on a jumping problem for quasilinear elliptic equations let us denote by k the eigenvalues of the operator with homogeneous dirichlet condition first studied in 1 when 1 2 this problem has been widely investigated in the case that some eigenvalue k belongs to the interval see e g 14 13 10 and references therein the methods used are often a combination of topological and variational techniques in this paper we will be concerned with a jumping problem associated to a quasilinear elliptic equation of the form n

on a simple triangular reissner mindlin plate element based on incompatible modes and discrete constraints in this paper the formulation of a new triangular element based on the reissner mindlin plate theory is presented the element has three nodes and three d o f per node only it is based on constant bending modes plus incompatible energy orthogonal higher order bending modes the transverse shear effects are represented using the moment equilibrium and the constitutive equations discrete collocation shear constraints are considered on each side to relate the kinematical and the independent shear strains the element has a proper rank is completely locking free passes all constant patch tests exactly the detailed numerical evaluation shows that the element called dst bk is a robust and high performance element for thick and thin plates

approximate solution of general high order linear nonhomogeneous difference equations by means of taylor collocation method in this paper a taylor collocation method is developed to find an approximate solution of general high order linear nonhomogenous difference equations with variable coefficients under the mixed conditions the solution is obtained in terms of taylor polynomials about any point also examples are presented which illustrate the pertinent features of the method

on elimination of shear locking in the element free galerkin method in this study a method for completely eliminating the presence of transverse shear locking in the application of the element free galerkin method efgm to shear deformable beams and plates is presented the matching approximation fields concept of donning and liu has shown that shear locking effects may be prevented if the approximate rotation fields are constructed with the innate ability to match the approximate slope first derivative of displacement fields and is adopted implementation of the matching fields concept requires the computation of the second derivative of the shape functions thus the shape functions for displacement fields and therefore the moving least squares mls weight function must be at least c1 continuous additionally the mls weight functions must be chosen such that successive derivatives of the mls shape function have the ability to exactly reproduce the functions from which they were derived to satisfy these requirements the quartic spline weight function possessing c2 continuity is used in this study to our knowledge this work is the first attempt to address the root cause of shear locking phenomenon within the framework of the element free galerkin method several numerical examples confirm that bending analyses of thick and thin beams and plates based on the matching approximation fields concept do not exhibit shear locking and provide a high degree of accuracy for both displacement and stress fields copyright 2001 john wiley sons ltd

approximate solution to linear complex differential equation by a new approximate approach the purpose of this study is to give a taylor polynomial approximation for the solution of high order linear complex differential equations with variable coefficients and mixed conditions about any point for this purpose a matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the complex differential equation and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately also examples that illustrate the pertinent features of the method are presented and the results are discussed

approximate solution of complex differential equations for a rectangular domain with taylor collocation method in this study taylor collocation method is investigated for the approximate computation of high order linear complex differential equations using the collocation points on any rectangular domain in the complex plane the method transforms the given complex differential equation and the mixed conditions to matrix equation with unknown taylor coefficients by means of the obtained matrix equations the taylor coefficients can be easily computed hence the finite taylor series approach is obtained also examples are presented and the results are discussed

multiplicity results for quasi linear problems abstract applying the minimax arguments and morse theory we establish some results on the existence of multiple nontrivial solutions for a class of p laplacian elliptic equations

a continuous method for nonlocal functional differential equations with delayed or advanced arguments abstract in the previous works the authors presented the reproducing kernel method rkm for solving various differential equations however to the best of our knowledge there exist no results for functional differential equations the aim of this paper is to extend the application of reproducing kernel theory to nonlocal functional differential equations with delayed or advanced arguments and give the error estimation for the present method some numerical examples are provided to show the validity of the present method

a taylor collocation method for the approximate solution of general linear fredholm volterra integro difference equations with mixed argument abstract in this paper a taylor collocation method is developed to find an approximate solution of high order linear fredholm volterra integro difference equations with variable coefficients under the mixed conditions the solution is obtained in terms of taylor polynomial near any point also examples are presented which illustrate the partinent features of the method and the results are discussed

projection schemes for one point quadrature shell elements abstract a new projection scheme to improve the performance of one point quadrature shell elements is presented the projection is performed elementwise and enforces invariance of the internal power to rigid body motion and drill rotation the projection is especially suitable for 6 degree of freedom per node explicit programs and can be easily implemented as an integral part of the subroutine that calculates the internal forces on the element level problems that could not be solved accurately with one point quadrature elements such as the twisted beam problem show excellent results even for very coarse meshes when the projection is used

morse theory for continuous functionals abstract using the notion of weak slope introduced by m degiovanni and m marzocchi 1994 ann mat pura appl 167 73 100 and also independently by c katriel 1994 ann inst h poincare anal non lineaire11 189 209 we obtain some results in morse theory for continuous functionals having isolated critical points the main tools are deformation properties generalizing classical ones for smooth functionals and refining those of j n corvellec m degiovanni and m marzocchi 1993 topol methods nonlinear anal 1 151 171 we obtain for example generalized morse relations through the notion of critical group which has a natural link with the weak slope these results are developed in view of applications to non smooth variational problems

the approximate solution of high order linear difference equations with variable coefficients in terms of taylor polynomials this paper presents a numerical method for the approximate solution of mth order linear difference equations with variable coefficients under the mixed conditions in terms of taylor polynomials about any point in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

extensions of the mountain pass theorem on considere une fonctionnelle c 1 i x r ou x est un espace de banach reel et ou i satisfait la condition de compacite de palais smale on presente une serie de theoremes qui generalisent le theoreme du col

a nondifferentiable extension of a theorem of pucci and serrin and applications abstract we study the multiplicity of critical points for functionals which are only differentiable along some directions we extend to this class of functionals the three critical point theorem of pucci and serrin and we apply it to a one parameter family of functionals j i r under suitable assumptions we locate an open subinterval of values in i for which j possesses at least three critical points applications to quasilinear boundary value problems are also given

properties of analytic solution and numerical solution of multi pantograph equation this paper deals with the properties of the analytic solution and the numerical solution of the multi pantograph equationu t email protected t email protected i 1 email protected iu q it 0

error through the constitutive relation for beam and c0 plate finite elements static and vibration analyses abstract applications of the so called error through the constitutive relation are presented for the static and modal analyses of beams and plates the error associated to the performed computation is estimated from the lack of verification by two admissible fields of the constitutive relation one is an admissible stress resultant field and the other is the generalised strain given by the finite element analysis admissible stress resultants are derived from the finite element stress solution following two main ideas firstly loads and moments ensuring the element equilibrium are calculated locally using an on average equilibrium property of the finite element stress solution secondly the stress resultants are determined within a set large enough to allow an optimisation in the way of rendering the error estimation more accurate for the normal mode analyses an error estimation is made for each eigenvector associated to a frequency within a given range an automatic procedure is described in case of beam structures that ensures a given accuracy for all modes in a given frequency range a set of examples is presented wherein the estimated error is compared to the error obtained from the analytical solution theses examples show that the beam and plate analyses are areas well adapted to the method

an approximate solution for a neutral functional differential equation with proportional delays in this paper a novel algorithm based on reproducing kernel theory for neutral functional differential equation with proportional delays is proposed the advantages of the presented method are the establishment of complete approximate solution theory and high precision of absolute error some examples are given to demonstrate the validity and applicability of the new method and some comparisons are made with the existing results

periodic boundary value problems for neutral multi pantograph equations in this paper a new method is proposed for solving nonlinear neutral multi pantograph equations with periodic boundary conditions by combining the advantages of the homotopy perturbed method hpm and the reproducing kernel method rkm the numerical results obtained show that the present method is reliable

assumed strain formulation for triangular c0 plate elements based on a weak form of the kirchhoff constraints application of a weak form of the kirchhoff constraints in the formulation of triangular assumed shear strain c0 plate element is described it is shown that compared to a similar formulation of the quadrilateral elements formulation of the triangles requires some additional considerations which lead to additional terms in the energy expression with those terms performance of the assumed shear strain triangular elements is comparable to that of the quadrilateral elements

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

new method for generating general solution of abel differential equation abstract a new method for generating a general solution of the nonlinear first kind abel type differential equation from a particular one is presented

solutions generating technique for abel type nonlinear ordinary differential equations abstract a second order nonlinear differential equation describing the dynamics of a bulk viscous fluid filled general relativistic friedmann robertson walker universe is considered by means of appropriate transformations the evolution equation of the space time is transformed into a third order polynomial abel differential equation with the use of some particular solutions of this equation repeated classes of exact solutions are generated

oscillation results of certain higher order linear differential equations with periodic coefficients in the complex plane abstract 1 has given a characterization of homogeneous linear differential equations with certain analytic periodic coefficients which admits a solution with finite exponent of convergence however her method seems too general and in most cases too complicated for applications we give in this paper a direct approach to the problem and obtain several such characterizations which do not seem to follow from those of baesch in particular an explicit necessary and sufficient condition to the problem is given for certain third order equations the results again do not seem to follow from those of baesch our method is based on that of 8 which in turn depends on basic representations of solutions given by bank laine and langley

chebyshev polynomial solutions of systems of high order linear differential equations with variable coefficients a chebyshev collocation method has been presented for numerically solving systems of high order linear ordinary differential equations with variable coefficients using the chebyshev collocation points this method transforms the ode system and the given conditions to matrix equations with unknown chebyshev coefficients by means of the obtained matrix equations a new system of equations which corresponds to the system of linear algebraic equations is gained hence by finding the chebyshev coefficients easily the finite chebyshev series approach is obtained

a rotation free shell formulation using nodal integration for static and dynamic analyses of structures summary this paper proposed a rotation free thin shell formulation with nodal integration for elastic static free vibration and explicit dynamic analyses of structures using three node triangular cells and linear interpolation functions the formulation is based on the classic kirchhoff plate theory in which only three translational displacements are treated as the filed variables based on each node the integration domains are further formed where the generalized gradient smoothing technique and green divergence theorem that can relax the continuity requirement for trial function are used to construct the curvature filed with the aid of strain smoothing operation and tensor transformation rule the smoothed strains in the integration domain can be finally expressed by constants the principle of virtual work is then used to establish the discretized system equations the translational boundary conditions are imposed same as the practice of standard finite element method while the rotational boundary conditions are constrained in the process of constructing the smoothed curvature filed to test the performance of the present formulation several numerical examples including both benchmark problems and practical engineering cases are studied the results demonstrate that the present method possesses better accuracy and higher efficiency for both static and dynamic problems copyright 2015 john wiley sons ltd

numeric solutions for the pantograph type delay differential equation using first boubaker polynomials a numerical method is applied to solve the pantograph equation with proportional delay under the mixed conditions the method is based on the truncated first boubaker series the solution is obtained in terms of first boubaker polynomials also illustrative examples are included to demonstrate the validity and applicability of the technique the results obtained are compared by the known results

unbounded critical points for a class of lower semicontinuous functionals for a general class of lower semicontinuous functionals we prove existence and multiplicity of critical points which turn out to be unbounded solutions to the associated euler equation we apply a nonsmooth critical point theory developed in 10 12 13 and applied in 8 9 20 to treat the case of continuous functionals

the embedded discontinuous galerkin method application to linear shell problems a new discontinuous galerkin method for elliptic problems which is capable of rendering the same set of unknowns in the final system of equations as for the continuous displacement based galerkin method is presented those equations are obtained by the assembly of element matrices whose structure in particular cases is also identical to that of the continuous displacement approach this makes the present formulation easily implementable within the existing commercial computer codes the proposed approach is named the embedded discontinuous galerkin method it is applicable to any system of linear partial differential equations but it is presented here in the context of linear elasticity an application of the method to linear shell problems is then outlined and numerical results are presented

three critical points theorem and its application to quasilinear elliptic equations abstract in this paper we prove a pucci serrin type three critical points theorem for continuous functionals and study its application to quasilinear elliptic equations with natural growth

a collocation method to find solutions of linear complex differential equations in circular domains in this study we introduce a collocation approach for solving high order linear complex differential equations in circular domain by using collocation points defined in a circular domain and bessel functions of the first kind this method transforms the linear complex differential equations into a matrix equation the matrix equation corresponds to a system of linear equations with the unknown bessel coefficients proposed method gives the analytic solution when the exact solutions are polynomials numerical examples are given to demonstrate the validity and applicability of the technique and the comparisons are made with existing results the results obtained from the examples demonstrate the efficiency and accuracy of the present work all of the numerical computations have been computed on computer using a code written in matlab

variational iteration method for solving a generalized pantograph equation the variational iteration method is applied to solve the generalized pantograph equation this technique provides a sequence of functions which converges to the exact solution of the problem and is based on the use of lagrange multipliers for identification of optimal value of a parameter in a functional employing this technique it is possible to find the exact solution or an approximate solution of the problem some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results

on the solution of the riccati equation by the taylor matrix method in this paper we suggest a matrix method to solve the riccati differential equation in terms of taylor polynomials this method is based on first taking the truncated taylor series of the function in equations and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately the solution is calculated in the form of a series with easily computable components the numerical results show the effectiveness of the method for this type of equation comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate

a new polynomial approach for solving difference and fredholm integro difference equations with mixed argument the purpose of this study is to give a taylor polynomial approximation for the solution of high order linear fredholm difference equations with variable coefficients and mixed argument under the mixed conditions about any point for this purpose a taylor matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately also examples that illustrate the pertinent features of the method are presented and the results are discussed

a three node mindlin plate element with improved transverse shear abstract a displacement methodology for mindlin elements recently employed in the development of an efficient four node quadrilateral min4 is the basis for a three node explicitly integrated triangular element min3 the approach herein referred to as anisoparametric is founded upon two kinematic requirements that are auxiliary to the standard convergence criteria the initial complete quadratic deflection field is constrained by continuous shear edge constraints to achieve an isoparametric like three node configuration much success of the element is due to an innovative shear correction factor concept several numerical experiments are performed in the linear elasto static regime to ascertain the element performance

multiple quadrature underintegrated finite elements new multiple quadrature point underintegrated finite elements with hourglass control are developed the elements are selectively underintegrated to avoid volumetric and shear locking and save computational time an approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co ordinates so that the elements require no stabilization parameter to improve accuracy over the traditional one point quadrature elements several quadrature points are used to integrate the internal forces especially for tracing the plastic fronts in the mesh during loading and unloading in elastic plastic analysis two and four point quadrature elements are proposed for use in the two and three dimensional elements respectively other multiple quadrature points can also be employed several numerical examples such as thin beam plate and shell problems are presented to demonstrate the applicability of the proposed elements

walsh stretch matrices and functional differential equations this correspondence presents a new operational matrix for stretch via walsh functions and discusses some interesting properties of this matrix and its role in the solution of certain functional differential equations the solutions of differential equations are obtained in a series of walsh functions the results are piecewise constant with minimal mean square error the single term approximation combined with the technique of extension 8 reduces the computational effort to a minimum

stochastic approach for the solution of multi pantograph differential equation arising in cell growth model in this paper a computational technique is introduced for the solution of the first order multi pantograph differential equation mpde through some well known optimization algorithms like sequential quadratic programming sqp and active set technique ast furthermore artificial neural network ann is used for networking of the first order multi pantograph differential equation in used to provide mathematical model based on unsupervised error for equation moreover mathematical modeling has been performed perfectly through multi runs for simulation to justify the better convergence of the solutions also two examples are presented to exhibit the aptitude of the method sqp and ast the comparative study will be made with reported techniques such as variational iteration technique vit 6 and collocation based on bernstein polynomial method bcm 6

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

a kirchhoff mode method for c0 bilinear and serendipity plate elements abstract a new formulation for the c0 quadrilateral and serendipity plate elements is presented it is an extension of the mode decomposition approach that has been successfully applied in evaluating the c0 triangular linear plate element in contrast to the triangular element the concept of an equivalent discrete kirchhoff configuration is utilized the elements are applied to several examples and the performance of these elements is shown to be quite good

solution of difference equations by using differential transform method abstract this work presents semi numerical analytical solutions to both linear and nonlinear difference equations with variable coefficients by applying differential transform method dtm and extending it with the related theorems and their proofs to make clear and illustrate the features and capabilities of the presented method examples are carried out and numerical results with comparison to exact solutions and other methods are given

a new uniqueness criterion for the number of periodic orbits of abel equations abstract a solution of the abel equation x a t x 3 b t x 2 such that x 0 x 1 is called a periodic orbit of the equation our main result proves that if there exist two real numbers a and b such that the function a a t b b t is not identically zero and does not change sign in 0 1 then the abel differential equation has at most one non zero periodic orbit furthermore when this periodic orbit exists it is hyperbolic this result extends the known criteria about the abel equation that only refer to the cases where either a t 0 or b t 0 does not change sign we apply this new criterion to study the number of periodic solutions of two simple cases of abel equations the one where the functions a t and b t are 1 periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials finally we give an upper bound for the number of isolated periodic orbits of the general abel equation x a t x 3 b t x 2 c t x when a t b t and c t satisfy adequate conditions

a multiple quadrature eight node hexahedral finite element for large deformation elastoplastic analysis abstract a multiple quadrature underintegrated hexahedral finite element which is free of volumetric and shear locking and has no spurious singular modes is described and implemented for nonlinear analysis finite element formulations are derived in the corotational coordinate system the use of consistent tangent operators for large deformation elastoplasticity with nonlinear isotropic kinematic hardening rules preserves the quadratic rate of convergence of the newton s iteration method in static analysis test problems studied demonstrate the efficiency and effectiveness of this element in solving a wide variety of problems including sheet metal forming processes

a taylor collocation method for the numerical solution of complex differential equations with mixed conditions in elliptic domains an approximate method for solving higher order linear complex differential equations in the elliptic domains is proposed the approach is based on a taylor collocation method which consists of the matrix representation of expressions in the differential equation and the collocation points defined in the elliptic domain illustrative examples are included to demonstrate the validity and applicability of the technique and performed on the computer using a program written in maple9

a numerical approach for solving generalized abel type nonlinear differential equations in this paper a numerical power series algorithm which is based on the improved taylor matrix method is introduced for the approximate solution of abel type differential equations and also riccati differential equations the technique is defined and illustrated with some numerical examples the obtained results reveal that the method is very effective simple and valid high accuracy the method can be easily extended to other nonlinear equations

a new collocation method for solution of mixed linear integro differential difference equations numerical solution of mixed linear integro differential difference equation is presented using chebyshev collocation method the aim of this article is to present an efficient numerical procedure for solving mixed linear integro differential difference equations our method depends mainly on a chebyshev expansion approach this method transforms mixed linear integro differential difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system maple10

on the solutions of a system of linear retarded and advanced differential equations by the bessel collocation approximation in this study we present a numerical approximation for the solutions of the system of high order linear retarded and advanced differential equations with variable coefficients under the mixed conditions this method is based on taking the truncated bessel expansion of the functions in the retarded and advanced differential equation system by the aid of the matrix operations and the collocation points the problem is transformed into a matrix equation with the unknown bessel coefficients by solving this matrix equation the unknown coefficients of the approximate solutions are computed in addition examples that illustrate the pertinent features of the method are presented and the results of this investigation are discussed

stability of the discretized pantograph differential equation in this paper we study discretizations of the general pantograph equation y t ay t by 6 t cy o t t 0 y o yo where a b c and yo are complex numbers and where 0 and 0 are strictly increasing functions on the nonnegative reals with 6 0 q o 0 and 0 t t 0 t t for positive t our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations and we will identify conditions on a b c and the stepsize which imply that the solution sequence yn i is bounded or that it tends to zero algebraically as a negative power of n

c0 finite element discretization of kirchhoff s equations of thin plate bending an alternative formulation of kirchhoff s equations is given which is amenable to a standard c 0 finite element discretization in this formulation the potential energy of the plate is formulated entirely in terms of rotations whereas the deflections are the outcome of a subsidiary problem the nature of the resulting equations is such that c 0 interpolation can be used on both rotations and deflections in particular general classes of triangular and quadrilateral isoparametric elements can be used in conjunction with the method unlike other finite element methods which are based on three dimensional or mindlin formulations the present approach deals directly with kirchhoff s equations of thin plate bending excellent accuracy is observed in standard numerical tests using both distorted and undistorted mesh patterns

e approximate polynomial solutions for the multi pantograph equation with variable coefficients abstract in this paper we present a new method to obtain analytical approximate polynomial solutions for the multi pantograph equation this method allows us to determine an approximate polynomial solution of the nonlinear multi pantograph equation with variable coefficients and to specify the maximal error of the approximation relative to the exact solution of the problem two numerical examples are given to illustrate the validity and the applicability of the method a comparison with previous results is also presented

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

a mixed formulation of c0 linear triangular plate shell element the role of edge shear constraints a simple and effective linear c0 triangular element for plates and shells is developed on the basis of the hellinger reissner mixed variational principle with independently assumed stress and displacement fields two main features are emphasized in this development concerning the assumed transverse shear stress field firstly this assumption results in edge type penalty constraints in the thin plate shell regimes secondly the element is used in the form of a four triangle cross diagonal macroelement mesh as a result the element is shown to be free from shear locking in thin plate shell applications and it satisfied all the appropriate patch tests required for kirchhoff plate models in addition the element exhibits good overall convergence properties in a variety of test problems for plates and shells finally from the computational standpoint the element is very efficient since all stiffness component matrices are derived in explicit forms

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

on the attainable order of collocation methods for pantograph integro differential equations for the pantograph integro differential equation pide with nonhomogeneous term y t ay t 01 y q t d q 01 y q t dv q f t t 0 y 0 y0 with proportional delays q t and q t 0 q q 1 0 q 1 we consider the attainable order of m stage implicit collocation based runge kutta methods at the first mesh point t h and give conditions on the collocation polynomials mm t of degree m to u th t 0 1 such that u h y h o h2m 1 where y t is the solution and u t is the collocation solution of pide if m 2 or f t is a polynomial of t whose degree is less than or equal to m then such conditions of mm t are simplified a numerical example is also included

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

a taylor polynomial approach for solving differential difference equations the purpose of this study is to give a taylor polynomial approximation for the solution of mth order linear differential difference equations with variable coefficients under the mixed conditions about any point for this purpose taylor matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the differential difference equations and then substituting their matrix forms into the equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed also we have discussed the accuracy of the method we use the symbolic algebra program maple to prove our results

finite rotation analysis and consistent linearization using projectors abstract a systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis this procedure is a generalization of the work presented in computers and structures volume 30 pp 257 267 the basis of our approach is the element independent co rotational algorithm where the element rigid body motion translations and rotations is separated from the deformational part of its total motion the variation of this co rotational relation results in a projector matrix with the property that a consistent internal force vector is invariant under its action the consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector this results in an unsymmetric tangent matrix whose anti symmetric part depends on the out of balance force vector in this paper we prove that using the symmetric part of the tangent matrix the newton iteration retains its quadratic rate of convergence this approach has been used to solve a number of large rotation test example problems the results demonstrate that it is possible to analyze structures undergoing large rotations within a general co rotational framework using simple and economical finite elements the resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre and post processors to the element routines

approximate solution of multi pantograph equation with variable coefficients this paper deals with the approximate solution of multi pantograph equation with nonhomogenous term in terms of taylor polynomials the technique we have used is based on a taylor matrix method in addition some numerical examples are presented to show the properties of the given method and the results are discussed

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

the pantograph equation in the complex plane abstract the subject matter of this paper focuses on two functional differential equations with complex lag functions we address ourselves to the existence and uniqueness of solutions and to their asymptotic behaviour

a c0 triangular plate element with one point quadrature on developpe un element de plaque triangulaire du type de mindlin qui utilise des champs lineaires pour les rotations et les deflexions transversales l element ne necessite qu un seul point de quadrature ce qui est tres interessant pour l analyse non lineaire

a hermite reproducing kernel approximation for thin plate analysis with sub domain stabilized conforming integration a hermite reproducing kernel rk approximation and a sub domain stabilized conforming integration ssci are proposed for solving thin plate problems in which second order differentiation is involved in the weak form although the standard rk approximation can be constructed with an arbitrary order of continuity the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems by imposing the kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously it is demonstrated that the minimum normalized support size coverage of kernel functions can be significantly reduced with this proposed approximation the galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived subsequently an ssci method is developed to achieve the exact pure bending solution as well as to maintain spatial stability numerical examples demonstrate that the proposed formulation offers superior convergence rates accuracy and efficiency compared with those based on higher order gauss quadrature rule copyright 2007 john wiley sons ltd

constant sign and sign changing solutions for nonlinear eigenvalue problems abstract we prove the existence of multiple constant sign and sign changing solutions for a nonlinear elliptic eigenvalue problem under dirichlet boundary condition involving the p laplacian more precisely we establish the existence of a positive solution of a negative solution and of a nontrivial sign changing solution when the eigenvalue parameter is greater than the second eigenvalue 2 of the negative p laplacian extending results by ambrosetti lupo ambrosetti mancini and struwe our approach relies on a combined use of variational and topological tools such as e g critical points mountain pass theorem second deformation lemma variational characterization of the first and second eigenvalue of the p laplacian and comparison arguments for nonlinear differential inequalities in particular the existence of extremal nontrivial constant sign solutions plays an important role in the proof of sign changing solutions

numerical solutions of the nonlinear volterra fredholm integral equations by using homotopy perturbation method abstract a numerical method for solving the nonlinear volterra fredholm integral equations is presented the method is based upon homotopy perturbation theory the results reveal that the homotopy perturbation method is very effective and simple and gives the exact solution

a taylor method for numerical solution of generalized pantograph equations with linear functional argument this paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument in this paper we introduce a numerical method based on the taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case the method is illustrated by studying the initial value problems the results obtained are compared by the known results

a new collocation method for approximate solution of the pantograph functional differential equations with proportional delay the paper presents a new numerical method for solving functional differential equations with proportional delays of the first and higher orders the method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of functional differential equations with proportional delays including equations of neutral type the method is applicable to both initial and boundary value problems

two dimensional dynamical system associated with abel s nonlinear differential equation a two dimensional dynamical system is proposed that is described by a pair of nonlinear ordinary differential equations odes with a complex parameter it reduces to abel s nonlinear ode of the first kind by an appropriate transformation using this fact the properties of solutions are investigated in detail with the aid of numerical computations it is found that various types of bifurcation phenomena occur depending on the values of the parameter in particular the solution is shown to blow up in finite time under certain conditions in order to visualize the behaviors of dynamical motions the trajectories of solutions are depicted in the plane finally a discussion is made on some generalizations of the proposed system