Perturbation Iteration Method Research Articles

ON THE BEHAVIOUR OF CURRENT-CARRYING WIRE-CONDUCTORS AND BUCKING OF A COLUMN

This paper applies approximate analytical methods namely Iteration Perturbation Method (IPM), variational approach (VA) and Parameter Expanding Method (PEM) to Single-Degree-Of-Freedom (SDOF) nonlinear oscillation systems. Some numerical cases as dynamic behavior of current-carrying wire-conductors and bucking of a column as well as their comparisons with the exact solutions are presented. Different specific parameters and initial values comprising the mass and stiffness are studied within the current research and excellent accuracy which is the most significant feature of the proposed solutions, is reported for the whole range of oscillation amplitude values.DOI: http://dx.doi.org/10.5755/j01.mech.19.3.4659

A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches

This paper intends to promote the application of modern analytical approaches to the governing equation of transversely vibrating quintic nonlinear beams. Four new studied methods are Stiffness analytical approximation method, Homotopy Perturbation Method with an Auxiliary Term, Max–Min Approach (MMA) and Iteration Perturbation Method (IPM). The powerful analytical approaches are used to obtain the nonlinear frequency–amplitude relationship for dynamic behavior of vibrating beams with quintic nonlinearity. It is demonstrated that the first terms in series expansions of all methods are sufficient to obtain a highly accurate solution. Finally, a numerical example is conducted to verify the integrity of the asymptotic methods.

On the accuracy of homotopy perturbation and variational iteration methods for lateral broadening of a monoenergetic proton beam

In this article, dispersion of a 60MeV proton pencil beam at various depths in a muscle tissue was numerically investigated via solving a three dimensional Fokker–Planck equation using homotopy perturbation method (HPM) and variational iteration method (VIM). The accuracy of these methods was benchmarked by comparison the radial flux distribution of protons traversing different depths in the tissue with the data of the High Charge and Energy Transport (HZETRN) model and Monte Carlo simulations. Furthermore, the computed depth dose distributions obtained from the HPM and VIM for monoenergetic protons passing through a medium were compared with the results of GEANT4.5.2 code as well as the experimental data reported in the literature. The satisfactory agreement obtained from our computations shows the reliability and applicability of the HPM and VIM in our analysis.

Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation

In this paper, we use the fractional variational homotopy perturbation iteration method (FVHPIM) with modified Riemann–Liouville derivative to solve a time-fractional diffusion equation. Using this method, a rapid convergent sequence tending to the exact solution of the equation can be obtained. To show the efficiency of the considered method, some numerical examples are presented.

New Perturbation Iteration Solutions for Fredholm and Volterra Integral Equations

In this paper, recently developed perturbation iteration method is used to solve Fredholm and Volterra integral equations. The study shows that the new method can be applied to both types of integral equations. Some numerical examples are given, and results are compared with other studies to illustrate the efficiency of the method.

Perturbation-Iteration Method for First-Order Differential Equations and Systems

The previously developed new perturbation-iteration algorithm has been applied to differential equation systems for the first time. The iteration algorithm for systems is developed first. The algorithm is tested for a single equation, coupled two equations, and coupled three equations. Solutions are compared with those of variational iteration method and numerical solutions, and a good agreement is found. The method can be applied to differential equation systems with success.

Investigating the Dynamic Behavior of Two Mechanical Structures Via Analytical Methods

In this paper, two new analytical techniques named max–min approach (MMA) and iteration perturbation method (IPM) are applied for solving nonlinear equations of two oscillatory systems. One case consists of a mass grounded by linear and nonlinear springs in series and the other case is the rigid rod rocks on a circular surface. The main aim of the work is obtaining highly accurate analytical solutions for nonlinear free vibration of conservative oscillations and investigates the dynamic behavior of the systems. Results reveal that the nonlinear frequency of oscillatory systems remarkably affected with the initial conditions. In contrast to the time marching solution results, the present analytical methods are effective and convenient. It is predictable that the MMA and the IPM can apply for various problems in engineering specially vibration equations.

New perturbation‐iteration solutions for nonlinear heat transfer equations

PurposeThe purpose of this paper is to apply, for the first time, the authors' newly developed perturbation iteration method to heat transfer problems. The effectiveness of the new method in nonlinear heat transfer problems will be tested.Design/methodology/approachNonlinear heat transfer problems are solved by perturbation iteration method. They are also solved by the well‐known technique variational iteration method in the literature.FindingsIt is found that perturbation iteration solutions converge faster to the numerical solutions. More accurate results can be achieved with this new method for nonlinear heat transfer problems.Research limitations/implicationsA few iterations are actually sufficient. Further iterations need symbolic packages to calculate the solutions.Practical implicationsThis new technique can practically be applied to many heat and flow problems.Originality/valueThe new perturbation iteration technique is successfully implemented to nonlinear heat transfer problems. Results show good agreement with the direct numerical simulations and the method performs better than the existing variational iteration method.

Nonlinear vibration of an elastically restrained tapered beam

This paper presents the analytical simulation of an elastically restrained tapered cantilever beam using the energy balance method (EBM) and the iteration perturbation method (IPM). To assess the accuracy of solutions, we compare the results with the harmonic balance method (HBM). The obtained results from EBM and IPM are in excellent agreement with HBM results. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering and physics.

ITERATIVE FEATURE PERTURBATION AS A GENE SELECTOR FOR MICROARRAY DATA

Gene-expression microarray datasets often consist of a limited number of samples with a large number of gene-expression measurements, usually on the order of thousands. Therefore, dimensionality reduction is critical prior to any classification task. In this work, the iterative feature perturbation method (IFP), an embedded gene selector, is introduced and applied to four microarray cancer datasets: colon cancer, leukemia, Moffitt colon cancer, and lung cancer. We compare results obtained by IFP to those of support vector machine-recursive feature elimination (SVM-RFE) and the t-test as a feature filter using a linear support vector machine as the base classifier. Analysis of the intersection of gene sets selected by the three methods across the four datasets was done. Additional experiments included an initial pre-selection of the top 200 genes based on their p values. IFP and SVM-RFE were then applied on the reduced feature sets. These results showed up to 3.32% average performance improvement for IFP across the four datasets. A statistical analysis (using the Friedman/Holm test) for both scenarios showed the highest accuracies came from the t-test as a filter on experiments without gene pre-selection. IFP and SVM-RFE had greater classification accuracy after gene pre-selection. Analysis showed the t-test is a good gene selector for microarray data. IFP and SVM-RFE showed performance improvement on a reduced by t-test dataset. The IFP approach resulted in comparable or superior average class accuracy when compared to SVM-RFE on three of the four datasets. The same or similar accuracies can be obtained with different sets of genes.

Reconsideration of the Iwasaki–Waggener iterative perturbation method for reconstructing high-energy X-ray spectra

We have reviewed applicable ranges for attenuating media and off-axis distances regarding the high-energy X-ray spectra reconstructed via the Iwasaki-Waggener iterative perturbation method for 4-20 MV X-ray beams. Sets of in-air relative transmission data used for reconstruction of spectra were calculated for low- and high-Z attenuators (acrylic and lead, respectively) by use of a functional spectral formula. More accurate sets of spectra could be reconstructed by dividing the off-axis distances of R = 0-20 cm into two series of R = 0-10 cm and R = 10-20 cm, and by taking into account the radiation attenuation and scatter in the buildup cap of the dosimeter. We also incorporated in the reconstructed spectra an adjustment factor (f (adjust) ≈ 1) that is determined by the attenuating medium, the acceleration voltage, and the set of off-axis distances. This resulted in calculated in-air relative transmission data to within ±2 % deviation for the low-Z attenuators water, acrylic, and aluminum (Al) with 0-50 cm thicknesses and R = 0-20 cm; data to within ±3 % deviation were obtained for high-Z attenuators such as iron (Fe), copper (Cu), silver (Ag), tungsten (W), platinum (Pt), gold (Au), lead (Pb), thorium (Th), and uranium (U) having thicknesses of 0-10 cm and R = 0-20 cm. By taking into account the radiation attenuation and scatter in the buildup cap, we could analyze the in-air chamber response along a line perpendicular to the isocenter axis.

Elastic solutions with arbitrary elastic inhomogeneity and anisotropy

An efficient numerical algorithm is proposed to accurately compute the elastic fields in two-dimensional (2D) or three-dimensional (3D) microstructures with arbitrary elastic inhomogeneity and anisotropy. It combines the equivalent inclusion method of Eshelby, the microelasticity theory of Khachaturyan, and the spectral iterative perturbation method of Hu and Chen. Its efficiency is compared with those of existing approaches in the literature. The method can be conveniently implemented in phase-field modeling of stress-dependent microstructure evolution and/or of mass/electrical transport.

Optimal homotopy perturbation method for solving partial differential equations with large solution domain

In this paper, an efficient modification of homotopy perturbation method, namely optimal homotopy perturbation method, is introduced for solving linear and nonlinear partial differential equations with large solution domain based on a new homotopy perturbation method and Pade approximation method. We compare the performance of the method with those of new homotopy perturbation and optimal variational iteration methods via three partial differential equations with large solution domain. Numerical results explicitly reveal that the suggested technique is highly capable to control the convergence region of approximate solution. Key words: New homotopy perturbation method, Pade´ approximation, optimal variational iteration method.

A Spectral Iterative Method for the Computation of Effective Properties Of Elastically Inhomogeneous Polycrystals

AbstractWe report an efficient phase field formalism to compute the stress distribution in polycrystalline materials with arbitrary elastic inhomogeneity and anisotropy The dependence of elastic stiffness tensor on grain orientation is taken into account, and the elastic equilibrium equation is solved using a spectral iterative perturbation method. We discuss its applications to computing residual stress distribution in systems containing arbitrarily shaped cavities and cracks (with zero elastic modulus) and to determining the effective elastic properties of polycrystals and multilayered composites.

Application of Homotopy Perturbation and Variational Iteration Methods for Fredholm Integrodifferential Equation of Fractional Order

This paper presents the application of homotopy perturbation and variational iteration methods as numerical methods for Fredholm integrodifferential equation of fractional order with initial‐boundary conditions. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented.

An Optimal Iteration Method for Strongly Nonlinear Oscillators

We introduce a new method, namely, the Optimal Iteration Perturbation Method (OIPM), to solve nonlinear differential equations of oscillators with cubic and harmonic restoring force. We illustrate that OIPM is very effective and convenient and does not require linearization or small perturbation. Contrary to conventional methods, in OIPM, only one iteration leads to high accuracy of the solutions. The main advantage of this approach consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. A very good agreement was found between approximate and numerical solutions, which prove that OIPM is very efficient and accurate.

A periodic solution for friction drive microrobots based on the iteration perturbation method

The friction drive principle, which is based on the superposition of two synchronized perpendicular vibrations at the interface of the robot and the work floor, plays a fundamental role in the locomotion of miniaturized robots. In this paper, the iteration perturbation method proposed by He is used to generate a periodic solution for this type of friction drive microrobot. The equation of motion for the system reveals a parametrically excited oscillator with discontinuity, the elastic force term for which is proportional to a signum function. The obtained solutions are in excellent agreement with those achieved from numerical integration and experiments reported in the literature. Results show that for a wide range of variation of system parameters, a phase shift near 90°between the perpendicular vibrations leads to the maximum achievable velocity of the slider. Besides, we have shown that driving frequency and the relative amplitude of vibrations are the best parameters for the control purposes of this type of microrobot.

A phase-field model of stress effect on grain boundary migration

We developed a phase-field model to study the stress-driven grain boundary migration in elastically inhomogeneous polycrystalline materials with arbitrary elastic inhomogeneity and anisotropy. The dependence of elastic stiffness tensor on grain orientation is taken into account, and the elastic equilibrium equation is solved using the Fourier spectral iterative-perturbation method. We studied the migration of planar and curved grain boundaries under an applied stress. The relation between grain boundary migration velocity and driving force is found to be linear in the steady-state regime. Our study shows that the stress distribution depends on the relative misorientation between the grains and the nature of the applied load. As a consequence, the mechanism of grain boundary migration is different when the load is applied parallel or perpendicular to a grain boundary. The bulk mechanical driving force for grain boundary migration is provided by the difference in the level of stress in the adjoining grains which arise due to difference in elastic moduli. We further show that under certain conditions an applied stress may act as a precursor to abnormal grain growth.

Nonlinear vibration of beams under nonideal boundary conditions

In this paper, we investigate the influence of nonideal boundary conditions on the nonlinear vibration of damped Euler–Bernoulli beams subjected to harmonic loads. These nonidealities allow for small deflections and/or moments at the supports of the beam. Using the iteration perturbation method, analytical expressions for the case of simply supported beams with immovable end conditions are provided. The results reveal that the first order of approximation obtained by the proposed method is more accurate than the perturbation solutions. Moreover, compared with the previous studies, some interesting phenomenon is predicted. We have shown that in some special combinations of the nonidealities the nonlinear frequency of vibration as well as the frequency–response curves would be unchanged.

Numerical and analytical simulation of peristaltic flows of generalized Oldroyd‐B fluids

AbstractIn this paper, we study the peristaltic flows of generalized Oldroyd‐B fluids through the gap between concentric uniform tubes under the assumption of large wavelength and low Reynolds number approximations. The inner tube is rigid and the outer tube has a sinusoidal wave travelling down its wall. Homotopy perturbation and variational iteration methods are used for solution of the problem. The obtained solution is then used to discuss various interesting features of peristalsis. The effects of relaxation time, retardation time and radii of the tubes on pressure rise and friction forces (per wavelength on the inner and outer tubes) are discussed with illustrations. It is found that pressure rise diminishes with increase in relaxation time or the ratio of radii of inner and outer tubes. It increases with increasing retardation time. The effects of both time parameters on friction forces have the opposite behavior to that of pressure rise. Copyright © 2010 John Wiley & Sons, Ltd.