numerical methods for the solution of special and general sixth order boundary value problems with applications to benard layer eigenvalue problems a family of numerical methods is developed for the solution of special nonlinear sixth order boundary value problems methods with second fourth sixth and eighth order convergence are contained in the family the problem is also solved by writing the sixth order differential equation as a system of three second order differential equations a family of second and fourth order convergent methods is then used to obtain the solution a second order convergent method is discussed for the numerical solution of general nonlinear sixth order boundary value problems this method with modifications where necessary is applied to the sixth order eigenvalue problems associated with the onset of instability in a benard layer numerical results are compared with asymptotic estimates appearing in the literature

b spline collocation for solution of two point boundary value problems a numerical method based on b spline is developed to solve the general nonlinear two point boundary value problems up to order 6 the standard formulation of sextic spline for the solution of boundary value problems leads to non optimal approximations in order to derive higher orders of accuracy high order perturbations of the problem are generated and applied to construct the numerical algorithm the error analysis and convergence properties of the method are studied via green s function approach o h 6 global error estimates are obtained for numerical solution of these classes of problems numerical results are given to illustrate the efficiency of the proposed method results of numerical experiments verify the theoretical behavior of the orders of convergence

time phase bispectral analysis bispectral analysis a technique based on high order statistics is extended to encompass time dependence for the case of coupled nonlinear oscillators it is applicable to univariate as well as to multivariate data obtained respectively from one or more of the oscillators it is demonstrated for a generic model of interacting systems whose basic units are the poincare oscillators their frequency and phase relationships are explored for different coupling strengths both with and without gaussian noise the distinctions between additive linear or quadratic and parametric frequency modulated interactions in the presence of noise are illustrated

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

a modification of the adomian decomposition method for a class of nonlinear singular boundary value problems in this paper the adomian decomposition method is modified to solve a class of nonlinear singular boundary value problems which arise as nonlinear normal modal equations in nonlinear conservative vibratory systems the effectiveness of the modified method is verified by three examples

the modified decomposition method for eighth order boundary value problems a modified adomian decomposition method is developed for the numerical solution of eighth order boundary value problems bvps the calculated numerical results show reliability and efficiency of the algorithm given the numerical procedure is tested on linear and nonlinear problems

the solution of multipoint boundary value problems by the optimal homotopy asymptotic method in this work we solve multipoint boundary value problems using the optimal homotopy asymptotic method oham the proposed method is tested upon several problems from the literature and the results are compared with the available exact solution this method provides easy tools to control the convergence region of approximating solution series where ever necessary

parametric septic splines approach for the solution of linear sixth order two point boundary value problems parametric splines which reduces to seven degree polynomial splines have been used to develop a class of numerical methods for the solution of linear sixth order two point boundary value problems convergence analysis of the methods is discussed through standard procedure it is shown that the present methods give approximations which are better than those produced by other splines and domain decomposition methods three numerical examples are given to illustrate the practical usefulness of the new approach

homotopy perturbation method for solving sixth order boundary value problems in this paper we apply the homotopy perturbation method for solving the sixth order boundary value problems by reformulating them as an equivalent system of integral equations this equivalent formulation is obtained by using a suitable transformation the analytical results of the integral equations have been obtained in terms of convergent series with easily computable components several examples are given to illustrate the efficiency and implementation of the homotopy perturbation method comparisons are made to confirm the reliability of the homotopy perturbation method we have also considered an example where the homotopy perturbation method is not reliable

wavelet bispectral analysis for the study of interactions among oscillators whose basic frequencies are significantly time variable bispectral analysis recently introduced as a technique for revealing time phase relationships is extended to make use of wavelets rather than fourier analysis it is thus able to encompass instantaneous phase time dependence for the case of two or more coupled nonlinear oscillators the method is demonstrated and evaluated by use of test signals from a pair of coupled poincare oscillators it promises to be useful in a wide range of scientific contexts for studies of interacting oscillators whose basic frequencies are significantly time variable

septic spline solutions of sixth order boundary value problems septic spline is used for the numerical solution of the sixth order linear special case boundary value problem end conditions for the definition of septic spline are derived consistent with the sixth order boundary value problem the algorithm developed approximates the solution and their higher order derivatives the method has also been proved to be second order convergent three examples are considered for the numerical illustrations of the method developed the method developed in this paper is also compared with that developed in m el gamel j r cannon j latour a i zayed sinc galerkin method for solving linear sixth order boundary value problems mathematics of computation 73 247 2003 1325 1343 as well and is observed to be better

on computing and implementing the running bispectra the paper extends recursion to bispectrum estimation problems and presents the systolic array implementation of the recursive higher order spectrum in which the bispectrum estimate is updated every data sample forward and reverse sequence running fourier transforms are first systolically realized the results are then used to drive a second systolic array whose outputs represent the ft of the data third order moment the proposed systolic arrays have no global communications with a number of processing elements independent of the size of the employed 2d lag window

consistent quaternion interpolation for objective finite element approximation of geometrically exact beam we explore an isoparametric interpolation of total quaternion for geometrically consistent strain objective and path independent finite element solutions of the geometrically exact beam this interpolation is a variant of the broader class known as slerp the equivalence between the proposed interpolation and that of relative rotation is shown without any recourse to local bijection between quaternions and rotations we show that for a two noded beam element the use of relative rotation is not mandatory for attaining consistency cum objectivity and an appropriate interpolation of total rotation variables is sufficient the interpolation of total quaternion which is computationally more efficient than the one based on local rotations converts nodal rotation vectors to quaternions and interpolates them in a manner consistent with the character of the rotation manifold this interpolation unlike the additive interpolation of total rotation corresponds to a geodesic on the rotation manifold for beam elements with more than two nodes however a consistent extension of the proposed quaternion interpolation is difficult alternatively a quaternion based procedure involving interpolation of relative rotations is proposed for such higher order elements we also briefly discuss a strategy for the removal of possible singularity in the interpolation of quaternions proposed in i romero the interpolation of rotations and its application to finite element models of geometrically exact rods comput mech 34 2004 121 133 the strain objectivity and path independence of solutions are justified theoretically and then demonstrated through numerical experiments this study being focused only on the interpolation of rotations uses a standard finite element discretization as adopted by simo and vu quoc j c simo l vu quoc a three dimensional finite rod model part ii computational aspects comput methods appl mech engrg 58 1986 79 116 the rotation update is achieved via quaternion multiplication followed by the extraction of the rotation vector nodal rotations are stored in terms of rotation vectors and no secondary storages are required

computationally lag invariant recursive spectrum estimators this paper is concerned with fourier based nonparametric spectral estimation methods for time varying environments it presents a class of interactive spectrum estimators in which the number of computations required to update the estimate at every data sample is independent of the number of the autocorrelation lags employed to produce this estimate these estimators are termed lag invariant and form an important class due to their computational simplicity because of the lag invariant property resolution enhancement is achieved without additional cost in computations per frequency sample the proposed class of recursive spectral estimators is based on specific selection of the lagged product windows used in the autocorrelation function estimation in this paper the conditions necessary to produce the lag invariance property are delineated

a new algorithm for solving classical blasius equation in this paper a reliable algorithm is employed to investigate the classical blasius equation the algorithm is based mainly on applying the adomian decomposition method to the transformation of the blasius equation the results demonstrate reliability and efficiency of the proposed algorithm

a comparison between the sinc galerkin and the modified decomposition methods for solving two point boundary value problems one of the new techniques used in solving boundary value problems involving ordinary differential equations is the sinc galerkin method this method has been shown to be a powerful numerical tool for finding fast and accurate solutions a less known technique that has been around for almost two decades is the decomposition method in this paper we solve boundary value problems of higher order using these two methods and then compare the results it is shown that the sinc galerkin method in many instances gives better results

non polynomial splines approach to the solution of sixth order boundary value problems non polynomial splines which are equivalent to seven degree polynomial splines are used to develop a class of numerical methods for computing approximations to the solution of sixth order boundary value problems with two point boundary conditions second fourth and sixth order convergence is obtained by using standard procedure it is shown that the present methods give approximations which are better than those produced by other spline and domain decomposition methods numerical examples are given to illustrate practical usefulness of the new approach

adomian decomposition method with green s function for sixth order boundary value problems in this paper the adomian decomposition method with green s function is applied to solve linear and nonlinear sixth order boundary value problems the numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the decomposition method is of high accuracy more convenient and efficient for solving high order boundary value problems

a new convergence proof of the adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws in this paper we propose a new convergence proof of the adomian s decomposition method adm applied to the generalized nonlinear system of partial differential equations pde s based on new formula for adomian polynomials the decomposition scheme obtained from the adm yields an analytical solution in the form of a rapidly convergent series for a system of conservation laws systems of conservation laws is presented we obtain the stability of the approximate solution when the system changes type we show with an explicit example that the latter property is true for general cauchy problem satisfying convergence hypothesis the results indicate that the adm is effective and promising

the numerical solution of sixth order boundary value problems by the modified decomposition method a fast and accurate algorithm is developed for the solution of sixth order boundary value problems bvps with two point boundary conditions a modified form of the adomian decomposition method is applied to construct the numerical solution for such problems the scheme is tested on one linear and two nonlinear problems the results demonstrate reliability and efficiency of the algorithm developed

an algorithm for adomian decomposition method in this paper a reliable technique for calculating adomian polynomials for nonlinear operators is developed the algorithm is fulfilled with symbolic computation so that the adomian decomposition methods for solving differential equations can be mechanized in maple ten examples are presented to illustrate the implementation of the program

convergence of adomian s decomposition method periodic temperatures abstract the convergence of adomian s decomposition method 1 as applied to periodic temperature fields in heat conductors is investigated

analytical approximations and pade approximants for volterra s population model in this paper an analytic approximation for volterra s model for population growth of a species in a closed system is presented the nonlinear integro differential model includes an integral term that characterizes accumulated toxicity on the species in addition to the terms of the logistic equation the series solution method and the decomposition method are implemented independently to the model and to a related ode the pade approximants that often show superior performance over series approximations are effectively used in the analysis to capture the essential behavior of the population u t of identical individuals

a kalman filtering approach to short time fourier analysis the problem of estimating time varying harmonic components of a signal measured in noise is considered the approach used is via state estimation two methods are proposed one involving pole placement of a state observer the other using quadratic optimization techniques the result is the development of a new class of filters akin to recursive frequency sampling filters for inclusion in a parallel bank to produce sliding harmonic estimates kalman filtering theory is applied to effect the good performance in noise and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate these filters can be seen as generalizing the dft

a new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations abstract in this paper we propose a new modified recursion scheme for the resolution of multi order and multi point boundary value problems for nonlinear ordinary and partial differential equations by the adomian decomposition method adm our new approach including duan s convergence parameter provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series we utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components whereas several prior recursion schemes have done so this modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots which is required to complete calculation of the solution by several prior modified recursion schemes using the adm

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

quintic spline solution of linear sixth order boundary value problems abstract there are few techniques available to numerically solve sixth order boundary value problems with two point boundary conditions in this paper quintic spline method is developed for the numerical solutions of linear special case sixth order boundary value problems two examples are considered for the numerical illustration of the method developed the method is also compared with those developed by el gamel et al mohamed el gamel john r cannon ahmed i zayed sinc galerkin method for solving sixth order bvps math comput 73 247 2003 and wazwaz a wazwaz the numerical solution of sixth order boundary value problems by the modified decomposition method appl math comput 118 2001 311 325 the comparison shows that the quintic spline method is more efficient and effective tool and yields better results

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

sinc galerkin method for solving linear sixth order boundary value problems mmthere are few techniques available to numerically solve sixth order boundary value problems with two point boundary conditions in this paper we show that the sinc galerkin method is a very effective tool in numerically solving such problems the method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made it is shown that the sinc galerkin method yields better results