scholarly journals Approximate analytical solutions for a class of nonlinear differential equations

2007 ◽  
Vol 56 (3) ◽  
pp. 1549
Author(s):  
Zheng Lian-Cun ◽  
Feng Zhi-Feng ◽  
Zhang Xin-Xin
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Approximate Analytical Solutions

scholarly journals Dirichlet series and analytical solutions of MHD viscous flow with suction / blowing

2017 ◽  
Vol 2 (2) ◽  
pp. 341-350 ◽  
Author(s):  
Vishwanath B. Awati
Keyword(s):  
Differential Equations ◽  
Viscous Flow ◽  
Dirichlet Series ◽  
Analytical Solutions ◽  
Similarity Transformation ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Mhd Flow ◽  
Incompressible Viscous Flow ◽  
Approximate Analytical Solutions

AbstractThis paper presents Dirichlet series and approximate analytical solutions of magnetohydrodynamic (MHD) flow due to a suction / blowing caused by boundary layer of an incompressible viscous flow. The governing nonlinear partial differential equations of momentum equations are reduced into a set of nonlinear ordinary differential equations (ODE) by using a classical similarity transformation along with appropriate boundary conditions. Both nonlinearity and infinite interval demand novel mathematical tools for their analysis. We use elegant fast converging Dirichlet series and approximate analytical solutions (method of stretching of variables) of these nonlinear differential equations. These methods have advantages over pure numerical methods for obtaining derived quantities accurately for various values of the parameters involved at a stretch and also they are valid in much larger parameter domains as compared with DTM-Padé and classical numerical schemes.

scholarly journals Analytical solitons for the space-time conformable differential equations using two efficient techniques

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmad Neirameh ◽  
Foroud Parvaneh
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Space Time ◽  
Nonlinear Phenomena ◽  
Approximate Analytical Solutions ◽  
Coupled Burgers Equation ◽  
Mathematics And Physics

AbstractExact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

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.

The Method of Multiple Scales in Solving Nonlinear Dynamic Differential Equations of Gear Systems

2013 ◽  
Vol 274 ◽  
pp. 324-327
Author(s):  
J.F. Nie ◽  
M.L. Zheng ◽  
G.B. Yu ◽  
J.M. Wen ◽  
B. Dai
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Transmission Error ◽  
Gear System ◽  
Approximate Analytical Solutions ◽  
Response Equation ◽  
Gear Systems

To obtain exact analytical solutions of differential equations of gear system dynamics due to the difficulty of solving complicated differential equations. Only the approximate analytical solutions can be determined. The method of multiple scales is one of the most powerful, popular perturbation methods. The dynamic model which describes the torsional vibration behaviors of gear system has been introduced accurately in this paper. The differential equation of gear system nonlinear dynamics exhibiting combined nonlinearity influence such as time-varying stiffness, tooth backlash and dynamic transmission error (DTE) has been proposed. The theory of multiple scales method has been presented in solving nonlinear differential equations of gear systems and the frequency response equation has been obtained. The fact that the approximate analytical solution by using the method of multiple scales is in good agreement with the exact solutions by numerically integrating differential equations has proved that the method of multiple scales is one of the most frequently used methods in solving differential equations, especially for large and complicated differential equations.

scholarly journals Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Least Squares Method ◽  
Nonlinear Differential Equations ◽  
Fractional Order System ◽  
New Method ◽  
Order System ◽  
Approximate Analytical Solutions

The paper presents a new method, called the Polynomial Least Squares Method (PLSM). PLSM allows us to compute approximate analytical solutions for the Brusselator system, which is a fractional-order system of nonlinear differential equations.

Electronic Analog Computer Solutions of Nonlinear Vibratory Systems of Two Degrees of Freedom

1956 ◽  
Vol 23 (4) ◽  
pp. 629-634
Author(s):  
C. P. Atkinson
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Harmonic Approximation ◽  
Degree Of Freedom ◽  
Single Term ◽  
Approximate Analytical Solutions ◽  
Two Degree Of Freedom ◽  
Differential Analyzer ◽  
Vibrating Systems

Abstract The main contribution of this paper is its account of the use of an electronic differential analyzer for solving the exact differential equations of certain two-degree-of-freedom nonlinear vibrating systems, and the comparison of the differential-analyzer solutions with the approximate analytical solutions obtained from a single-term harmonic approximation (Ritz approximation). The results of these two methods for solving nonlinear differential equations compare favorably over much of the ranges of variables. Where discrepancies arise between the results of the two methods the value of the differential-analyzer approach can be practically appreciated. For example, in the region where superharmonics might be expected, the single-term harmonic approximation ignores them, while the analyzer solutions contain the superharmonic components of the exact solution. A secondary contribution of the paper is the account of the use of the differential analyzer for verification of suspected stability criteria for two-degree-of-freedom nonlinear vibrating systems. Analytical solutions that were reproducible on the analyzer were considered stable; those that were not reproducible were considered unstable. This paper also can be considered as a call for further application of modern computer techniques to the problems of nonlinear mechanics. Since the computer solves the exact or complete differential equations, the results are the complete solutions including transient phenomena, steady state, subharmonics, super-harmonics, and so on. These exact solutions are produced as functions of time which are analogous to the actual vibrations of the physical system studied.

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.

Sign in / Sign up

Export Citation Format

Share Document