the generalized singular value decomposition and the method of particular solutions a powerful method for solving planar eigenvalue problems is the method of particular solutions mps which is also well known under the name point matching method the implementation of this method usually depends on the solution of one of three types of linear algebra problems singular value decomposition generalized eigenvalue decomposition or generalized singular value decomposition we compare and give geometric interpretations of these different variants of the mps it turns out that the most stable and accurate of them is based on the generalized singular value decomposition we present results to this effect and demonstrate the behavior of the generalized singular value decomposition in the presence of a highly ill conditioned basis of particular solutions

the method of fundamental solutions for the numerical solution of the biharmonic equation abstract the method of fundamental solutions mfs is a relatively new technique for the numerical solution of certain elliptic boundary value problems it falls in the class of methods generally called boundary methods and like the well known boundary integral equation method is applicable when a fundamental solution of the differential equation is known in the mfs the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem the locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible in many applications these quantities are determined by a least squares fit of the boundary conditions a nonlinear problem which is solved using standard software in this paper the mfs is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow

the numerical solution of laplace s equation in three dimensions we consider the dirichlet problem for laplace s equation on a simply connected three dimensional region with a smooth boundary this problem is easily converted to the solution of a fredholm integral equation of the second kind based on representing the harmonic solution as a double layer potential function we solve this integral equation formulation by using galerkin s method with spherical harmonics as the basis functions this approach leads to small linear systems and once the galerkin coefficients for the region have been calculated the computation time is small for any particular boundary function the major disadvantage of the method is the calculation of the galerkin coefficients each of which is a four fold integral with a singular integrand theoretical and computational details of the method are presented

integration of thiele continued fractions and the method of fundamental solutions for solving unconfined seepage problems unconfined seepage through an earth dam or a levee is recognized as a challenging problem this complexity is mainly due to the fact that determination of the phreatic line through the dam levee body is not straightforward for simulation purposes mesh generation as well as accurate and smooth alteration of the phreatic line at the junction with the downstream slope referred to as exit point for an unconfined seepage problem with complex geometry generally makes cumbersome numerical solutions this study presents an innovative boundary type meshfree method to determine the phreatic line location in unconfined seepage problems the current study explicitly addresses the problem that alternative methods commonly face to deal with the exit point the method is developed based upon integrating the method of fundamental solutions mfs particle swarm optimization pso algorithm and thiele continued fractions tcf to accurately estimate the phreatic line location the proposed framework uses mfs to solve the flow continuity equation tcf to generate the phreatic line and pso to optimize the phreatic line location generated by tcf as a boundary method mfs only deals with the boundaries of the domain and consequently it only takes the exact position of phreatic line as a variable boundary the proposed approach employs tcf to guarantee that the phreatic line is tangent to the downstream slope at the exit point a characteristics which is important especially for the cases where abrupt changes occur in the phreatic line near the exit point for comparison and validation purposes the phreatic lines determined by the proposed approach for two unconfined seepage problems are compared and verified against those obtained from alternative analytical and numerical methods as well as a set of experimental results an excellent agreement is demonstrated upon comparison of the proposed method to the results attained from the analytical solutions and experimental tests

approximation by solutions of elliptic equations in semilocal spaces in the present work we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain by solutions of the same equations in a larger domain we construct an abstract framework which allows us to deal with such density questions simultaneously for various norms more specifically we study approximations with respect to the norms of semilocal banach spaces of distributions these spaces are required to satisfy certain postulates we establish density results for elliptic operators with constant coefficients which unify and extend previous results in our density results may possess holes and it is required to satisfy the segment condition we observe that analogous density results do not hold in spaces where the infinitely smooth functions are not dense finally we provide applications related to the method of fundamental solutions

applicability and applications of the method of fundamental solutions in the present work we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems more specifically we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration in our study the singularities of the fundamental solutions lie on a prescribed pseudo boundary the boundary of a domain which embraces the domain of the problem under consideration we extend previous density results of kupradze and aleksidze and of bogomolny to more general domains and partial differential operators and with respect to more appropriate norms our domains may possess holes and their boundaries are only required to satisfy a rather weak boundary requirement namely the segment condition our density results are with respect to the norms of the spaces c l analogous density results are obtainable with respect to holder norms we have studied approximation by fundamental solutions of the laplacian biharmonic and m harmonic and modified helmholtz and poly helmholtz operators in the case of elliptic systems we obtain analogous density results for the cauchy navier operator as well as for an operator which arises in the linear theory of thermo elasticity we also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space finally we show that linear combinations of fundamental solutions of operators of order m 4 with singularities lying on a prescribed pseudo boundary are not in general dense in the corresponding solution space

stability and convergence of the method of fundamental solutions for helmholtz problems on analytic domains the method of fundamental solutions mfs is a popular tool to solve laplace and helmholtz boundary value problems its main drawback is that it often leads to ill conditioned systems of equations in this paper we investigate for the interior helmholtz problem on analytic domains how the singularities charge points of the mfs basis functions have to be chosen such that approximate solutions can be represented by the mfs basis in a numerically stable way for helmholtz problems on the unit disc we give a full analysis which includes the high frequency short wavelength limit for more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued which in turn depends on both domain shape and boundary data using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10 1 1 on a wide variety of analytic domains at high frequencies of order only 3 points per wavelength are needed which compares very favorably to boundary integral methods

convergence of the method of fundamental solutions for solving the boundary value problem of modified helmholtz equation in this paper we derive the rate of convergence for the method of fundamental methods in solving the boundary value problem of a homogeneous modified helmholtz equation by using the fourier sine expansion to find particular solutions we establish the convergence results for solving the boundary value problem of a non homogeneous equation

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

on the range of the first two dirichlet and neumann eigenvalues of the laplacian in this paper we study the set of points in the plane defined by x y lambda 1 omega lambda 2 omega omega 1 where lambda 1 omega lambda 2 omega are either the two first eigenvalues of the dirichlet laplacian or the two first non trivial eigenvalues of the neumann laplacian we consider the case of general open sets together with the case of convex open domains we give some qualitative properties of these sets show some pictures obtained through numerical computations and state several open problems

domain decomposition methods with fundamental solutions for helmholtz problems with discontinuous source terms abstract the direct application of the classical method of fundamental solutions mfs is restricted to homogeneous linear partial differential equations pdes the use of fundamental solutions with different frequencies allowed the extension of the mfs to non homogeneous pdes in particular for poisson or helmholtz equations and for elastostatic or elastodynamic problems this method has been called method of fundamental solutions for domains mfs d but it faces an approximation problem when the non homogeneous term presents discontinuities because the fundamental solutions are analytic functions outside the source point set in this paper we analyze two domain decomposition techniques for overcoming this approximation problem the problem is set in the context of the modified helmholtz equation and we also establish the missing density results that justify both the mfs and the mfs d approximations numerical results are presented comparing a direct and an iterative domain decomposition technique with simulations in non trivial domains

stable and unstable manifolds of the henon mapping by using a parametric representation of the stable and unstable manifolds we prove that for some given values of the parameter in particular in the case first investigated by henon the henon mapping has a transversal homoclinic orbit

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

diffusion problems using transient discrete source superposition discrete source superposition techniques seek sets of discrete singular source strengths which satisfy the boundary conditions of a particular problem and necessarily the governing differential equation the sum of the fields produced by such sources thus provides an approximation to the required solution in many respects these techniques are a special case of conventional boundary integral equation methods where density distributions of smeared sources are sought in this paper an attempt is made to extend the discrete source superposition technique to transient problems the example addressed is the transient diffusion equation the method developed does seem to have some promise it is found that even coarse spatial discretization can yield quite accurate results and that surprisingly large timesteps can be employed and accuracy maintained further although no direct comparisons are made the absence of spatial integrations indicates that the computational effort per timestep is likely to be less than for conventional bie

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3

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

fast polynominal transforms for two dimensional convolution in a recent paper 1 truong et al have presented a new method for computing two dimensional convolutions it is shown that a modification of their algorithm can lead to a slightly more efficient method i e a gain in efficiency of 5 10 percent can be attained

numerical treatment of acoustic problems with boundary singularities by the singular boundary method abstract the singular boundary method sbm is a novel boundary type meshless method based on the fundamental solution of the given governing equation the sbm employs the origin intensity factors to circumvent the singularities resulting from the fundamental solutions in this paper we investigate the acoustic problems with boundary singularities using the sbm this is achieved by combining the sbm with the singularity subtraction techniques where the solution is decomposed into the singular solution and the regular solution the singular solution is derived analytically which satisfies the governing equation and the corresponding boundary conditions containing the singularities then the regular solution is obtained by the sbm numerical examples show the excellent performance of the proposed technique

an efficient wave based method for 2d acoustic problems containing corner singularities abstract the wave based method is an efficient alternative prediction technique based on a trefftz approach which as compared to element based methods can be applied in a wider frequency range at a reasonable cost this paper discusses the use of this method for the particular case of 2d acoustic problems where singularities are present in the corners of the acoustic domain the conventional set of expansion functions used to approximate the dynamic pressure field is extended with special purpose functions that incorporate the singular behaviour in the region around the singular corners this enrichment comes with a beneficial convergence rate while only a small increase of computational effort is incurred this is illustrated for various problem settings including different combinations of both homogeneous and non homogeneous boundary conditions and bounded as well as unbounded problems are considered

the method of fundamental solutions for inverse boundary value problems associated with the steady state heat conduction in anisotropic media in this paper the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady state heat conduction in an anisotropic medium since the resulting matrix equation is severely ill conditioned a regularized solution is obtained by employing the truncated singular value decomposition while the optimal regularization parameter is chosen according to the l curve criterion numerical results are presented for both two and three dimensional problems as well as exact and noisy data the convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries and decreasing the amount of noise added into the input data respectively are analysed a sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed the numerical results obtained show that the proposed numerical method is accurate convergent stable and computationally efficient and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady state heat conduction copyright 2005 john wiley sons ltd

two projection methods for the solution of signorini problems abstract two iterative methods based on projection and boundary element methods are considered for signorini problems the regularized problem with the projection boundary condition is first deduced from the signorini problem by using the equivalence between the signorini boundary condition and the projection fixed point problem our methods formulate the signorini boundary condition into a sequence of robin boundary conditions in the new boundary condition there is a penalty parameter the convergence speed of the first method is greatly influenced by the value of which is difficult to choose for individual problems to improve the performance of this method we present a self adaptive projection method which adjusts the parameter automatically per iteration based on the iterative data the main result of this work is to provide the convergence of the methods under mild assumptions as the iteration process is given by the potential and its derivative on the boundary of the domain the unknowns of the problem are computed explicitly by using the boundary element method both theoretical results and numerical experiments indicate efficiency of the methods proposed

efficient implementation of the mfs the three scenarios in this study we investigate the approximation of the solutions of harmonic problems subject to dirichlet boundary conditions by the method of fundamental solutions mfs in particular we study the application of the mfs to dirichlet problems in a disk the mfs discretization yields systems which possess special features which can be exploited by using fast fourier transform fft based techniques we describe three possible formulations related to the ratio of boundary points to sources namely when the number of boundary points is equal larger and smaller than the number of sources we also present some numerical experiments and provide an efficient matlab implementation of the resulting algorithms

convolutions of long integer sequences by means of number theoretic transforms over residue class polynomial rings in a recent paper dubois and venetsanopoulos 6 have derived methods for convolving sequences of numbers belonging to a ring s using number theoretic transforms ntt s over an extension ring r of s in this paper we obtain more explicit expressions for some of their results and more important improve the efficiency of their methods attention is focused on the case r s z p z that is r is the quotient ring of s z modulo the principal ideal generated by a polynomial p z of degree n

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

a meshfree method for elasticity problems with interfaces in this work we develop a meshfree method based on fundamental solutions basis functions for a transmission problem in linear elasticity the addressed problem consists in given the displacement field on the boundary compute the corresponding displacement field of an elastic object which has piecewise constant lame coefficients the lame coefficients are assumed to be constant in non overlapping subdomains and on the corresponding interface interior boundaries non homogeneous jump conditions on the displacement and on the traction vectors are considered the main properties of the method are analyzed and illustrated with several numerical simulations in 2d domains

the newhouse set has a positive hausdorff dimension the newhouse phenomenon of infinitely many coexisting periodic attractors is studied in its simplest form one shows that the corresponding parameter set the newhouse set jn has a strictly positive hausdorff dimension this result is stronger than that of tedeschini lalli and yorke commun math phys 106 635 1986 concerning the lebesgue measure of the newhouse set and is complementary to our knowledge on the topological properties ofjn namely it is a residual set hence uncountable and everywhere dense in a parameter interval

on the application of a fast polynomial transform and the chinese remainder theorem to compute a two dimensional convolution in this paper a fast algorithm is developed to compute two dimensional convolutions of an array of d 1 d 2 complex number points where d 2 2mand d 1 2m r 1for some 1 leq r leq m this new algorithm requires fewer multiplications and about the same number of additions as the conventional fft method for computing the two dimensional convolution it also has the advantage that the operation of transposing the matrix of data can be avoided

the effective condition number applied to error analysis of certain boundary collocation methods abstract the boundary collocation method bcm is widely used in the engineering community for the numerical solution of linear boundary value problems it is commonly stated that the results computed by this method are affected by large rounding errors this conclusion is based on a simple worst case analysis using the conventional condition number for linear systems in this paper we present a refined error analysis for a class of boundary value problems using the concept of effectively well conditioned systems this analysis leads to much smaller a priori bounds for the rounding error in the computed bcm solution these smaller bounds imply that the numerically computed solution expresses much more accurately than previously supposed properties of the ideal bcm solution computed with infinite precision therefore the intrinsic properties of the bcm can be investigated through numerical computations to a far larger extent than previously supposed thus allowing for a numerical diagnostic investigation of the bcm for example when the bcm is applied to problems with certain geometries it is observed that the approximation does not converge using our new smaller error bounds we conclude that the lack of convergence of the bcm solution for these geometries is due to the bcm itself and not to rounding errors

asymptotic stability of the fundamental solution method abstract in a previous paper 1988 we studied the numerical stability of the fundamental solution method applied to the dirichlet problem of laplace s equation and presented a practical scheme for examining the stability of the method using the singular value decomposition in the present paper we give an asymptotic estimate of the stability with respect to the number of collocation points

a meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks the numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods based on global interpolation by analytic functions show low accuracy in order to circumvent this issue a meshfree modification of the method of fundamental solutions is developed where the approximation basis is enriched by an extra span of corner adapted non smooth shape functions the high accuracy of the new method is illustrated by solving several boundary value problems for the helmholtz equation modelling physical phenomena from the fields of room acoustics and acoustic resonance

on computing the entropy of the henon attractor in a recent article d ruelle inlecture notes in physics no 80 springer berlin 1978 has conjectured that for the henon attractor its measure theoretic entropy should be equal to its characteristic exponent this result is known to be true for systems which satisfy smale s axiom a in this article we report the results of our computations which suggest that ruelle s conjecture may be true for the henon attractor further in our study we are confronted with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view

mathematical and numerical studies on meshless methods for exterior unbounded domain problems the method of fundamental solutions mfs is an efficient meshless method for solving boundary value problems in an exterior unbounded domain the numerical solution obtained by the mfs is accurate while the corresponding matrix equation is ill conditioned a modified mfs mmfs with the proper basis functions is proposed by the introduction of the modified trefftz method mtm the concrete expressions of the corresponding condition numbers are given in mathematical forms and the solvability by these methods is mathematically proven thereby the optimal parameter minimizing the condition number is also mathematically given numerical experiments show that the condition numbers of the matrices corresponding to the mtm and the mmfs are reduced and that the numerical solution by the mmfs is more accurate than the one by the conventional method

an integral equations method combined minimum norm solution for 3d elastostatics cauchy problem abstract in this paper we establish new density results for the equilibrium equations based on the denseness result of the elastic potential functions the cauchy problem for the equilibrium equations is investigated for this ill posed problem we construct a regularizing solution using the single layer potential function the well posedness of the regularizing solution as well as the convergence property is rigorously analyzed the advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented the method combines minimum norm solution with morozov discrepancy principle to solve an inverse problem convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method the numerical convergence accuracy and stability of the method with respect to the discretisation about the integral equations on pseudo boundary and the distance between the pseudo boundary and the real boundary of the solution domain and decreasing the amount of noise added into the input data respectively are also analysed with some examples

analysis of the method of fundamental solutions for the modified helmholtz equation the method of fundamental solutions mfs was first proposed by kupradze in 1963 since then the mfs has been extensively applied for solving various kinds of problems in science and engineering however few theoretical works have been reported in the literature in this paper we devote our work to the error analysis and stability of the mfs for the case of the modified helmholtz equation for disk domains a convergence analysis of the mfs was provided by li j comput appl math 159 113125 2004 for solving the modified helmholtz equation in this paper the error bounds of the mfs for bounded and simply connected domains are derived for smooth solutions of the modified helmholtz equation the exponential convergence rates can be achieved for analytic solutions the bounds of condition numbers of the mfs are derived for both disk domains and the bounded and simply connected domains to give the exponential growth via the number of fundamental solutions used numerical experiments are carried out to support the theoretical analysis moreover the analysis of this paper is applied to parabolic equations and some reviews of proof techniques and the analytical characteristics of the mfs are addressed

high order finite point method for the solution to the sound propagation problems abstract in this paper we present an accuracy improvement of the meshfree finite point method this high order method has been used to solve the sound propagation problems which can be modelled by linearized euler equations high accuracy has been obtained using polynomial reconstruction of variables involved in the riemann solver the order of the meshfree method will be verified on 2d acoustic pulse problem which serves as a benchmark problem with known analytical solution

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

on solving boundary value problems of modified helmholtz equations by plane wave functions plane wave functions e x w in r2 where 0 x x y w cos sin and x w x cos y sin are used as basis functions to solve boundary value problems of modified helmholtz equations u x 2u x 0 x u x h x x where is the laplace operator and a bounded and simply connected domain in r2 approximations of the exact solution of the above problem by plane wave functions are explicitly constructed for the case that is a disc and the order of approximations is derived a computational algorithm by collocation methods based on a simple singular decomposition of circular matrices is proposed and numerical examples are shown to demonstrate the efficiency of the methods

simple models for bifurcations creating horseshoes we present a class of simple models for global bifurcations creating horseshoes some properties known for henon mappings are easily obtained for these models such as e g the existence of nontrivial hyperbolic sets kneading sequences techniques allow us to exhibit explicit differences with the global bifurcation diagram for maps of the interval explicit examples displaying wild hyperbolic sets and infinitely many sinks are also given as an illustration of the simplicity of these models

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

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

a meshless method for some inverse problems associated with the helmholtz equation in this paper a new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the helmholtz equation including the cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data and the convergence and stability of the scheme are investigated these results show that the proposed scheme is highly accurate computationally efficient stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries and could be considered as a competitive alternative to existing methods for these problems

on the inverse spectral problem for euclidean triangles we consider the inverse spectral problem for the laplace operator on triangles with dirichlet boundary conditions providing numerical evidence to the effect that the eigenvalue triplet 1 2 3 is sufficient to determine a triangle uniquely on the other hand we show that other combinations such as 1 2 4 will not be enough and that there will exist at least two triangles with the same values on these triplets

study on connections of the mfs trefftz method indirect biem and invariant mfs in the three dimensional laplace problems containing spherical boundaries following the success of a study on the method of fundamental solutions using an image concept 13 we extend to solve the three dimensional laplace problems containing spherical boundaries by using the three approaches the case of eccentric sphere for the laplace problem is considered the optimal locations for the source distribution to include the foci in the mfs are also examined by using the image concept in the 3d problems whether a free constant is required or not in the mfs is also studied the error distribution is discussed after comparing with the analytical solution derived by using the bispherical coordinates besides the relationship between the trefftz bases and the singularity in the mfs for the three dimensional laplace problems is also addressed it is found that one source of the mfs contains several interior and exterior trefftz sets through a degenerate kernel on the contrary one single trefftz base can be superimposed by some lumped sources in the mfs through an indirect biem based on this finding the relationship between the fictitious boundary densities of the indirect biem and the singularity strength in the mfs can be constructed due to the fact that the mfs is a lumped version of an indirect biem

the method of fundamental solutions for annular shaped domains this paper consists of two parts in the first part we combine the analysis of bogomolny a bogomolny fundamental solutions method for elliptic boundary value problems siam journal on numerical analysis 22 1985 644 669 comodi and mathon m i comodi r mathon a boundary approximation method for fourth order problems mathematical models and methods in applied sciences 1 1991 437 445 and li et al z c li r mathon p sermer boundary methods for solving elliptic equations with singularities siam journal on numerical analysis 24 1987 487 498 for the method of fundamental solutions mfs and new error bounds are derived for the bounded simply connected in a readable approach a factor of o n is removed in the error bounds of bogomolny 1985 in the second part we extend the analysis for the annular domain s a by the mfs the other fundamental solutions are needed whose source nodes may also be located uniformly on a circle inside the domain s a as in the above reference error bounds are derived in detail to display the polynomial convergence rates

source field superposition analysis of two dimensional potential problems this method for problems governed by laplace s equation involves representing the potential field in a given solution domain as a superposition of fields generated by a number of sousces

the method of fundamental solutions for solving options pricing models abstract this paper provides a foundation of the method of fundamental solutions mfs for the options pricing models governed by the black scholes equation in which both the european option and american options are considered in the solution procedure no artificial boundary conditions are imposed for both datum and infinite sides of the stock prices in the cases of the european options no time marching procedures are required and numerical results are verified with the exact solutions since the free boundary conditions are considered for the american options boundary update procedure is thus applied at the same time numerical results are compared with the results in the literatures these numerical results indicate the mfs is an effective and robust meshless numerical solution for solving the options pricing models

isospectral shapes with neumann and alternating boundary conditions the isospectrality of a well known pair of shapes constructed from two arrangements of seven congruent right isosceles triangles with the neumann boundary condition is verified numerically to high precision equally strong numerical evidence for isospectrality is presented for the eigenvalues of this standard pair in new boundary configurations with alternating dirichlet and neumann boundary conditions along successive edges good agreement with theory is obtained for the corresponding spectral staircase functions strong numerical evidence is also presented for isospectrality in an example of a different pair of shapes whose basic building block triangle is not isosceles some possible confirmatory experiments involving fluids are suggested

the method of fundamental solutions applied to boundary value problems on the surface of a sphere abstract in this work we propose using the method of fundamental solutions mfs to solve boundary value problems for the helmholtz beltrami equation on a sphere we prove density and convergence results that justify the proposed mfs approximation several numerical examples are considered to illustrate the good performance of the method

the mfs for the solution of harmonic boundary value problems with non harmonic boundary conditions we investigate applications of the method of fundamental solutions mfs for the numerical solution of two dimensional boundary value problems in complex geometries governed by the laplace equation and subject to dirichlet boundary conditions which are not harmonic such problems can be very challenging because of the appearance of boundary singularities we consider several ways of choosing the boundary collocation points as well as the source points in the mfs we show that with an appropriate such choice the mfs yields highly accurate results

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

stress intensity factor computation using the method of fundamental solutions mixed mode problems the method of fundamental solutions is applied to the computation of stress intensity factors in linear elastic fracture mechanics the displacements are approximated by linear combinations of the fundamental solutions of the cauchy navier equations of elasticity and the leading terms for the displacement near the crack tip two algorithms are developed one using a single domain and one using domain decomposition numerical results are given copyright 2006 john wiley sons ltd

on the application of the method of fundamental solutions to boundary value problems with jump discontinuities abstract two meshfree methods are proposed for the numerical solution of boundary value problems bvps for the laplace equation coupled with boundary conditions with jump discontinuities in the first case the bvp is solved in two steps using a subtraction of singularity approach here the singular subproblem is solved analytically while the classical method of fundamental solutions mfs is applied for the solution of the regular subproblem in the second case the total bvp is solved using a variant of the mfs where its approximation basis is enriched with a set of harmonic functions with singular traces on the boundary of the domain the same singularity capturing functions motivated by the boundary element method bem are used for the singular part of the solution in the first method and for augmenting the mfs basis in the second method comparative numerical results are presented for 2d problems with discontinuous dirichlet boundary conditions in particular the inappropriate oscillatory behavior of the classical mfs solution due to the gibbs phenomenon is shown to vanish

the almansi method of fundamental solutions for solving biharmonic problems a new fundamental solutions method for the numerical solution of two dimensional biharmonic problems is described in this method which is based on the almansi representation of a biharmonic function in the plane the approximate solution is expressed in terms of fundamental solutions of laplace s equation and is determined by a least squares fit of the boundary conditions the results of numerical experiments which demonstrate the efficacy of the method are presented

numerical analysis of acoustic modes using the linear least squares method of fundamental solutions the method of fundamental solutions mfs has been proved to be an accurate and efficient meshless numerical method to solve acoustic eigenproblems traditionally the technique of the singular value decomposition svd is employed to obtain the corresponding contours of acoustical modes after the eigenvalues are solved however it is found that the mode shapes are sensitive to the source locations of the mfs in this paper we try to derive a robust meshless numerical scheme to obtain the contours of acoustical modes based on the linear least squares method of fundamental solutions lsmfs by specifying an additional normalized dual boundary condition the failure for determining the mode shapes by specifying a normalized data at boundary locations near or on the nodes are examined moreover it is demonstrated that the mode shapes of degenerate eigenmodes can be distinguished by specifying the boundary data at different boundary points furthermore a normalization procedure is introduced for degenerate eigenmodes three numerical experiments with regular and irregular boundaries are carried out to validate the proposed method mode shapes obtained by the linear lsmfs are in good agreement with the analytical solutions and also the results obtained by the finite element method in addition the robustness and accuracy of the eigenvalues obtained with respect to different locations of source points by the linear lsmfs in conjunction with direct determinant search method are also revisited

the method of fundamental solutions for helmholtz type equations in composite materials in this paper we introduce and develop the method of fundamental solutions mfs for solving helmholtz type elliptic partial differential equations in composite materials this study builds upon the previous developments and applications of the mfs to linear and nonlinear heat conduction elasticity and functionally graded composite layered materials numerical results are presented and discussed for four examples involving both the modified helmholtz and the helmholtz equations in two dimensional or three dimensional bounded or unbounded smooth or non smooth composite domains it was found that the method produces numerical results which are in good agreement with the analytical solutions where available

error analysis of the method of fundamental solutions for linear elastostatics for linear elastostatics in 2d the trefftz methods i e the boundary methods using the particular solutions and the fundamental solutions satisfying the cauchy navier equation lead to the method of particular solutions mps and the method of fundamental solutions mfs respectively in this paper the mixed types of the displacement and the traction boundary conditions are dealt with and both the direct collocation techniques and the lagrange multiplier are used to couple the boundary conditions the former is just the mfs and the mps and the latter is also called the hybrid trefftz method htm in jirousek 1978 1992 1996 1 3 in bogomolny 1985 4 and li 2009 5 the error analysis of the mfs is given for laplace s equation and in li 2012 6 the error bounds of both mps and htm using particular solutions ps are provided for linear elastostatics in this paper our efforts are devoted to explore the error analysis of the mfs and the htm using fundamental solutions fs the key analysis is to derive the errors between fs and ps of the linear elastostatics where the expansions of the fs in li et al 2011 7 are a basic tool in analysis then the optimal convergence rates can be achieved for the mfs and the htm using fs recently the mfs has been developed with numerous reports in computation the analysis is behind the analysis of the mfs for linear elastostatics in this paper may narrow the existing gap between computation and theory of the mfs

the method of fundamental solutions in solving coupled boundary value problems for m eeg the estimation of neuronal activity in the human brain from electroencephalography eeg and magnetoencephalography meg signals is a typical inverse problem whose solution process requires an accurate and fast forward solver in this paper the method of fundamental solutions is for the first time proposed as a meshfree boundary type and easy to implement alternative to the boundary element method bem for solving the m eeg forward problem the solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three dimensional laplace equation numerical accuracy convergence and computational load are investigated the proposed method is shown to be a competitive alternative to the state of the art bem for m eeg forward solving

the method of fundamental solutions and quasi monte carlo method for diffusion equations the laplace transform is applied to remove the time dependent variable in the diffusion equation for non harmonic initial conditions this gives rise to a non homogeneous modified helmholtz equation which we solve by the method of fundamental solutions to do this a particular solution must be obtained which we find through a method suggested by atkinson to avoid costly gaussian quadratures we approximate the particular solution using quasi monte carlo integration which has the advantage of ignoring the singularity in the integrand the approximate transformed solution is then inverted numerically using stehfest s algorithm two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2 d and 3 d 1998 john wiley sons ltd

fmm accelerated source model technique for many scatterer problems motivated by the lack of scalability of the source model technique smt to many scatterer problems a new smt based method for 3 d many scatterer problems is developed the large time complexity of the smt is reduced by deriving an implicit compact representation of the scattering problem via a fast multipole formulation which divides the interactions into far and near ones while grouping and approximating the far interactions this results in significant acceleration of classical smt which is demonstrated in a variety of 3 d many body scattering problems the improvement is notable already at the minimal number of unknowns necessary and grows with the size of the problem

mathematical foundation of the mfs for certain elliptic systems in linear elasticity the method of fundamental solutions mfs is a trefftz type technique in which the solution of an elliptic boundary value problem is approximated by a linear combination of translates of fundamental solutions with singularities placed on a pseudo boundary i e a surface embracing the domain of the problem under consideration in this work we develop a mathematical framework for the numerical implementation of the mfs in elliptic systems we obtain density results with respect to the c l norms which establish the applicability of the method in certain systems arising from the theory of elastostatics and thermo elastostatics the domains in our density results may possess holes and they satisfy the segment condition

an adaptive method of fundamental solutions for solving the laplace equation abstract in this paper we propose a residual type adaptive method of fundamental solutions amfs for solving the two dimensional laplace equation an error estimator is defined only on the boundary of the domain initial distributions of source points and collocation points are determined by using approaches proposed in chen et al 2006 the adding removing and stopping strategies are designed so that the required accuracy can be satisfied within finite steps numerical experiments reveal that amfs improves the accuracy of the mfs approximation obtained from uniformly distributed sources and collocation points which makes the mfs more practical for non harmonic and non smooth boundary conditions moreover it is shown that the error estimator becomes equidistributed after an adaptive iteration a detailed comparison between amfs and mfs using uniformly distributed points is also presented for each numerical example

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