discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

finite difference preconditioning cubic spline collocation method of elliptic equations we discuss a finite difference preconditioner for the c 1 interpolatory cubic spline collocation method for a uniformly elliptic operator a defined by au delta u a 0 u in omega the unit square with homogeneous dirichlet boundary conditions using the generalized field of values arguments we discuss the eigenvalues of the preconditioned matrix hat l n 2 1 hat a n 2 where hat a n 2 is the matrix of the collocation discretization operator a n 2 corresponding to a and hat l n 2 is the matrix of the finite difference operator l n 2 corresponding to the uniformly elliptic operator l given by l v delta v v in omega with homogeneous dirichlet boundary conditions finally we mention a bound of h 1 singular values of hat l 1 n 2 hat a n 2 for a general elliptic operator au delta u a 1 u x a 2 u y a 0 u in omega

tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

discrete singular convolution and its application to the analysis of plates with internal supports part 2 applications part 2 of this series of two papers presents the applications of the discrete singular convolution dsc algorithm the main purpose of this paper is to explore the utility test the accuracy and examine the convergence of the proposed approach for the vibration analysis of rectangular plates with internal supports both partial internal line supports and complex internal supports are considered for 21 square plates of various combinations of edge support conditions the effects of different size shape and topology of the internal supports and different boundary conditions on the vibration response of plates are investigated the partial internal line supports may vary from a central point support to a full range of cross or diagonal line supports several closed loop supports such as ring square and rhombus and their combinations are studied for complex internal supports convergence and comparison studies are carried out to establish the correctness and accuracy of the dsc algorithm the dsc results are compared with those in the available literature obtained by using other methods numerical results indicate that the dsc algorithm exhibits controllable accuracy for plate analysis and shows excellent flexibility in handling complex geometries boundary conditions and support conditions copyright 2002 john wiley sons ltd

an analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports abstract a general analytical solution is obtained for the free vibration of rectangular plates supported on their lateral surface by symmetrically distributed point supports highly accurate eigenvalues are provided for a square plate supported at various points along the diagonal results are compared with earlier findings obtained by a finite element method it is shown that the method may be extended to solve various other problems the tabulation of eigenvalues appears to be the most comprehensive yet made for a square plate with point supports on the diagonals

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

natural frequencies of rectangular plates using characteristic orthogonal polynomials in rayleigh ritz method natural frequencies of rectangular plates are obtained by employing a set of beam characteristic orthogonal polynomials in the rayleigh ritz method the orthogonal polynomials are generated by using a gram schmidt process after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods the method yields superior results for lower modes particularly when plates have some of the edges free

on the synthesis of digital transmit filters a simple method for calculating the coefficients of a digital transversal filter when cascaded with analog filters is proposed the optimization criterion takes into account both the spectral characteristics of the signal and the residual amount of intersymbol interference at the decision device input in contrast with standard methods no a priori constraints are applied to the filter in the time or in the frequency domain

analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables optimal order error estimates are derived for the continuous time collocation method and a discrete time collocation method the crank nicolson collocation method for approximating the solution of a semilinear parabolic initial boundary value problem in r times 0 t where r is the unit square in each case at each time level the approximation is a c 1 piecewise polynomial of degree r geqq 3 defined by collocation at gauss points in r optimal order error estimates are also derived for a family of discrete time collocation methods for a semilinear second order hyperbolic initial boundary value problem in r times 0 t

spline collocation differentiation matrices spline collocation differentiation matrices arising when solving pdes with periodic boundary conditions have block circulant structure in this paper we use this fact to present an eigenvalue analysis for a large class of such spline collocation differentiation matrices we use this to discuss the stability of collocation algorithms for parabolic and hyperbolic problems and fast collocation algorithms for elliptic pdes

minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by rayleigh ritz method abstract the rayleigh ritz method is employed to determine the minimum stiffness location of the elastic point support for raising the fundamental natural frequency of a rectangular plate to the second frequency of the unsupported plate which usually is the upper limit of the first frequency for a single support based on the optimal design of the support position the minimum stiffness can be obtained numerically by solving a characteristic eigenvalue sub problem in the rayleigh ritz procedure the boundary characteristic orthogonal polynomials are used for the admissible functions several typical examples of plate structures with the additional point supports are analyzed in detail and the results prove that the proposed method is very effective in the solution to the optimal design of the supports it will be shown that elastic supports can be designed like rigid ones for the purpose of increasing the fundamental natural frequency of a rectangular plate

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

on the use of orthogonal polynomials in the rayleigh ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates series composed of recently proposed orthogonal polynomial functions are used in the rayleigh ritz method to generate results for a number of flexural vibration and buckling problems for rectangular isotropic and orthotropic plates it is demonstrated that the approach yields excellent results not only for natural frequencies and buckling loads but also for the more sensitive mode shape bending moment and shear force distribution problem

quadratic spline collocation methods for two point boundary value problems a new collocation method based on quadratic splines is presented for second order two point boundary value problems first o h4 approximations to the first and second derivative of a function are derived using a quadratic spline interpolant of u then these approximations are used to define an o h4 perturbation of the given boundary value problem second the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal o h3 j global estimate for the jth derivative of the error is derived further o h4 j error bounds for the jth derivative are obtained for certain superconvergence points it should be observed that standard collocation at midpoints gives o h2 j bounds results from numerical experiments are reported that verify the theoretical behaviour of the method

fast numerical solution of the biharmonic dirichlet problem on rectangles a new method for the numerical solution of the first biharmonic dirichlet problem in a rectangular domain is presented for an n times n mesh the complexity of this algorithm is on the order of

a newtonian procedure for the solution of the kron characteristic value problem abstract in this paper a quadratically convergent newtonian procedure is presented to determine the latent roots of the kron matrix the roots are found in ascending order with the certainty that none has been missed since the procedure is allied to a sturm sequence algorithm the procedure has been programmed and it performs at least 50 faster than the bisection scanning method available for the kron matrix examples of successful use of the procedure to solve for the natural frequencies of frames with large numbers of degrees of freedom are presented

free vibration of rectangular plates with internal column supports a direct method is proposed to derive the exact solution for the free vibration of a thin rectangular plate with two opposite edges simply supported and with internal column supports considering the compatibility of displacements and rotations between the plate and the columns the coupled vibration of a plate column system is derived the accuracy and correctness of the present method are demonstrated through the comparison of the results obtained from the proposed method and from finite element analysis then the effects of column sizes and column models on the natural frequencies of a plate column system are investigated in detail a parametric study is focused on fully simply supported rectangular plates with a single internal column it is shown that the effect of column flexibility in plate column structures should be considered the solution provided in the paper is general and includes several particular solutions for fixed point supports pinned point supports and elastic point supports results with high accuracy have been obtained and these can be used as the benchmark for the further investigation and for other approximate methods

free vibration of thick layered rectangular plates with point supports by finite layer method in this paper the free vibration of thick isotropic and laminated composite rectangular plates with point supports is analyzed by the finite layer method a new set of two dimensional basic functions which satisfies the kinematic boundary conditions at the edges of the plate and the zero displacement conditions at point supports is developed to describe the variation of three dimensional displacements in the plane of a thin finite layer one dimensional linear or quadratic shape functions are adopted to describe the variation of the displacements through the thickness layer the governing eigenvalue equation of the plate is derived via the conventional displacement method numerical results for the three dimensional vibration of rectangular plates with point supports are presented herein for the first time the eigenfrequencies of simply supported rectangular plates with a central point support are studied in detail by considering the variations of aspect ratio side to thickness ratio properties of materials number of laminates and stacking sequences comparison with known thin plate and mindlin plate solutions is carried out to verify the applicability and accuracy of the proposed method

free vibration analysis by the superposition method of rectangular mindlin plates with internal columns resting on uniform elastic edge supports free vibration analysis of rectangular mindlin plates is carried out via the superposition method and is shown to produce accurate results the analytical method proposed by the authors in this paper solves the problem for the case where the plate has simultaneous elastic edge and internal supports the conditions of edge supports are uniform lateral rotational and torsional elastic supports whereas the internal supports are column supports with finite area compatibility between the plate and column is achieved by requiring that the column and plate rotations be equal the results presented herein are verified through comparison with results presented by others numerical examples presented in this study confirm that the analytical method is able to model the property of the elastic edge supports while simultaneously considering the effect of column restraint an effect which increases with column number and area

basis spline collocation method for solving the schrodinger equation in axillary symmetric systems we use basis spline collocation to solve the schrodinger equation on a lattice for axially symmetric systems with and without spin the focus of the present work is on systems which have continuum states weakly bound states or strong spin orbit coupling since these are the most difficult to solve on the lattice a brief overview of the basis spline collocation method is included which concentrates primarily on those aspects of the theory which are relevant to its application in cylindrical coordinates to demonstrate the method we solve several model problems selected from the fields of atomic and nuclear physics

matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions abstract bialecki b and g fairweather matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions journal of computational and applied mathematics 46 1993 369 386 in recent years several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite difference finite element galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle the success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two point boundary value problems in one space dimension the primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation that is spline collocation at gauss points all of the methods discussed in this paper are modular and possess a great deal of natural parallelism

stiff string approximations in rayleigh ritz method for rotating beams the governing differential equation of the rotating beam reduces to that of a stiff string when the centrifugal force is assumed as constant the solution of the static homogeneous part of this equation is enhanced with a polynomial term and used in the rayleighs method numerical experiments show better agreement with converged finite element solutions compared to polynomials using this as an estimate for the first mode shape higher mode shape approximations are obtained using gram schmidt orthogonalization estimates for the first five natural frequencies of uniform and tapered beams are obtained accurately using a very low order rayleigh ritz approximation

vibration of complex structures the modal constraint method abstract a method is described for establishing the natural frequencies of an arbitrary structure with arbitrary supports the method is based on the modal constraint technique described in a previous paper 1 as shown in the present paper weinstein s theory for the intermediate problem can be regarded as equivalent to the lagrangian multiplier method i e both methods result in the same eigenvalue equations weinstein s theory deals with modifications of base differential operators whereas the lagrangian multiplier method deals with modifications of base energy functionals the modal constraint technique is an extension of weinstein s theory or in energy terms the generalized fourier expansion of the lagrangian multiplier the merits of this method lie in the fact that the eigenvalues and eigenfunctions of structures are used as base structures the coupling of these structures are taken into account by lagrangian generalized forces of the constraint acting on the base structures some examples are given and the results compared with known solutions

an alternating direction implicit method for orthogonal spline collocation linear systems an alternating direction implicit method is analyzed for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to some separable second order linear elliptic partial differential equations in rectangles on annxn partition with jordan s selection of the acceleration parameters the method requireso n 2 ln 2 n arithmetic operations to produce an approximation whose accuracy in theh 1 norm is that of the collocation solution

matrix decomposition algorithms for modified spline collocation for helmholtz problems we consider the solution of various boundary value problems for the helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal fourth order approximations for the solution of the collocation equations we formulate matrix decomposition algorithms fast direct methods which employ fast fourier transforms and require o n2 log n operations on an n n uniform partition of the unit square a computational study confirms the published analysis for the dirichlet problem and indicates that similar results hold for neumann mixed and periodic boundary conditions the numerical results also exhibit superconvergence phenomena not reported in earlier studies

discrete time orthogonal spline collocation methods for vibration problems discrete time orthogonal spline collocation schemes are formulated and analyzed for vibration problems involving various boundary conditions each problem is written as a schrodinger type system which is then approximated by crank nicolson and or alternating direction implicit orthogonal spline collocation schemes these schemes are shown to be second order accurate in time and of optimal order accuracy in space in the hm norm m 1 2

block iterative algorithms for solving hermite bicubic collocation equations in this paper we present a so called local elimination technique by which a nonsymmetric system arising in the discretization of the poisson equation with a hermite bicubic collocation approximation is reduced to a block tridiagonal and block symmetric system a class of block iterative algorithms is developed for solving the resulting system the convergence rates of algorithms are discussed

multilevel additive and multiplicative methods for orthogonal spline collocation problems multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise hermite bicubics is applied to the dirichlet boundary value problem for a self adjoint elliptic partial differential equation on a rectangle additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space a general theory of additive and multiplicative schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the laplacian with the spectral constants independent of the fine partition stepsize and the number of levels the preconditioned conjugate gradient and preconditioned orthomin methods are considered for the solution of collocation problems an implementation of the methods is discussed and the results of numerical experiments are presented

second order splitting combined with orthogonal cubic spline collocation method for the kuramoto sivashinsky equation in this paper a second order splitting method is applied to the kuramoto sivashinsky equation and then an orthogonal cubic spline collocation procedure is employed to the approximate resulting system this semidiscrete method yields a system of differential algebraic equations daes of index 1 error estimates in l 2 and l c norms are obtained for the semidiscrete approximation for the time discretization the time integrator radau5 is used the results of numerical experiments are presented to validate the theoretical findings

optimization of binary transversal filters this paper presents a method for optimization with respect to bandwidth efficiency for a binary transversal filter when the input pulse format is nrz the objective is to concentrate as much as possible of the energy in the emitted signal within certain frequency bands the filter coefficients are given by a symmetric eigenvector for a toeplitz matrix and the corresponding eigenvalue equals the optimum value

on the flexural vibration of rectangular plates approached by using simple polynomials in the rayleigh ritz method abstract sets of simple polynomials are proposed for use as admissible functions in the rayleigh ritz method for the study of the flexural vibration of rectangular plates the plates may be subject to a number of different complicating factors including the presence of in plane forces concentrated masses point and line spring supports and special orthotropy of the plate material the sets of polynomials are shown to be equivalent to the orthogonally generated polynomials used very successfully in earlier studies but are more easily generated and the subsequent evaluation of the integrals in the resulting eigenvalue problem are more simply performed numerical results are presented for several problems illustrating the applicability of the method and its accuracy

modified nodal cubic spline collocation for poisson s equation we present a new modified nodal cubic spline collocation scheme for solving the dirichlet problem for poisson s equation on the unit square we prove existence and uniqueness of a solution of the scheme and show how the solution can be computed on an n 1 times n 1 uniform partition of the square with cost o n 2 mbox log n using a direct fast fourier transform method using two comparison functions we derive an optimal fourth order error bound in the continuous maximum norm we compare our scheme with other modified nodal cubic spline collocation schemes in particular the one proposed by houstis vavalis and rice in siam j numer anal 25 1988 pp 54 74 we believe that our paper gives the first correct convergence analysis of a modified nodal cubic spline collocation for solving partial differential equations

on the free transverse vibration of annular and circular thin sectorial plates subject to certain complicating effects abstract the rayleigh ritz method with orthogonally generated polynomials as admissible functions is used to obtain an eigenvalue equation for the free transverse vibration of thin annular sectorial plates circular sectorial plates are treated with this equation by permitting the inner radius to become negligibly small the plates may be of isotropic or polar orthotropic material of uniform or radially and or circumferentially varying thickness and may include interior circumferential or radial line supports numerical values for the natural frequency parameters of several different plates are presented and compared with previously published values where available

transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the rayleigh ritz method abstract natural frequencies and mode shapes of a rotating uniform cantilever beam with a tip mass are studied by using beam characteristic orthogonal polynomials in the rayleigh ritz method the set of orthogonal polynomials which satisfy the geometrical boundary conditions are generated by using the gram schmidt process the results are compared with those obtained by the myklestad method the extended galerkin method and finite element methods the variation of natural frequencies with the speed of rotation is plotted for several parameter combinations such as setting angle tip mass moment of inertia of tip mass etc mode shapes at different rotational speeds are also plotted use of orthogonal polynomials for the deflection shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any numerical difficulties which is not the case when arbitrary polynomial expressions are used

alternating direction collocation for rectangular regions abstract an alternating direction procedure for finite element collocation equations can be formulated on rectangular regions finite element collocation offers a great savings over finite element techniques in the matrix formation stage of the numerical process moreover with an alternating direction procedure multidimensional problems can be solved as a series of one dimensional problems this greatly reduces the amount of computer time and storage required to solve large problems

convergence of o h 4 cubic spline collocation methods for elliptic partial differential equations this paper presents a new class of collocation methods using cubic splines for solving elliptic partial differential equations pdes the error bounds obtained for these methods are optimal the methods are formulated and a convergence analysis is carried out for a broad class of elliptic pdes experimental results confirm the optimal convergence and indicate that these methods are computationally more efficient than methods based on either collocation with hermite cubics or on the galerkin method with cubic splines various direct and iterative methods have been applied for the solution of cubic spline collocation methods

on the lateral vibration of thin annular and circular composite plates subject to certain complicating effects abstract series comprising orthogonally generated polynomial functions are used in the rayleigh ritz method to obtain an eigenvalue equation for the flexural vibration of thin flat annular and circular plates the plates may be of isotropic or polar orthotropic material of uniform or radially varying thickness and may pass over intermediate concentric ring supports numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

optimal superconvergent one step nodal cubic spline collocation methods we formulate new optimal fourth order one step nodal cubic spline collocation methods for the solution of various elliptic boundary value problems in the unit square these methods are constructed so that the respective collocation equations can be solved using matrix decomposition algorithms mdas mdas are fast direct methods which employ fast fourier transforms and require o n2 log n operations on an n times n uniform partition of the unit square the results of numerical experiments exhibit expected global optimal orders of convergence as well as desired superconvergence phenomena

on the eigenvectors of symmetric toeplitz matrices this paper presents a number of results concerning the eigenvectors of a symmetric toeplitz matrix and the location of the zeros of the filters eigenfilters whose coefficients are the elements of the eigenvectors one of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues if distinct have their zeros on the unit circle while the zeros of the other eigenfilters may or may not have their zeros on the unit circle even if the zeros of the eigenfilters of a matrix are all on the unit circle the matrix need not be toeplitz examples are given to illustrate the different properties

convergence analysis of orthogonal spline collocation for elliptic boundary value problems existence uniqueness and optimal order h2 h1 and l2 error bounds are established for the orthogonal spline collocation solution of a dirichlet boundary value problem on the unit square the linear elliptic nonself adjoint partial differential equation is given in nondivergence form the approximate solution which is a tensor product of continuously differentiable piecewise polynomials of degree r geq 3 is determined by satisfying the partial differential equation at the nodes of a composite gauss quadrature

efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles piecewise hermite bicubic orthogonal spline collocation laplace modified and alternating direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed the schemes are shown to be unconditionally stable and of optimal order accuracy in the h 1 and discrete maximum norms for space and time respectively implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms

basis spline collocation method for the lattice solution of boundary value problems abstract we study a particular utilization of the basis spline collocation method bscm for the lattice solution of boundary value problems we demonstrate the implementation of a general set of boundary conditions among the selected problems are the schrodinger equation in radial coordinates the poisson and the generalized helmholtz equations in radial and three dimensional cartesian coordinates

the flexural vibration of rectangular plates with point supports abstract orthogonally generated polynomial functions are used in the lagrangian multiplier method to study the free flexural vibration problem of point supported thin flat rectangular plates the analysis applies to isotropic and specially orthotropic plates having any combination of clamped simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges numerical results are presented for a number of specific problems illustrating the accuracy and versatility of the approach and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports

adi methods for cubic spline collocation discretizations of elliptic pde this paper presents the formulation analysis and implementation of alternating direction implicit adi methods for solving the linear system of algebraic equations that arise from the discretization of multidimensional linear elliptic partial differential equations pdes the theoretical analysis is carried out for a broad class of pde problems numerical experiments confirm the theoretically determined characteristics of the adi iterative schemes the computational effectiveness of the proposed schemes is shown through a detailed theoretical complexity analysis confirmed with our experimental data

spectral analysis of hermite cubic spline collocation systems in this paper we present an eigenvalue analysis of the first order and second order hermite cubic spline collocation differentiation matrices with arbitrary collocation points some important features are explored and compared with some other discrete methods such as finite difference methods

free vibration analysis of rectangular plates on elastic intermediate supports a procedure using the finite strip element method in combination with a spring system is proposed to treat the free vibration analysis of plates on elastic intermediate supports results indicate that the spring system can successfully simulate elastic intermediate supports such as point supports line supports local uniformly distributed supports and mixed edge supports from the results it is also evident that support stiffness and support areas have significant influence on the free vibration response of plates on line supports and local uniformly distributed supports

modified successive overrelaxation msor and equivalent 2 step iterative methods for collocation matrices abstract we consider a class of consistently ordered matrices which arise from the discretization of boundary value problems bvps when the finite element collocation method with hermite elements is used through a recently derived equivalence relationship for the asymptotic rates of convergence of the modified successive overrelaxation msor and a certain 2 step iterative method we determine the optimum values for the parameters of the msor method as it pertains to collocation matrices a geometrical algorithm which utilizes capturing ellipse arguments has been successfully used the fast convergence properties of the optimum msor method are revealed after its comparison to several well known iterative schemes numerical examples which include the solution of poisson s equation are used to verify our results

iterative line cubic spline collocation methods for elliptic partial differential equations in several dimensions this paper presents a new class of second and fourth order line cubic spline collocation methods the lcsc methods for multidimensional linear elliptic partial differential equations with no cross derivative terms the lcsc methods approximate the differential operator along lines in each dimension independently and then combine the results into one large linear system expressed in terms of discretization stencils for the operator these methods have nonzero entries only in the coordinate directions the advantage of this approach is that the discretization is much simpler to derive and analyze further iterative methods are easily applied to the resulting linear systems especially on parallel computers the disadvantage is that the resulting linear system is k times larger in k dimensions using the simplicity of the methods iterative schemes are analyzed and formulated in order to solve the resulting lcsc linear systems in the case of helmholtz problems block jacobi extrapolated jacobi ej and

the minimum support stiffness required to raise the fundamental natural frequency of plate structures the minimum stiffness and optimum position of a flexible point support is calculated that raises the fundamental natural frequency of plate structures for a single support the maximum fundamental natural frequency of the supported structure is equal to the second natural frequency of the unsupported structure the structure is modelled using finite element analysis allowing a wide range of applications and boundary conditions if a support is positioned within a finite element then the shape functions are used to calculate the contribution of the support to the system stiffness and also the slope of the mode at the support efficient methods are used to calculate the minimum support stiffness required numerical results demonstrate that a system with flexible supports may be designed with the same fundamental natural frequency as that of a system with rigid supports examples of plate structures with one or two point supports are analysed and show that the method is efficient in these cases

free vibration analysis of rectangular plates with variable thickness and point supports a discrete method is developed for analyzing the free vibration problem of rectangular plates with point supports the fundamental differential equations involving dirac s delta function are established for the bending problem of the plate with point supports by transforming these differential equations into integral equations and using numerical integration the solution of these equations is obtained and used as green function to obtain the characteristic equation of the free vibration the effects of the point support the boundary condition the variable thickness and aspect ratio on the frequencies are considered by comparing the numerical results obtained by the present method with those previously published the efficiency and accuracy of the present method are investigated