Fixed Point Analytical Method for Nonlinear Differential Equations

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Ding Xu ◽  
Xin Guo
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Nonlinear Differential Equations ◽  
Perturbation Technique ◽  
Linear Equations ◽  
Nonlinear Problems ◽  
Weakly Nonlinear ◽  
Linear Differential

In this paper, a new analytical method, which is based on the fixed point concept in functional analysis (namely, the fixed point analytical method (FPM)), is proposed to acquire the explicit analytical solutions of nonlinear differential equations. The key idea of this approach is to construct a contractive map which replaces the nonlinear differential equation into a series of linear differential equations. Usually, the series of linear equations can be solved relatively easily and have explicit analytical solutions. The FPM is different from all existing analytical methods, such as the well-known perturbation technique applied in weakly nonlinear problems, because it is independent of any small physical or artificial parameters at all; thus, it can handle more nonlinear problems, including strongly nonlinear ones. Two typical cases are investigated by FPM in detail and the comparison with the numerical results shows that the present method is one of high accuracy and efficiency.

scholarly journals Highly accurate analytical solution for free vibrations of strongly nonlinear Duffing oscillator

2021 ◽  
pp. 146134842110340
Author(s):  
Gamal Mohamed Ismail ◽  
Mahmoud Abul-Ez ◽  
Mohra Zayed ◽  
Hijaz Ahmad ◽  
Maha El-Moshneb
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Perturbation Technique ◽  
Duffing Oscillator ◽  
Free Vibrations ◽  
Strongly Nonlinear ◽  
Present Technique ◽  
Analytical Perturbation

Based on the suggested parameter, a new analytical perturbation technique is presented to obtain highly ordered accurate analytical solutions for nonlinear Duffing oscillator with nonlinearity of high order. Comparing the obtained results with the numerical and other previously published results reveals the usefulness and correctness of the present technique. It is shown that the results are valid for small and large amplitudes. Indeed, it is found that our proposed technique produces more accurate and computationally results than the rival known methods. The obtained results show the efficiency and capability of the present perturbation technique to be applied to various strongly nonlinear differential equations.

scholarly journals A Review of Some Recent Results for the Approximate Analytical Solutions of Nonlinear Differential Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-34 ◽  
Author(s):  
Serdal Pamuk
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Research Literature ◽  
Science And Engineering ◽  
Adomian’S Decomposition Method ◽  
Approximate Analytical Solutions ◽  
Recent Developments

This paper features a survey of some recent developments in techniques for obtaining approximate analytical solutions of some nonlinear differential equations arising in various fields of science and engineering. Adomian's decomposition method is applied to some nonlinear problems, and some mathematical tools such as He's homotopy perturbation method and variational iteration method are introduced to overcome the shortcomings of Adomian's method. The results of some comparisons of these three methods appearing in the research literature are given.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Jifeng Chu ◽  
Kateryna Marynets
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Topological Degree ◽  
Antarctic Circumpolar Current ◽  
Fixed Point Theorems ◽  
Nonlinear Differential Equations ◽  
Existence Results ◽  
Circumpolar Current ◽  
The Antarctic

AbstractThe aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

Marine Pipeline Analysis Based on Newton’s Method With an Arctic Application

1970 ◽  
Vol 92 (4) ◽  
pp. 827-833 ◽  
Author(s):  
D. W. Dareing ◽  
R. F. Neathery
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Newton’S Method ◽  
Finite Differences ◽  
Newton's Method ◽  
Nonlinear Differential Equations ◽  
Bending Moments ◽  
Iterative Solutions ◽  
Linear Differential ◽  
Pipe Deflection

Newton’s method is used to solve the nonlinear differential equations of bending for marine pipelines suspended between a lay-barge and the ocean floor. Newton’s method leads to linear differential equations, which are expressed in terms of finite differences and solved numerically. The success of Newton’s method depends on initial trial solutions, which in this paper are catenaries. Iterative solutions converge rapidly toward the exact solution (pipe deflection) even though large bending moments exist in the pipe. Example calculations are given for a 48-in. pipeline suspended in 300 ft of water.

scholarly journals Existence and Multiplicity Results for Nonlinear Differential Equations Depending on a Parameter in Semipositone Case

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Hailong Zhu ◽  
Shengjun Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Multiplicity Results ◽  
Second Order Differential Equations ◽  
Existence And Multiplicity ◽  
Nonlinear Alternative ◽  
Krasnosel’Skii’S Fixed Point Theorem ◽  
Alternative Principle

The existence and multiplicity of solutions for second-order differential equations with a parameter are discussed in this paper. We are mainly concerned with the semipositone case. The analysis relies on the nonlinear alternative principle of Leray-Schauder and Krasnosel'skii's fixed point theorem in cones.

scholarly journals The variational iteration method for solving n-th order fuzzy differential equations

Open Physics ◽  
2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Mohammad Saeidy ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Analytical Method ◽  
Iteration Method ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Linear Differential Equations ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
Linear Differential

AbstractThe variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

scholarly journals On non-homogeneous canonical third-order linear differential equations

1994 ◽  
Vol 57 (2) ◽  
pp. 138-148 ◽  
Author(s):  
N. Parhi
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Linear Differential Equations ◽  
Third Order ◽  
Order Linear ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

AbstractIn this paper sufficient conditions have been obtained for non-oscillation of non-homogeneous canonical linear differential equations of third order. Some of these results have been extended to non-linear equations.

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