Classical Perturbation Techniques Research Articles

Heat Transfer of Casson Fluid in Poiseuille Flow of Carbon Nanotubes: A Power Series Approach

Combination of Carbon Nanotubes (CNTs) with Human blood as base fluid indicates the enhancement of heat transport in Poiseuille flow and this physical phenomenon could not possible in normal liquids. Usually, when blood cells get touch with external surfaces, the platelets of blood become activated and form blood clots. Therefore, we have considered blood compatible CNT, so that chosen base fluid (blood) can easily pass through it. Due to the higher thermal conductivity, CNTs play an important role to enhance the heat transport in blood flow. These features lead to the novelty of this investigation to study the heat transport of Casson fluid through CNTs in unsteady MHD free convective Poiseuille flow with thermal radiation. Such study consigns practical applications in manufacturing of drugs, biomedical and Tissue engineering, biosensor and other applications in myocardial therapy, neuronal and muscle regeneration. The non-dimensional governing equations are formulated and solved analytically through classical Perturbation Technique and the analysis of results are drawn in smooth curves via MATLAB code. Significant results for different implanted parameters are compared between SWCNT and MWCNT and their significant behaviours are plotted graphically. The obtained results indicate that Casson parameter accelerates the flow velocity for MWCNT and SWCNT. Furthermore, interesting behavior on the outlines of velocity for SWCNT and MWCNT due to the presence of Schmidt number, Peclet number and Reynolds number are detected. Comparison with previously published work has been inspected and originated excellent agreement.

On two approaches to the third-order solution of surface gravity waves

A third-order approximate solution for surface gravity waves in the finite water depth is studied in the context of potential flow theory. This solution corresponds to the steady-state part of a multidirectional irregular wave field and provides explicit expressions for the surface elevation, free-surface velocity potential, and velocity potential. The amplitude dispersion relation is also provided. Two approaches are used to derive the third-order analytical solution, resulting in two types of approximate solutions: the perturbation solution and the Hamiltonian solution. The perturbation solution is obtained by a classical perturbation technique in which the time variable is expanded in multiscale to eliminate secular terms. The Hamiltonian solution is derived from the canonical transformation in the Hamiltonian theory of water waves. By comparing the two types of solutions, it is found that they are completely equivalent for the first- and second-order solutions and the nonlinear dispersion, but for the third-order part only the sum–sum terms are the same. Due to the canonical transformation that could completely separate the dynamic and bound harmonics, the Hamiltonian solutions break through the difficulty that the perturbation theory breaks down due to singularities in the transfer functions when the quartet resonance criterion is satisfied. Furthermore, it is also found that some time-averaged quantities based on the Hamiltonian solution, such as mean potential energy and mean kinetic energy, are equal to those in the initial state in which the sea surface is assumed to be a Gaussian random process. This is because there are associated conserved quantities in the Hamiltonian form. All of these show that the Hamiltonian solution is more reasonable and accurate to describe the third-order steady-state wave field. Finally, based on the Hamiltonian solution, some statistics are given such as the volume flux, skewness, excess kurtosis, and non-uniqueness of the induced mean flow and mean surface.

Analytical approximation solution for logistic delay differential equation

Although the non-linear analytical techniques are fast developing, they still do not entirely satisfy mathematicians and engineers. Many researchers have conducted the study to find the analytical solution for the logistic delay differential equation. However, for the time lags occasion, it is quite hard and tough to achieve analytical solution due to its limitation, and thus, we can only expect the approximate analytical solution. This paper describes the approximate analytical techniques, homotopy analysis method (HAM), and homotopy perturbation method (HPM) in order to indicate their ability in solving the logistic delay differential equation. HAM is one of the better approaches that can be used for solving this equation. The use of HAM will lead to obtaining the series solution that contains an auxiliary parameter that can help to adjust and control the convergence and rate approximation for the series solution. Meanwhile, HPM is an analytical method with a combination of homotopy in topology and classical perturbation technique. Using the HPM technique, the logistic delay differential equation is reduced to a sufficiently simplified form, which usually becomes a linear equation that is easy to be solved. The comparison of numerical solution with -values of HAM has shown the influence of parameter in the convergence of series solution. Using HAM and HPM, the relationship between the time-delay τ and the population size is obtained. As a result, the higher the value of the steeper the gradient of the population size . It is concluded that the parameter helps to adjust and control the convergence and rate approximation for the series solution of HAM. Laterally, the comparison between HAM and HPM with numerical method is done to show that both methods are relatively approximate to the exact solution. Moreover, homotopy perturbation method (HPM) is a special case of homotopy analysis method (HAM) when and . Hence, using HAM and HPM techniques, two different kinds of series solutions of logistic delay differential equation are obtained.

Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves

Abstract In this work, we aim to apply a numerical approach based on Homotopy perturbation transform method (HPTM) for derive the exact and approximate solutions of nonlinear fifth order KdV equations for study magneto-acoustic waves in plasma. The approach is a mixed form of the standard Laplace transform with the classical Homotopy perturbation technique. Nonlinear term can be handled with the aid of He's Polynomials. In this technique, the solution is calculated in the form of a convergent series and convergence of the HPTM solutions to the exact solutions is shown. The Homotopy perturbation transform method presents a wide applicability to handling nonlinear wave equations in science and engineering. As a novel application of HPTM, the present work shows some essential difference with the existing similar method. Several examples are provided for illustrate the simplicity and reliability of the method and highlighted the significance of this work.

Generalised perturbation techniques for strongly nonlinear oscillators with a positive, zero or negative linear stiffness term

This study is concerned with nonlinear oscillators whose equations of motion contain a linear stiffness term with a positive, zero or negative coefficient and a power-form nonlinear term whose coefficient is not necessarily small. The case of free conservative oscillations and forced non-conservative oscillations are treated separately. A new small parameter is introduced, enabling one to transform original oscillators with not necessarily small nonlinearity into weakly nonlinear systems. Two classical perturbation techniques (the Lindstedt–Poincare method and the method of multiple scales) are used subsequently to find an amplitude–frequency equation and approximations to motion.

Is homotopy perturbation method the traditional Taylor series expansion

The search for a better and easy to use tool for the solution of nonlinear equations illuminating the nonlinear phenomena of our life is continuing. Various methods therefore were proposed to find approximate solutions. One of the most recent popular technique is the homotopy perturbation method, which is a combination of the classical perturbation technique and homotopy concept as used in topology. In the homotopy perturbation method, which requires neither a small parameter nor a linear term in a differential equation, a homotopy with an embedding parameter p ∈ [0, 1] is constructed. In [1] a basic idea of homotopy perturbation method for solving nonlinear differential equations was presented. A numerous nonlinear problems were recently treated by the method. We in the present paper investigate the homotopy perturbation technique from a mathematical point of view. The aim is to analyze the method and to show that under certain circumstances, by particular choice of auxiliary linear operator and initial approximation, the homotopy perturbation method simply

Singular Dynamics in Gene Network Models

Gene regulatory networks are commonly modeled by steep sigmoid interactions or by their limiting step functions. This leads to difficulties in dealing with singular dynamics, i.e., when some of the gene expressions are close to their thresholds. Two methods have been proposed to analyze this situation: the steep sigmoid framework based on singular perturbation techniques and the Filippov theory of differential inclusions. However, these lead to different concepts of solutions. Here we revisit these approaches and show their relationship. For the Filippov framework, we emphasize the use of two different solution concepts, namely, with a convex and a not necessarily convex right-hand side of the differential inclusion. We show that the second case is identical, under certain assumptions, to the solution concept used in the steep sigmoid framework. Even without these assumptions we obtain an existence result in the 2-dimensional case. We present 3-dimensional examples that do not fit the classical singular perturbation theory and for which even Filippov solutions may not be meaningful. We therefore suggest a generalization of the classical singular perturbation techniques that is based on the Artstein theory of dynamic (nonstationary) limits of the fast flow.

Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm

A general, nonlinear, multi-d.o.f. structure, excited by harmonic external force in 1:1 resonance with one of the modes of the system, is considered. The structure is attached to an essentially nonlinear oscillator, with small mass and damping (Nonlinear Energy Sink, NES). The scope of the NES is to passively control the amplitude of vibrations of the main structure. A mixed Multiple Scale/Harmonic Balance Method (MSHBM) is proposed to get the differential equations describing the slow- and fast-flow dynamics of the whole structure. The main advantage of the procedure is that no complexification-averaging is required, so that the analysis is reconducted in the framework of the classical perturbation techniques.

An efficient iterated method for mathematical biology model

The purpose of this study is to introduce an efficient iterated homotopy perturbation transform method (IHPTM) for solving a mathematical model of HIV infection of CD4+ T cells. The equations are Laplace transformed, and the nonlinear terms are represented by He’s polynomials. The solutions are obtained in the form of rapidly convergent series with elegantly computable terms. This approach, in contrast to classical perturbation techniques, is valid even for systems without any small/large parameters and therefore can be applied more widely than traditional perturbation techniques, especially when there do not exist any small/large quantities. A good agreement of the novel method solution with the existing solutions is presented graphically and in tabulated forms to study the efficiency and accuracy of IHPTM. This study demonstrates the general validity and the great potential of the IHPTM for solving strongly nonlinear problems.

Periodic solutions of multi-degree-of-freedom strongly nonlinear coupled van der Pol oscillators by homotopy analysis method

The mathematical models of multi-degree-of-freedom (MDOF) strongly nonlinear dynamical systems are described by coupled second-order differential equations. In general, the exact solutions of MDOF strongly nonlinear dynamical systems are frequently unavailable. Therefore, efforts have been mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic tool for solving strongly nonlinear dynamical systems, and it provides a simple way to ensure the convergence of solution series by means of a convergence-control parameter \({\hbar}\). Unlike the classical perturbation techniques, this method is independent of the presence of small parameters in the governing equations of motion. In this paper, the HAM is applied to formulate the analytical approximate periodic solutions of MDOF strongly nonlinear coupled van der Pol oscillators. Within this research framework, the frequency and the displacements of two-degree-of-freedom (2-DOF) strongly nonlinear systems can be explicitly obtained. For authentication, comparisons are carried out between the results obtained by the homotopy analysis and numerical integration methods. It is shown that the fourth-order or eighth-order solutions of the present method provide excellent accuracy. Illustrative examples of three-degree-of-freedom (3-DOF) strongly nonlinear coupled van der Pol oscillators are also presented and discussed. Finally, the optimal HAM approach is used to accelerate the convergence of the solutions.

Applicability of a Classical Perturbation Technique for Perturbation Parameters with Large Values

instead of being estimated by using Eq. (4). Equations (1) and (2) possess an exact solution, and hence a comparison of accuracy of Eqs. (4) and (5) can be carried out. Hu has shown numerically that Eq. (3), contrary to traditional application of Eq. (5), yields suitable results even for 0 ≤ e ≤ ∞. Hu claims that he has derived a new perturbation technique that is valid for large parameters.1 However, this should be treated rather as a particular case, and such a general statement for any other dynamical systems remains invalid. In order to explain the result obtained by Hu1 we will recall the exact formula in what follows:

A perturbation method for estimation of dynamic systems

A novel framework called the Perturbed Jth Moment Extended Kalman Filter (PJMEKF), based on a classical perturbation technique is proposed for estimating the states of a nonlinear dynamical system from sensor measurements. This method falls under a class of architectures under investigation primarily to study the interplay of major issues in nonlinear estimation such as nonlinearity, measurement sparsity, and initial condition uncertainty in an environment with low levels of process noise. Taylor series expansion of the departure motion dynamics about the best estimate is used to derive a series representation of the unforced motion. It is found that such series representation evolves as a set of differential equations that force each other in a cascade manner, adding up to give the unforced motion (in a so-called “triangular” structure). This formal perturbation solution for the departure motion dynamics is used in deriving the differential equations governing the time evolution of the high order statistical moments of the estimation error. These tensor differential equations are found to possess a similar high order triangular structure in addition to being symmetric (in N tensorial dimensions and we appropriately term the evolution equations as Tensor Lyapunov Equations of statistical moment perturbations). Elegance of the tensor differential equations thus derived is accompanied by the computational advantages due to symmetry in all tensorial dimensions. A vector matrix representation of tensors is proposed with which the representation and solution of the tensor differential equations can be carried out effectively. Approximations are introduced to incorporate low levels of process noise forcing function in the propagation phase of the moment equations. The statistics thus propagated are used in a filtering framework to estimate the state vector of a nonlinear system from noisy measurements, within the traditional Kalman update paradigm. The Kalman gain thus determined is utilized in updating all high order moments in preparation for the subsequent propagation phase leading to improved estimation accuracy. The filter developed is applied to an orbit estimation problem and comparisons are presented with classical extended Kalman filter.

Nonlinear Study of Magnetohydrodynamic Laminar Flow Between Permeable Disks Using CESS

A two-dimensional, steady, laminar flow of an electrically conducting incompressible, viscous fluid between two parallel permeable disks in the presence of a transverse magnetic field is studied using the computer-extended series solution (CESS). It is shown, using suitable analysis that the radius of convergence (ROC) is greatly influenced by the parameters of the problem, which renders the classical regular perturbation technique (RPT) ineffective for values of the perturbation parameter beyond the ROC. The ROC is shown to decrease with an increase in the value of the interaction parameter N. The skin friction coefficient and coefficient of pressure distribution are evaluated for different values of suction/injection Reynolds number R and magnetic interaction parameter N. The advantages of using the CESS method over the RPT and numerical techniques are discussed. © 2009 Begell House, Inc.

Approximate period calculation for some strongly nonlinear oscillation by He’s parameter-expanding methods

In this paper, He’s modified Lindstedt–Poincare method and bookkeeping parameter method, also known as He’s parameter-expanding method, is applied to various kinds of strongly nonlinear oscillators. We obtained sufficiently accurate solutions with one iteration which is valid for whole domain to the contrary of classical perturbation techniques.

Smoothed “Slack Cable” Models for Large Amplitude Oscillations of Suspension Bridges

This paper proposes a technique to study the dynamics of suspension bridges in those cases where the use of traditional models is not possible, i.e., when the deformed shape of the structure allows the simultaneous presence of taut and slack hangers. The typical unilateral behavior of these suspension elements, which are unable to transmit compressive forces, introduces a discontinuity in the stiffness that implies strong computational difficulties in the solution of the equations governing the motion; as a consequence, the response of the continuous or discrete models proposed may be evaluated only through step by step time integration. In this paper an alternative technique is proposed, which replaces the distribution of unilateral hangers with a smooth nonlinear regular system made equivalent to the effective one by means of an energy criterion. The proposed technique, referred to as “equivalent nonlinearization,” allows description of the motion through partial differential equations that are nonlinear but regular and, therefore, can be solved through classical perturbation techniques. The paper also describes two models formulated on the basis of the previously mentioned technique and some classes of solutions corresponding to particular characteristics of the forcing action. Dedicated to Prof. Giuliano Augusti for his imminent 70th birthday.

Nonlinear Flow Between Permeable Disks Using Computer‐Extended Series Method

The problem of two‐dimensional, steady, nonlinear flow of an incompressible, viscous fluid between two parallel permeable disks is studied using the computer‐extended series solution (CESS). The limitation of the classical regular perturbation technique (RPT) in solving this problem is highlighted and the CESS method in conjunction with Padé approximation is advocated to analyze the problem for much larger values of suction/injection Reynolds number R and to achieve higher accuracy. The skin‐friction coefficient and coefficient of pressure distribution are evaluated for different values of R. The advantages of using CESS method over the RPT and numerical technique are discussed.

Erratum: “Secular Evolution of Hierarchical Triple Star Systems” (ApJ, 535, 385 [2000

We derive octupole-level secular perturbation equations for hierarchical triple systems, using classical Hamiltonian perturbation techniques. By extending previous work done to leading (quadrupole) order to octupole level (i.e., including terms of order alpha^3, where alpha = a_1/a_2 < 1 is the ratio of semimajor axes) we obtain expressions that are applicable to a much wider range of parameters. For triple systems containing a close inner binary, we also discuss the possible interaction between the classical Newtonian perturbations and the general relativistic precession of the inner orbit. In some cases we show that this interaction can lead to resonances and a significant increase in the maximum amplitude of eccentricity perturbations. We establish the validity of our analytic expressions by providing detailed comparisons with the results of direct numerical integrations of the three-body problem obtained for a large number of representative cases. We also discuss applications of the theory in the context of several observed triple systems of current interest, including the millisecond pulsar PSR B1620-26 in M4, the giant planet in 16 Cygni, and the protostellar binary TMR-1. [short version]

Smoothed ?Slack Cable? Models for Large Amplitude Oscillations of Suspension Bridges

Abstract This paper proposes a technique to study the dynamics of suspension bridges in those cases where the use of traditional models is not possible, i.e., when the deformed shape of the structure allows the simultaneous presence of taut and slack hangers. The typical unilateral behavior of these suspension elements, which are unable to transmit compressive forces, introduces a discontinuity in the stiffness that implies strong computational difficulties in the solution of the equations governing the motion; as a consequence, the response of the continuous or discrete models proposed may be evaluated only through step by step time integration. In this paper an alternative technique is proposed, which replaces the distribution of unilateral hangers with a smooth nonlinear regular system made equivalent to the effective one by means of an energy criterion. The proposed technique, referred to as “equivalent nonlinearization,” allows description of the motion through partial differential equations that are nonlinear but regular and, therefore, can be solved through classical perturbation techniques. The paper also describes two models formulated on the basis of the previously mentioned technique and some classes of solutions corresponding to particular characteristics of the forcing action.

A classical perturbation technique that works even when the linear part of restoring force is zero

A classical perturbation technique that works even when the linear part of restoring force is zero

A classical perturbation technique which is valid for large parameters

There are various perturbation techniques for constructing analytical approximations to the oscillatory solutions of second order, non-linear differential equations [1,2]. But many of them apply to weakly non-linear cases only. To overcome the limitations, many novel techniques have been proposed in recent years. For example, Cheung et al. [3] proposed a modified Lindstedt– Poincare method, and Lim et al. [4] presented a modified Mickens procedure for certain non-linear oscillators. Recently, He [5] proposed a perturbation technique, which is valid for large parameters. But unfortunately, o1 and B in Ref. [5] are the functions of Z which is unknown. The main purpose of this paper is to point out that there already exists a classical perturbation technique, which is valid for large parameters. In this paper, the following well-known Duffing equation is considered: