scholarly journals Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Excellent Agreement ◽  
Analytical Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Volterra Equations ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Small Parameters

We apply an analytical method called the Optimal Parametric Iteration Method (OPIM) to multispecies Lotka-Volterra equations. By using initial values, accurate explicit analytic solutions have been derived. The method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement has been demonstrated between the obtained solutions and the numerical ones. This new approach, which can be easily applied to other strongly nonlinear problems, is very effective and yields very accurate results.

Optimal Homotopy Asymptotic Approach to Self-Excited Vibrations

2013 ◽  
Vol 430 ◽  
pp. 27-31 ◽  
Author(s):  
Nicolae Herisanu ◽  
Vasile Marinca
Keyword(s):  
Excellent Agreement ◽  
Nonlinear Oscillation ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Asymptotic Approach ◽  
Analytical Treatment ◽  
General Validity ◽  
Strongly Nonlinear ◽  
Convergence Of Solutions ◽  
Excited System

This paper is concerned with analytical treatment of nonlinear oscillation of a self-excited system. An analytic approximate technique, namely OHAM is employed for this purpose. Our procedure provides us with a convenient way to optimally control the convergence of solutions, such that the accuracy is always guaranteed. An excellent agreement of the approximate solutions with the numerical ones has been demonstrated. Three examples are given and the results reveal that the procedure is very effective and accurate, demonstrating the general validity and the great potential of the OHAM for solving strongly nonlinear problems.

An optimal iteration method with application to the Thomas-Fermi equation

Open Physics ◽  
2011 ◽  
Vol 9 (3) ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Approximate Solution ◽  
Approximate Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Analytical Approximate Solution ◽  
Thomas Fermi ◽  
Strongly Nonlinear ◽  
Good Agreement

AbstractThe aim of this paper is to introduce a new approximate method, namely the Optimal Parametric Iteration Method (OPIM) to provide an analytical approximate solution to Thomas-Fermi equation. This new iteration approach provides us with a convenient way to optimally control the convergence of the approximate solution. A good agreement between the obtained solution and some well-known results has been demonstrated. The proposed technique can be easily applied to handle other strongly nonlinear problems.

scholarly journals A New Approach for the Construction of Long-Periodic Perturbations

1978 ◽  
Vol 41 ◽  
pp. 57-64
Author(s):  
Rudolf Dvorak
Keyword(s):  
Analytical Method ◽  
Specific Problem ◽  
Computer Time ◽  
Lagrange Equations ◽  
New Approach ◽  
Precise Method ◽  
Periodic Perturbations ◽  
Small Parameters ◽  
New Form ◽  
Specific Formulation

AbstractThe aim of this work is to study perturbations of planets of a period of some thousands of years. The use of an analytical method allows us to separate all different influences, e.g. near resonances and is combined with the very precise method of the numerical integration. The truncation to low orders can be avoided which is made by analytical methods in using developments with respect to the small parameters inclinations and eccentricities. For this purpose a special form of the Lagrange Equations is used where the terms containing the inverse distancefrom the planet to the perturbing one are separated as it is the most difficult to compute. To develop this a specific formulation has been found where the short periodic terms can precisely be determined. Although the development seems to be of a certain complexity the small numbers of quantities used can be tabulated once and for all in a specific problem. It should be possible to integrate the new form of the Lagrange Equations within a reasonable computer-time to determine the long periodic perturbations.

Analytical approximate solutions to the Thomas-Fermi equation

Open Physics ◽  
2014 ◽  
Vol 12 (7) ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene
Keyword(s):  
Excellent Agreement ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Thomas Fermi ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method

AbstractThe purpose of this paper is to show how to use the Optimal Homotopy Asymptotic Method (OHAM) to solve the nonlinear differential Thomas-Fermi equation. Our procedure does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

scholarly journals A modified averaging operator with some applications

2020 ◽  
Vol 42 (3) ◽  
pp. 343-356
Author(s):  
Anh Tay Nguyen ◽  
N. D. Anh
Keyword(s):  
Approximate Solutions ◽  
Boundary Values ◽  
Remarkable Feature ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Averaging Operator ◽  
Boundary Regulation ◽  
Two Parameters ◽  
Local Value

The paper presents a new approach to the conventional averaging in which the role of boundary values is considered in a more detailed way. It results in a new weighted local averaging operator (WLAO) taking into account the particular role of boundary values. A remarkable feature of WLAO is that this operator contains a parameter of boundary regulation p and depends on a local value $h$ of the integration domain. By varying these two parameters one can regulate the obtained approximate solutions in order to get more accurate ones. It has been shown that the combination of WLAO with Galerkin method can lead to an effective approximate tool for the buckling problem of columns and for the frequency analysis of free vibration of strongly nonlinear systems. 

A Nonlinear Model Arising in the Buckling Analysis and its New Analytic Approximate Solution

2013 ◽  
Vol 68 (5) ◽  
pp. 355-361 ◽  
Author(s):  
Yasir Khan ◽  
Waleed Al-Hayani
Keyword(s):  
Approximate Solution ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Nonlinear Ordinary Differential Equation ◽  
Function Technique ◽  
Nonlinear Buckling ◽  
New Approach ◽  
Adomian Polynomials ◽  
Adomian Decomposition ◽  
Small Parameters

An analytical nonlinear buckling model where the rod is assumed to be an inextensible column and prismatic is studied. The dimensionless parameters reduce the constitutive equation to a nonlinear ordinary differential equation which is solved using the Adomian decomposition method (ADM) through Green’s function technique. The nonlinear terms can be easily handled by the use of Adomian polynomials. The ADM technique allows us to obtain an approximate solution in a series form. Results are presented graphically to study the efficiency and accuracy of the method. To the author’s knowledge, the current paper represents a new approach to the solution of the buckling of the rod problem. The fact that ADM solves nonlinear problems without using perturbations and small parameters can be judged as a lucid benefit of this technique over the other methods

scholarly journals Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 44-64 ◽  
Author(s):  
Remus-Daniel Ene ◽  
Vasile Marinca ◽  
Valentin Bogdan Marinca
Keyword(s):  
Thin Film ◽  
Differential Equation ◽  
Constant Velocity ◽  
Basic Equation ◽  
Film Flow ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Thin Film Flow ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method

AbstractIn this paper the thin film flow of an Oldroyd 6-constant fluid on a vertically moving belt is investigated. The basic equation of a non-Newtonian fluid in a container with a wide moving belt which passes through the container moving vertically upward with constant velocity, is reduced to an ordinary nonlinear differential equation. This equation is solved approximately by means of the Optimal Homotopy Asymptotic Method (OHAM). The solutions take into account the behavior of Newtonian and non-Newtonian fluids. Our procedure intended for solving nonlinear problems does not need small parameters in the equation and provides a convenient way to control the convergence of the approximate solutions.

scholarly journals Numerical approach for approximating the Caputo fractional-order derivative operator

2021 ◽  
Vol 6 (11) ◽  
pp. 12743-12756
Author(s):  
Ramzi B. Albadarneh ◽  
◽  
Iqbal Batiha ◽  
A. K. Alomari ◽  
Nedal Tahat ◽  
...  
Keyword(s):  
Fractional Order ◽  
Truncation Error ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Analytic Solutions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Linear And Nonlinear ◽  
Caputo Fractional Order Derivative

<abstract><p>This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator $ D_{a}^{\alpha}y(t) $ of order $ m-1 &lt; \alpha &lt; m $, where $ m\in\mathbb{N} $. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.</p></abstract>

scholarly journals Simple and accurate approach for solving of nonlinear heat convective-radiative equation in fin by using the collocation method and comparison with HPM and VIM

2014 ◽  
Vol 41 (3) ◽  
pp. 159-176
Author(s):  
Petroudi Rahimi ◽  
D.D. Ganji ◽  
Y. Rostamiyan ◽  
E. Rahimi ◽  
Nejad Khazayi
Keyword(s):  
Heat Transfer ◽  
Collocation Method ◽  
Analytical Techniques ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Mathematical Tool ◽  
Science And Engineering ◽  
Analytical Technique ◽  
Transfer Equations ◽  
Small Parameters

Collocation Method (CM) such as analytical technique, which does not need small parameters is here used to evaluate the analytical approximate solutions of the nonlinear heat transfer equation. The obtained results from Collocation Method are compared with other analytical techniques such as Homotopy Perturbation Method (HPM) and Variation Iteration Method (VIM). Also, boundary value problem (BVP) is applied as a numerical method for validation. The results reveal that the Collocation Method is very effective, simple and more accurate than other techniques. Also, it is found that this method is a powerful mathematical tool and can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering especially at some heat transfer equations.

scholarly journals Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

2022 ◽  
Author(s):  
Penghui Song ◽  
Wenming Zhang ◽  
Lei Shao
Keyword(s):  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Analytical Approximation ◽  
Homotopy Method ◽  
Initial Guess ◽  
Convergence Region ◽  
Strongly Nonlinear ◽  
Base Functions ◽  
Frequency Components ◽  
And Performance

Abstract It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach, which combines the homtopy concept with a “residue-regulating” technique to construct a continuous homotopy from an initial guess solution to a high-accuracy analytical approximation of the nonlinear problems, namely the residue regulating homotopy method (RRHM). In our method, the analytical expression of each order homotopy-series solution is associated with a set of base functions which are pre-selected or generated during the previous order of approximations, while the corresponding coefficients are solved from deformation equations specified by the nonlinear equation itself and auxiliary residue functions. The convergence region, rate and final accuracy of the homotopy are controlled by a residue-regulating vector and an expansion threshold. General procedures of implementing RRHM are demonstrated using the Duffing and Van der Pol-Duffing oscillators, where approximate solutions containing abundant frequency components are successfully obtained, yielding significantly better convergence rate and performance stability compared to the other conventional homotopy-based methods.

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