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 Homotopy Perturbation Method with Chebyshev Polynomials to Nonlinear Problems

2010 ◽  
Vol 65 (1-2) ◽  
pp. 65-70
Author(s):  
Changbum Chun
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Chebyshev Polynomials ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Efficiency And Reliability

AbstractIn this paper, we present an efficient modification of the homotopy perturbation method by using Chebyshev’s polynomials and He’s polynomials to solve some nonlinear differential equations. Some illustrative examples are given to demonstrate the efficiency and reliability of the modified homotopy perturbation method.

scholarly journals Solving nonlinear boundary value problems by the Galerkin method with sinc functions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Sinc Approximation ◽  
Sinc Functions ◽  
The Galerkin Method

AbstractIn this paper, the sinc-Galerkin method is used for numerically solving a class of nonlinear differential equations with boundary conditions. The importance of this study is that sinc approximation of the nonlinear term is stated as a new theorem. The method introduced here is tested on some nonlinear problems and is shown to be a very efficient and powerful tool for obtaining approximate solutions of nonlinear ordinary differential equations.

scholarly journals A Novel Homotopy Perturbation Algorithm Using Laplace Transform for Conformable Partial Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Sajad Iqbal ◽  
Mohammed K. A. Kaabar ◽  
Francisco Martínez
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Homotopy Perturbation Method ◽  
Approximate Analytical Solution ◽  
Partial Differential ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Different Types

In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.

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.

scholarly journals A Laplace decomposition algorithm applied to a class of nonlinear differential equations

2001 ◽  
Vol 1 (4) ◽  
pp. 141-155 ◽  
Author(s):  
Suheil A. Khuri
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Decomposition Method ◽  
Iterative Scheme ◽  
Nonlinear Differential Equations ◽  
Decomposition Algorithm ◽  
Numerical Examples ◽  
Numerical Laplace Transform

In this paper, a numerical Laplace transform algorithm which is based on the decomposition method is introduced for the approximate solution of a class of nonlinear differential equations. The technique is described and illustrated with some numerical examples. The results assert that this scheme is rapidly convergent and quite accurate by which it approximates the solution using only few terms of its iterative scheme.

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.

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.

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