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

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

characteristic boundary layers in real vanishing viscosity limits abstract in this paper we study the vanishing viscosity limit of initial boundary value problems for one dimensional mixed hyperbolic parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems in particular we assume that an eigenvalue of the inviscid system vanishes uniformly we prove the stability of boundary layers expansions in small time i e before shocks for the inviscid system as long as the amplitude of the boundary layers remains sufficiently small in particular by using lagrangian coordinates we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous dirichlet condition for the velocity at the boundary

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

spatiotemporal chaos in mixed linear nonlinear two dimensional coupled logistic map lattice abstract we investigate a new spatiotemporal dynamics with mixing degrees of nonlinear chaotic maps for spatial coupling connections based on 2dcml here the coupling methods are including with linear neighborhood coupling and the nonlinear chaotic map coupling of lattices and the former 2dcml system is only a special case in the proposed system in this paper the criteria such kolmogorov sinai entropy density and universality bifurcation diagrams space amplitude and snapshot pattern diagrams are provided in order to investigate the chaotic behaviors of the proposed system furthermore we also investigate the parameter ranges of the proposed system which holds those features in comparisons with those of the 2dcml system and the mlncml system theoretical analysis and computer simulation indicate that the proposed system contains features such as the higher percentage of lattices in chaotic behaviors for most of parameters less periodic windows in bifurcation diagrams and the larger range of parameters for chaotic behaviors which is more suitable for cryptography

the boundary element method applied to a moving free boundary problem in this paper a boundary problem is considered for which the boundary is to be determined as part of the solution a time dependent problem involving linear diffusion in two spatial dimensions which results in a moving free boundary is posed the fundamental solution is introduced and green s theorem is used to yield a non linear system of integral equations for the unknown solution and the location of the boundary the boundary element method is used to obtain a numerical solution to this system of integral equations which in turn is used to obtain the solution of the original problem graphical results for a two dimensional problem are presented published in 1999 by john wiley sons ltd

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

short term memories in lattices of inductively coupled ac driven circuits short term memory formation is a collective e ect displayed by driven spatiotemporal dynamical systems in which the injectedsignal is echoedback in many d ways d to the coupling e ect the existence of multiple short term memories enables us to use these systems to encode signals we present numerical evidence of short term memory generation in a lattice of ac driven circuits with inductive coupling the multiple con4gurations exhibited by this oscillator chain can be usedto encod e graphical characters c 2002 elsevier science b v all rights reserved

synchronization and coherence in thermodynamic coupled map lattices with intermediate range coupling in spatially extended systems intermediate range interactions arise naturally in some physical contexts to study them we investigate a model of coupled map lattices cml s with intermediate range coupling and derive analytic conditions for its synchronization we find that in these cml s if the range of coupling is fixed the law of large numbers applies for the mean field the total normalized power in nonzero components of the power spectrum of the mean field goes to zero in the thermodynamic limit we also show that in the same limit the relevant parameter for synchronization and coherence is the fraction of sites coupled and not their number s1063 651x 99 07410 3 pacs number s 05 45 2a the past decade has seen a surge of interest in synchronization and coherence in coupled map lattices cml s 1 10 most work has focused on systems with finite degrees of freedom and short range global or random coupling in this paper we study the synchronization of one and two dimensional 1d and 2d cml s with intermediaterange couplings in the thermodynamic limit this study is related to systems with finite degrees of freedom studied by sinha et al 2 and kozma 10 in these works each site is coupled to a range of neighbors on either side rather than just its nearest neighbors this may be a valid approximation in systems where interaction strength decays slowly with distance just as cml s with nearest neighbor coupling could be considered to be a discretization of partial differential equation cml s with intermediate range coupling could be seen as a discretization of a partial integro differential equation in fact such integro differential equations have often been used in modeling certain physico chemical reactions 11 elimination of rapidly diffusing components in a system of diffusion coupling can also lead to effectively nonlocal coupling in resultant equations 12 cml s in which coupling between two sites separated by distance r decays as power law 1 r a with exponent a have also been considered as models for biological neural networks 6 in the present work we derive the analytic conditions for synchronization in our cml s we find that in the thermodynamic limit the relevant parameter for synchronization is the fraction of sites coupled and not their number we present numerical results which show that for the mean field in the cml s with a fixed range of coupling sites for each site the law of large numbers applies and the total normalized power in nonzero components of the power spectrum goes to zero in the thermodynamic limit this is in sharp contrast with the conjecture of sinha et al 2

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

the boundary riemann solver coming from the real vanishing viscosity approximation we study the limit of the hyperbolic parabolic approximation left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr tilde rm ss v varepsilon t 0 equiv bar g cr v varepsilon 0 x equiv bar v 0 end array right the function tilde ss is defined in such a way as to guarantee that the initial boundary value problem is well posed even if tilde b is not invertible the data bar g and bar v 0 are constant when tilde b is invertible the previous problem takes the simpler form left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr v varepsilon t 0 equiv bar v b cr v varepsilon 0 x equiv bar v 0 end array right again the data bar v b and bar v 0 are constant the conservative case is included in the previous formulations convergence of the v varepsilon smallness of the total variation and other technical hypotheses are assumed and a complete characterization of the limit is provided the most interesting points are the following first the boundary characteristic case is considered that is one eigenvalue of tilde a can be 0 second as pointed out before we take into account the possibility that tilde b is not invertible to deal with this case we take as hypotheses conditions that were introduced by kawashima and shizuta relying on physically meaningful examples we also introduce a new condition of block linear degeneracy we prove that if this condition is not satisfied then pathological behaviors may occur

on the riemann problem for non conservative hyperbolic systems we consider the construction and the properties of the riemann solver for the hyperbolic system

asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions under the shizuta kawashima condition we show that these solutions approach constant equilibrium state in the lp norm at a rate o t m 2 1 1 p as t tends to infty for p in min m 2 infty moreover we can show that we can approximate with a faster order of convergence theconservative part of the solution in terms of the linearized hyperbolic operator for m 2 and by a parabolic equation in the spirit of chapman enskog expansion the main tool is given by a detailed analysis of the green function for the linearized problem

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

associative memory based on parametrically coupled chaotic elements abstract in this paper we propose an associative memory system based on parametrically coupled chaotic elements the proposed system is obtained by adding a new parameter control to our previously proposed system a chaotic activity in an early association stage makes an efficient association over the memories that are stored by means of autocorrelational learning when the system successfully recalls the target memory the system s motion is dominated by a spatially coherent oscillation while unstable motions remain when the system fails to make the association in addition the system has a large memory capacity a comparison between the proposed system and an approach with a nonmonotonic transfer function is also shown

mean field model of a game for power abstract our aim is to model a game for power equivalent to total energy as a dynamical process where an excess of power possessed by a player allows him to gain even more power such a positive feedback is often termed as the matthew effect analytical and numerical methods allow to identify a set of stationary states i e fixed points of the model dynamics the positions of the unstable fixed points give an insight on the basins of attraction of the stable fixed points the results are interpreted in terms of modeling of coercive power

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

boundary layers in weak solutions of hyperbolic conservation laws this paper is concerned with the initial boundary value problem for a nonlinear hyperbolic system of conservation laws we study the boundary layers that may arise in approximations of entropy discontinuous solutions we consider both the vanishing viscosity method and finite difference schemes lax friedrichs type schemes and the godunov scheme we demonstrate that different regularization methods generate different boundary layers hence the boundary condition can be formulated only if an approximation scheme is selected first assuming solely uniform l infty bounds on the approximate solutions and so dealing with l infty solutions we derive several entropy inequalities satisfied by the boundary layer in each case under consideration a young measure is introduced to describe the boundary trace when a uniform bound on the total variation is available the boundary young measure reduces to a dirac mass from the above analysis we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not each formulation is based on a set of admissible boundary values following the terminology of dubois lefloch 15 the local structure of these sets and the well posedness of the corresponding initial boundary value problem are investigated the results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics

one dimensional dynamics for traveling fronts in coupled map lattices multistable coupled map lattices typically support traveling fronts separating two adjacent stable phases we show how the existence of an invariant function describing the front profile allows a reduction of the infinitely dimensional dynamics to a one dimensional circle homeomorphism whose rotation number gives the propagation velocity the mode locking of the velocity with respect to the system parameters then typically follows we study the behavior of fronts near the boundary of parametric stability and we explain how the mode locking tends to disappear as we approach the continuum limit of an infinite density of sites

wave interactions and variation estimates for self similar zero viscosity limits in systems of conservation laws we consider the problem of self similar zero viscosity limits for systems ofn conservation laws first we give general conditions so that the resulting boundary value problem admits solutions the obtained existence theory covers a large class of systems in particular the class of symmetric hyperbolic systems second we show that if the system is strictly hyperbolic and the riemann data are sufficiently close then the resulting family of solutions is of uniformly bounded variation and oscillation third we construct solutions of the riemann problem via self similar zero viscosity limits and study the structure of the emerging solution and the relation of self similar zero viscosity limits and shock profiles the emerging solution consists ofn wave fans separated by constant states each wave fan is associated with one of the characteristic fields and consists of a rarefaction a shock or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side at shocks the solutions of the self similar zero viscosity problem have the internal structure of a traveling wave

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

validity of numerical trajectories in the synchronization transition of complex systems we investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems in the neighborhood of the critical transition to strongly nonhyperbolic behavior the system undergoes on off intermittency with respect to the synchronization state there are potentially severe consequences of these facts on the validity of the computer generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties

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

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

existence and stability of noncharacteristic boundary layers for the compressible navier stokes and viscous mhd equations for a general class of hyperbolic parabolic systems including the compressible navier stokes and compressible mhd equations we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical navier stokes boundary conditions our first main result using the abstract framework established by the authors in the companion work gues et al in j differ equ 244 309 387 2008 is to show that existence and stability of arbitrary amplitude exact boundary layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an evans function uniform evans stability whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with residual boundary conditions our second is to show that uniform evans stability for small amplitude layers is equivalent to evans stability of the limiting constant layer which in turn can be checked by a linear algebraic computation finally for a class of symmetric dissipative systems including the physical examples mentioned above we carry out energy estimates showing that constant and thus small amplitude layers always satisfy uniform evans stability this yields existence of small amplitude multi dimensional boundary layers for the compressible navier stokes and mhd equations for both equations these appear to be the first such results in the compressible case

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

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

trefftz method fitting boundary conditions the paper presents various ways of fitting the boundary conditions in the t complete functions method the authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit the convergence of the collatz error measures and the conditioning of the solution matrices are investigated in detail

self organized memories in coupled map lattices abstract spatially extended dynamical systems subjected to a periodic external input are able to store short term memories as a result of self organization this phenomenon has been used to interpret sliding charge density wave experiments and has been found in lattices of coupled maps and oscillators in this paper we aim to describe the influence of a stochastic perturbation in the self organized memories in a lattice of coupled linear and weakly nonlinear maps we consider the case in which the coupling between maps depends on the lattice distance as a power law

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

turing instability in oscillator chains with nonlocal coupling we investigate analytically and numerically the conditions for the turing instability to occur in a one dimensional chain of nonlinear oscillators coupled nonlocally in such a way that the coupling strength decreases with the spatial distance as a power law a range parameter makes it possible to cover the two limiting cases of local nearest neighbor and global all to all couplings we consider an example from a nonlinear autocatalytic reaction diffusion model

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

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

existence theory by front tracking for general nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n quasilinear system in one space dimension u t a u u x 0 qquad u 0 x bar u x 1 where u u t x u 1 t x dots u n t x u mapsto a u is a smooth matrix valued map and the initial data overline u is assumed to have small total variation we present a front tracking algorithm that generates piecewise constant approximate solutions converging in mathbb l loc 1 to the vanishing viscosity solution of 1 which by the results in 6 is the unique limit of solutions to the artificial viscous parabolic approximation u t a u u x mu u xx qquad u 0 x bar u x as mu to 0 in the conservative case where a u is the jacobian matrix of some flux function f u with values in mathbb r n the limit of front tracking approximations provides a weak solution of the system of conservation laws ut f u x 0 satisfying the liu admissibility conditions

treatment of corners in bie analysis of potential problems at corners or edges in the boundary of the domain of a potential problem the local normal gradient of potential is double valued when dirichlet boundary conditions are specified there are thus two unknowns at a single nodal point and the sets of equations resulting from the usual bie discretization are rendered indeterminate we discuss here earlier approaches to the resolving of this problem and describe a further approach which appears to offer some advantages both normal gradients can be approximated directly from local potential boundary conditions showing the problem indeed to be formally overdetermined this ability is discarded in favour of yielding a robust and well conditioned relationship between the two gradients this in conjunction with the bie analysis permits solutions of considerable accuracy to be found including the gradients at such corner nodes illustrative calculations are presented for rectilinear and curvilinear domains these show that even with as few elements as there are corners and thus one and a half times as many unknowns as there are nodal points good approximations to the gradients can be obtained the need for progressively finer discretization as a corner is approached is thus much reduced

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

invariant manifolds for a singular ordinary differential equation we study the singular ordinary differential equation 0 1 dudt 1 u s u ns u where u rn the functions s rn and ns rn are of class c2 and is a real valued c2 function the equation is singular because u can attain the value 0 we focus on the solutions of 0 1 that belong to a small neighborhood of a point u such that s u ns u 0 and u 0 we investigate the existence of manifolds that are locally invariant for 0 1 and that contain orbits with a prescribed asymptotic behavior under suitable hypotheses on the set u u 0 we extend to the case of the singular ode 0 1 the definitions of center manifold center stable manifold and of uniformly stable manifold we prove that the solutions of 0 1 lying on each of these manifolds are regular this is not trivial since we provide examples showing that in general a solution of 0 1 is not continuously differentiable finally we show a decomposition result for a center stable manifold and for the uniformly stable manifold an application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic parabolic systems in one space variable such a class includes the compressible navier stokes equation

lyapunov spectrum and synchronization of piecewise linear map lattices with power law coupling we study the synchronization properties of a lattice of chaotic piecewise linear maps the coupling strength decreases with the lattice distance in a power law fashion we obtain the lyapunov spectrum of the coupled map lattice and investigate the relation between spatiotemporal chaos and synchronization of amplitudes and phases using suitable numerical diagnostics

stability of small amplitude boundary layers for mixed hyperbolic parabolic systems we consider an initial boundary value problem for a symmetrizable mixed hyperbolic parabolic system of conservation laws with a small viscosity e u e t f u e x e b u e u e x x when the boundary is noncharacteristic for both the viscous and the inviscid system and the boundary condition dissipative we show that u e converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small this generalizes previous results obtained for b invertible and the linear study of serre and zumbrun obtained for a pure dirichlet s boundary condition

a method for computing acoustic fields based on the principle of wave superposition a method for computing the acoustic fields of arbitrarily shaped radiators is described that uses the principle of wave superposition the superposition integral which is shown to be equivalent to the helmholtz integral is based on the idea that the combined fields of an array of sources interior to a radiator can be made to reproduce a velocity prescribed on the surface of the radiator the strengths of the sources that produce this condition can in turn be used to compute the corresponding surface pressures the results of several numerical experiments are presented that demonstrate the simplicity of the method also the advantages that the superposition method has over the more commonly used boundary element methods are discussed these include simplicity of generating the matrix elements used in the numerical formulation and improved accuracy and speed the latter two being due to the avoidance of uniqueness and singularity problems inherent in the boundary element formulation

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

hermitian cubic boundary elements for two dimensional potential problems a hermite interpolation based formulation is presented for the boundary element analysis of two dimensional potential problems two three noded hermitian cubic elements hce are introduced for the modelling of corners or points with non unique tangents on the boundary these elements along with the usual two noded hce are used in numerical examples the results obtained show that faster convergence can be achieved using hce compared with using lagrange interpolation type quadratic elements qe for about the same amount of computing resources

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

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

stable viscosity matrices for systems of conservation laws abstract a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension u 1 f u x 0 u r m is studied these matrices are admissible in the sense that small amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t f u x v du x x as ifv 0 if d is in this class the class is determined by a linearized stability requirement the cauchy problem for the equation u 1 f u 0 u x vdu xx should be well posed in l 2 uniformly in v as v 0 previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion

boundary layer stability in real vanishing viscosity limit in the previous paper 20 an evans function machinery for the study of boundary layer stability was developed there the analysis was restricted to strongly parabolic perturbations that is to an approximation of the form ut f u x b u ux x 1 with an elliptic matrix b however real models like the navier stokes approximation of the euler equations for a gas flow involve incompletely parabolic perturbations b is not invertible in general

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

analytical results for coupled map lattices with long range interactions we obtain exact analytical results for lattices of maps with couplings that decay with distance as r ensuremath ensuremath alpha we analyze the effect of the coupling range on the system dynamics through the lyapunov spectrum for lattices whose elements are piecewise linear maps we get an algebraic expression for the lyapunov spectrum when the local dynamics is given by a nonlinear map the lyapunov spectrum for a completely synchronized state is analytically obtained the critical line characterizing the synchronization transition is determined from the expression for the largest transversal lyapunov exponent in particular it is shown that in the thermodynamical limit such transition is only possible for sufficiently long range interactions namely for ensuremath alpha l ensuremath alpha c d where d is the lattice dimension

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

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

modified method of regularized sources for potential flow abstract this paper describes the development of the method of regularized sources for potential flow problems it is based on the modification of the fundamental solution near the source point by replacing the singularity with a blob in form of a steep rational function this allows to solve the problems in the same way as with method of fundamental solutions however without an artificial boundary method of regularized sources gives excellent results for dirichlet boundary conditions however it fails for neumann boundary conditions to overcome this problem are the source point positions on the segments of the boundary with neumann boundary positions placed close to the collocation points this approach somehow represents a blending of the method of regularized sources and the classical method of fundamental solutions the novel approach is characterized by two free parameters the blob thickness and the artificial boundary displacement position a two dimensional numerical example of potential flow around circle is analyzed in detail regarding these two free parameters the modified method of regularized sources gives even more accurate results for potential and derivatives than the method of fundamental solutions the source point can be placed 2 25 times closer to the boundary collocation points than with the classical method of fundamental solutions and thus reduces the problem of the placement of the artificial boundary

inducing coherence in networks of bistable maps by varying the interaction range ordinarily two different topologies have been used to model spatiotemporal chaos and to study complexity in networks of maps one where sites interact only with nearest neighbors e g the standard diffusive coupling and one where sites interact with all sites in the network global coupling here we investigate intermediate regimes considering the interaction range as a free tunable parameter the synchronization behavior normally seen in globally coupled maps is found to set in for interaction ranges considerably smaller than the system size in addition we analytically derive stability conditions for the onset of coherent states full synchronization from which the minimum interaction range needed to induce coherence in homogeneously coupled maps can be determined such conditions are also obtained for inhomogeneous situations when the coupling strength decreases linearly with the distance the characteristic range for the onset of coherence is studied in detail as a function of model parameters

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

short time memories in a network with randomly distributed connections coupled map lattices are able to store short term memories when an external periodic input is applied we consider short term memory formation in networks with both regular nearest neighbor and randomly chosen connections the regimes under which single or multiple memorized patterns are stored are studied in terms of the coupling and nonlinear parameters of the network

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

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