Series Solution Approach Research Articles

Eleventh Degree Polynomial Series Solution Approach of Special Non-linear Fourth Order Boundary Value Problems

In this paper a new eleventh degree polynomial series solution approach is developed for the solution of special non-linear fourth order boundary value problems. The method consist first of obtaining a linear differentials system of twelve equations from the boundary conditions, governing equation and its three successive derivatives which were evaluated at boundary points. Secondly we assume the approximate solution in the form of a polynomial of degree eleventh with twelve unknown coefficients. To determine the unknown coefficients we incorporate the assumed solution into linear differentials systems of twelve equations which results into a linear systems of twelve equations with twelve unknown and which can be solve uniquely. It is clear from the tables and figures that the method is in good agreement with the exact and with some existing results in the literatures. Also it can be seen from example 3.3 that the exact solution is reproduced which is an added advantage of the method.

Modeling of FGM contacts using piecewise linear variation in Young’s modulus

Functionally graded materials (FGM) are used as coatings at contact interfaces to tailor the desired surface properties while avoiding sudden changes in material properties. This work investigates the full-sliding frictional contact between a rigid indenter and FGM-coated half-space under plane-strain conditions. The FGM-coating is divided into a number of sublayers, with Young’s modulus varying linearly in each sublayer, effectively making the overall variation piece-wise linear. This approach overcomes the disadvantage of discontinuity of material properties at the interfaces of sublayers that exists in multi-layered piecewise constant modeling. Series solution approach based on the solution to the singular integral equation that characterizes the 2D contact is considered. In this approach, pressure distribution at the interface is expressed as a series expansion. This approach can be used to analyze any arbitrarily shaped contacts. The results show that the piecewise linear FGM modeling approach with tractions expressed as modified Fourier series is a more accurate and efficient way of representing FGM material property as compared to piecewise constant material properties, which are used in most of the existing literature. The number of layers needed to converge in continuous variation is also very less compared to the piece-wise constant and hence the computational time is less.

Nonlinear flexure mechanics of beams: stress gradient and nonlocal integral theory

In order to study the intrinsic size-effects, the stress gradient theory is implemented to a nano-scale beam model in nonlinear flexure. The nonlocal integral elasticity model is considered as a suitable counterpart to examine the softening behavior of nano-beams. Reissner variational principle is extended consistent with the stress gradient theory and applied to establish the differential, constitutive and boundary conditions of a nano-sized beam in nonlinear flexure. The nonlinear integro-differential and boundary conditions of inflected beams in the framework of the nonlocal integral elasticity are determined utilizing the total elastic strain energy formulation. A practical series solution approach in terms of Chebyshev polynomials is introduced to appropriately estimate the kinematic and kinetic field variables. A softening structural behavior is observed in the flexure of the stress gradient and the nonlocal beam in terms of the characteristic parameter and the smaller-is-softer phenomenon is, therefore, confirmed. The flexural response associated with the stress gradient theory is demonstrated to be in excellent agreement with the counterpart results of the nonlocal elasticity model equipped with the Helmholtz kernel function. The nonlocal elasticity theory endowed with the Error kernel function is illustrated to underestimate the flexural results of the stress gradient beam model. Detected numerical benchmark can be efficiently exploited for structural design and optimization of pioneering nano-engineering devices broadly utilized in advanced nano-electro-mechanical systems.

A Series Solution Approach to the Circular Restricted Gravitational Three-Body Dynamical Problem

The present manuscript examines the circular restricted gravitational three-body problem (CRGTBP) by the introduction of a new approach through the power series method. In addition, certain computational algorithms with the aid of Mathematica software are specifically designed for the problem. The algorithms or rather mathematical modules are established to determine the velocity and position of the third body’s motion. In fact, the modules led to accurate results and thus proved the new approach to be efficient.

Boundary observer design for a class of semi-linear hyperbolic PDE systems with recycle loop

ABSTRACT The present manuscript considers the observer design problem for a class of scalar semi-linear hyperbolic partial differential equation (PDE) systems with a recycle loop through the boundary point. The design method of Kazantzis and Kravaris [(1998). Nonlinear observer design using Lyapunov's auxiliary theorem. Systems & Control Letters, 34(5), 241–247] developed for the nonlinear finite dimensional systems observer design is extended to semi-linear hyperbolic PDE systems. The observer design problem is tackled through a first-order associated PDE. Due to the existence of spatial partial derivative operator, Lyapunov's Auxiliary Theorem originated from finite-dimensional systems can be no longer applied to seek conditions ensuring solvability of the associated PDE. In this manuscript, a new theorem is formulated and proved to ensure the solvability. The solution for the associated PDE is locally analytic nonlinear coordinate transformation which provides foundation for observer realization and a series solution approach is developed. Simulation examples are presented to study the performance of the proposed observer.

General Series Solutions for Stresses and Displacements in an Inner-fixed Ring

The general series solution approach is provided to get the stress and displacement fields in the inner-fixed ring. After choosing an Airy stress function in series form, stresses are expressed by infinite coefficients. Displacements are obtained by integrating the geometric equations. For an inner-fixed ring, the arbitrary loads acting on outer edge are extended into two sets of Fourier series. The zero displacement boundary conditions on inner surface are utilized. Then the stress (and displacement) coefficients are expressed by loading coefficients. A numerical example shows the validity of this approach.

Analytical Solutions for Stress and Displacement Fields in Disks under Arbitrarily Distributed Loads

The general series solution (GSS) approach is presented, in order to determine the stress and displacement fields in disks under arbitrarily distributed normal and tangential loads. An Airy stress function in series form is selected. Stresses are expressed by infinite coefficients. Thus displacements are expressed by the infinite stress coefficients. And self-equilibrated loads acting on the side edge are extended to Fourier series. Stress coefficients are related to loading coefficients by stress boundary conditions. Then five examples show the validity of this approach. The GSS approach might lead to industrial applications in rock mechanics, petroleum and mining engineering, etc.

Bright and Dark Optical Solitons with Kerr and Parabolic Law Nonlinearities by Series Solution Approach

Bright and Dark Optical Solitons with Kerr and Parabolic Law Nonlinearities by Series Solution Approach

Physical optics solution for the scattering of a partially coherent wave from a circular cylinder

The physical optics solution for the scattering of a partially coherent wave from a circular cylinder is presented. The derivations of the far-zone scattered fields are presented and discussed. Scattering width relations are developed from the physical optics fields that are functions of the coherence of the incident field. The scattering width relations are evaluated asymptotically to show the behavior of the backscattered and forward scattered energy in the incoherent- and coherent-field limits. Results are presented comparing and contrasting the physical optics results to those obtained using a previously published exact series solution approach. A physical discussion of the results is included.

Soliton Solutions of the Long-Short Wave Equation with Power Law Nonlinearity

This paper studies the generalized long-short wave equation with power law nonlinearity. There are several approaches that are used to solve this coupled system nonlinear evolution equations. The series solution approach yields the topological 1-soliton solution or shock wave solution. The ansatz method and the semiinverse variational principle leads to the non-topological 1-soliton of the equation. Additionally, the variational iteration method is used to study the equation. Finally, numerical simulations are also given to this equation.

Numerical method of slope failure probability based on Bishop model

Based on Bishop’s model and by applying the first and second order mean deviations method, an approximative solution method for the first and second order partial derivatives of functional function was deduced according to numerical analysis theory. After complicated multi-independent variables implicit functional function was simplified to be a single independent variable implicit function and rule of calculating derivative for composite function was combined with principle of the mean deviations method, an approximative solution format of implicit functional function was established through Taylor expansion series and iterative solution approach of reliability degree index was given synchronously. An engineering example was analyzed by the method. The result shows its absolute error is only 0.78% as compared with accurate solution.

Mathematical analysis on a model for the rectangular cantilever beam

The mechanical model for the rectangular cantilever beam proposed by Zhang et al is solved analytically by the series solution with mathematical properties investigated in detail. The derived series solution is proved convergent, and restrained only by the small deflection presumed by the Euler–Bernoulli beam theorem, and is applicable for calculating the deflection and curvature for any value of the exerted axial stress. The formulae estimating the accuracy of the coefficients and the series solution are developed from Stirling's approximation for the gamma function. The condition on the axial stress is developed, by which the genuinely nonlinear curvature can be approximated by a linear function and the deflection can be calculated from the boundary condition by a cubic polynomial. The additional redundant boundary condition used in Zhang's work is discussed, which should be removed since it fails to fit the model by inducing errors for calculating the deflection and the curvature. The present series solution approach provides formal deflection–stress and curvature–stress relations for the design of a MEMS micro-cantilever system as a bio-detection device. For self-assembly applications, the adsorbing material can be identified by solving the exerted axial stress from the series solution.

General Analytical Solutions for Merton's-Type Consumption-Investment Problems

We solve analytically the Merton's problem of an investor with timeadditive power utility. For general state dynamics, we prove existence of two power series representations of the relevant optimal policies and value functions, which hold for all admissible risk aversion parameters. We characterize all terms in the power series by a recursive formula, allowing analytical computations to arbitrary order. Some applications to explicit model settings highlight a very satisfactory accuracy of nite order approximations provided by our power series solution approach.

Bending of Sector Plates on Elastic Foundations by Bessel Functions

A series solution approach based on the complex Bessel functions is presented in this paper for the analysis of bending of annual sector plates that are simply supported along their straight (or radial) edges, resting on Winkler-type elastic foundations. After a real general solution to a homogeneous differential equation of the bending plates is derived from the complex Bessel functions, the solution to the nonhomogeneous differential equation may be formulated by the general solution. Finally, some numerical examples are given to show the feasibility of this approach.

A review and critique of theories for piezoelectric laminates

A review and critique of different laminate theories used for the modeling and analysis of laminated composite beams or plate structures is presented. Many finite-element models use classical laminate theory (CLT), also known as first-order shear deformation theory (FSDT), for the numerical simulation of active structures. The basic assumptions of this model have evolved from those proposed for composite laminate models and are based on thin-plate theory with resulting approximations for the elastic displacement, stress and strain components. In the case of piezoelectric laminates, the approximations spill over into the electric potential and electric field components. No studies and simulations have been documented for the dynamical electromechanical field variations through the thickness of the laminate structure at the resonant frequencies of the structure. This is essential to the understanding of the validity and range of applicability of thin-plate assumptions for active vibration control of structures. On the one hand, thin-plate models result in a computationally tractable model for smart structures, but they should not compromise on the electromechanical coupling effect, which is at the basis of active control. This paper first presents a three-dimensional (3D) complete field solution for active laminates based on a modal, Fourier series solution approach that is used to compute all the through-thickness electromechanical fields near the dominant resonance frequency of a beam plate with two piezoelectric (sensor and actuator) and one structural layers. Then a detailed review of the extant laminate models used for piezoelectric laminates, emphasizing the underlying assumptions in each case, is presented. The non-zero, through-thickness field components are computed under these assumptions. The results of the 3D model and FSDT model are compared for two aspect ratios ((ARs) - thickness-to-width of the layers). An AR of 20 is at the limit of the FSDT and an AR of 50 well within the assumptions of the FSDT. It is concluded that for moderate ARs, several of the approximations of the FSDT are questionable at resonance frequencies. A detailed set of pertinent and general references to papers dealing with piezoelectric laminates is also included. It is hoped that this study will be a reference source for those who want to use FSDT and for those want to understand the dynamical behavior of the internal fields in a smart laminate.

A series solution approach to an analytical load-deflection relation for the measurement of mechanical properties of thin films

The load-deflection technique is a popular technique to measure mechanical properties such as internal stress and Young's modulus of thin films. The measurement of internal stress and Young's modulus depends on the determination of the analytical load-deflection relation. In this paper, a series solution approach is presented to obtain the analytical load-deflection relation for a circular membrane. To determine the number of terms required for the results to settle to a stable value, a convergence study is carried out. The present study found that to obtain the analytical load-deflection relation using a series solution approach, the transverse displacement can only admit one term solution, while the in-plane displacement can have a series solution. The present approach has been verified with a finite-element solution obtained using Macneal-Schwendler Corporation Nastran by comparing maximum deflections as well as deflection curves. The agreement is excellent.

A review and critique of theories for piezoelectric laminates

This paper presents a three-dimensional (3-D) complete field solution for active laminates based on a modal, Fourier series solution approach that is used to compute all the through thickness electromechanical fields near the dominant resonance frequency of a beam plate with two piezoelectric (sensor and actuator) and one structural layers. Then a detailed review of the extant laminate models used for piezoelectric laminates emphasizing the underlying assumptions in each case is presented. The nonzero, through thickness field components are computed under these assumptions. The results of the 3-D model and FSDT model are compared for two aspect ratios (AR thickness to width of layers). An AR of 20 is at the limit of FSDT and an AR of 50 well within the assumptions of FSDT. It is concluded that for moderate aspect ratios, several of the approximations of FSDT are questionable at resonance frequencies, resulting in large errors in the estimation of the elastic and electric field components.

A novel approach to the solution of shear flexible rectangular plates with arbitrary laminations

A novel analytical solution to shear flexible rectangular plates with arbitrary laminations is presented. The deformation formulation of the plate characterized by the equivalent single layer theory that considers the effects of transverse shear deformations, suitable for moderately thick plates, results in highly coupled five second-order partial differential equations in five unknowns. A novel boundary continuous double Fourier series solution approach has been developed to solve these coupled equations in conjunction with the admissible boundary conditions. The analytical results presented should serve as bench-marks for future comparisons of such popular approximate methods as finite element, finite difference, Rayleigh-Ritz, Galerkin, and collocation methods.

A series solution of the nonlinear wind-driven ocean circulation and its inertial limit

A series solution approach to the forced and damped quasi-geostrophic barotropic vorticity equation is considered in order to examine the strongly forced and inertial limits of ocean gyre dynamics. The strongly forced limit is the limit investigated numerically by Veronis (1966b). It is shown that this limit, although superficially having the same symmetry properties as the inertial limit, is distinguishably different from the latter. After isolating the inertial limit in an appropriate way it is shown that our series solution method is able to find the ‘free mode’ and its ‘almost free correction’, that the ‘free mode’ obeys the integral criteria of Niiler (1966) and Pierrehumbert & Malguzzi (1984) and that the relationship between the streamfunction and the absolute vorticity is in general a nonlinear one.

SOLUTION OF INVERSE STEFAN PROBLEMS BY A SOURCE‐AND‐SINK METHOD

A method has been developed for the solution of inverse heat diffusion problems to find the initial condition, boundary condition, and the source and sink function in the heat diffusion equation. The method has been used in the development of a source‐and‐sink method to find the boundary conditions in inverse Stefan problems. Green's functions have been used in the solution, and the problems are solved by using two approaches: a series solution approach, and a time incremental approach. Both can be used to find the boundary conditions without reliance on the flux information to be supplied at both sides of the interface. The methods are efficient in that they require less equations to be solved for the conditions. The numerical results have shown to be accurate, convergent, and stable. Most of all, the results do not degrade with time as in other time marching schemes reported in the literature. Algorithms can also be easily developed for the solution of the conditions.