Traditional Perturbation Technique Research Articles

Soliton Solutions of Kaup-Kupershimdt Equation Using HPTM

In this paper, we employ the idea of homotopy perturbation transform method(HPTM), which is a combination of Laplace transformation and homotopy perturbation method(HPM) for solving nonlinear ordinary and partial differential equations. The equations are Laplace transformed and nonlinear terms are represented by He’s polynomials. The solutions are obtained in the series form which converges at fast rate with easily computable terms. Comparison of standard perturbation method shows that HPTM is appropriate even for system without any small/large parameters and therefore it can be applied more extensively than traditional perturbation techniques. HPTM is used to obtain soliton solution of Kaup-kupershimdt(KK) equation, which is modified form of fifth order KdV equation. Numerical results are presented in tabular form to show the accuracy, efficiency and reliability of this method.

Possibility of the existence of the rogue wave and the super rogue wave in granular matter.

By using the traditional perturbation technique, a focusing nonlinear Schrödinger equation (NLSE) for the one-dimensional bead chain with the initial prestress is first obtained. The Peregrine soliton, called the rogue wave in the present paper, and the super rogue wave are investigated both numerically and analytically. It is noted that both the rogue wave and the super rogue wave do exist in the one-dimensional bead chain. The solutions from the NLSE can correctly describe the real rogue wave as well as the real super rogue wave in the limiting case of small amplitude. Both the rogue wave and the super rogue wave propagate in the granular bead chain as if they are solitary waves.

Simplified Liénard Equation by Homotopy Analysis Method

The homotopy analysis method (HAM) has been shown to approximate solutions to nonlinear problems of various types. The method is useful because of the flexibility in defining the auxiliary parameter $$\hbar $$ and that no assumption concerning relative size of parameters is required. We use the HAM to approximate frequency and solution of a conservative nonlinear oscillator which has many applications including lasers, epidemics, and microparasites. The oscillator is modeled by the simplified Lienard Equation $$\ddot{x}+\epsilon {\dot{x}}x+x=0$$ . We compare our results for the weakly-nonlinear case to those gained from a more traditional perturbation technique, the Poincare–Lindstedt method. We then show results from the HAM in the strongly-nonlinear regime and discuss some benefits and limitations of the method including conditions for convergence.

Rogue wave in coupled electric transmission line

Distributed electrical transmission lines that consist of a large number of identical sections have been theoretically studied in the present paper. The rogue wave is analyzed and predicted using the nonlinear Schrodinger equation (NLSE). The results indicate that, in the continuum limit, the voltage for the transmission line is described in some cases by the NLSE that is obtained using the traditional perturbation technique. The dependences of the characteristics of the rouge wave parameters on the coupled electric transmission line are shown in the paper. As is well known, rogue waves can be found for a large number of oceanic disasters, and such waves may be disastrous. However, the results of the present paper for coupled electric transmission lines may be useful.

Remarks on the treatments of nonsolvable potentials

A recently introduced scheme is extended to propose an algebraic nonperturbative approach for the analytical treatment of Schrodinger equations with nonsolvable potentials involving an exactly solvable potential form together with an additional potential term. As an illustration the procedure is successfully applied to the Cornell potential by means of very simple algebraic manipulations. However, instead of providing numerical eigenvalues for the only consideration of the small strength of the related linear potential as in the previous reports, the present model puts forward a clean route to interpret related experimental or precise numerical results involving a wide range of the linear potential strengths. We hope this new technique will shed some light on the questions concerning the limitations of the traditional perturbation techniques.

Improved Analysis for Squeezing of Newtonian Material between Two Circular Plates

This article presents a scheme for the analysis of an unsteady axisymmetric flow of incompressible Newtonian material in the form of liquid squeezed between two circular plates. The scheme combines traditional perturbation technique with homotopy using an adaptation of the Laplace Transform. The proposed method is tested against other schemes such as the Regular Perturbation Method (RPM), Homotopy Perturbation Method (HPM), Optimal Homotopy Asymptotic Method (OHAM), and the fourth-order Explicit Runge-Kutta Method (ERK4). Comparison of the solutions along with absolute residual errors confirms that the proposed scheme surpasses HPM, OHAM, RPM, and ERK4 in terms of accuracy. The article also investigates the effect of Reynolds number on the velocity profile and pressure variation graphically.

Advanced perturbation technique for digital backward propagation in WDM systems

An improved digital backward propagation (DBP) is proposed to compensate inter-nonlinear effects and dispersion jointly in WDM systems based on an advanced perturbation technique (APT). A non-iterative weighted concept is presented to replace the iterative in analytical recursion expression, which can dramatically simplify the complexity and improve accuracy compared to the traditional perturbation technique (TPT). Furthermore, an analytical recursion expression of the output after backward propagation is obtained initially. Numerical simulations are executed for various parameters of the transmission system. The results indicate that the advanced perturbation technique will relax the step size requirements and reduce the oversampling factor when launch power is higher than -2 dBm. We estimate this technique will reduce computational complexity by a factor of around seven with respect to the conventional DBP.

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.

Infiltration in unsaturated soils – An analytical approach

An approximate analytical solution has been established for the well known Richards’ equation for unsaturated flow of transports in soils. Despite the importance of Richards’ equation in geotechnical and geoenvironmental applications, most solutions to the problem are generally based on numerical methods often having limitations in application. It is therefore advantageous to explore analytical solutions to the problem. The proposed solution is based on homotopy analysis method (HAM). The method is superior over traditional perturbation techniques in a sense that it is not dependent on the assumption of a small parameter as the initial step. Several simple cases have been explored to describe the application of HAM to Richards’ equation. The results obtained by HAM are also compared with existing solutions to demonstrate the accuracy of the proposed method.

The application of homotopy analysis method to thin film flows of a third order fluid

The aim of the current article is to provide the analytic solutions to two thin film flows of a third order fluid. These are: (i) when the fluid moves on a belt and (ii) when the fluid moves down an inclined plane. Both problems have been solved using homotopy analysis method (HAM). These problems were already solved by Siddiqui et al. [Siddiqui AM, Mahmood R, Ghori QK. Thin film flow of a third grade fluid on a moving belt by He’s homotopy perturbation method. Int J Non-Linear Sci Numer Simul 2006;7:1–8, Siddiqui AM, Mahmood R, Ghori QK. Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos, Solitons & Fractals in press] using homotopy perturbation method (HPM) and traditional perturbation technique. With the help of two examples, it is shown that HPM is a special case of HAM. It has been noted that the solution up to second order is not enough in the case of flow on a moving belt. It is explicitly proved that the solutions of the flow down an inclined plane given in reference [Siddiqui AM, Mahmood R, Ghori QK. Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos, Solitons & Fractals in press] are divergent and hence have no meanings. The variation of velocity field corresponding to pertinent flow parameters is graphically presented and discussed.

Application of He's Homotopy Perturbation Method to Sumudu Transform

Abstract In this work, an application of He’s homotopy perturbation method is proposed to compute Sumudu transform. The method, in contrast of usual methods which need integration, requires simple differentiation. The results reveal that the method is very effective and simple. Keywords: Perturbation methods; Homotopy perturbation method; Sumudu transform 1. Introduction Various perturbation methods [1,2] have been widely applied by scientists and engineers to solve nonlinear problems. The traditional perturbation techniques are based on the existence of small parameter. These techniques are so powerful that sometimes small parameter is introduced artificially into a problem having no parameter and then finally set equal to unity to recover the solution of the original problem. He [3-6] proposed a new method called homotopy perturbation method (HPM) in 1998. The HPM, in fact, is a coupling of the traditional perturbation method and homotopy in topology. The HPM method, without demanding a small parameter in equations, deforms continuously to a simple problem which is easily solved. This method yields a very rapid convergence of the solution series in most cases, usually only a few iterations leading to very accurate solutions. This new method was further developed and improved by He and applied to non-linear oscillators with discontinuity [7], asymptotology [8], non-linear wave equations [9], bifurcation of nonlinear problems [10], limit cycle [11], delay-differential equations [[12], and boundary values problems [13]. He’s method is a universal one which can solve various types of non-linear problems. For example, it was applied to the non-Newtonian flow by Siddiqui et al. [14-15], to Volterra’s integro-differential equation by El-Shahed [16], to Helmholtz equation and fifth-order KdV equation by Rafei et al. [17], to nonlinear oscillator by Cai et al. [18], and to compute the Laplace transform by Abbasbandy [19]. A complete review on HPM’s applications is given in [20-21]. Sumudu transform was probably first time introduced by Watugala in his work [22]. Its simple formulation and direct applications to ordinary differential equations immediately sparked interest in this new tool. This new transform was further developed and applied to many problems by various workers. Asiru [23,24] applied to integro-differential equations, Watugala [25,26] extended the transform to two variables with emphasis on solution to partial differential equations and applications to engineering control problem, and its fundamentals properties were established by Belgacem et al. [27-29]. The Sumudu transform has very special and useful properties and can

Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane

The present paper analyses the thin film flow problem with a third grade fluid on an inclined plane. The governing non-linear equation is solved for the velocity field using the traditional perturbation technique as well as the recently introduced homotopy perturbation method and the results are compared. The expressions for volume flux and average film velocity are also given.

Linearized perturbation technique and its applications to strongly nonlinear oscillators

A new perturbation technique called linearized perturbation method is proposed. Contrary to the traditional perturbation techniques, the unperturbed equations is obtained by linearizing the original nonlinear equation, not by setting ϵ = 0. Therefore, the obtained results are valid not only for small parameter, but also for very large values of ϵ. The present theory is processed as simple as the straightforward expansion, while omits the secular terms completely.

Linearization and correction method for nonlinear problems

A new perturbation-like technique called linearization and correction method is proposed. Contrary to the traditional perturbation techniques, the present theory does not assume that the solution is expressed in the form of a power series of small parameter. To obtain an asymptotic solution of nonlinear system, the technique first searched for a solution for the linearized system, then a correction was added to the linearized solution. So the obtained results are uniformly valid for both weakly and strongly nonlinear equations.

A Modified Perturbation Technique Depending Upon an Artificial Parameter

In this paper, a modified perturbation method is proposed to search for analytical solutions of nonlinear oscillators without possible small parameters. An artificial perturbation equation is carefully constructed by embedding an artificial parameter, which is used as expanding parameter. It reveals that various traditional perturbation techniques can be powerfully applied in this theory. Some examples, such as the Duffing equation and the van der Pol equation, are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weak nonlinear systems, but also for strongly nonlinear systems. In applying the new method, some special techniques have been emphasized for different problems.

Analytical solution of a nonlinear oscillator by the linearized perturbation technique

In this paper, a nonlinear oscillator with no possible small parameters is considered. The traditional perturbation techniques can not be applied to such a problem. By linearizing the nonlinear system, and embedding an artificial parameter, we obtain an approximate solution which is valid on the whole solution domain, and the results are in remarkable agreement with the exact one even in the case where the amplitude of the oscillator tends to infinite.

Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parametrized perturbation technique

For the strongly nonlinear equations without small parameter, a transformation u = βv can be introduced for the nonlinear equation L( u) + N( u) = 0, where L and N are general linear and nonlinear differential operators respectively, β is sufficiently small. Therefore, a strongly non-linear system is transformed into a small parameter system with respect to the new introduced parameter β, thus various traditional perturbation techniques can be applied. By Lindstedt-Poincare method, a perturbation solution for Dufing equation with 5th order nonlinearity is obtained, which is valid not only for the small parameter ε in the equation, but also for very large values of ε.

Some new approaches to duffing equation with strongly and high order nonlinearity (I) linearized perturbation technique

A new perturbation technique called linearized perturbation method is proposed. Contrary to the traditional perturbation techniques, the unperturbed equation is obtained by linearizing the original nonlinear equation, not by setting ε = 0. The obtained results are valid not only for small parameters, but also for very large values of ε. The period for the Duffing equation with 5th order nonlinearity is of very high accuracy, the maximal relative error in the total region 0 ≤ ε < ∞ is less than 0.08%.