processes across moving fronts we develop a radial basis function finite collocation method rbf fc that utilises an adaptive quadtree dataset to cluster nodes around critical features in the domain for the solution of transport processes occurring in moving front problems an adaptive quadtree dataset is used to generate an improved distribution of solution centres in the domain around which local hermitian collocation systems are formed in these local systems the governing pde and boundary operators of the problem are enforced by collocation globally the systems are linked via reconstruction of the solution variable in terms of the solution value at neighbouring nodes producing a sparse global matrix with a solution cost that scales linearly with the number of nodes in the domain by generating an adaptive set of nodal points with the quadtree dataset we can reduce the global solution error in the rbf fc method whilst maintaining low solution times the proposed method is validated on a steady state boundary layer capture problem and a transient advection problem infinite peclet number in a 2d domain we then couple the method with an interface tracking technique to solve the transient heat transfer in a hele shaw cell viscous fingering problem we demonstrate the effectiveness of the proposed method for the solution of convection dominated transport in problems across moving fronts

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

multidimensional linearized nonsteady infiltration toward a shallow water table the green s function method is used to derive a general analytical model pertaining to multidimensional nonsteady infiltration toward a shallow water table with arbitrary initial conditions boundary conditions and root uptake forcing functions and for various simple source geometries the general green s function solution allows the derivation of particular analytical solutions pertaining to cases of surface and subsurface irrigation evaporation root uptake and moisture redistribution the model assumes that the hydraulic conductivity function is an exponential function of the water pressure and a linear function of the moisture content one dimensional solutions are obtained for time dependent distributions of the flux at the land surface and of root uptake in the subsurface two and three dimensional solutions are also derived for point and line sources in the domain for a strip a rectangle and a disc source at the land surface and for two and three dimensional moisture redistribution and root uptake most of the solutions are a product of two terms one term is a function of depth while the other term is a function of time and the lateral coordinates numerical results show the effect of the water table and soil type on the two dimensional moisture movement from a strip source

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

optimal constant shape parameter for multiquadric based rbf fd method radial basis functions rbfs have become a popular method for interpolation and solution of partial differential equations pdes many types of rbfs used in these problems contain a shape parameter and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter in this paper we focus on pde problems solved with a multiquadric based rbf finite difference rbf fd method we propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error the algorithm is based on analytical approximations to the local rbf fd error derived in 1 we show through several examples in 1d and 2d both with structured and unstructured nodes that very accurate solutions compared to finite differences can be achieved using the optimal value of the constant shape parameter

analytical solution to transient richards equation with realistic water profiles for vertical infiltration and parameter estimation a general analytical model for one dimensional transient vertical infiltration is presented the model is based on a combination of the brooks and corey soil water retention function and a generalized hydraulic conductivity function this leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index accordingly the proposed analytical solution covers many existing realistic models in the literature the general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time it can be used to model infiltration through semi infinite dry soils with prescribed water content or flux boundary conditions some mathematical expressions of practical importance are also derived the general form solution is useful for comparison between models validation of numerical solutions and for better understanding the effect of some hydraulic parameters based on the analytical expression a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented

meshfree explicit local radial basis function collocation method for diffusion problems this paper formulates a simple explicit local version of the classical meshless radial basis function collocation kansa method the formulation copes with the diffusion equation applicable in the solution of a broad spectrum of scientific and engineering problems the method is structured on multiquadrics radial basis functions instead of global the collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node the computational effort thus grows roughly linearly with the number of the nodes the developed approach thus overcomes the principal large scale problem bottleneck of the original kansa method two test cases are elaborated the first is the boundary value problem nafems test associated with the steady temperature field with simultaneous involvement of the dirichlet neumann and robin boundary conditions on a rectangle the second is the initial value problem associated with the dirichlet jump problem on a square the accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes number of nodes in the domain of influence multiquadrics free parameter and timestep length on uniform and nonuniform node arrangements the developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the dirichlet jump where the approximation properties appear similar

an oligopoly model with best response and imitation rules abstract imitation based behaviors are considered in economics with significant contributions in reference to homogeneous populations where players are characterized by the same decisional processes see for example 42 48 however the presence of imitation behaviors is detected in experimental oligopolies coexisting with rational like rules this motivates us to consider an heterogeneous population where best responders and imitators coexist and compete in a deterministic oligopoly framework the model we formulate is characterized by two stationary states specifically the cournot nash equilibrium and a further production level at which best responders are inactive and imitators produce at the marked clearing price the heterogeneities among players give to the model a nonlinear structure influence the stability properties of the cournot nash equilibrium and give rise to complex dynamic scenarios we found that the imitators relative fraction have an ambiguous role in determining the stability properties of the cournot nash equilibrium and provided intermediate values of the population size its variations may cause the occurrence of both flip and neimark sacker bifurcations chaotic dynamics and coexistent attractors characterized by not connected basins may also be observed we finally note that certain dynamic regimes described by the model are provided by analogous features as those characterizing experimental outcomes and several experiments can be reproduced with different parameters sets

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

radial basis function partition of unity method for modelling water flow in porous media abstract water flow in variably saturated porous media is modelled by using the highly nonlinear parabolic richards equation the nonlinearity is due to the hydraulic conductivity and moisture content variables the latter were estimated by using experimental models including gardner burdine mualem and van genuchten models the aim of this work is to develop a new technique based on the radial basis function partition of unity method rbfpum and gardner model in order to solve richards equation in one and two dimensions we have used gardner model to handle the nonlinearity issue and the rbfpum is used to approximate the solution of the linearized richards equation our proposed algorithm is based on testing many configurations of the partitions number and selecting the optimal shape parameter for each case then we pick up the optimal configuration partitions number shape parameter that yields the best solution in terms of error and conditioning number by following this procedure an optimal solution is ensured for our given problem as numerical tests we consider the vertical infiltration of water in soils in order to validate our proposed method

a new hybrid boundary node method based on taylor expansion and the shepard interpolation method summary a novel meshless method based on the shepard and taylor interpolation method stim and the hybrid boundary node method hbnm is proposed based on the shepard interpolation method and taylor expansion the stim is developed to construct the shape function of the hbnm in the stim the shepard shape function is used as the basic function which is the zero level shape function and the high power basic functions are constructed through taylor expansion four advantages of the stim are the interpolation property the arbitrarily high order consistency the absence of inversion for the whole process of shape function construction and the low computational expense these properties are desirable in the implementation of meshless methods by combining the stim and the hbnm a much more effective meshless method is proposed to solve the elasticity problems compared with the traditional hbnm the stim can improve accuracy because of the use of high power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process numerical examples are given to demonstrate the accuracy and efficiency of the proposed method copyright 2015 john wiley sons ltd

a meshfree local rbf collocation method for anti plane transverse elastic wave propagation analysis in 2d phononic crystals in this paper a meshfree or meshless local radial basis function rbf collocation method is proposed to calculate the band structures of two dimensional 2d anti plane transverse elastic waves in phononic crystals three new techniques are developed for calculating the normal derivative of the field quantity required by the treatment of the boundary conditions which improve the stability of the local rbf collocation method significantly the general form of the local rbf collocation method for a unit cell with periodic boundary conditions is proposed where the continuity conditions on the interface between the matrix and the scatterer are taken into account the band structures or dispersion relations can be obtained by solving the eigenvalue problem and sweeping the boundary of the irreducible first brillouin zone the proposed local rbf collocation method is verified by using the corresponding results obtained with the finite element method for different acoustic impedance ratios various scatterer shapes scatterer arrangements lattice forms and material properties numerical examples are presented and discussed to show the performance and the efficiency of the developed local rbf collocation method compared to the fem for computing the band structures of 2d phononic crystals

a localized approach for the method of approximate particular solutions the method of approximate particular solutions maps has been recently developed to solve various types of partial differential equations in the maps radial basis functions play an important role in approximating the forcing term coupled with the concept of particular solutions and radial basis functions a simple and effective numerical method for solving a large class of partial differential equations can be achieved one of the difficulties of globally applying maps is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points thereby affecting our ability to solve large scale science and engineering problems in this paper we develop a localized scheme for the method of approximate particular solutions lmaps the new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation in this paper this local numerical scheme is used for solving large scale problems up to one million interpolation points three numerical examples in two dimensions are used to validate the proposed numerical scheme

circulant matrix and conformal mapping for solving partial differential equations in recent years a fast radial basis function rbf solver for surface interpolation has been developed by karageorghis et al 2007 in this paper we follow up their work by extending the idea of fast evaluation of the surface interpolation to efficiently solve various types of partial differential equations pdes we look into the possibility of solving pdes with conformal mapping to map the rbf circular points on a disk to the interior of an irregular domain we also compare the pros and cons for these two approaches in both approaches the circulant matrix formulation and the matrix decomposition method are the central ideas of fast computation in numerical experiments we test these two approaches for solving pdes using up to 160000 rbf interpolation points with greater accuracy and efficiency

an element free smoothed radial point interpolation method efs rpim for 2d and 3d solid mechanics problems abstract this paper presents a novel element free smoothed radial point interpolation method efs rpim for solving 2d and 3d solid mechanics problems the idea of the present technique is that field nodes and smoothing cells scs used for smoothing operations are created independently and without using a background grid which saves tedious mesh generation efforts and makes the pre process more flexible in the formulation we use the generalized smoothed galerkin gs galerkin weak form that requires only discrete values of shape functions that can be created using the rpim by varying the amount of nodes and scs as well as their ratio the accuracy can be improved and upper bound or lower bound solutions can be obtained by design the scs can be of regular or irregular polygons in this work we tested triangular quadrangle n sided polygon and tetrahedron as examples stability condition is examined and some criteria are found to avoid the presence of spurious zero energy modes this paper is the first time to create gs galerkin weak form models without using a background mesh that tied with nodes and hence the efs rpim is a true meshfree approach the proposed efs rpim is so far the only technique that can offer both upper and lower bound solutions numerical results show that the efs rpim gives accurate results and desirable convergence rate when comparing with the standard finite element method fem and the cell based smoothed fem cs fem

a perturbative view on the subsurface water pressure response at hillslope scale 1 this work deals with a rearrangement of richards equation to obtain a better understanding of the components of the subsurface water and pressure flows and their interactions in a hillslope the basic idea proposed in this paper is the normalization of the spatial and temporal coordinates according to the physical knowledge of the processes and the geometry of a hillslope the pressure head and all the other quantities contained in richards equation are rescaled to obtain a dimensionless equation the pressure head is also split into an equilibrium component and a transient component and expanded in a perturbation series subsequently richards equation itself is conveniently split into two equations connected by a source sink term results of these manipulations show that for the first order approximation the water and pressure flow is slope normal while slope parallel effects only occur at successive orders integrating the perturbed equation obtained for the long term pressure head component results in the well known boussinesq equation for the water table level further integrations then lead to the hillslope storage boussinesq model and finally to a generalization of the o loughlin formula for the slope parallel long term subsurface flow the introduced source sink term is estimated at the first order of approximation to be proportional to the temporal variation of water pressure head at the bottom of the integration domain the treatment of the equations clarifies some of the mechanisms acting in water pressure redistribution at hillslope scale even without solving the equation