SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS

Authors

Ji‐Huan He

Type: Article

Publication Date: 2006-04-19

Citations: 2188

DOI: https://doi.org/10.1142/s0217979206033796

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Abstract

This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this field in China so the references, therefore, are not exhaustive.

Topics

Locations

International Journal of Modern Physics B
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ON THE RECENT DEVELOPMENT OF SOME ASYMPTOTIC METHODS FOR NONLINEAR VIBRATION EQUATIONS

2012-07-10

Mahmoud Bayat

NORMAL 0 FALSE FALSE FALSE EN-US X-NONE AR-SA /* STYLE DEFINITIONS */ TABLE.MSONORMALTABLE {MSO-STYLE-NAME:TABLE NORMAL; MSO-TSTYLE-ROWBAND-SIZE:0; MSO-TSTYLE-COLBAND-SIZE:0; MSO-STYLE-NOSHOW:YES; MSO-STYLE-PRIORITY:99; MSO-STYLE-QFORMAT:YES; MSO-STYLE-PARENT:; MSO-PADDING-ALT:0IN 5.4PT 0IN 5.4PT; MSO-PARA-MARGIN:0IN; MSO-PARA-MARGIN-BOTTOM:.0001PT; MSO-PAGINATION:WIDOW-ORPHAN; FONT-SIZE:11.0PT; FONT-FAMILY:CALIBRI,SANS-SERIF; … NORMAL 0 FALSE FALSE FALSE EN-US X-NONE AR-SA /* STYLE DEFINITIONS */ TABLE.MSONORMALTABLE {MSO-STYLE-NAME:TABLE NORMAL; MSO-TSTYLE-ROWBAND-SIZE:0; MSO-TSTYLE-COLBAND-SIZE:0; MSO-STYLE-NOSHOW:YES; MSO-STYLE-PRIORITY:99; MSO-STYLE-QFORMAT:YES; MSO-STYLE-PARENT:; MSO-PADDING-ALT:0IN 5.4PT 0IN 5.4PT; MSO-PARA-MARGIN:0IN; MSO-PARA-MARGIN-BOTTOM:.0001PT; MSO-PAGINATION:WIDOW-ORPHAN; FONT-SIZE:11.0PT; FONT-FAMILY:CALIBRI,SANS-SERIF; MSO-ASCII-FONT-FAMILY:CALIBRI; MSO-ASCII-THEME-FONT:MINOR-LATIN; MSO-FAREAST-FONT-FAMILY:TIMES NEW ROMAN; MSO-FAREAST-THEME-FONT:MINOR-FAREAST; MSO-HANSI-FONT-FAMILY:CALIBRI; MSO-HANSI-THEME-FONT:MINOR-LATIN; MSO-BIDI-FONT-FAMILY:ARIAL; MSO-BIDI-THEME-FONT:MINOR-BIDI;} THIS PAPER FEATURES A SURVEY OF SOME RECENT DEVELOPMENTS IN ASYMPTOTIC TECHNIQUES, WHICH ARE VALID NOT ONLY FOR WEAKLY NONLINEAR EQUATIONS, BUT ALSO FOR STRONGLY ONES. FURTHER, THE ACHIEVED APPROXIMATE ANALYTICAL SOLUTIONS ARE VALID FOR THE WHOLE SOLUTION DOMAIN. THE LIMITATIONS OF TRADITIONAL PERTURBATION METHODS ARE ILLUSTRATED, VARIOUS MODIFIED PERTURBATION TECHNIQUES ARE PROPOSED, AND SOME MATHEMATICAL TOOLS SUCH AS VARIATIONAL THEORY, HOMOTOPY TECHNOLOGY, AND ITERATION TECHNIQUE ARE INTRODUCED TO OVER-COME THE SHORTCOMINGS. IN THIS PAPER THE FOLLOWING CATEGORIES OF ASYMPTOTIC METHODS ARE EMPHASIZED: (1) VARIATIONAL ITERATION METHOD (2) HOMOTOPY PERTURBATION METHOD (3) HE'S VARIATIONAL ITERATION METHOD FOR NONLINEAR OSCILLATORS (4) ENERGY BALANCE METHOD (5) PARAMETER-EXPANSION METHOD (6) PARAMETERIZED PERTURBATION METHOD (7) AMPLITUDE-FREQUENCY FORMULATION FOR NONLINEAR OSCILLATORS BASED ON ANCIENT CHINESE MATHEMATICS (8) ITERATION PERTURBATION METHOD. KEYWORDS : NONLINEAR VIBRATION ; VARIATIONAL ITERATION METHOD(VIM); HOMOTOPY PERTURBATION METHOD (HPM); ENERGY BALANCE METHOD (EBM); ITERATION PERTURBATION METHOD; PARAMETER-EXPANSION METHOD(PEM); ANCIENT CHINESE MATHEMATICS METHOD; PARAMETERIZED PERTURBATION METHOD(PPM) .

Asymptotic Methods in Non Linear Dynamics

2016-01-01

Ritabrata Dutta

This paper features and elaborates recent developments and modifications in asymptotic techniques in solving differential equation in non linear dynamics. These methods are proved to be powerful to solve weakly … This paper features and elaborates recent developments and modifications in asymptotic techniques in solving differential equation in non linear dynamics. These methods are proved to be powerful to solve weakly as well as strongly non linear cases. Obtained approximate analytical solutions are valid for the whole solution domain. In this paper, limitations of traditional perturbation methods are illustrated with various modified techniques. Mathematical tools such as variational approach, homotopy and iteration technique are discussed to solve various problems efficiently. Asymptotic methods such as Variational Method, modified Lindstedt-Poincare method, Linearized perturbation method, Parameter Expansion method, Homotopy Perturbation method and Perturbation-Iteration methods(singular and non singular cases) have been discussed in various situations. Main emphasis is given on Singular perturbation method and WKB method in various numerical problems.

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Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review

2012-04-01

Mahmoud Bayat , Iman Pakar , Ganji Domairry

This review features a survey of some recent developments in asymptotic techniques and new developments, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, … This review features a survey of some recent developments in asymptotic techniques and new developments, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the achieved approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to over-come the shortcomings.In this review we have applied different powerful analytical methods to solve high nonlinear problems in engineering vibrations. Some patterns are given to illustrate the effectiveness and convenience of the methodologies.

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SOME HIGH ORDER ITERATIVE METHODS FOR NONLINEAR EQUATIONS BASED ON THE MODIFIED HOMOTOPY PERTURBATION METHODS

2010-09-01

Bilian Chen , Yajun Xie , Changfeng Ma

In this paper, we present some efficient iterative methods for solving nonlinear equation (systems of nonlinear equations, respectively) by using modified homotopy perturbation methods. We also discuss the convergence criteria … In this paper, we present some efficient iterative methods for solving nonlinear equation (systems of nonlinear equations, respectively) by using modified homotopy perturbation methods. We also discuss the convergence criteria of the present methods. Some numerical examples are given to illustrate the performance and efficiency of the proposed methods.

Homotopy analysis method - A new analytic approach for highly nonlinear problems

2015-01-01

Shijun Liao

In this talk, we briefly describe the basic ideas and applications of the homotopy analysis method (HAM), an analytic technique for highly nonlinear problems. Compared to other analytic approximation methods, … In this talk, we briefly describe the basic ideas and applications of the homotopy analysis method (HAM), an analytic technique for highly nonlinear problems. Compared to other analytic approximation methods, the HAM has some advantages. First, unlike perturbation techniques, the HAM has nothing to do with any small/large physical parameters so that it works for more problems, especially for those without small/large physical parameters. Besides, unlike all other methods, the HAM provides us a simple way to guarantee the convergence of solution series. In addition, the HAM provides us great freedom to choose equation-type and solution expression of high-order equations so that it is easy to obtain approximations at rather high order. Due to these advantages, the HAM have been successfully applied to solve lots of nonlinear problems in science, engineering, finance and so on. By means of the HAM, some classical problems have been solved with much better results. Especially, some new concepts have been proposed and some new solutions have been found by means of the HAM, mainly because a truly new method always brings us something new/different!

Homotopy Analysis Method

2022-01-01

Xiaoyan Yang , Zhiliang Lin

In this entry, we describe the basic ideas of the homotopy analysis method (HAM), an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests … In this entry, we describe the basic ideas of the homotopy analysis method (HAM), an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers. HAM fundamentally overcomes the excessive dependence on small parameters in the framework of perturbation theory, and its validity has nothing to do with small/large physical parameter of the studied nonlinear problem. So, it has a wide range of applications. In addition, HAM provides us great freedom to choose the base function, so that a better base function can be selected to approximate the solution of the problem more effectively. Furthermore, unlike all other analytical approximation methods, we can guarantee the convergence of solution series in a simple way in the framework of HAM. In general, HAM provides a new way of thinking that how to get the analytical approximate solution for nonlinear problems and opens up a new way for solving nonlinear problems (especially strong nonlinear problems without small parameters).

The asymptotic solution of a class of weakly nonlinear perturbation equations

2013-01-01

Cheng Ouyang

A class of general weakly nonlinear perturbation equations is studied.The nonhomogeneous term of the equation is expressed as Fourier series.The uniformly valid asymptotic solutions are obtained using the direct expansion … A class of general weakly nonlinear perturbation equations is studied.The nonhomogeneous term of the equation is expressed as Fourier series.The uniformly valid asymptotic solutions are obtained using the direct expansion method,the re-normalization method,the multiple scale method,respectively for three different cases.The expression of the solution of a special weakly nonlinear perturbation equation in the related reference is given concretely and exactly by the result with the Fourier coefficients.For the first two cases,the results are consistent with the those in the reference,and the nature of the solution is revealed with a higher point of view.For the third case,the asymptotic solution with first-order is obtained,which was not discussed in the references.

BEYOND PERTURBATION:THE BASIC CONCEPTS OF THE HOMOTOPY ANALYSIS METHOD AND ITS APPLICATIONS

2008-01-01

Shijun Liao

A new and rather general analytic method for strongly nonlinear problems,namely the homotopy analysis method (HAM),is reviewed.Different from perturbation techniques,the homotopy analysis method is totally independent of small physical parameters,and … A new and rather general analytic method for strongly nonlinear problems,namely the homotopy analysis method (HAM),is reviewed.Different from perturbation techniques,the homotopy analysis method is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.Besides, different from all other analytic techniques,it provides us a simple way to ensure the convergence of solution series,so that one can always get accurate enough analytic approximations.Furthermore,different from all other analytic methods,it provides us a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The homotopy analysis method provides us a completely new way and a different approach to solve nonlinear problems,especially those without small physical parameters.In this review paper,the basic concepts of the homotopy analysis method and its applications in nonlinear mechanics,physics,chemistry,biology,finance,engineering,computational mathematics and so on are discussed,together with its difference and relationship to perturbation techniques,Lyapunov artificial small parameter method,δ-expansion method,Adomian decomposition method,and the so-called homotopy perturbation method.

New Higher Order Iterative Methods for Solving Nonlinear Equations

2017-01-01

Shuliang Huang , Arif Rafiq , M. R. Shahzad +1 more

In this paper, using the system of coupled equations involving an auxiliary function, we introduce some new efficient higher order iterative methods based on modified homotopy perturbation method.We study the … In this paper, using the system of coupled equations involving an auxiliary function, we introduce some new efficient higher order iterative methods based on modified homotopy perturbation method.We study the convergence analysis and also present various numerical examples to demonstrate the validity and efficiency of our methods.

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Homotopy perturbation method for nonlinear dynamic equations based on precise integration technology

2005-01-01

Shuli Mei

A new asymptotic numerical method for nonlinear dynamic equations is proposed in this paper by combining the precise integration method(PIM) with the homotopy perturbation method(HPM).For solving nonlinear dynamic equations in … A new asymptotic numerical method for nonlinear dynamic equations is proposed in this paper by combining the precise integration method(PIM) with the homotopy perturbation method(HPM).For solving nonlinear dynamic equations in PIM,the nonlinear term should be expanded in Taylor series to the time parameter.The computational accuracy is sensitive to the time step if the series is truncated at the first order or second order,and if the series is truncated at the higher order,the computational format will be more complex.Correspondingly,the format derived from the homotopy perturbation method is simpler,but its applicability is limited to one or two dimensional nonlinear differential equations and the computational accuracy is lower.The new asymptotic numerical method obtained by combining above two methods possesses all their merits,that is,not only extend the applicability of the homotopy perturbation method to high dimensional nonlinear dynamic equations,but also simplify the computational format of PIM in solving nonlinear problems.The numerical example shows that the numerical accuracy and the computational efficiency of the new method is higher.

Homotopy perturbation method and variational iteration method for solving the nonlinear equations with variable coefficients in applied sciences

2022-01-01

R. Swaminathan , B. Manimegalai , K. Venugopal +1 more

In this article, the nonlinear differential equations with variable coefficients are adopted to solve using a homotopy perturbation method and a variational iteration method. The results obtained by these methods … In this article, the nonlinear differential equations with variable coefficients are adopted to solve using a homotopy perturbation method and a variational iteration method. The results obtained by these methods are very useful and convenient, as shown by comparing their results. We show the method's efficiency by examining several scientific models with various orders and variable coefficients. We compare the results to exact or numerical solutions to determine the accuracy of the solutions, indicating that both methods can be used to solve several nonlinear differential equations with high accuracy.

Iteration Perturbation Method for Strongly Nonlinear Oscillations

2001-07-01

Ji‐Huan He

In this paper, the author proposes a new perturbation technique coupling with iteration method, yielding a powerful mathematical tool for an analytical solution of nonlinear equations. The obtained results are … In this paper, the author proposes a new perturbation technique coupling with iteration method, yielding a powerful mathematical tool for an analytical solution of nonlinear equations. The obtained results are valid not only for weakly nonlinear problems but also for strongly nonlinear ones. Furthermore, the approximate solutions are valid for the whole solution domain, and even the first-step iteration leads to high accuracy Some examples are given to illustrate its effectiveness.

LINEARIZATION AND CORRECTION METHOD FOR NONLINEAR PROBLEMS

2002-01-01

He , Ji-huan

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 … 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.

New homotopy perturbation method for solving nonlinear differential equations and fisher type equation

2017-09-01

Nahid Fatima

In this paper, we apply the new Homotopy Perturbation Method (NHPM) to get accurate results for solving systems of nonlinear equations, we indicate that our method (NHPM) is equivalent to … In this paper, we apply the new Homotopy Perturbation Method (NHPM) to get accurate results for solving systems of nonlinear equations, we indicate that our method (NHPM) is equivalent to the Variational Iteration Method (VIM) with a specific convex. Four examples are given to depict our proposed methods. The method is easy to carry out and gives very accurate solutions for solving linear and nonlinear equations.

Comparative Study between Optimal Homotopy Asymptotic Method and Perturbation-Iteration Technique for Different Types of Nonlinear Equations

2016-06-04

Necdet Bildik , Sinan Deni̇z

An improved optimal homotopy analysis algorithm for nonlinear differential equations

2020-03-27

Zaid Odibat

High Order Iterative methods without derivatives for solving non- linear equations

2019-01-01

ZabihUllah Zahir

The new second-order and third-order iterative methods without derivatives are presented for solving nonlinear equa- tions; the iterative formulae based on the homotopy perturbation method are deduced and their convergences … The new second-order and third-order iterative methods without derivatives are presented for solving nonlinear equa- tions; the iterative formulae based on the homotopy perturbation method are deduced and their convergences are pro- vided. Finally, some numerical experiments show the eﬃciency of the theoretical results for the above methods.

The Application of Homotopy Perturbation Method: An Overview

2022-02-24

Xintong Yang

As an effective method for solving linear and nonlinear equations, the homotopy perturbation method is usually applied to solving relevant problems. We analyze 74 studies on the application of the … As an effective method for solving linear and nonlinear equations, the homotopy perturbation method is usually applied to solving relevant problems. We analyze 74 studies on the application of the homotopy perturbation method and present a comprehensive review of them with the conclusion obtained: (1) Homotopy perturbation method is generally applied to solving the problems of ordinary differential equations; (2) Homotopy perturbation method is usually combined with the technology of transform when it is used to solve more complicated equations; (3) By comparing homotopy perturbation method with other similar methods, many researchers sought that homotopy perturbation method is rapidly convergent, highly accurate, computational simple; (4) Some studies point out that when homotopy perturbation method is applied, some parameters including the number of terms, time span, time step must be prescribed carefully. Finally, two suggestions on the further study of the application of the HPM are provided.

Homotopy perturbation method coupled with the enhanced perturbation method

2018-09-23

Xiaoxia Li , Chun‐Hui He

The enhanced perturbation method is used to improve a governing equation to a higher order, followed by the classic perturbation method. This paper adopts the basic idea of the method … The enhanced perturbation method is used to improve a governing equation to a higher order, followed by the classic perturbation method. This paper adopts the basic idea of the method to construct a homotopy equation with a higher order. The results show that this modification of the homotopy perturbation method leads to a very high accuracy of the obtained solution. Its solution process is elucidated by using Duffing oscillator as an example, and the obtained frequency is valid for all amplitudes from an extremely small value to infinity with a maximal error of 3.34%. The main merit of this higher order homotopy perturbation method is that the obtained frequency is valid for the whole solution domain.

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Homotopy Asymptotic Method and Its Application

2017-06-14

Baojian Hong

As we all know, perturbation theory is closely related to methods used in the numerical analysis fields. In this chapter, we focus on introducing two homotopy asymptotic methods and their … As we all know, perturbation theory is closely related to methods used in the numerical analysis fields. In this chapter, we focus on introducing two homotopy asymptotic methods and their applications. In order to search for analytical approximate solutions of two types of typical nonlinear partial differential equations by using the famous homotopy analysis method (HAM) and the homotopy perturbation method (HPM), we consider these two systems including the generalized perturbed Kortewerg-de Vries-Burgers equation and the generalized perturbed nonlinear Schrödinger equation (GPNLS). The approximate solution with arbitrary degree of accuracy for these two equations is researched, and the efficiency, accuracy and convergence of the approximate solution are also discussed.

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Laplace substitution method for solving Poisson equation

2024-01-01

Hasan Ali Khawwan , Ali Al-Fayadh

In this article, A new method to achieve the exact solution of a special type of linear partial differential equations (LPDEs) will be introduce which involving mixed partial derivatives. This … In this article, A new method to achieve the exact solution of a special type of linear partial differential equations (LPDEs) will be introduce which involving mixed partial derivatives. This method is called the Laplace substitution method (LSM), This method is used to solve Poisson equation and the results are compared with other methods. Three examples are presented to explain the simplicity and reliability of this method.

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Chebyshev wavelets method for solving Bratu’s problem

2013-06-02

Changqing Yang , Jianhua Hou

A numerical method for one-dimensional Bratu's problem is presented in this work. The method is based on Chebyshev wavelets approximates. The operational matrix of derivative of Chebyshev wavelets is introduced. … A numerical method for one-dimensional Bratu's problem is presented in this work. The method is based on Chebyshev wavelets approximates. The operational matrix of derivative of Chebyshev wavelets is introduced. The matrix together with the collocation method are then utilized to transform the differential equation into a system of algebraic equations. Numerical examples are presented to verify the efficiency and accuracy of the proposed algorithm. The results reveal that the method is accurate and easy to implement.

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Heat transfer analysis for squeezing flow of a Casson fluid between parallel plates

2015-10-26

Umar Khan , Sheikh Irfanullah Khan , Naveed Ahmed +2 more

Heat transfer analysis for the squeezing flow of a Casson fluid between parallel circular plates has been presented. Viable mathematical model has been constructed by using conservation laws coupled with … Heat transfer analysis for the squeezing flow of a Casson fluid between parallel circular plates has been presented. Viable mathematical model has been constructed by using conservation laws coupled with suitable similarity transforms. This model ends up on a set of two highly nonlinear ordinary differential equations. Resulting equations have been solved by using a well-known analytical technique homotopy perturbation method (HPM). A numerical solution using forth order Runge–Kutta method has also been sought to support our analytical solution and the comparison shows an excellent agreement. Flow behavior under altering involved physical parameters is also discussed and explained in detail with graphical aid. For the presented problem, values of parameters are restricted. Analysis is carried out using the following ranges of parameters; squeeze number (-4⩽S⩽4), Casson fluid parameter (0.1⩽β⩽∞), Prandtl number (0.1⩽Pr⩽0.7), Eckert number (0.1⩽Ec⩽0.7) and 0.1⩽δ⩽0.4. Increase in velocity for squeeze number and Casson fluid parameter is observed. Temperature profile is found to be decreasing function of squeeze number and Casson fluid parameter and increasing function of Pr, Ec and δ.

On the oscillations in a nonextensive complex plasma by improved differential transformation method: An application to a damped Duffing equation

2023-03-14

Noufe H. Aljahdaly , Maram A. Alharbi , Abrar Alharbi +2 more

In this study, the nonlinear damping oscillations in a complex non-Maxwellian plasma are investigated. For this purpose, the set of fluid equations of the present plasma model is reduced to … In this study, the nonlinear damping oscillations in a complex non-Maxwellian plasma are investigated. For this purpose, the set of fluid equations of the present plasma model is reduced to the Burger-modified Korteweg De Vries equation (BmKdV) equation using a reductive perturbation technique. Using the traveling wave transformation, the BmKdV equation can be reduced to a damped Duffing equation. The numerical solutions to the damped Duffing equation are obtained using multistage differential transformation method (MsDTM). Also, we compared the obtained results to the semi-analytical approximations using the Padé differential transformation (PDTM) method and numerical solution, by the 4th-order Rung Kutta (RK4) method and analytical solution by He’s frequency method. The impact of relevant plasma parameters, namely, negative dust concentrations and ion kinematic viscosity on the profile of dust ion-acoustic oscillations are examined. The suggested mathematical approaches can help many authors for explaining the mystery of their laboratory results. Moreover, the suggested numerical method can be applied for solving higher order nonlinearity oscillations for a long domain.

Effect of Endometriosis to fallopian tube of the peristaltic-ciliary flow of third grade fluid in a finite narrow tube

2020-05-01

Pro..Dr.Ahmad Mawlood , Malath Sagban Abied

Abstract The present prospective theoretical investigation deals with analysis of the peristaltic-ciliary transport of a developing embryo within the fallopian tubal fluid in the human fallopian tube under the effect6 … Abstract The present prospective theoretical investigation deals with analysis of the peristaltic-ciliary transport of a developing embryo within the fallopian tubal fluid in the human fallopian tube under the effect6 of Endometriosis. This disease make the peristalsis ciliary flow become to peristalsis flow. A mathematical model induced flow of viscoelastic fluid characterized by the third grade fluid in a finite two dimensional narrow tube. That research is study the effect of couple stress to peristaltic –ciliary flow to Non-Newtonian fluids. Non-linear partial differential equations are solved by perturbation method. Flow variables like axial and radial velocities, appropriate residue time over tube length, pressure difference over have been derived under the assumption of long wavelength and low Reynolds number approximation and the expression for pressure rise is obtained by using wavelength and stream function are analysed for embedded parameter. This study is done through by the “MATHEMATICA”

Fundamental solutions for the long–short-wave interaction system

2020-12-29

Mustafa İnç , S. Z. Hassan , Mahmoud A. E. Abdelrahman +2 more

Abstract In this article, the system for the long–short-wave interaction (LS) system is considered. In order to construct some new traveling wave solutions, He’s semi-inverse method is implemented. These solutions … Abstract In this article, the system for the long–short-wave interaction (LS) system is considered. In order to construct some new traveling wave solutions, He’s semi-inverse method is implemented. These solutions may be applicable for some physical environments, such as physics and fluid mechanics. These new solutions show that the proposed method is easy to apply and the proposed technique is a very powerful tool to solve many other nonlinear partial differential equations in applied science.

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Construction of solitonary and periodic solutions to some nonlinear equations using EXP-function method

2007-04-01

Mei Zhang , Wenjing Zhang

Soliton and Periodic Wave Solutions to the Osmosis<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>K</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>Equation

2009-01-01

Jiangbo Zhou , Lixin Tian , Xinghua Fan

Two types of traveling wave solutions to the osmosis K (2, 2) equation are investigated. They are characterized by two parameters. The expresssions for the soliton and periodic wave solutions … Two types of traveling wave solutions to the osmosis K (2, 2) equation are investigated. They are characterized by two parameters. The expresssions for the soliton and periodic wave solutions are obtained.

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Generalized Variational Principle for Long Water‐Wave Equation by He′s Semi‐Inverse Method

2009-01-01

Wei-Min Zhang

Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. However, it is well known that not every nonlinear partial differential equation … Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. However, it is well known that not every nonlinear partial differential equation admits a variational formula. In this paper, He′s semi‐inverse method is used to construct a family of variational principles for the long water‐wave problem.

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Modified variational iteration method for solving Helmholtz equations

2009-01-01

Muhammad Aslam Noor , Syed Tauseef Mohyud‐Din

New analytical methods for cleaning up the solution of nonlinear equations

2009-04-09

Ji‐Huan He , Eric Wai Ming Lee

An efficient approach to converting the damping fractal models to the traditional system

2022-11-29

Yusry O. El‐Dib , Nasser S. Elgazery

An efficient hybrid method for solving fredholm integral equations using triangular functions

2017-03-19

Mohamed Ramadan , Mohamed R. Ali

In this paper the orthogonal triangular function (TF) based method is first applied to transform the Fredholm integral equations and Fredholm system of integral equations to a coupled system of … In this paper the orthogonal triangular function (TF) based method is first applied to transform the Fredholm integral equations and Fredholm system of integral equations to a coupled system of matrix algebraic equations.The obtained system is a variant of coupled Sylvester matrix equations.A finite iterative algorithm is then applied to solve this system to obtain the coefficients used to get the form of approximate solution of the unknown functions of the integral problems.Some numerical examples are solved to illustrate the accuracy and the efficiency of the proposed hybrid method.The obtained numerical results are compared with other numerical methods and the exact solutions.

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Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

2022-12-08

Weaam Alhejaili , Alvaro H. Salas , S. A. El-Tantawy

Abstract In this investigation, some different approaches are implemented for analyzing a generalized forced damped complex Duffing oscillator, including the hybrid homotopy perturbation method (H-HPM), which is sometimes called the … Abstract In this investigation, some different approaches are implemented for analyzing a generalized forced damped complex Duffing oscillator, including the hybrid homotopy perturbation method (H-HPM), which is sometimes called the Krylov-Bogoliubov-Mitropolsky (KBM) method and the multiple scales method (MSM). All mentioned methods are applied to obtain some accurate and stable approximations to the proposed problem without decoupling the original problem. All obtained approximations are discussed graphically using different numerical values to the relevant parameters. Moreover, all obtained approximate solutions are compared with the 4th-order Runge-Kutta (RK4) numerical approximation. The maximum residual distance error (MRDE) is also estimated, in order to verify the high accuracy of the obtained analytic approximations.

Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

2022-01-01

A. A. Elmandouh , Mamdouh Elbrolosy

The Ivancevic option pricing model comes as an alternative to the Black‐Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of … The Ivancevic option pricing model comes as an alternative to the Black‐Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of this model have been studied by many researchers, but the analytical approach has been virtually absent from the literature. This study intends to examine some dynamic features of this model. By using the well‐known ARS algorithm, it is demonstrated that this model is not integrable in the Painlevé sense. He’s variational method is utilized to create new abundant solutions, which contain the bright soliton, bright‐like soliton, kinky‐bright soliton, and periodic solution. The bifurcation theory is applied to investigate the phase portrait and to study some dynamical behavior of this model. Furthermore, we introduce a classification of the wave solutions into periodic, super periodic, kink, and solitary solutions according to the type of the phase plane orbits. Some 3D‐graphical representations of some of the obtained solutions are displayed. The influence of the model’s parameters on the obtained wave solutions is discussed and clarified graphically.

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Application of Homotopy Perturbation Method with an Auxiliary Term for Nonlinear Dropping Equations Arisen in Polymer Packaging System

2013-01-01

Xiang Hong , Jun Wang , Lixin Lu

The homotopy perturbation method (HPM) with an auxiliary term was applied to obtain approximate analytical solutions of polymer cushioning packaging system. The second-order solution of the equation of motion was … The homotopy perturbation method (HPM) with an auxiliary term was applied to obtain approximate analytical solutions of polymer cushioning packaging system. The second-order solution of the equation of motion was obtained and compared with the numerical simulation solution solved by the Runge-Kutta algorithm. The results showed the high accuracy of this modified HPM with convenient calculation.

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Comparison the solutions for some kinds of differential equations using iterative methods

2020-10-07

Huda Omran Altaie

This manuscript presents several applications for solving special kinds of ordinary and partial differential equations using iteration methods such as Adomian decomposition method (ADM), Variation iterative method (VIM) and Taylor … This manuscript presents several applications for solving special kinds of ordinary and partial differential equations using iteration methods such as Adomian decomposition method (ADM), Variation iterative method (VIM) and Taylor series method. These methods can be applied as well as to solve nonperturbed problems and 3rd order parabolic PDEs with variable coefficient. Moreover, we compare the results using ADM, VIM and Taylor series method. These methods are a commination of the two initial conditions.

Orthogonal Basis to Solve Linear Fredholm Integral Equations via Galerkin Method

2020-11-01

Ahmed Flayih Jaber , Adil Mohamad Hassan Rizak Al-Rammahi

Abstract in this paper, the approximate solution of linear Fredholm integral equations of the second type on a closed interval is studied. The Galerkin method enhanced with Chebyshev polynomials was … Abstract in this paper, the approximate solution of linear Fredholm integral equations of the second type on a closed interval is studied. The Galerkin method enhanced with Chebyshev polynomials was used to improve the approximate solution of our study, the related linear system was solved by the Boole’s numerical integrated rule. for testing the efficiency of the proposed method, the comparison with related updated methods was executed.

Computing the luminosity distance via optimal homotopy perturbation method

2020-10-14

Bo Yu , Zihua Wang , Dezi Liu +1 more

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Laplace transform: Making the variational iteration method easier

2019-01-18

Naveed Anjum , Ji‐Huan He

Application of perturbation–iteration method to Lotka–Volterra equations

2016-03-05

Yiğit Aksoy , Ünal Göktaş , Mehmet Pakdemi̇rli̇ +1 more

Perturbation–iteration method is generalized for systems of first order differential equations. Approximate solutions of Lotka–Volterra systems are obtained using the method. Comparisons of our results with each other and with … Perturbation–iteration method is generalized for systems of first order differential equations. Approximate solutions of Lotka–Volterra systems are obtained using the method. Comparisons of our results with each other and with numerical solutions are given. The method is implemented in Mathematica, a major computer algebra system. The package PerturbationIteration.m automatically carries out the tedious calculations of the method.

Double exp-function method for multisoliton solutions of the Tzitzeica-Dodd-Bullough equation

2016-04-28

Alaattin Esen , Nuri Murat Yağmurlu , Orkun Taşbozan

A note on soliton solutions of Klein-Gordon-Zakharov equation by variational approach

2016-01-01

Najeeb Alam Khan , Fatima Riaz , Asmat Ara

Abstract In this paper, an investigation has been made to validate the variational approach to obtain soliton solutions of the Klein-Gordon-Zakharov (KGZ) equations. It is evident that to resolve the … Abstract In this paper, an investigation has been made to validate the variational approach to obtain soliton solutions of the Klein-Gordon-Zakharov (KGZ) equations. It is evident that to resolve the non-linear partial differential equations are quite complex and difficult. The presented approach is capable of achieving the condition for continuation of the solitary solution of KGZ equation as well as the initial solutions selected in soliton form including various unknown parameters can be resolute in the solution course of action. The procedure of attaining the solution reveals that the scheme is simple and straightforward.

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A Closed Form Solution for Nonlinear Oscillators Frequencies Using Amplitude-Frequency Formulation

2012-01-01

Amin Barari , A. Kimiaeifar , M.G. Nejad +2 more

Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due … Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due to this nonlinearity, most of the studies on oscillation systems are numerically carried out while an analytical approach with a closed form expression for system response would be very useful in different applications. Some analytical techniques have been presented in the literature for the solution of strong nonlinear oscillators as well as approximate and numerical solutions. In this paper, Amplitude-Frequency Formulation (AFF) approach is applied to analyze some periodic problems arising in classical dynamics. Results are compared with another approximate analytical technique called Energy Balance Method developed by the authors (EBM) and also numerical solutions. Close agreement of the obtained results reveal the accuracy of the employed method for several practical problems in engineering.

An efficient accurate scheme for solving the three-dimensional Bratu-type problem

2023-09-08

Helmi Temimi , Mohamed Ben-Romdhane , Mahboub Baccouch

Some Numerical Methods for Solving the Korteweg-de Vries Equation (KdV)

2009-01-01

Mustafa İnç

Homotopy analysis method for systems of integro-differential equations

2014-01-01

M. Matinfar , M. Saeidy , B. Gharahsuflu

In this paper, the application of the homotopy analysis method (HAM) for solving linear and nonlinear systems of integro-differential equations is presented. We used the above method for some nonlinear … In this paper, the application of the homotopy analysis method (HAM) for solving linear and nonlinear systems of integro-differential equations is presented. We used the above method for some nonlinear examples to show how implementation and efficien

New exact solutions for the Boiti-Leon-Manna-Pempinelli equation

2024-01-01

Rasime Kübra YILMAZ

Phenomena in physics, plasma physics, optical fibers, chemical physics, fluid mechanics, and many fields are often described by the nonlinear evolution equations.The analytical solutions of these equations are very important … Phenomena in physics, plasma physics, optical fibers, chemical physics, fluid mechanics, and many fields are often described by the nonlinear evolution equations.The analytical solutions of these equations are very important to understand the evaluation of the physical models.In this paper, the Boiti-Leon-Manna-Pempinelli (BLMP) nonlinear partial differential equation, which can be used to describe the incompressible fluid flow, is analytically studied by using the five different techniques which are direct integration, (G' / G)-expansion method, different form of the (G' / G)-expansion method, two variable (G' / G, 1 / G)-expansion method, and (1 / G')-expansion method.Hyperbolic, trigonometric and rotational forms of solutions are obtained.Our solutions are reduced to the well-known solutions found in the literature by assigning the some special values to the constants appeared in the analytic solutions.Moreover, we have also obtained the new analytic solutions of the BLMP equation.

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Non-linear vibration of Euler-Bernoulli beams

2011-06-01

Amin Barari , H. D. Kaliji , M. Ghadimi +1 more

In this paper, variational iteration (VIM) and parametrized perturbation (PPM) methods have been used to investigate non-linear vibration of Euler-Bernoulli beams subjected to the axial loads. The proposed methods do … In this paper, variational iteration (VIM) and parametrized perturbation (PPM) methods have been used to investigate non-linear vibration of Euler-Bernoulli beams subjected to the axial loads. The proposed methods do not require small parameter in the equation which is difficult to be found for nonlinear problems. Comparison of VIM and PPM with Runge-Kutta 4th leads to highly accurate solutions.

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A wavelet method for solving a class of nonlinear boundary value problems

2012-12-27

Xiaojing Liu , Youhe Zhou , Xiaomin Wang +1 more

Variational approach to the nonlinear oscillator of a mass attached to a stretched wire

2008-03-20

Dahua Shou

In this paper, we apply a novel variational approach to the nonlinear oscillator of a mass attached to a stretched wire. Comparison with the exact results is made, revealing that … In this paper, we apply a novel variational approach to the nonlinear oscillator of a mass attached to a stretched wire. Comparison with the exact results is made, revealing that the present method is efficient, straightforward and easy. The method can also be easily extended to other nonlinear oscillations.

Approximate periodic solutions to microelectromechanical system oscillator subject to magnetostatic excitation

2020-11-29

Ji‐Huan He , Piotr Skrzypacz , Yanni Zhang +1 more

A way to obtain approximate periodic solutions to nonlinear oscillators arising in a microelectromechanical system (MEMS) is presented for the case of zero initial conditions and magnetostatic excitation. The frequency–amplitude … A way to obtain approximate periodic solutions to nonlinear oscillators arising in a microelectromechanical system (MEMS) is presented for the case of zero initial conditions and magnetostatic excitation. The frequency–amplitude relationship is derived by adopting He′s frequency formulation. The obtained analytical results are illustrated graphically. The proposed simple approach gives the fast insight into the dynamics of the singular oscillator and can be useful for design of MEMS devices.

Nonlinear Vibration of Oscillation Systems using Frequency-Amplitude Formulation

2012-01-01

Abdolhossein Fereidoon , M. Ghadimi , Amin Barari +2 more

In this paper we study the periodic solutions of free vibration of mechanical systems with third and fifth-order nonlinearity for two examples using He's Frequency-Amplitude Formulation (HFAF).The effectiveness and convenience … In this paper we study the periodic solutions of free vibration of mechanical systems with third and fifth-order nonlinearity for two examples using He's Frequency-Amplitude Formulation (HFAF).The effectiveness and convenience of the method is illustrated in these examples. It will be shown that the solutions obtained with current method have a fabulous conformity with those achieved from time marching solution. HFAF is easy with powerful concepts and the high accuracy, so it can be found widely applicable in vibrations, especially strong nonlinearity oscillatory problems.

Abundant soliton solutions of the resonant nonlinear Schrödinger equation with time-dependent coefficients by ITEM and He’s semi-inverse method

2017-09-11

Jalil Manafian , Peyman Bolghar , Atefeh Mohammadalian

Analytical investigation of beam deformation equation using perturbation, homotopy perturbation, variational iteration and optimal homotopy asymptotic methods

2011-01-01

F. Farrokhzad , Pejman Mowlaee , Amin Barari +2 more

The beam deformation equation has very wide applications in structural engineering. As a differential equation, it has its own problem concerning existence, uniqueness and methods of solutions. Often, original forms … The beam deformation equation has very wide applications in structural engineering. As a differential equation, it has its own problem concerning existence, uniqueness and methods of solutions. Often, original forms of governing differential equations used in engineering problems are simplified, and this process produces noise in the obtained answers. This paper deals with solution of second order of differential equation governing beam deformation using four analytical approximate methods, namely the Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM) and Optimal Homotopy Asymptotic Method (OHAM). The comparisons of the results reveal that these methods are very effective, convenient and quite accurate to systems of non-linear differential equation.

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Solution of the heat equation in the cast‐mould heterogeneous domain using a weighted algorithm based on the homotopy perturbation method

2013-04-11

S. Karimi Vanani , Ahmet Yıldırım , Fazlollah Soleymani +2 more

Purpose The purpose of this paper is to present a weighted algorithm based on the homotopy perturbation method for solving the heat transfer equation in the cast‐mould heterogeneous domain. Design/methodology/approach … Purpose The purpose of this paper is to present a weighted algorithm based on the homotopy perturbation method for solving the heat transfer equation in the cast‐mould heterogeneous domain. Design/methodology/approach A weighted algorithm based on the homotopy perturbation method is used to minimize the volume of computations. The authors show that this technique yields the analytical solution of the desired problem in the form of a rapidly convergent series with easily computable components. Findings The authors illustrate that the proposed method produces satisfactory results with respect to Adomian decomposition method and standard homotopy perturbation method. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability. Originality/value This research presents, for the first time, a new modification of the proposed technique, for aforementioned problems and some interesting results are obtained.

Periodic solution for strongly nonlinear vibration systems by He's variational iteration method

2009-04-14

Sayyid H. Hashemi Kachapi , D.D. Ganji , A. G. Davodi +1 more

Abstract In this paper, we use variational iteration method for strongly nonlinear oscillators. This method is a combination of the traditional variational iteration and variational method. Some examples are given … Abstract In this paper, we use variational iteration method for strongly nonlinear oscillators. This method is a combination of the traditional variational iteration and variational method. Some examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy. The method can be easily extended to other nonlinear oscillations and hence widely applicable in engineering and science. Copyright © 2009 John Wiley & Sons, Ltd.

Construction of Target Controllable Image Segmentation Model Based on Homotopy Perturbation Technology

2013-01-01

Shuli Mei

Based on the basic idea of the homotopy perturbation method which was proposed by Jihuan He, a target controllable image segmentation model and the corresponding multiscale wavelet numerical method are … Based on the basic idea of the homotopy perturbation method which was proposed by Jihuan He, a target controllable image segmentation model and the corresponding multiscale wavelet numerical method are constructed. Using the novel model, we can get the only right object from the multiobject images, which is helpful to avoid the oversegmentation and insufficient segmentation. The solution of the variational model is the nonlinear PDEs deduced by the variational approach. So, the bottleneck of the variational model on image segmentation is the lower efficiency of the algorithm. Combining the multiscale wavelet interpolation operator and HPM, a semianalytical numerical method can be obtained, which can improve the computational efficiency and accuracy greatly. The numerical results on some images segmentation show that the novel model and the numerical method are effective and practical.

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

2010-04-07

Maziyar Jalaal , D.D. Ganji , Farah Mohammadi

Abstract Heat 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 … Abstract Heat 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

Solitary Wave Solutions of Nonlinear Conformable Time-Fractional Boussinesq Equations

2019-07-01

Özkan Güner , Ahmet Bekir , K. Hosseini

In this paper, the ( G G )-expansion method based on conformable fractional derivative is proposed to solve time fractional partial differential equations in mathematical physics.To illustrate the validity of … In this paper, the ( G G )-expansion method based on conformable fractional derivative is proposed to solve time fractional partial differential equations in mathematical physics.To illustrate the validity of this method, we solve the Boussinesq equation, coupled time-fractional Boussinesq equations and a variety of exact solutions for them are successfully established.With the help of Maple software, three-dimensional solution graphs are presented.

Differential Equations and Their Applications: An Introduction to Applied Mathematics

1977-09-01

David L. Farnsworth , Martin Braun

1 First-order differential equations.- 2 Second-order linear differential equations.- 3 Systems of differential equations.- 4 Qualitative theory of differential equations.- 5 Separation of variables and Fourier series.- Appendix A Some … 1 First-order differential equations.- 2 Second-order linear differential equations.- 3 Systems of differential equations.- 4 Qualitative theory of differential equations.- 5 Separation of variables and Fourier series.- Appendix A Some simple facts concerning functions of several variables.- Appendix B Sequences and series.- Appendix C Introduction to APL.- Answers to odd-numbered exercises.

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Asymptotic Methods in the Theory of Nonlinear Oscillations

1961-01-01

N. N. Bogolyubov , Yu. A. Mitropol’skii

Abstract : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It … Abstract : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended for the wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes. Contents include the following: Natural oscillations in quasi-linear systems; The method of the phase plane; The influence of external periodic forces; The method of the mean; Justification of the asymptotic methods.

Introduction to Perturbation Methods

2012-12-03

Mark H. Holmes

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Nonlinear Differential Equations and Dynamical Systems

1990-01-01

Ferdinand Verhulst

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An introduction to nonlinear oscillations

1981-01-01

Ronald E. Mickens

Nonlinear Oscillations

1981-12-01

Peter Hagedorn , W. Stadler , C. S. Hsu

Part 1 The mathematical pendulum as an illustration of linear and non-linear oscillations - systems which are similar to a simple linear oscillator: Undamped free oscillations of the pendulum damped … Part 1 The mathematical pendulum as an illustration of linear and non-linear oscillations - systems which are similar to a simple linear oscillator: Undamped free oscillations of the pendulum damped free oscillations forced oscillations. Part 2 Liapounov stability theory and bifurcations: The concept of Liapounov stability the direct method of Liapounov stability by the first approximation the Poincare map the critical case of a conjugate pair of eigenvalues simple bifurcation of equilibria and the Hopf bifurcation. Part 3: Self-excited oscillations in mechanical and electrical systems analytical approximation methods for the computation of self-excited oscillations analytical criteria for the existence of limit cycles forced oscillations in self-excited systems self-excited oscillations in systems with several degrees of freedom Part 4 Hamiltonian systems: Hamiltonian differential equations in mechanics canonical transformations the Hamilton-Jacobi differential equation canonical transformations and the motion perturbation theory Part 5 Introduction to the theory of optimal control: Control problems, controllability the Pontryagin maximum principle transversality conditions and problems with target sets canonical perturbation theory in optimal control.

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Modified Lindstedt–Poincare methods for some strongly non-linear oscillations

2002-03-01

Ji‐Huan He

None

2000-01-01

Ji‐Huan He

Problems in Perturbation

1985-08-07

Ali H. Nayfeh

Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free Oscillations of Systems with Quadratic Nonlinearities General Systems with Odd Nonlinearities … Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free Oscillations of Systems with Quadratic Nonlinearities General Systems with Odd Nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equations with Variable Coefficients Differential Equations with a Large Parameter Solvability Conditions Index

Modified Lindstedt–Poincare methods for some strongly non-linear oscillations

2002-03-01

Ji‐Huan He

Bifurcations of a Generalized Camassa-Holm Equation

2005-01-01

Zhengdi Zhang , Qinsheng Bi

A generalized F-expansion method to find abundant families of Jacobi Elliptic Function solutions of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

2005-06-27

Yujie Ren , Hongqing Zhang

Approximate analytical solution for seepage flow with fractional derivatives in porous media

1998-12-01

Ji‐Huan He

Approximate solution of nonlinear differential equations with convolution product nonlinearities

1998-12-01

Ji‐Huan He

Stability and bifurcation of a discrete red blood cell survival model

2005-08-03

Chunrui Zhang , Yuangang Zu , Baodong Zheng

A perturbation-incremental method for strongly non-linear oscillators

1996-01-01

H.S.Y. Chan , K.W. Chung , Zhengji Xu

Dynamic complexities of a food chain model with impulsive perturbations and Beddington–DeAngelis functional response

2005-06-20

Shuwen Zhang , Dejun Tan , Lansun Chen

Variational iteration method – a kind of non-linear analytical technique: some examples

1999-07-01

Ji‐Huan He

Linearized perturbation technique and its applications to strongly nonlinear oscillators

2003-01-01

Ji‐Huan He

Determination of limit cycles for a modified van der Pol oscillator

2005-07-28

Mario D’Acunto

Comment on “Determination of Limit Cycles for Strongly Nonlinear Oscillators”

2004-08-02

Surjeet Rajendran , S. N. Pandey , M. Lakshmanan

A Comment on the Letter by Ji-Huan He, Phys. Rev. Lett. 90, 174301 (2003).Received 23 July 2003DOI:https://doi.org/10.1103/PhysRevLett.93.069401©2004 American Physical Society A Comment on the Letter by Ji-Huan He, Phys. Rev. Lett. 90, 174301 (2003).Received 23 July 2003DOI:https://doi.org/10.1103/PhysRevLett.93.069401©2004 American Physical Society

The homotopy perturbation method for nonlinear oscillators with discontinuities

2003-07-07

Ji‐Huan He

Variational approach to the Lane–Emden equation

2003-06-02

Ji‐Huan He

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Solution of nonlinear equations by an ancient Chinese algorithm

2003-06-30

Ji‐Huan He

Homotopy perturbation method for solving boundary value problems

2005-10-14

Ji‐Huan He

Periodic solutions and bifurcations of delay-differential equations

2005-08-16

Ji‐Huan He

Homotopy perturbation method: a new nonlinear analytical technique

2002-10-07

Ji‐Huan He

Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method

2004-06-22

Hongmei Liu

Determination of Limit Cycles for Strongly Nonlinear Oscillators

2003-05-02

Ji‐Huan He

An innovative approach to finding limit cycles is proposed and illustrated on the van der Pol equation. The technique developed in this Letter is similar to the Ritz's method in … An innovative approach to finding limit cycles is proposed and illustrated on the van der Pol equation. The technique developed in this Letter is similar to the Ritz's method in variational theory. The present theory can be applied to not only weakly nonlinear equations, but also strongly nonlinear ones, and the obtained results are valid for the whole solution domain.

A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method

2005-12-19

S. Abbasbandy

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Application of He’s homotopy perturbation method to functional integral equations

2005-12-19

S. Abbasbandy

Variational principles for some nonlinear partial differential equations with variable coefficients

2003-08-07

Ji‐Huan He