an operator splitting radial basis function method for the solution of transient nonlinear poisson problems abstract this paper presents an operator splitting radial basis function os rbf method as a generic solution procedure for transient nonlinear poisson problems by combining the concepts of operator splitting radial basis function interpolation particular solutions and the method of fundamental solutions the application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit adams bashforth time marching for half the time step this leaves a nonhomogeneous modified helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step the resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions radial basis functions are used to construct approximate particular solutions and a grid free dimension independent method with high computational efficiency is obtained this method is demonstrated for some prototypical nonlinear poisson problems in heat and mass transfer and for a problem of transient convection with diffusion the results obtained by the os rbf method compare very well with those obtained by other traditional techniques that are computationally more expensive the new os rbf method is useful for both general irregular two and three dimensional geometry and provides a mesh free technique with many mathematical flexibilities and can be used in a variety of engineering applications

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

least squares finite element formulation in the time domain for the dual reciprocity boundary element method in heat conduction abstract this paper presents a least squares finite element scheme applied to the time domain in the dual reciprocity bem as an alternative to two point finite difference or weighted residual schemes which have been used so far in the literature to integrate the system of ordinary differential equations in time arising from the spatial boundary element discretization the proposed scheme obtains the desired recurrence relations via a least squares formulation in the context of one linear time element representing the entire time domain global boundary errors are used to obtain a measure of solution accuracy and convergence behavior of the proposed scheme through detailed and systematic numerical experiments results are presented for four representative problems and compared with those obtained using three weighted residual schemes viz the fully implicit the galerkin and the crank nicolson schemes the least squares scheme obtains in general the most accurate results at about the same computational cost as the weighted residual schemes and exhibits superior convergence behavior with linear time elements the proposed scheme shows nearly quadratic rate of convergence for all the problems considered it indicates exponential convergence for the problems involving dirichlet boundary conditions further this least squares scheme gives very accurate large time solutions

hermite biehler routh hurwitz and total positivity simple proofs of the hermite biehler and routh hurwitz theorems are presented the total nonnegativity of the hurwitz matrix of a stable real polynomial follows as an immediate corollary

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

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

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

zeta functions of three dimensional p adic lie algebras we give an explicit formula for the subalgebra zeta function of a general three dimensional lie algebra over the p adic integers mathbb z p to this end we associate to such a lie algebra a ternary quadratic form over mathbb z p the formula for the zeta function is given in terms of igusa s local zeta function associated to this form

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

representation zeta functions of nilpotent groups and generating functions for weyl groups of type b we study representation zeta functions of finitely generated torsion free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields using the kirillov orbit method and frak p adic integration we prove rationality and functional equations for almost all local factors of the euler products of these zeta functions we further give explicit formulae in terms of dedekind zeta functions for the zeta functions of class 2 nilpotent groups obtained from three infinite families of group schemes generalizing the integral heisenberg group as an immediate corollary we obtain precise asymptotics for the representation growth of these groups and key analytic properties of their zeta functions such as meromorphic continuation we express the local factors of these zeta functions in terms of generating functions for finite weyl groups of type b this allows us to establish a formula for the joint distribution of three functions or statistics on such weyl groups finally we compare our explicit formulae to frak p adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces

matrices and spectra satisfying the newton inequalities abstract an n by n real matrix is called a newton matrix and its eigenvalues a newton spectrum if the normalized coefficients of its characteristic polynomial satisfy the newton inequalities a number of basic observations are made about newton matrices including closure under inversion and then it is shown that a newton matrix with nonnegative coefficients remains newton under right translations those matrices that become and stay newton under translation are characterized in particular newton spectra in low dimensions are characterized

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

computing local zeta functions of groups algebras and modules we develop a practical method for computing local zeta functions of groups algebras and modules in fortunate cases using our method we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six we also determine the generic local subalgebra zeta functions associated with gl2 q finally we introduce and compute examples of graded subobject zeta functions

the method of fundamental solutions for the cauchy problem in two dimensional linear elasticity in this paper the application of the method of fundamental solutions to the cauchy problem in two dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularised by employing the first order tikhonov functional while the choice of the regularisation parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries as well as for constant and linear stress states the convergence and the 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

families of newton like inequalities for sets of self conjugate complex numbers abstract we derive families of newton like inequalities involving the elementary symmetric functions of sets of self conjugate complex numbers in the right half plane these are the first known inequalities of this type which are independent of the proximity of the complex numbers to the real axis

functional equations for zeta functions of groups and rings we introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings the specific form of these formulae enables us to deduce local functional equations more precisely we prove local functional equations for the subring zeta functions associated to rings the subgroup conjugacy and representation zeta functions of finitely generated torsion free nilpotent or t groups and the normal zeta functions of t groups of class 2 in particular we solve the two problems posed in 9 section 5 we deduce our theorems from a blueprint result on certain p adic integrals which generalises work of denef and others on igusa s local zeta function the malcev correspondence and a kirillov type theory developed by howe are used to linearise the problems of counting subgroups and representations in t groups respectively

a meshless method for the numerical solution of the cauchy problem associated with three dimensional helmholtz type equations in this paper the application of the method of fundamental solutions to the cauchy problem associated with three dimensional helmholtz type equations is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for under equally and over determined cauchy problems in a piecewise 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

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

methods of fundamental solutions for time dependent heat conduction problems a time dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified helmholtz equation or the fundamental solution to the heat equation this paper presents solutions using both formulations in terms of initial and boundary conditions such formulations enable calculation of errors and variance which indicates sensitivities of solutions to uncertainties in initial and boundary conditions both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods copyright 2005 john wiley sons ltd

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