Quasi-analytical method for solving nonlinear differential equations for turbulent self-confined magneto-plasmas

1986 ◽  
Vol 66 (1) ◽  
pp. 151-172 ◽  
Author(s):  
M Maurer ◽  
A Hayd ◽  
H.J Kaeppeler
Keyword(s):  
Differential Equations ◽  
Analytical Method ◽  
Nonlinear Differential Equations

Analytical solution of nonlinear differential equations two oscillators mechanism using Akbari–Ganji method

2021 ◽  
pp. 2150462 ◽  
Author(s):  
Saman Hosseinzadeh ◽  
Kh. Hosseinzadeh ◽  
M. Rahai ◽  
D. D. Ganji
Keyword(s):  
Differential Equations ◽  
Analytical Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Approximate Analytic Solution ◽  
Energy Balance Method ◽  
Wide Range ◽  
Vibration Problems ◽  
Mechanical Oscillators

In the last decade, many potent analytical methods have been utilized to find the approximate solution of nonlinear differential equations. Some of these methods are energy balance method (EBM), homotopy perturbation method (HPM), variational iteration method (VIM), amplitude frequency formulation (AFF), and max–min approach (MMA). Besides the methods mentioned above, the Akbari–Ganji method (AGM) is a highly efficient analytical method to solve a wide range of nonlinear equations, including heat transfer, mass transfer, and vibration problems. In this study, it was constructed the approximate analytic solution for movement of two mechanical oscillators by employing the AGM. In the derived analytical method, both oscillator motion equations and the sensitivity analysis of the frequency were included. The AGM was validated through comparison against Runge–Kutta fourth-order numerical method and an excellent agreement was achieved. Based on the results, the highest sensitivity of the oscillation frequency is related to the mass. As [Formula: see text] and [Formula: see text] increase, the slope of the system velocity and acceleration will increase.

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 Elementary Quadrature for the Riccati equation

2020 ◽  
Author(s):  
Ji-Xiang Zhao
Keyword(s):  
Differential Equations ◽  
Riccati Equation ◽  
Analytical Method ◽  
Nonlinear Differential Equations ◽  
Integrability Condition ◽  
Type Equation ◽  
Abel Equation ◽  
General Solutions ◽  
Suitable Transformation ◽  
Elementary Quadrature

Abstract Using suitable transformation in combination with a specific Riccati-type equation solvable, the problem of solving Riccati equation can be transformed into that of a quasi-Abel equation of the second kind. By the extended Julia’s integrability condition, the general solutions of Riccati equation in the form of elementary quadrature are obtained, which contains numerous. This method opens up a new prospect for the study of nonlinear differential equations by analytical method.

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 Solving nonlinear differential equations with differentiable quantum circuits

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Oleksandr Kyriienko ◽  
Annie E. Paine ◽  
Vincent E. Elfving
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Quantum Circuits

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.

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