three dimensional vibration analysis of rectangular plates based on differential quadrature method abstract this paper presents the formulation and numerical analysis of the three dimensional elasticity plate model using the differential quadrature dq method the governing equations in terms of displacement stress displacement relations and boundary conditions for the three dimensional plate model are first presented these equations are then normalized and discretized using the dq procedure example problems on the free vibration of rectangular plates with generic boundary conditions are selected two types of mesh pattern the uniform mesh pattern and the cosine mesh pattern were used and their convergence characteristics were studied the cosine mesh was then chosen as the better mesh pattern pertaining to the problems solved the solutions calculated using the cosine mesh pattern were then compared where possible with the exact or the numerical or the analytical solutions it is found that the differential quadrature method yields accurate results for the plate problems under the current investigation

sharp bounds for generalized elliptic integrals of the first kind abstract in this paper we prove that the double inequality 1 r 2 k a r sin a log e r a 2 r 1 r 2 holds for all a 0 1 2 and r 0 1 if and only if r a sin a 1 and a 1 a where r 1 r 2 k a r is the generalized elliptic integral of the first kind and r x is the ramanujan constant function besides as the key tool the series expression for the ramanujan constant function r x is given

static and stability analyses of arbitrary straight sided quadrilateral thin plates by dqm abstract a differential quadrature dq methodology is employed for the static and stability analysis of irregular quadrilateral straight sided thin plates a four noded super element is used to map the irregular physical domain into a square computational domain second order transformation schemes with relative ease and low computational effort are employed to transform the fourth order governing equations of thin plates between the domains within the domain the displacements are the only degrees of freedom whereas along the boundaries the displacements as well as the second order derivatives of the displacements with respect to the associated normal coordinate variables in the computational domain are the two sets of degrees of freedom the implementation procedures for different boundary conditions including free edge boundaries are formulated to demonstrate the accuracy convergency and stability of the methodology detailed studies of skewed and trapezoidal plates for different geometries under different boundary and loading conditions are made good agreement is achieved between the results of the present methodology and those of other dq methodologies or other comparable numerical algorithms

free vibration analysis of plates using least square based finite difference method in this paper we apply the two dimensional least square based finite difference lsfd method for solving free vibration problems of isotropic thin arbitrarily shaped plates with simply supported and clamped edges using the chain rule we show how the fourth order derivatives of the plate governing equation can be discretized in two or three steps as well as how the boundary conditions can be implemented directly into the governing equation by analyzing vibrating plates of various shapes and comparing the solutions obtained against existing results we clearly demonstrate the effectiveness of lsfd as a mesh free method for computing vibration frequencies of generally shaped plates accurately

a generalized approach for implementing general boundary conditions in the gdq free vibration analysis of plates abstract a new approach for implementing general boundary conditions in the gdq free vibration analysis of rectangular plates is presented in this paper the proposed approach directly couples the boundary conditions with the governing equations and is referred to as the cbcge approach it can be applied to any combination of simply supported clamped and free boundary conditions the generality and accuracy of the cbcge approach are demonstrated through its application to the vibration analysis of a rectangular plate with various combinations of free edges and corners in particular the effect of grid point distribution on the numerical results for plates with free boundary conditions is discussed and a new grid point distribution is suggested

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

monotonicity properties of generalized elliptic integrals with respect to the parameter abstract for a 0 1 2 and r 0 1 let k a r and e a r k r and e r denote the generalized elliptic integrals the complete elliptic integrals respectively of the first and second kinds respectively in this paper the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a 0 1 2 by these results sharp bounds expressed in terms of k r e r and elementary functions are obtained for k a r and e a r thus showing the dependence on the parameter a of k a r and e a r and substantially improving the known related results

vibration analysis of non circular curved panels by the differential quadrature method free vibration characteristics of cantilever non circular curved panels are analyzed by using the differential quadrature method dqm in this paper the equations of motion of a curved panel are based on the love s hypothesis and are expressed in an orthogonal curvilinear co ordinate system by applying the differential quadrature formulation and the proposed modified relationships for specified boundary conditions the free vibration equations of motion of the curved panel are transformed to a set of algebraic equations natural frequencies of a cantilever flat plate and a circular curved panel are obtained for verifying the applicability of the present approach good convergent trend and accuracy are observed effects of shallowness thickness and aspect ratios on the natural frequencies of a cantilever curved panel are also investigated furthermore natural frequencies of parabolic curved panels are obtained in all cases studied the efficiency and convenience of the dqm are illustrated

some properties of the difference between the ramanujan constant and beta function abstract the authors present the power series expansions of the function r a b a at a 0 and at a 1 2 show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so called ramanujan constant r a and the beta function b a b a 1 a and obtain asymptotically sharp lower and upper bounds for r a in terms of b a and polynomials in addition some properties of the riemann zeta function n n n and its related sums are derived

stability and accuracy of differential quadrature method in solving dynamic problems in this paper the differential quadrature method is used to solve dynamic problems governed by second order ordinary differential equations in time the legendre radau chebyshev chebyshev gauss lobatto and uniformly spaced sampling grid points are considered besides two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated the stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices it is found that higher order accurate solutions can be obtained at the end of a time step if the gauss and radau sampling grid points are used however the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used unfortunately the commonly used chebyshev gauss lobatto sampling grid points would not generate unconditionally stable algorithms

solving high order partial differential equations with indirect radial basis function networks this paper reports a new numerical method based on radial basis function networks rbfns for solving high order partial differential equations pdes the variables and their derivatives in the governing equations are represented by integrated rbfns the use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high order derivatives the proposed rbfn method is verified successfully through the solution of thin plate bending and viscous flow problems which are governed by biharmonic equations for thermally driven cavity flows the solutions are obtained up to a high rayleigh number

a dqem for transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions in this paper a differential quadrature element method dqem for free transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions is proposed governing equations the compatibility conditions at the damaged cross sections and implementation of the external boundary conditions are derived and formulated by the differential quadrature analogue the accuracy convergence and versatility of the proposed method are confirmed by the exact solution of the uniform beam which has been presented by other authors and 2d finite element method fem numerical results for non uniform beam after the validation of the presented method the effect of quantity depth and location of the cracks on the frequency values of vibrations are investigated the achieved results show that the existence of the crack leads to a decrease in the frequencies of the vibrations through decrease in the stiffness of the beam meanwhile the compatibility conditions at the damaged section is considered as a discontinuity in slope and vertical displacement where the effect of the discontinuity in the slope is more considerable as many authors have neglected the discontinuity in the vertical displacement as it will be shown consideration of the discontinuity in the vertical displacement causes more decrease in the frequencies

a new approach to the differential quadrature method for fourth order equations a generalized and more complete methodology for treating boundary conditions in the differential quadrature method dqm is presented this improved approach eliminates the deficiencies of the type grid arrangement which represents an approximation by applying the boundary conditions exactly two kinds of basis functions chebyshev and lagrange are used for concept demonstration it is found that the new approach cures most deficiencies of the current dqm 1997 by john wiley sons ltd

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

vibrations of rectangular plates with arbitrary non uniform elastic edge restraints abstract arbitrary non uniform elastic edge restraints represent the most general class of boundary conditions for plate problems and are encountered in many real world applications the vibrations of plates with this kind of boundary conditions however are rarely studied in the literature perhaps because there is a lack of suitable analytical or numerical techniques in this investigation a general analytical method is derived for the vibration analysis of rectangular plates with elastic edge restraints of varying stiffness both rotational and translational restraints can be arbitrarily applied to an edge and their stiffness distributions are generally described in terms of a set of invariants cosine functions the displacement solution is sought simply as a linear combination of several one and two dimensional fourier cosine series expansions all the unknown fourier coefficients are treated equally as a set of independent generalized coordinates and solved directly from the rayleigh ritz formulation unlike the existing techniques the current method does not require any special procedures or schemes to deal with different boundary conditions a few classical problems involving non uniform rotational restraints are first solved and used to check the current solution against some of the existing techniques the modal results are also presented for plates with more complicated boundary conditions in which an edge is no longer completely restrained in the translational direction the accuracy and reliability of the current method are repeatedly demonstrated through all these examples

dual reciprocity bem applied to transient elastodynamic problems with differential quadrature method in time this paper is concerned with numerical analysis of the transient elastodynamic problems by the dual reciprocity bem drbem in space combined with the differential quadrature method dqm in time emphasis is placed on a comparative study of various time marching schemes three numerical examples considered are for the longitudinal forced vibration of plates and the transverse free vibration of membranes to authors best knowledge this is the first attempt to apply the dqm to the second order time derivative in the drbem elastodynamic formulations a recent approach using boundary conditions in the dqm was here extended to handle the initial conditions of elastodynamic problems the resulting algebraic formulation is a lyapunov matrix equation which can be very efficiently solved by the bartels stewart algorithm it is revealed that the dqm is an unconditionally stable algorithm and gives much better accuracy than the standard finite difference schemes such as the wilson newmark and houbolt methods for the same time step size for the cases considered

asymptotic green s functions for time fractional diffusion equation and their application for anomalous diffusion problem abstract asymptotic green s functions for short and long times for time fractional diffusion equation derived by simple and heuristic method are provided in case if fractional derivative is presented in caputo sense the applicability of the asymptotic green s functions for solving the anomalous diffusion problem on a semi infinite rod is demonstrated the initial value problem for longtime solution of the time fractional diffusion equation by green s function approach is resolved

fractional schrodinger equation with noninteger dimensions the spatial and time dependent solutions of the schrodinger equation incorporating the fractional time derivative of distributed order and extending the spatial operator to noninteger dimensions are investigated they are obtained by using the green function approach in two situations the free case and in the presence of a harmonic potential the results obtained show an anomalous spreading of the wave packet which may be related to an anomalous diffusion process

vibration analysis of elastically supported turbomachinery blades by the modified differential quadrature method abstract the modified differential quadrature method mdqm is proposed for vibration analysis of elastically supported turbomachinery blades a pre twisted blade with varying cross section is modelled as a timoshenko beam the blade is supported by two translational springs and three rotational springs at each end and has a shroud that is modelled as a mass at the tip of the blade the equations of motion and the boundary conditions for the coupled flexural and torsional vibration of the blade are obtained by using hamilton s principle numerical results of elastically supported blades with or without a shroud are obtained by the mdqm and are validated by comparing with analytical solutions campbell diagrams for a pre twisted blade with an airfoil cross section are constructed the accuracy and efficiency of the present approach have been demonstrated

an efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates abstract in this paper a simple and efficient mixed ritz differential quadrature dq method is presented for free vibration and buckling analysis of orthotropic rectangular plates the mixed scheme combines the simplicity of the ritz method and high accuracy and efficiency of the dq method the accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature it is shown that highly accurate results can be obtained using a small number of ritz terms and dq sample points the proposed method is suitable for the problem considered due to its simplicity and potential for further development

differential quadrature for static and free vibration analyses of anisotropic plates abstract the differential quadrature method is used to analyse the deflection buckling and free vibration behavior of anisotropic rectangular plates under various boundary conditions the roots of chebyshev polynomials are used to obtain grid point locations and a new approach is used to apply boundary conditions results compare very well with existing numerical data and less computational effort is required for the problems considered

high accuracy mixed finite element ritz formulation for free vibration analysis of plates with general boundary conditions abstract a high order accurate mixed finite element ritz method is introduced and developed to study the vibration problem of plates with general boundary conditions the finite element method fem with higher order interpolation functions is first used to discretize the spatial partial derivatives with respect to a co ordinate direction of the plate the ritz method is then employed to analogize the resulting system of ordinary differential equations a novel technique is also presented to exactly satisfy the mixed natural boundary conditions the proposed method is applied here to solve some benchmark vibration problems of plates including isotropic and anisotropic rectangular plates variable thickness rectangular plates multi span rectangular plates and skew plates comparisons with existing numerical and analytical solutions show that the proposed mixed method can yield highly accurate results for vibration problem of plates involving free edges free corners and irregular boundaries using a small number of finite elements and ritz terms

on optimal selection of interior points for applying discretized boundary conditions in dq vibration analysis of beams and plates the direct implementation of derivative boundary conditions at edge points was demonstrated to be simple flexible and computationally efficient in the vibration analysis of beams and plates by using the differential quadrature dq method in the approach the discretized governing equations at certain interior points need to be replaced by the discretized boundary conditions naturally one may have questions as to whether the replacement points can be selected arbitrarily and how to choose the optimal replacement points this paper tries to answer these questions and its focus is to investigate systematically the effect of the replacement location on the accuracy of the numerical solution from the error distribution analysis of derivative approximation it is found that the optimal position for the replacement is the interior point just adjacent to the boundary this conclusion is confirmed by the numerical experiments of free vibration analysis for beams and plates with various boundary conditions

sharp landen transformation inequalities for hypergeometric functions with applications abstract the authors present sharp landen transformation inequalities for the hypergeometric functions f 1 2 a b a b x and f 1 2 a b a b 1 2 x by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear or rational functions thus giving complete solutions of the problem on extending the well known landen transformation identities for the complete elliptic integrals of the first kind to these two hypergeometric functions and substantially improving the related known results as applications of these results sharp landen transformation inequalities are obtained for the generalized grotzsch ring functions and the modular functions which appear in ramanujan s modular equations some other properties of hypergeometric functions and several properties of the ramanujan constant are obtained too

comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates this paper presents a comprehensive comparison study between the discrete singular convolution dsc and the well known global method of generalized differential quadrature gdq for vibration analysis so as to enhance the understanding of the dsc algorithm the dsc method is implemented through lagrange s delta sequence kernel dsc lk which utilizes local lagrange polynomials to calculate weighting coefficients whereas the gdq requires global ones moreover it is shown that the treatments of boundary conditions and the use of grid systems are different in the two methods comparison study is carried out on 21 rectangular plates of different combinations of simply supported s clamped c and transversely supported with nonuniform elastic rotational restraint e edges and five rectangular plates of mixed supporting edges some of which with a range of aspect ratios and rotational spring coefficients all the results of the dsc lk agree very well with both those in the literature and newly computed gdq results furthermore it is observed that the dsc lk performs much better for plates vibrating at higher order eigenfrequencies unlike the gdq the dsc lk is numerically stable for problems which require a large number of grid points

free vibration analysis of curvilinear quadrilateral plates by the differential quadrature method a methodology for applying the differential quadrature dq method to the free vibration analysis of arbitrary quadrilateral plates is developed in our approach the irregular physical domain is transformed into a rectangular domain in the computational space the governing equation and the boundary conditions are also transformed into relevant forms in the computational space then all the computations are based on the computational domain as compared to the approach proposed by c w bert and m malik int j mech sci 38 589 1996 the present approach requires much less computational effort and virtual storage in addition the present work uses a simple and convenient way to implement clamped and simply supported boundary conditions an exact mapping technique is used to perform the coordinate transformation in this study some numerical examples are provided to show the computational efficiency of the present scheme

two new approximate methods for analyzing free vibration of structural components two approximate methods which have not previously been used for structural dynamics problems are applied to the free vibration analysis of various structural components the first method is a new version of the complementary energy method it is shown to be considerably more accurate than the conventional rayleigh and rayleigh schmidt methods when applied to spatially one dimensional free vibration problems prismatic and tapered bars prismatic beams and axisymmetric motion of circular membranes the second method is the differential quadrature method introduced by bellman and his associates it is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates

a new differential quadrature methodology for beam analysis and the associated differential quadrature element method this paper presents a computationally efficient and an accurate new methodology in differential quadrature analysis of beam elements the methodology would overcome the difficulties in boundary conditions implementations of fourth order differential equations encountered in such problems the methodology benefited from defining the second order derivatives along the boundaries as independent degrees of freedom would enable the differential quadrature method to exactly satisfy some types of boundary conditions where by most other conventional algorithms have to be satisfied approximately the weighting coefficients employed are not exclusive and any accurate and efficient method such as the generalized differential quadrature method may be used to produce the methods weighting coefficients by solving some typical stability deflection and frequency analysis beam problems and by comparing the results with those of exact solutions and or those of other methodologies accuracy convergency and efficiency of the methodology is asserted in order to generalize its application to large scale beam structures and to the cases with the discontinuity in loading conditions and geometry a new one dimensional differential quadrature element method formulation is presented and implemented

three dimensional elasticity solutions for free vibrations of rectangular plates by the differential quadrature method this paper presents accurate three dimensional elasticity solutions for free vibrations of six types of plates having free lateral surfaces two opposite sides simply supported and two other sides having combinations of simply supported clamped and free boundary conditions the solution methodology adopted in the present work first reduces the spatial variables in the governing elasticity equations to two by using displacement functions in the forms which satisfy the boundary conditions of the assumed simply supported sides the eigenvalue equations are then formulated from the reduced governing equations and the boundary conditions via the differential quadrature method the numerical results include the first nine natural frequencies of the six plate configurations for combinations of three aspect ratios and two thickness ratios these results are supported by appropriate convergence studies and comparisons with the results of other authors

a pb 2 ritz formulation for flexural vibration of shallow cylindrical shells of rectangular planform abstract this paper presents a free flexural vibration analysis of shallow cylindrical shells of rectangular planform with arbitrary combinations of free simply supported and clamped boundaries the pb 2 ritz method previously used in the plate analysis by one of the authors is now extended to shallow shell studies the ritz approach along with the in plane and transverse displacements assumed in the form of orthogonal polynomials is used to formulate the problems numerical results for the vibration frequencies for wide ranges of aspect ratio shallowness ratio and thickness ratio are presented several selected contour plots of the mode shapes for six different combinations of boundary conditions are presented as usual the accuracy of the present method is illustrated by convergence tests and comparison with the well established results from the literature

out of plane static analysis of circular arches by dqm the differential quadrature dq methodology introduced by the authors see comput methods appl mech engng 191 2002a 3509 int j solids struct 39 19 2002b 4927 int j numer methods engng 54 3 2003a 847 j sound vibrat 263 2 2003b 415 is employed for out of plane static analysis of circular arches under a wide spectrum of boundary conditions in addition to the classical boundary conditions elastic restraints against translation and rotation are also considered different loading conditions are examined several examples of arches with uniform continuous or stepped varying cross sections are presented to demonstrate the accuracy of the methodology the domain decomposition technique in conjunction with the present dq methodology is examined for certain cases the results are compared with those of exact solutions for several uniform or stepped sections arches and also for arches on elastic foundations accurate converged numerical solutions are obtained with only few grid points

non axisymmetric bending of thin annular plates due to circumferentially distributed moments abstract the non linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied a von karman plate model is adopted to formulate the equations of motion free free boundary conditions have been applied at both inner and outer edges the matrix formulation of the differential quadrature method is used to discretise and solve the governing equations linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome however the proposed non linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non axisymmetric cylindrical deformation this is the case when at least one of two parameters reaches a critical value these parameters are the non dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius furthermore it is noted that free free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk hence the principal axes of curvature in the deformed configuration do not have a preferred orientation therefore the present work describes a shell possessing infinite identical equilibria having different yet no favoured direction and hence links to previous researches on neutrally stable structures

application of a new differential quadrature methodology for free vibration analysis of plates a new methodology is introduced in the differential quadrature dq analysis of plate problems the proposed approach is distinct from other dq methods by employing the multiple boundary conditions in a different manner for structural and plate problems the methodology employs the displacement within the domain as the only degree of freedom whereas along the boundaries the displacements as well as the second derivatives of the displacements with respect to the co ordinate variable normal to the boundary in the computational domain are considered as the degrees of freedom for the problem employing such a procedure would facilitate the boundary conditions to be implemented exactly and conveniently in order to demonstrate the capability of the new methodology all cases of free vibration analysis of rectangular isotropic plates in which the conventional dq methods have had some sort of difficulty to arrive at a converged or accurate solution are carried out excellent convergence behaviour and accuracy in comparison with exact results and or results obtained by other approximate methods were obtained the analogous dq formulation for a general rectangular plate is derived and for each individual boundary condition the general format for imposing the given conditions is devised it must be emphasized that the computational efforts of this new methodology are not more than for the conventional differential quadrature methods copyright 2002 john wiley sons ltd

differential quadrature analysis of free vibration of rhombic plates with free edges an accurate free vibration analysis of rhombic plates with free edges is presented by using the modified differential quadrature method dqm fourteen combinations of boundary conditions with free edges and five skew angles for each plate are analyzed it is found for the first time that the problem is extremely sensitive to grid spacing and the method to applying the multiple boundary conditions when the skew angle is very large various existing non uniform grid spacing and methods for applying the multiple boundary conditions are tested it is surprising to see that dqm with only one grid spacing can yield accurate and reliable solutions besides applying boundary conditions correctly at free corners is also important for success for verifications the dq results are compared with either existing accurate solutions or data obtained by finite element method due to bending stress singularities occurred at corners having obtuse angles the rate of convergence is low for a few cases and quite different from rhombic plates without free edges some new results are presented and waiting for comparison with future data obtained by other investigators

concavity with respect to holder means involving the generalized grotzsch function the generalized grotzsch function has numerous applications in geometric function theory and analytic number theory and its properties have been investigated by many authors in this paper we study the concavity of the generalized grotzsch function with respect to holder means

vibrations of completely free shallow shells of rectangular planform abstract although much has been written about the free vibrations of rectangular plates having completely free boundaries very little has appeared for the case when the plates have curvature i e shallow shells a solution of the problem is presented here for shells having arbitrary but constant curvature the ritz method is used with displacement functions assumed in the form of polynomials convergence studies were made to determine the number of terms required for reasonable solution accuracy numerical results were obtained for the frequencies and mode shapes of three types of shells circular cylindrical spherical and hyperbolic paraboidal and these are compared with those of a flat plate

implementation of clamped and simply supported boundary conditions in the gdq free vibration analysis of beams and plates abstract in this paper a new methodology for implementing the clamped and simply supported boundary conditions is presented for the free vibration analysis of beams and plates using the generalized differential quadrature gdq method the proposed approach directly substitutes the boundary conditions into the governing equations and is referred to as sbcge approach the sbcge approach is presented to overcome the drawbacks of previous approaches in treating the boundary conditions a comparison of the sbcge approach with the method of modifying weighting coefficient matrices mwcm is made by their application to the vibration analysis of beams and plates with combinations of simply supported and clamped boundary conditions some details of the gdq method are also described in the paper

monotonicity criterion for the quotient of power series with applications abstract in this paper we present the necessary and sufficient condition for the monotonicity of the quotient of power series as applications some gaps and misquotations in certain published articles are pointed out and corrected and some known results involving the landen inequalities for zero balanced hypergeometric functions are improved

precise bounds for the weighted holder mean of the complete p elliptic integrals abstract in the article we present the best possible bounds for the weighted holder mean of the complete p elliptic integrals of the first and second kinds which are the generalizations of the previously results for the complete elliptic integrals

numerical study of grid distribution effect on accuracy of dq analysis of beams and plates by error estimation of derivative approximation the accuracy of global methods such as the differential quadrature dq approach is usually sensitive to the grid point distribution this paper is to numerically study the effect of grid point distribution on the accuracy of dq solution for beams and plates it was found that the stretching of grid towards the boundary can improve the accuracy of dq solution especially for coarse meshes the optimal grid point distribution corresponding to optimal stretching parameter depends on the order of derivatives in the boundary condition and the number of grid points used the optimal grid distribution may not be from the roots of orthogonal polynomials this differs somewhat from the conventional analysis this paper also proposes a simple and effective formulation for stretching the grid towards the boundary the error distribution of derivative approximation is also studied and used to analyze the effect of grid point distribution on accuracy of numerical solutions copyright 2001 john wiley sons ltd

imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method one of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions there may be multiple boundary conditions involving higher order derivatives at the boundary points the boundary conditions can be imposed by modifying the weighting coefficient matrices directly however the existing method is not robust and is known to have many limitations in this paper a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations the given boundary conditions are imposed exactly furthermore it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established the other sampling grid points with no associated boundary conditions are not essential copyright 2002 john wiley sons ltd

accurate free vibration analysis of the completely free orthotropic rectangular plate by the method of superposition abstract the method of superposition is exploited to obtain highly accurate analytical type solutions for the free vibration of completely free orthotropic rectangular plates eigenvalues for 12 modes are provided for a particular set of problems representative of reinforced concrete and similar materials for a wide range of plate aspect ratios while it is recognized that data covering the entire range of possible problems cannot be tabulated it is hoped that the data provided will prove useful to some designers and provide a reference for other researchers to compare their results this appears to be the first study devoted to this problem to appear in the literature

fractional diffusion equation in a confined region surface effects and exact solutions surface effects on a diffusion process governed by a fractional diffusion equation in a confined region with spatial and time dependent boundary conditions are investigated first we consider the one dimensional case with the boundary conditions rho 0 t phi0 t and rho a t phia t subsequently the two dimensional case in the cylindrical symmetry with rho a theta t phia theta t and rho b theta t phib theta t is investigated for these cases we also obtain exact solutions for an arbitrary initial condition by using the green s function approach

application of differential quadrature to static analysis of structural components the numerical technique of differential quadrature for the solution of linear and non linear partial differential equations first introduced by bellman and his associates is applied to the equations governing the deflection and buckling behaviour of one and two dimensional structural components separate transformations are used for higher order derivatives as suggested by mingle thus extending the method to treat fourth order equations and to include multiple boundary conditions in the respective co ordinate directions results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods the application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort

high accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain summary based on the differential quadrature dq rule the gauss lobatto quadrature rule and the variational principle a dq finite element method dqfem is proposed for the free vibration analysis of thin plates the dqfem is a highly accurate and rapidly converging approach and is distinct from the differential quadrature element method dqem and the quadrature element method qem by employing the function values themselves in the trial function for the title problem the dqfem without using shape functions essentially combines the high accuracy of the differential quadrature method dqm with the generality of the standard finite element formulation and has superior accuracy to the standard fem and fdm and superior efficiency to the p version fem and qem in calculating the stiffness and mass matrices by incorporating the reformulated dq rules for general curvilinear quadrilaterals domains into the dqfem a curvilinear quadrilateral dq finite plate element is also proposed the inter element compatibility conditions as well as multiple boundary conditions can be implemented simply and conveniently as in fem through modifying the nodal parameters when required at boundary grid points using the dq rules thus the dqfem is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter element compatibilities a series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the dqfem copyright q 2009 john wiley sons ltd

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

a center box method for radially symmetric solution of fractional subdiffusion equation in this paper a center box difference method is considered for the radially symmetric solution of fractional subdiffusion equation by method of order reduction the derivative boundary condition is transformed into dirichlet boundary condition and thus the geometrical singularity is successfully removed from the original problem as a matter of course a natural discretization scheme is obtained to investigate the stability and convergence of the method we define a new norm with a weight r d 1 thus the usual sobolev inequality is not suitable to the new norm therefore we prove three new sobolev like embedding inequalities which can also be applied to the other problems in polar coordinates then the scheme is proved to be unconditionally stable and convergent in maximum norm with the help of the new sobolev like embedding inequalities some illustrative examples are provided to demonstrate the theoretical results by some comparisons it can be seen that the natural discretization scheme is accurate and effective in physical simulations and it can be used to both long time and short time computation

on the solution of the thomas fermi equation by differential quadrature on utilise la methode de quadrature differentielle pour resoudre l equation de thomas fermi

dynamic analysis of a spinning timoshenko beam by the differential quadrature method thedynamiccharacteristicsofaspinningtimoshenkobeamareinvestigatedbyusingthedifferentialquadrature method dqm the beam is subject to any combination of free simply supported clamped and elastically supported boundary conditions the weighting matrices in the differential quadrature formulation are modie ed to incorporate classical boundary conditions of the beam whereas boundary conditions of elastic supports are incorporated into the discretized equations numerical results of the spinning timoshenko beam obtained by the dqm are compared with the exact solutions orresults obtained by the e nite element method the results show the high accuracy and efe ciency of the differential quadrature method nomenclature a cross sectional area of beam c b damping matrix of elastic supports cxx cyy damping coefe cients e young s modulus g shear modulus g gyroscopic matrix i transverse moment of inertia jp polar mass moment of inertia k stiffness matrix k b stiffness matrix of elastic supports kxx kyy spring constants l

some results for a fractional diffusion equation with radial symmetry in a confined region abstract we investigate an n dimensional fractional diffusion equation with radial symmetry by taking a spatial and time dependent diffusion coefficient into account i e d r t d t r with d t d t d t the equation is considered in a confined region and subjected to time dependent boundary conditions which may be related to inhomogeneous characteristics of the surfaces confining the system the results show an anomalous spreading of the solutions and an unusual behavior of the survival probability