fast calculation of laurent expansions for matrix inverses previously described algorithms for calculating the laurent expansion of the inverse of a matrix valued analytic function become impractical already for singularity orders as low as around p 6 since they require over o 28p matrix multiplications and correspondingly large amounts of memory in place of using mathematically exact recursions we show that for floating point calculations a rational approximation approach can avoid this cost barrier without any significant loss in accuracy

an efficient numerical method for the solution of sliding contact problems in this paper an efficient numerical method to solve sliding contact problems is proposed explicit formulae for the gauss jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with cauchy kernels are derived the resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a jacobi polynomial gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n 1 where n is the number of nodes from this gauss jacobi quadrature the existing gauss chebyshev quadrature formulas can be easily derived another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact also this analysis shows that once if the total normal load and the friction coefficient are given the external moment m and contact eccentricity e for incomplete contact in fully sliding contact are uniquely determined finally numerical solutions are computed for two typical contact cases including sliding hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi infinite elastic solid these results provide a demonstration of the validity of the proposed method copyright 2005 john wiley sons ltd

a meshfree approach to non newtonian free surface ice flow numerical models of glacier and ice sheet dynamics traditionally employ finite difference or finite element methods although these are highly developed and mature methods they suffer from some drawbacks such as inability to handle complex geometries finite differences or a costly assembly procedure for nonlinear problems finite elements additionally they are mesh based and therefore moving domains become a challenge in this paper we introduce a novel meshfree approach based on a radial basis function rbf method the meshfree nature of rbf methods enables efficient handling of moving margins and free ice surface rbf methods are also accurate easy to implement and allow for reduction the computational cost associated with the linear system assembly since stated in strong form to demonstrate the global rbf method we model the velocity field of ice flow in the haut glacier d arolla which is governed by the nonlinear stokes equations we test the method for different basal conditions and for a free moving surface we also compare the global rbf method with its localized counterpart the rbf partition of unity method rbf pum that allows for a significant gain in the computational efficiency both rbf methods are compared with the classical finite element method in terms of accuracy and efficiency we find that the rbf methods are more efficient than the finite element method and well suited for ice dynamics modeling especially the partition of unity approach

the overlapped radial basis function finite difference rbf fd method a generalization of rbf fd abstract we present a generalization of the rbf fd method that computes rbf fd weights in finite sized neighborhoods around the centers of rbf fd stencils by introducing an overlap parameter 0 1 such that 1 recovers the standard rbf fd method and 0 results in a full decoupling of stencils we provide experimental evidence to support this generalization and develop an automatic stabilization procedure based on local lebesgue functions for the stable selection of stencil weights over a wide range of values we provide an a priori estimate for the speedup of our method over rbf fd that serves as a good predictor for the true speedup we apply our method to parabolic partial differential equations with time dependent inhomogeneous boundary conditions neumann in 2d and dirichlet in 3d our results show that our method can achieve as high as a 60 speedup in 3d over existing rbf fd methods in the task of forming differentiation matrices

on numerical cubatures of singular surface integrals in boundary element methods we present and analyze methods for the accurate and efficient evaluation of weakly cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3 the class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three dimensional domains by the boundary integral equation method the possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point this rules out weighted quadrature approaches with a fixed singular weight for weakly singular integrals it is shown that duffy s triangular coordinates leadalways to a removal of the kernel singularity also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero these are based on the rabinowitz richter estimates in connection with an asymptotic estimate of domains of analyticity in 2 it is further shown that the modified extrapolation approach due to lyness is in the weakly singular case always applicable corresponding error and asymptotic work estimates are presented for the weakly singular as well as for cauchy and hypersingular integrals which e g arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates these rules are hierarchically constructed from finite part integration formulas in radial and gaussian formulas in angular direction again we show how the rabinowitz richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size

a new method based on haar wavelet for the numerical solution of two dimensional nonlinear integral equations a new numerical method based on haar wavelet is proposed for two dimensional nonlinear fredholm volterra and volterra fredholm integral equations of first and second kind the proposed method is an extension of the haar wavelet method aziz and siraj ul islam 2013 siraj ul islam et al 2013 and siraj ul islam et al 2014 from one dimensional nonlinear integral equations fredholm and volterra to two dimensional nonlinear integral equations fredholm volterra and volterra fredholm the main characteristic of the method is that unlike several other methods it does not involve numerical integration which results in an improved accuracy of the method in order to show the effectiveness of the method it is applied to several benchmark problems the numerical results are compared with other methods existing in the recent literature

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

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

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

moments in infinite channel queues we consider a new approach to the problem of evaluating the moments of the steady state number of customers in system b for an infinite channel queue having general input our new approach uses the rate conservation law as a consequence we show that for a variety of g gi queues var b as long as service times have finite first moment in addition we give some ordering relations for the variance of b when comparing two alternative systems with convex ordered inter arrival time distributions

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

global well posedness for the phan thein tanner model in critical besov spaces without damping in this paper we mainly investigate the cauchy problem for the phan thein tanner ptt model the ppt model can be viewed as navier stokes equations coupled with a nonlinear transport system this model is derived from network theory of the polymeric fluid we study about the global well posedness of the ptt model in critical besov space when the initial data are a small perturbation around the equilibrium the strong solution in critical besov space is proved to be globally existed in this paper we mainly investigate the cauchy problem for the phan thein tanner ptt model the ppt model can be viewed as navier stokes equations coupled with a nonlinear transport system this model is derived from network theory of the polymeric fluid we study about the global well posedness of the ptt model in critical besov space when the initial data are a small perturbation around the equilibrium the strong solution in critical besov space is proved to be globally existed

a high order multivariate approximation scheme for scattered data sets we present a high order multivariate approximation scheme for scattered data sets each data point is represented as a taylor series and the high order derivatives in the taylor series are treated as random variables the approximation coefficients are then chosen to minimize an objective function at each point by solving an equality constrained least squares the approximation is an interpolation when the data points are given as exact or a nonlinear regression function when nonzero measurement errors are associated with the data points using this formulation the gradient information on each data point can be used to significantly reduce the approximation error all parameters of the approximation scheme can be computed automatically from the data points an uncertainty bound of the approximation function is also produced by the scheme numerical experiments demonstrate that although this method is more computationally intensive than traditional methods it produces more accurate approximation functions

on the use of interpolative quadratures for hypersingular integrals in fracture mechanics the implementation of finitepart integration of hypersingular boundary integrals is discussed in the context of the applications in engineering fracture mechanics the approach uses a formulation o

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions the development of numerical methods for the solution of partial differential equations pdes with nonlocal boundary conditions is important since such type of problems arise as mathematical models of various real world processes we use radial basis function rbf collocation technique for the solution of a multidimensional linear elliptic equation with classical dirichlet boundary condition and nonlocal integral conditions rbf based meshless methods are easily implemented and efficient especially for multidimensional problems formulated on complexly shaped domains in this paper properties of the method are investigated by studying two and three dimensional test examples with manufactured solutions we analyze the influence of the rbf shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix

enforcing energy helicity and strong mass conservation in finite element computations for incompressible navier stokes simulations abstract we study a finite element scheme for the 3d navier stokes equations nse that globally conserves energy and helicity and through the use of scott vogelius elements enforces pointwise the solenoidal constraints for velocity and vorticity a complete numerical analysis is given including proofs for conservation laws unconditional stability and optimal convergence we also show the method can be efficiently computed by exploiting a connection between this method its associated penalty method and the method arising from using grad div stabilized taylor hood elements finally we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme

fast and accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights an efficient algorithm for the accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights is presented the algorithm is based on newton s root finding method with initial guesses and function evaluations computed via asymptotic formulae the n point quadrature rule is computed in mathcal o n operations to an accuracy of essentially double precision for any n geq 100

frictional contact problem for a rigid cylindrical stamp and an elastic layer resting on a half plane abstract the frictional contact problem for a layer resting on a homogeneous half plane is handled using linear elasticity theory in this study the layer is in contact with a rigid cylindrical stamp that is on the layer and applies a concentrated force in the normal and tangential directions friction between the component couples of layer stamp and layer half plane is taken into account the problem is reduced to a system of singular integral equations in which the contact pressures and the contact areas are the unknowns and it is treated using fourier transforms and the boundary conditions for the problem the system of singular integral equations is solved numerically using the gauss jacobi integration formula with equilibrium and consistency conditions numerical results for the contact pressures and the contact areas are given as a solution for both the frictional and the frictionless cases this work is the first study that investigates the effect of friction on the receding contact problem of a layer and a half plane with two contact areas

an efficient numerical scheme for a nonlinear integro differential equations with an integral boundary condition nonlinear functional integro differential equations with an integral boundary condition is appeared in chemical engineering underground water flow and population dynamics phenomena and other field of physics and mathematical chemistry so this paper presents a powerful numerical approach based on an iterative technique and sinc quadrature to estimate a solution for this equation then convergence of this technique is discussed by preparing a theorem which guarantees the applicability of that finally some numerical examples are given to confirm efficiency and accuracy of the numerical scheme

wavelet based iterative methods for a class of 2d partial integro differential equations abstract in this paper an iterative method based on quasilinearization is presented to solve a class of two dimensional partial integro differential equations that arise in nuclear reactor models and population models two different approaches based on haar and legendre wavelets are studied to develop numerical methods in the first approach time domain is approximated with the help of forward finite difference approach in the second approach both time as well as space domains are approximated by wavelets appropriate examples are solved using these methods and the obtained results are compared with the methods available in the recent literature

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

numerical solution of the heat equation with nonlocal boundary conditions abstract the method for the heat equation with nonlocal boundary conditions is discussed in this paper the unconditional stability is proved for 1 2 subject to a condition that is much weaker than the one assumed in a paper by ekolin due to the nonlocal boundary conditions the systems of linear equations generated by the method have a coefficient matrix that is tridiagonal except its first and last rows three efficient algorithms for solving this kind of linear systems are presented a simple numerical example is given to compare their efficiency

pseudospectral simulation of turbulent viscoelastic channel flow abstract the methodology and validation of direct numerical simulations of viscoelastic turbulent channel flow are presented here using differential constitutive models derived from kinetic and network theories numerical simulations have demonstrated drag reduction for various values of the parameters under conditions where there is a substantial increase in the extensional viscosity compared to the shear viscosity sureshkumar beris handler direct numerical simulation of turbulent channel flow of a polymer solution phys fluids 9 1997 743 755 and dimitropoulos sureshkumar beris direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction effect of the variation of rheological parameters j non newtonian fluid mech 79 1998 433 468 in this work new results pertaining to the reynolds stress and the pressure are presented and the convergence of the pseudospectral algorithm utilized in the simulations as well as its parallel implementation are discussed in detail it is shown that the lack of mesh refinement or the use of a larger value for the artificial stress diffusivity used to stabilize the conformation tensor evolution equations introduce small quantitative errors which qualitatively have the effect of lowering the drag reduction capability of the simulated fluid however an insufficient size of the periodic computational domain can also introduce errors in certain cases which albeit usually small can qualitatively alter various features of the solution

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

a comparative study of numerical integration based on haar wavelets and hybrid functions a quadrature rule based on uniform haar wavelets and hybrid functions is proposed to find approximate values of definite integrals the wavelet based algorithm can be easily extended to find numerical approximations for double triple and improper integrals the main advantage of this method is its efficiency and simple applicability error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method

superconvergence of the composite simpson s rule for a certain finite part integral and its applications the superconvergence phenomenon of the composite simpson s rule for the finite part integral with a third order singularity is studied the superconvergence points are located and the superconvergence estimate is obtained some applications of the superconvergence result including the evaluation of the finite part integrals and the solution of a certain finite part integral equation are also discussed and two algorithms are suggested numerical experiments are presented to confirm the superconvergence analysis and to show the efficiency of the algorithms

fast reliable and unrestricted iterative computation of gauss hermite and gauss laguerre quadratures methods for the computation of classical gaussian quadrature rules are described which are effective both for small and large degree these methods are reliable because the iterative computation of the nodes has guaranteed convergence and they are fast due to their fourth order convergence and its asymptotic exactness for an appropriate selection of the variables for gauss hermite and gauss laguerre quadratures local taylor series can be used for computing efficiently the orthogonal polynomials involved with exact initial values for the hermite case and first values computed with a continued fraction for the laguerre case the resulting algorithms have almost unrestricted validity with respect to the parameters full relative precision is reached for the hermite nodes without any accuracy loss and for any degree and a mild accuracy loss occurs for the hermite and laguerre weights as well as for the laguerre nodes these fast methods are exclusively based on convergent processes which together with the high order of convergence of the underlying iterative method makes them particularly useful for high accuracy computations we show examples of very high accuracy computations of up to 1000 digits of accuracy

on the weights of certain quadratures for the numerical evaluation of cauchy principal value integrals and their derivatives we consider interpolatory quadrature rules of the type begin gathered frac d k d lambda k int limits 1 1 1 x alpha 1 x beta frac f x x lambda dx sum limits i 1 n w ni k lambda f x ni r n k f hfill k 0 1 alpha beta 0 hfill end gathered where the nodes x ni are the zeros of the jacobi polynomialp n x in particular we prove that sum limits i 1 n left w ni k lambda right o n k log n uniformly in 1 e 1 e furthermore we show that whenk 1 f x c p 1 1 p 1 andf p x h 1 1 0 1 we have r n 1 f o n p mu 1 delta 0 small as we like uniformly in 1 e 1 e

flow of a generalized oldroyd b fluid due to a constantly accelerating plate the velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized oldroyd b fluid due to a constantly accelerating plate have been established using fourier sine and laplace transforms in order to avoid lengthy calculations of residues and contour integrals the discrete inverse laplace transform method has been used the solutions that have been obtained presented under integral and series forms in terms of the generalized g and r functions satisfy all imposed initial and boundary conditions the similar solutions for generalized maxwell fluids as well as those for oldroyd b maxwell and newtonian fluids are obtained as limiting cases of our general solutions

numerical evaluation of hypersingular integrals abstract boundary integral equations which employ integrals which exist only if defined in the cauchy principal value sense or as the hadamard finite part are currently used with success to solve many two and three dimensional problems of applied mechanics we will recall definitions and main properties of these integrals examine some numerical approaches for their evaluation and present several new results

testing viscoelastic numerical schemes using the oldroyd b fluid in newtonian kinematics abstract we focus here on using a newtonian velocity field to evaluate numerical schemes for two different formulations of viscoelastic flow the two distinct formulations we consider correspond to either using a fixed basis for the elastic stress or one that uses the flow directions or streamlines the former is the traditional cartesian stress formulation whilst the later may be referred to as the natural stress formulation of the equations we choose the oldroyd b fluid and three benchmarks in computational rheology the 4 1 contraction flow the stick slip and cross slot problems in the context of the contraction flow fixing the kinematics as newtonian actually gives a larger stress singularity at the re entrant corner the matched asymptotics of which are presented here numerical results for temporal and spatial convergence of the two formulations are compared first in this decoupled velocity and elastic stress situation to assess the performance of the two approaches this may be regarded as an intermediate test case before proceeding to the far more difficult fully coupled velocity and stress situation we also present comparison results between numerics and asymptotics for the stick slip problem finally the natural stress formulation is used to investigate the cross slot problem again in a newtonian velocity field

numerical modelling of viscoelastic cavity driven flow using finite difference simulations abstract a study of the unsteady flow of viscoelastic liquids contained in an cavity driven by a moving wall has been undertaken as a first step toward understanding heat and mass transport in polymer processing equipment the material was represented by the constitutive equation for a upper convected criminale ericson filbey cef model solution to the vorticity equation for various moving walls and different aspect ratios were obtained numerically the solutions were found to be stable and convergent for limited value of weissenberg numbers despite this fact some new results which are governed by inertia elasticity and aspect ratio were obtained and this has been documented first time

some new operational matrices and its application to fractional order poisson equations with integral type boundary constrains abstract enormous application of fractional order partial differential equations fpdes subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world the prime objective of this article is to find approximate solution of a general fpdes subject to nonlocal integral type boundary conditions on both ends of the domain the proposed method is based on spectral method we construct some new operational matrices which have the ability to handle integral type non local boundary constrains these operational matrices can be effectively applied to convert the fpdes together with its integral types boundary conditions to easily solvable matrix equation the accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems the proposed method has the ability to solve non local fpdes with high accuracy and low computational cost different aspects of presented approach are compared with two other recently developed methods haar wavelets collocation method and a family of collocation methods which are based on radial base functions it is observed that the proposed method is highly accurate robust efficient and stable as compared to these methods

numerical quadratures for singular and hypersingular integrals in boundary element methods a method is developed for the computation of the weights and nodes of a numerical quadrature which integrates functions containing singularities up to order 1 x 2 without the requirement to know the coefficients of the singularities exactly the work is motivated by the need to evaluate such integrals on boundary elements in potential problems and is a simplification of a previously published method but with the advantage of handling singularities at the endpoints of the integral the numerical performance of the method is demonstrated by application to an integral containing logarithmic first and second order singularities characteristic of the problems encountered in integrating a green s function in boundary element problems it is found that the quadrature is accurate to 11 12 decimal places when computed in double precision

blow up and global existence for the periodic phan thein tanner model abstract in this paper we mainly investigate the cauchy problem for the periodic phan thein tanner ptt model this model is derived from network theory for the polymeric fluid we prove that the strong solutions of ptt model will blow up in finite time if the trace of initial stress tensor tr 0 x is negative it is thus very different from the other viscoelastic model on the other hand we obtain the global existence result with small initial data when tr 0 x c 0 0 for some c 0 moreover we study about the large time behavior

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

taming hypersingular integrals using dimensional continuation we use the method of dimensional continuation to isolate singularities in integrals containing products of green s functions or their derivatives rules for the extraction of the finite part of so called hypersingular integrals are developed which should be useful in methods based on boundary integral techniques in science and engineering in applications to potential theory electromagnetic scattering and crack dynamics in continuum mechanics boundary integrals now can be readily evaluated using computational techniques without recourse to complex analysis or contour distortions since the hypersingularities occurring in intermediate steps of the computations can be isolated and ignored while taking the finite parts of the integrals into account in a consistent manner we have also identified new forms of the dirac ensuremath delta function in d dimensions which are useful and convenient in the calculations a summary of the integrable singular integrals is given in tabular form we extend the considerations to a wider class of green s functions and present a theorem with additional results arising from it that shows that hypersingular integrals associated with three dimensional potential problems can be reduced to one dimensional finite integrals rather than two dimensional integrals again leading to direct evaluations in such cases these calculations are compared with existing results to show the efficacy of the approach

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

local similarity solutions for boundary layer flow of a fene p fluid we study the boundary layer flow past a semi infinite plate of a viscoelastic fluid governed by the fene p model in the limit as the reynolds number re and the wessenberg number we is such that were 1 2 o 1 the stream function is governed by a generalized blasuis equation a local similarity solution is obtained by a shooting method the effect of extensibility on the skin friction and first normal stress coefficients are discussed we show that the skin friction increases with the extensibility parameter b as b approaches infinity the result reduces to blasius solution compared to blasius solution a reduction of up to 20 in the skin friction coefficient is possible for finite values of b

axisymmetric finite element solution of non isothermal parallel plate flow steady non isothermal parallel plate flow of a newtonian fluid with a temperature dependent viscosity is considered the viscosity is modelled by a nahme type law we apply axisymmetric q k finite elements to the coupled nonlinear system to obtain numerical solutions for a wide range of parameters

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

gauss radau and gauss labatto quadratures with double end points abstract we develop explicit formulae for generalized gauss radau and gauss lobatto quadrature rules having end points of multiplicity 2 and containing chebyshev weight functions of any of the four kinds

displacement discontinuity method for modeling axisymmetric cracks in an elastic half space this paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half space or full space the formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack the integral kernels which represent stress influence functions for ring dislocation dipoles are derived from available axisymmetric dislocation solutions the crack is discretized into constant strength displacement discontinuity elements where each element represents a slice of a cone the influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper and cauchy singularities the accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element the method is validated by a comparison with analytical and numerical reference solutions

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

improved radial basis function methods for multi dimensional option pricing in this paper we have derived a radial basis function rbf based method for the pricing of financial contracts by solving the black scholes partial differential equation as an example of a financial contract that can be priced with this method we have chosen the multi dimensional european basket call option we have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter for other non optimal choices of shape parameter values the resulting convergence rate is algebraic we propose an adapted node point placement that improves the accuracy compared with a uniform distribution compared with an adaptive finite difference method the rbf method is 20 40 times faster in one and two space dimensions and has approximately the same memory requirements

time space domain mesh free finite difference based on least squares for 2d acoustic wave modeling abstractfinite difference fd methods approximate derivatives through a weighted summation of function values from neighboring nodes traditionally these neighboring nodes are assumed to be distributed regularly such as in square or rectangular lattices to improve geometric flexibility one option is to develop fd in a mesh free discretization in which scattered nodes can be placed suitably with respect to irregular boundaries or arbitrarily shaped anomalies without coordinate transformation or forming any triangles or tetrahedra etc these mesh free fds have had successful applications especially in computational geoscience however they are all space domain fd schemes in which fd coefficients are derived by approximating spatial derivatives individually in the space domain for acoustic wave modeling it has been proven that space domain fd methods normally have higher dispersion error than time space domain fd methods which determine fd coefficients by approximating the time space domain disp

the asymptotic cardinal function of the multiquadratic ϕ r r2 c2 12as c a radial basis function approximation has the form s x j rd yj x xj 2 x rd where 0 r is some given function yj j zd are real coefficients and the centres xj j zd are points in rd it is known that radial basis function approximations using the multiquadric r r2 c2 12 possess many useful and interesting properties when the centres form an infinite regular lattice we analyse the limiting case as c and identify a class of functions that arise as uniform limits of the multiquadric interpolants in the univariate case we observe that the cardinal function for the multiquadratic becomes the sinc function as c the limit of the multivariate cardinal function is also identified

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

accurate computation of weights in classical gauss christoffel quadrature rules for many classical gauss christoffel quadrature rules there does not exist a method which guarantees a uniform level of accuracy for the gaussian quadrature weights at all quadrature nodes unless the nodes are known exactly more disturbing some algebraic expressions for these weights exhibit an excessive sensitivity to even the smallest perturbations in the node location this sensitivity rapidly increases with high order quadrature rules current uses of very high order quadratures are common with the advent of more powerful computers and a loss of accuracy in the weights has become a problem and must be addressed a simple but efficient and general method for improving the accuracy of the computation of the quadrature weights is proposed it ensures a high level of accuracy for these weights even though the nodes may carry a significant large error in addition a highly efficient root finding iterative technique with superlinear converging rates for computing the nodes is developed it uses solely the quadrature polynomials and their first derivatives a comparison of this method with the eigenvalue method of golub and welsh implemented in most standard software libraries is made the proposed method outperforms the latter from the point of view of both accuracy and efficiency the legendre lobatto radau hermite and laguerre quadrature rules are examined

laurent expansion of the inverse of perturbed singular matrices in this paper we describe a numerical algorithm to compute the laurent expansion of the inverse of singularly perturbed matrices the algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices the input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix the matrix coefficients of the laurent expansion of the inverse are computed using recursive analytical formulae we show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix but exponentially with the order of the singularity we apply this algorithm to several matrices that arise in applications we make special emphasis to interpolation problems with radial basis functions

on computing the points and weights for gauss legendre quadrature three methods are compared for computing n points xi and weights w xi for gauss legendre quadrature the eigensystem method computes the points as the eigenvalues of a symmetric tridiagonal matrix and the weights are computed from its eigenvectors the resulting error is o n for the points and o n 2 for the weights using an inverse taylor series computation for the points yakimiw j comput phys 129 1996 pp 406 430 reduced the point error growth to o 1 he also reduced the error in the weights to o n and obtained uniform pointwise relative accuracy however the points xi cluster quadratically near the end points x pm 1 where for some n nmax they coalesce or even exchange to within machine accuracy and computation fails although this has not been a problem in the past it could soon become so because weather models are now approaching resolutions with n 1000 here we compare these methods to the fourier newton method which consists of newton s method for computing the transformed points theta i arccos x i as the zeros of the fourier representations of the legendre polynomials p n theta the theta i are near equally spaced particularly in the neighborhood of x 0 for a fixed arithmetic precision this extends the range of computable points and weights to n n max 2 where nmax is the limit for methods based on xi both the accuracy and speed of the three methods are tabulated

numerical solution of two dimensional elliptic pdes with nonlocal boundary conditions in the present paper two numerical methods are analyzed for the solution of two dimensional poisson equation with two different types of nonlocal boundary conditions the first numerical method is a collocation method based on haar wavelet whereas the second numerical method is a meshless method based on different types of radial basis functions rbfs a two point boundary condition and an integral boundary condition are the two types of nonlocal boundary conditions considered in the present work for the collocation method based on haar wavelet a new approach is formulated which involves the approximation of a fourth order mixed derivative by a haar expansion which is integrated subsequently to get wavelet approximation of the solution for the meshless method based on rbfs the algorithm is implemented using two different splitting schemes with and without shape parameter splitting for numerical solution of the model the comparative analysis of the meshless methods with and without shape parameter splitting scheme is performed between themselves as well as with the haar wavelet accuracy and efficiency wise performance is confirmed through application of the algorithms on the benchmark tests

positive solutions for a system of fractional differential equations with coupled integral boundary conditions we investigate the existence of positive solutions for a system of nonlinear riemann liouville fractional differential equations with coupled integral boundary conditions

rbf loi augmenting radial basis functions rbfs with least orthogonal interpolation loi for solving pdes on surfaces abstract we present a new method for the solution of pdes on manifolds m r d of co dimension one using stable scale free radial basis function rbf interpolation our method involves augmenting polyharmonic spline phs rbfs with polynomials to generate rbf finite difference rbf fd formulas these polynomial basis elements are obtained using the recently developed least orthogonal interpolation technique loi on each rbf fd stencil to obtain local restrictions of polynomials in r 3 to stencils on m the resulting rbf loi method uses cartesian coordinates does not require any intrinsic coordinate systems or projections of points onto tangent planes and our tests illustrate robustness to stagnation errors we show that our method produces high orders of convergence for pdes on the sphere and torus and present some applications to reaction diffusion pdes motivated by biology

modeling and simulation of three dimensional planar contraction flow of viscoelastic fluids with ptt giesekus and fene p constitutive models abstract the planar contraction flow is a representative benchmark problem that is often used as a stringent test for the robustness of the numerical method to predict real flow patterns of viscoelastic fluids a numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three dimensional flow of viscoelastic fluids through planar contraction the viscoelastic rheological responses are described by using three kind differential constitutive models including the phan thien tanner ptt model the giesekus model and the finite extensible nonlinear elastic dumbbell with a peterlin closure approximation fene p model the discrete elastic viscous split stress devss formulation in cooperating with the inconsistent streamline upwind su scheme is employed to improve the computational stability the simulation results of flow velocity and stress on different sections of the flow field are compared with quinzani s experimental results that detected by laser doppler velocimetry and flow induced birefringence technologies it is found that the simulation results predicted with three kind differential constitutive models agree well with the experimental results the numerical methodology proposed in the study can be used successfully to predict complex flow patterns of viscoelastic fluids

an improved method based on haar wavelets for numerical solution of nonlinear integral and integro differential equations of first and higher orders in this paper a novel technique is being formulated for the numerical solution of integral equations ies as well as integro differential equations ides of first and higher orders the present approach is an improved form of the haar wavelet methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 the proposed modifications resulted in computational efficiency and simple applicability of the earlier methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 in addition to this the new approach is being extended from ides of first order to ides of higher orders with initial and boundary conditions unlike the methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 where the kernel function is being approximated by two dimensional haar wavelet the kernel function in the present case is being approximated by one dimensional haar wavelet the modified approach is easily extendable to higher order ides numerical examples are being included to show the accuracy and efficiency of the new method

smoothing of crank nicolson scheme for the two dimensional diffusion with an integral condition the parabolic partial differential equations pdes with nonlocal boundary conditions model various physical phenomena e g chemical diffusion thermoelasticity heat conduction process control theory and medicine science this paper deals with the smoothing of the crank nicolson numerical scheme for two dimensional parabolic pdes with nonlocal boundary conditions we use the numerical scheme based on pade approximations of the matrix exponential the graphs of numerical results demonstrate the successful smoothing of the crank nicolson numerical scheme

a two parameter defect correction method for computation of steady state viscoelastic fluid flow the numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter the weissenberg number increases specifically at a weissenberg number larger than a critical value the iterative nonlinear solver fails to converge in this paper a two parameter defect correction method for viscoelastic fluid flow is presented and analyzed in the defect step the weissenberg number is artificially reduced to solve a stable nonlinear problem the approximation is then improved in the correction step using a linearized correction iteration numerical experiments support the theoretical results and demonstrate the effectiveness of the method

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

limit problems for interpolation by analytic radial basis functions interpolation problems for analytic radial basis functions like the gaussian and inverse multiquadrics can degenerate in two ways the radial basis functions can be scaled to become increasingly flat or the data points coalesce in the limit while the radial basis functions stay fixed both cases call for a careful regularization which if carried out explicitly yields a preconditioning technique for the degenerating linear systems behind these interpolation problems this paper deals with both cases for the increasingly flat limit we recover results by larsson and fornberg together with lee yoon and yoon concerning convergence of interpolants towards polynomials with slight modifications the same technique can also handle scenarios with coalescing data points for fixed radial basis functions the results show that the degenerating local lagrange interpolation problems converge towards certain hermite birkhoff problems this is an important prerequisite for dealing with approximation by radial basis functions adaptively using freely varying data sites

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

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

on the optimal shape parameters of radial basis functions used for 2 d meshless methods abstract a radial point interpolation meshless or radial pim method was proposed by authors to overcome the possible singularity associated with only polynomial basis the radial pim used multiquadric mq or gaussian as basis functions these two radial basis functions all included shape parameters although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed how these shape parameters affect the accuracy of the radial pim has not been studied yet this paper studied the effect of shape parameters on the numerical accuracy of radial pim a range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix error of energy and irregularity of node distribution it is observed that the widely used shape parameters for mq and reciprocal mq basis are not even close to their optimums the optimal shape parameters are found in this paper to be simply q 1 03 and r 1 42 for mq basis and c 0 003 0 03 for gaussian basis

reliable computation of the zeros of solutions of second order linear odes using a fourth order method a fourth order fixed point method to compute the zeros of solutions of second order homogeneous linear odes is obtained from the approximate integration of the riccati equation associated with the ode the method requires the evaluation of the logarithmic derivative of the function and also uses the coefficients of the ode an algorithm to compute with certainty all the zeros in an interval is given which provides a fast reliable and accurate method of computation the method is illustrated by the computation of the zeros of gauss hypergeometric functions including jacobi polynomials and confluent hypergeometric functions laguerre polynomials hermite polynomials and bessel functions included among others the examples show that typically 4 or 5 iterations per root are enough to provide more than 100 digits of accuracy without requiring a priori estimations of the roots

solving pdes with the aid of two dimensional haar wavelets two dimensional haar wavelets are applied for solution of the partial differential equations pdes the proposed method is mathematically simple and fast to demonstrate the efficiency of the method two test problems solution of the diffusion and poisson equations are discussed computer simulation showed that the method guarantees the necessary exactness already for a small number of grid points

oscillating flow of a viscoelastic fluid in a pipe with the fractional maxwell model the unidirectional oscillating flow of a viscoelastic fluid with the fractional maxwell model is studied the flow is produced by a periodic pressure gradient in an infinite straight pipe exact solutions are obtained in the time and frequency domains by using fourier transform the fractional maxwell model exhibits resonance phenomena similar to that of the maxwell model however the number of the resonance peaks the amplitudes of the velocity enhancement and the frequencies where resonance occurs are totally different the amplitudes of resonance peaks decay rapidly with frequency in addition a critical dimensionless radius is found for the fractional maxwell model below the critical radius the resonance behavior is weakened but above this value the enhancement is strengthened

numerical simulation of viscoelastic flows using integral constitutive equations a finite difference approach this work presents a numerical technique for simulating incompressible isothermal viscoelastic flows of fluids governed by the upper convected maxwell ucm and k bkz kaye bernstein kearsley and zapas integral models the numerical technique described herein is an extension of the gensmac method to the solution of the momentum and mass conservation equations to include integral constitutive equations the governing equations are solved by the finite difference method on a staggered grid using a marker and cell approach the finger tensor b t t is computed in an eulerian framework using the ideas of the deformation fields method however improvements to the deformation fields method are introduced the finger tensor b t x t is obtained by a second order accurate method and the stress tensor t x t is computed by a second order quadrature formula the numerical method presented in this work is validated by comparing the predictions of velocity and stress fields in two dimensional fully developed channel flow of a maxwell fluid with the corresponding analytic solutions furthermore the flow through a planar 4 1 contraction is investigated and the numerical results were compared with the corresponding experimental data finally the ucm and the k bkz models were used to simulate the planar 4 1 contraction flow over a wide range of reynolds and weissenberg numbers and the numerical results obtained are in agreement with published data

fundamental formulation for frictional contact problems of coated systems abstract the plane problem about surface loading of an elastic layer perfectly bonded to an elastically dissimilar half plane is considered the fundamental solutions for concentrated forces acting perpendicular and parallel to the layer surface are obtained the stress and displacement fields in the coating layer and the substrate due to these concentrated forces are found on the basis of these expressions the fundamental integral equations are obtained which describe the frictional contact between an elastic body and a coated substrate from the fundamental integral equations a series of integral equations for special cases are deduced corresponding to practical contact situations finally the numerical results for a typical example are given to demonstrate the validity of the fundamental equations and numerical procedures given in this paper

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100