scholarly journals An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations

2013 ◽  
Vol 37 (5) ◽  
pp. 2702-2708 ◽  
Author(s):  
Y. Khan ◽  
H. Vázquez-Leal ◽  
N. Faraz
Keyword(s):  
Differential Equations ◽  
Laplace Transformation ◽  
Nonlinear Differential Equations ◽  
Parameter Method ◽  
Auxiliary Parameter ◽  
Adomian Polynomials

scholarly journals Numerical study of flow and heat transfer in a porous medium between two stretchable disks using quasi-linearization method

2019 ◽  
pp. 163-163
Author(s):  
Shaheen Akhter ◽  
Muhammad Ashraf
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Differential Equations ◽  
Numerical Study ◽  
Nonlinear Differential Equations ◽  
Linearization Method ◽  
Temperature Fields ◽  
Parameter Method ◽  
Heat Equations ◽  
Constant Density

In this study, the flow as well as heat transfer of a classical Newtonian fluid of constant density and viscosity in a porous medium between two radially stretching disks is explored. The role of the porosity of the medium, the stretching of the disks, the viscous dissipation and radiation on the flow and temperature fields is taken into account. The flow and heat equations are transformed into nonlinear ordinary differential equations by invoking the classical similarity transformations. These nonlinear differential equations were linearized using Quasi linearization method. Further the linearized equations were discretized by employing the finite differences which were then solved numerically using the successive over relaxation parameter method. Some features of the flow and temperature are discussed in detail in the form of tables and graphs. The present study may be beneficial in lubricants and computational storage devices as well as fluid flows and heat transmission in rotor-stator systems.

scholarly journals Error of small parameter methods in solving shortened equations of a synchronized oscillator

Radiotekhnika ◽  
2021 ◽  
pp. 113-117
Author(s):  
V.V. Rapin
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Relative Error ◽  
Nonlinear Differential Equations ◽  
Parameter Method ◽  
Frequency Detuning ◽  
Small Parameter Method ◽  
First Approximation ◽  
Linear Differential ◽  
Zero Frequency

The paper considers the use of recently appeared analytical methods for solving shortened equations of a synchronized oscillator. These are a quasi-small parameter method and a combined small parameter method. Both methods use the classic small parameter method. A peculiarity of their application is that in this case they are used for solving nonlinear differential equations that do not contain a small parameter. The difference between the above methods is in obtaining the equations of the first approximation. In the quasi-small parameter method, they are linear differential equations obtained by linearizing the original nonlinear differential equations in the area of the zero frequency detuning. In the combined small parameter method, the equations of the first approximation are obtained by approximating the original nonlinear differential equations. Of course, a number of transformations of these equations were made for this. The approximation made it possible to obtain better representation of the original nonlinear differential equations by means of linear differential equations. This representation provided a smaller error, which in both cases was presented as a discrepancy. The discrepancy does not allow obtaining a relative error and investigating its peculiarity. A study of the relative error of the quasi-small parameter method shows that this error is a continuous function of the frequency detuning with a zero value for a zero frequency detuning. A function representing relative error has a gap at zero frequency detuning for the combined small parameter method. However, this kind of gap can be eliminated by additional function definition.

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Umesh Umesh ◽  
Manoj Kumar
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
The Other ◽  
Step Size ◽  
Adomian Polynomials ◽  
Content Type ◽  
Adomian Decomposition

Purpose The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions. Design/methodology/approach First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed. Findings The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples. Originality/value Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.

scholarly journals Bertolino-Baksa stability at nonlinear vibrations of motor vehicles

2017 ◽  
Vol 44 (2) ◽  
pp. 271-291 ◽  
Author(s):  
Ljudmila Kudrjavceva ◽  
Milan Micunovic ◽  
Danijela Miloradovic ◽  
Aleksandar Obradovic
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Motor Vehicles ◽  
Numerical Code ◽  
Parameter Method ◽  
Spatial Stability ◽  
Linear Vibrations ◽  
Linear Differential

Research of vehicle response to road roughness is particularly important when solving problems related to dynamic vehicle stability. In this paper, unevenness of roads is considered as the source of non-linear vibrations of motor vehicles. The vehicle is represented by an equivalent spatial model with seven degrees of freedom. In addition to solving the response by simulating it within a numerical code, quasi-linearization of nonlinear differential equations of motion is carried out. Solutions of quasi-linear differential equations of forced vibrations are determined using the small parameter method and are indispensable for the study of spatial stability of the vehicle. An optimal stabilization for a simplified two-dimensional model was performed. Spatial stability and internal resonance are considered briefly.

scholarly journals Some Study Methods for Ordinary Differential Equation Integrability of the Second Order of a Certain Type

2019 ◽  
pp. 48-62
Author(s):  
O. V. Zadorozhnaya ◽  
V. K. Kochetkov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Practical Interest ◽  
Second Order ◽  
Parameter Method ◽  
Study Methods ◽  
Special Cases ◽  
General Conditions

The paper deals with treating some study methods of the equation integrability of a certain type that are little studied in the theory of differential equations. It is known that a significant part of the differential equations cannot be integrated. Then, to develop methods for their study is, certainly, of scientific interest. The obtained results, formulated as theorems and statements, are of scientific and practical interest because of their importance for applications in modern science.In the paper we present an alternative method for studying the integrability of both linear and nonlinear differential equations of the second order. An introduction of parameters allowed us to develop a study method for the integrability of ordinary differential equations of the second order. We also formulate the theorems describing some General conditions for the integrability of the second-order linear equation and consider special cases of integrability, which arise out of the above facts.Based on the obtained parameter method, some General conditions for the integrability of the nonlinear differential equation of the second order are given, and the consequences of these General conditions are indicated.We have obtained new results related to the construction and development of methods for studying the differential equation to which some types of differential equations are reduced and laid the foundations for a rigorous and systematic study of the introduced special nonlinear differential equation of the second order.

scholarly journals A Reliable Treatment for Nonlinear Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
H. R. Marasi ◽  
M. Sedighi ◽  
H. Aydi ◽  
Y. U. Gaba
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Convergent Series ◽  
Parameter Method ◽  
Numerical Techniques ◽  
He’S Polynomials ◽  
Computational Work ◽  
Different Types

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

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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