Classical Perturbation Method Research Articles

He's homotopy perturbation method for two‐dimensional heat conduction equation: Comparison with finite element method

AbstractHeat conduction appears in almost all natural and industrial processes. In the current study, a two‐dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy‐based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20292

Determination of competitive adsorption isotherms applying the nonlinear frequency response method: Part II. Experimental demonstration

This paper demonstrates an experimental application of the nonlinear frequency response (FR) method extension to determine adsorption isotherms of binary mixtures. This method, based on the analysis of the response of a chromatographic column subjected to the sinusoidal inlet concentration changes, is shown to be an alternative for isotherm determination. The critical issue related to the successful application of the method is to reach experimentally the low frequency asymptotic behaviour of the corresponding frequency response functions (FRFs). Although, there are different possibilities to perform periodical inlet concentration changes, in this paper only simultaneous changes for both components were considered. The adsorption of phenol and 2-phenylethanol on octadecyl silica was analyzed experimentally using a mixture of methanol and water as a solvent. Parameters of competitive isotherms were also estimated for comparison using the classical perturbation method. Despite certain differences between competitive isotherms estimated with the two methods that were found, the obtained results show the potential of the nonlinear FR method for measuring competitive isotherms.

Calculation of resonant short-crested waves in deep water

Solving the problem of resonant short-crested waves is very challenging because the appearance of small divisors causes the classical perturbation methods to fail. From a numerical point of view, the case of resonant gravity short-crested waves has been studied, but as far as we know, there are very few results in the case of resonant capillary-gravity short-crested waves. In fact, to the best of our knowledge, the most related study which has been made is the one of Craig and Nicholls [SIAM J. Math. Anal. 32, 323 (2000)] who gave existence theorems for the case where the surface tension is supposed not to be too small. There is a need for such an investigation, and the work considered herein therefore provides a calculation technique and presents new results on resonant short-crested gravity-capillary waves. We overcome the technical problems associated with small divisors by using a method derived from Whitham’s variational formulation of the classical problem of short-crested waves. Whitham’s method is...

An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem

The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically obtained solutions.

On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Delta Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Delta E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt-Miranker method indeed reflect the behavior of the exact solutions of the O Delta Es. For that reason, a version of the multiple scales perturbation method for O Delta Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Delta Es and to singularly perturbed, linear O Delta Es.

Parametric oscillations and stability of systems with large modulation coefficient

We study parametric oscillations of linear systems with one degree of freedom for large values of the modulation coefficient. We use the classical analytic Lyapunov-Poincare perturbation methods and an original numerically-analytic method of accelerated convergence to construct periodic solutions and the corresponding eigenvalues. We find the boundaries of stability and instability domains. We use specific models to illustrate the main properties of parametric oscillations of systems with singular character of the perturbation dependence on the modulation coefficient. We consider periodic boundary value problems for the modified Mathieu equation and the Kochin equation modeling crankshaft torsional vibrations and show that there are significant differences between weakly and essentially perturbed periodicmotions both for the lowest and arbitrary oscillation modes. We also describe the unusual properties of the boundaries in the domain of the system determining parameters.

Sodium Chloride in Supercritical Water as a Function of Density: Potentials of Mean Force and an Equation for the Dissociation Constant from 723 to 1073 K and from 0 to 0.9 g/cm3

The potential of mean force (PMF) of sodium chloride in water has been calculated by using the ab initio classical free-energy perturbation method at five state points: at 973 K with densities of 0.2796, 0.0935, and 0.0101 g/cm (3) and at 723 K with densities of 0.0897 and 0.0098 g/cm (3). The method is based on a QM-MM model in which Na-H 2O, Cl-H 2O, and Na-Cl interactions are calculated by ab initio methods. The water-water interactions are from the polarizable TIP4P-FQ model. The logarithm of the dissociation constant (log K c) has been calculated from the PMF. These predictions, together with experimental measurements, were used to derive an equation for log K c at densities from 0 to 0.9 g/cm (3) and temperatures from 723 to 1073 K, as well as from 600 to 1073 K for densities from 0.29 g/cm (3) to 0.9 g/cm (3). Extrapolation of the present equation below 723 K for densities less than 0.29 g/cm (3) does not fit the experimental results. This is attributed to long-range changes in the local dielectric constant due to the high compressibility. Comparisons with previous predictions and simulations are presented.

Application of He's parameter-expansion method to Jerk equation

The present work applies He's parameter-expansion method (PEM) to Jerk equation. The method, taking full advantage of simple solution procedure of the classical perturbation method, does not depend upon small parameter assumption. Comparison of the obtained results with the exact solution shows that the method is very effective, convenient and quite accurate. The method can be easily extended to other nonlinear equations.

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.

Performance of perturbation methods on orbit prediction

In this paper we analyze briefly the use of two classical perturbation methods vs. the formulation using cartesian coordinates in the numerical integration of satellite orbits. In particular, we compare with the variation-of-parameters (VOP) and Encke’s methods. The numerical tests performed with a realistic Earth potential model and using a well-recognized numerical ODE integrator ( dop853) permit us to establish the power of these alternative formulations on orbital problems, in spite of the affirmations of other researchers.

Statistical moments predictions for a moored floating body oscillating in random waves

Two statistical techniques are developed to predict the statistical moments of the horizontal motion of a floating moored dock, known as catenary anchor leg mooring (CALM), loaded by hydrodynamic random forces. The dock is represented by a lumped mass, the mooring cables by equivalent nonlinear springs and the hydrodynamic forces are modelled by a modified Morison equation. The model of the floating dock leads to a nonlinear ordinary differential equation. Although the problem could be approached by a direct numerical integration, e.g. by Monte-Carlo simulations, because of the stochastic nature of the excitation, this would imply a large number of runs to produce results of some statistical significance. In the present paper an alternative solution is based on the development of two more efficient techniques to predict the relevant statistical moments of the dock response. The first method, called CPSP (conventional perturbation–statistical perturbation), is based on the application of two subsequent perturbation techniques, the first relying on a classical perturbation method, the second on a statistical perturbation approach. The second method, called SLSP (statistical linearization–statistical perturbation), combines indeed a statistical linearization approach together with a statistical perturbation approach. The procedures allow the linearization of the cables restoring forces as well as of the hydrodynamic load and they can be easily generalized to be applied to different dock configurations or to systems of different physical nature. The results, compared with those obtained by a Monte-Carlo simulations, show, in terms of statistical moments of the dock response, a satisfactory agreement.

Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction

We develop a singular perturbation technique to study the existence of periodic traveling wave solutions with large wave speed for a class of reaction-diffusion equations with time delay and non-local response. Unlike the classical singular perturbation method, our approach is based on a transformation of the differential equations to integral equations in a Banach space that reduces the singular perturbation problem to a regular perturbation problem. The periodic traveling wave solutions then are obtained by the use of Liapunov-Schmidt method and a generalized implicit function theorem. The general result obtained has been applied to a non-local reaction-diffusion equation derived from an age-structured population model with a logistic type of birth function.

Interval finite element method for dynamic response of closed‐loop system with uncertain parameters

AbstractIn practical engineering, it is difficult to obtain all possible solutions of dynamic responses with sharp bounds even if an optimum scheme is adopted where there are many uncertain parameters. In this paper, using the interval finite element (IFE) method and precise time integration (PTI) method, we discuss the dynamic response of vibration control problem of structures with interval parameters. With matrix perturbation theory and interval arithmetic, the algorithm for estimating upper and lower bounds of dynamic response of the closed‐loop system is developed directly from the interval parameters. Two numerical examples are given to illustrate the application of the present method. The example 1 is used to show the applicability of the present method. The example 2 is used to show the validity of the present method by comparing the results with those obtained by the classical random perturbation method. Copyright © 2006 John Wiley & Sons, Ltd.

Higher-Order Approximate Solutions to a Strongly Nonlinear Duffing Oscillator

This paper deals with the Duffing oscillator, which is generally a strong nonlinear oscillation of a conservative and single-degree-of-freedom system with odd nonlinearity. By coupling linearization of governing equation with the method of harmonic balancing method, the paper presents new, accurate, higher-order approximate solutions for the general strong nonlinear systems. Unlike the classical harmonic balance method, analytical approximate solutions are possible because linearization of governing differential equation is conducted prior to harmonic balancing. Simple linear algebraic equations are obtained instead of nonlinear algebraic equations without analytical solution. Using the approach, general approximate analytical expressions for the exact period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method.

Interval eigenvalues of closed-loop systems of uncertain structures

Using the interval analysis, the vibration control problem of structures with interval parameters is discussed, which is approximated by a deterministic one. A method to solve the interval eigenvalues of the closed-loop system is presented. The expressions of the interval stiffness and interval mass matrix are developed directly from the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of the interval complex eigenvalue is developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. Two numerical examples are presented to illustrate the applications of the present method. The example 1 is used to show the applicability of the proposed method for the finite element model. The example 2 is used to show the validity of the present method by comparing the results with those obtained by the classical random perturbation method.

Accurate Higher-Order Analytical Approximate Solutions to Large-Amplitude Oscillating Systems with a General Non-Rational Restoring Force

A new approximate analytical approach for accurate higher-order nonlinear solutions of oscillations with large amplitude is presented in this paper. The oscillatory system is subjected to a non-rational restoring force. This approach is built upon linearization of the governing dynamic equation associated with the method of harmonic balance. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. This approach also explores large parameter regions beyond the classical perturbation methods which in principle are confined to problems with small parameters. It has significant contribution as there exist many nonlinear problems without small parameters. Through some examples in this paper, we establish the general approximate analytical formulas for the exact period and periodic solution which are valid for small as well as large amplitudes of oscillation.

Modified Two-Scale Model for Electromagnetic Scattering fromthe Non-Gaussian Oceanic Surface

Based on the skewness of sea waves, a modified two-scale model isdeveloped for the non-Gaussian sea surface scattering. In this newmodel, a complementary term is added to the first-order scatteringcoefficient of the classical small perturbation method (SPM), theadditional part is proportional to the surface bispectrum and it is thecritical part in explaining the scattering difference between upwindand downwind observations. Meanwhile, the effects of the shadowingfunction of the anisotropic surface, the curvature of the surface arealso taken into account. The numerical results show the theoreticalestimates obtained are consistent with the experimental result, and theinfluence of the wind speed, the trend and the incident frequency onthe backscattering coefficients from the non-Gaussian oceanicsurface is discussed in detail.

On analyticity of travelling water waves

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables. The travelling waves we find can be interpreted as resulting from the (nonlinear) interaction of two two-dimensional wavetrains, giving rise to a periodic travelling pattern. Our analyticity theorem extends the most sophisticated results known to date in the absence of resonance; ‘short crested waves’, which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case. Our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves. Indeed, we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations. These observations motivate the derivation and use of an alternative, stable, formulation which, in addition to providing our method of proof, suggests new stabilized implementations of boundary perturbation algorithms.

Stochastic modeling of complex nonstationary groundwater systems

Despite the intensive research over the past two decades in the field of stochastic subsurface hydrology, a substantial gap remains between theory and application. The most popular stochastic theories are still based on closed-form solutions that apply, strictly speaking, only for statistically uniform flows. In this paper, we present an efficient, nonstationary spectral approach for modeling complex stochastic flows in moderately heterogeneous media. Specifically, we reformulate the governing stochastic equations and introduce a new transfer function to characterize the propagation of system uncertainty. The new transfer function plays a similar role as the commonly used Green’s functions in classical stochastic perturbation methods but is more amenable to numerical solution. The compact transfer function can be used to construct efficiently various spatial statistics of interest, such as covariances, cross-covariances, variances, and mean closure fluxes. We demonstrate the advantages of the proposed approach by applying it to a number of nonstationary problems, including a large, complex problem that is difficult to solve by traditional methods. In particular, we focus in this paper on demonstrating the new approach’s ability to compute efficiently covariances and cross-covariances critical for measurement conditioning, monitoring network analyses, and stochastic transport modeling in the presence of complex mean flow nonstationarities (caused, e.g., by complex trends in aquifer properties, boundary conditions, and sources and sinks). This paper is an extension of our recent work that illustrated the basic approach for modeling nonlocal and nonstationary scale effects and uncertainty propagation in relatively simple situations.

A computationally practical method for stochastic groundwater modeling

Stochastic theories of subsurface flow and transport have changed the way we think about heterogeneity but have not had much impact on practical groundwater modeling. Most numerical models still provide no information on prediction uncertainty. This gap between theory and practice is due largely to the excessive computational demands of available numerical methods for solving stochastic problems. The two primary alternatives, Monte Carlo and classical perturbation methods, both require solving large numbers of equations on fine grid spacings that are smaller than the log conductivity correlation length, no matter how smooth the mean flow and variance dynamics of interest are. This results in computational times that are many orders-of-magnitude greater than those required for conventional deterministic simulations. In addition, classical perturbation methods need much more real memory than comparable deterministic simulations. In this paper, we present an innovative approach for stochastic groundwater modeling––one that finally allows taking advantage of the scale disparity between the mean distributions and small-scale process. The new approach, which is called the nonstationary spectral method, implicitly separates the problem into rapidly varying random (stationary) part and a slowly varying deterministic (nonstationary) part and only the latter needs to be solved numerically. A general analysis and a specific numerical example both demonstrate that the nonstationary spectral method is much more efficient than the available alternatives. The methodological innovation dramatically increases the size and range of the problems that can be solved using stochastic methods and represents a major step forward in our effort to make stochastic uncertainty analysis a routine part of groundwater modeling.