scholarly journals Detecting Optimal Leak Locations Using Homotopy Analysis Method for Isothermal Hydrogen-Natural Gas Mixture in an Inclined Pipeline

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1769
Author(s):  
Sarkhosh S. Chaharborj ◽  
Zuhaila Ismail ◽  
Norsarahaida Amin
Keyword(s):  
Natural Gas ◽  
Homotopy Analysis Method ◽  
Numerical Solutions ◽  
Gas Mixture ◽  
Analysis Method ◽  
Convergence Region ◽  
Reduced Order ◽  
Homotopy Analysis ◽  
And Control ◽  
Good Agreement

The aim of this article is to use the Homotopy Analysis Method (HAM) to pinpoint the optimal location of leakage in an inclined pipeline containing hydrogen-natural gas mixture by obtaining quick and accurate analytical solutions for nonlinear transportation equations. The homotopy analysis method utilizes a simple and powerful technique to adjust and control the convergence region of the infinite series solution using auxiliary parameters. The auxiliary parameters provide a convenient way of controlling the convergent region of series solutions. Numerical solutions obtained by HAM indicate that the approach is highly accurate, computationally very attractive and easy to implement. The solutions obtained with HAM have been shown to be in good agreement with those obtained using the method of characteristics (MOC) and the reduced order modelling (ROM) technique.

Homotopy Analysis Method

2017 ◽  
pp. 173-226
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Partial Differential Equations ◽  
Analytic Method ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
And Control

In this chapter, the analytic solution of nonlinear partial differential equations arising in heat transfer is obtained using the newly developed analytic method, namely the Homotopy Analysis Method (HAM). The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.

scholarly journals Approximate Analytical Solution for the Forced Korteweg-de Vries Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Vincent Daniel David ◽  
Mojtaba Nazari ◽  
Vahid Barati ◽  
Faisal Salah ◽  
Zainal Abdul Aziz
Keyword(s):  
Vries Equation ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
Convergence Region ◽  
Analytical Technique ◽  
De Vries ◽  
Homotopy Analysis ◽  
And Control ◽  
Good Agreement ◽  
Fkdv Equation

The forced Korteweg-de Vries (fKdV) equations are solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique which provides a novel way to obtain series solutions of such nonlinear problems. It has the auxiliary parameterℏ, where it is easy to adjust and control the convergence region of the series solution. Some examples of forcing terms are employed to analyse the behaviours of the HAM solutions for the different fKdV equations. Finally, this form of HAM solution is compared with the analytical soliton-type solution of fKdV equation as derived by Zhao and Guo. The results is found to be in good agreement with Zhao and Guo.

Homotopy Analysis Method for a Class of Holling II's Model

2013 ◽  
Vol 694-697 ◽  
pp. 2891-2894
Author(s):  
Xiu Rong Chen
Keyword(s):  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Solution Series ◽  
Homotopy Analysis ◽  
And Control

In this paper, the homotopy analysis method (HAM) is used for solving a class of Holling IIs model. The approximation solutions were obtained by HAM, and contain the auxiliary parameter h which provides us with a convenient way to adjust and control convergence region and rate of solution series. This results showed that this method is valid and feasible to the model.

scholarly journals Homotopy Analysis Method for Second-Order Boundary Value Problems of Integrodifferential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Ahmad El-Ajou ◽  
Omar Abu Arqub ◽  
Shaher Momani
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Convergent Series ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Analysis Method ◽  
Convergence Region ◽  
New Approach ◽  
Homotopy Analysis ◽  
And Control

In this paper, series solution of second-order integrodifferential equations with boundary conditions of the Fredholm and Volterra types by means of the homotopy analysis method is considered. The new approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. The homotopy analysis method provides us with a simple way to adjust and control the convergence region of the infinite series solution by introducing an auxiliary parameter. The proposed technique is applied to a few test examples to illustrate the accuracy, efficiency, and applicability of the method. The results reveal that the method is very effective, straightforward, and simple.

scholarly journals Series Solution of the System of Fuzzy Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
M. S. Hashemi ◽  
J. Malekinagad ◽  
H. R. Marasi
Keyword(s):  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Solution Series ◽  
Homotopy Analysis ◽  
Contour Plots ◽  
And Control

The homotopy analysis method (HAM) is proposed to obtain a semianalytical solution of the system of fuzzy differential equations (SFDE). The HAM contains the auxiliary parameterħ, which provides us with a simple way to adjust and control the convergence region of solution series. Concept ofħ-meshes and contour plots firstly are introduced in this paper which are the generations of traditionalh-curves. Convergency of this method for the SFDE has been considered and some examples are given to illustrate the efficiency and power of HAM.

Homotopy Analysis Method for a Prey-Predator System with Holling IV Functional Response

2014 ◽  
Vol 687-691 ◽  
pp. 1286-1291 ◽  
Author(s):  
Jia Shang Yu ◽  
Jia Ju Yu
Keyword(s):  
Functional Response ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Solution Series ◽  
Homotopy Analysis ◽  
And Control ◽  
Predator System

In this paper, the homotopy analysis method is used for solving a prey-predator system with holling IV functional response. The approximation solutions were obtained by homotopy analysis method, and contain the auxiliary parameter h which provides us with a convenient way to adjust and control convergence region and rate of solution series. This result showed that this method is valid and feasible for the system.

Analytic Solution of the Sharma-Tasso-Olver Equation by Homotopy Analysis Method

2010 ◽  
Vol 65 (4) ◽  
pp. 285-290 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Mahnaz Ashtiani ◽  
Esmail Babolian
Keyword(s):  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
And Control ◽  
Convergence Control Parameter

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the kink solution of the Sharma-Tasso-Olver equation. The homotopy analysis method is one of the analytic methods and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ħwhich gives us a simple way to adjust and control the convergence region of series solution. “Due to this reason, it seems reasonable to rename ħthe convergence-control parameter” [1].

scholarly journals COMPARISON OF A GENERAL SERIES EXPANSION METHOD AND THE HOMOTOPY ANALYSIS METHOD

2010 ◽  
Vol 24 (15) ◽  
pp. 1699-1706 ◽  
Author(s):  
CHENG-SHI LIU ◽  
YANG LIU
Keyword(s):  
Series Expansion ◽  
Homotopy Analysis Method ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Basis Functions ◽  
Analysis Method ◽  
Convergence Region ◽  
General Series ◽  
Homotopy Analysis ◽  
Series Expansion Method

A simple analytic tool, namely the general series expansion method, is proposed to find the solutions for nonlinear differential equations. A set of suitable basis functions [Formula: see text] is chosen such that the solution to the equation can be expressed by [Formula: see text]. In general, t0 can control and adjust the convergence region of the series solution such that our method has the same effect as the homotopy analysis method proposed by Liao, but our method is simpler and clearer. As a result, we show that the secret parameter h in the homotopy analysis methods can be explained by using our parameter t0. Therefore, our method reveals a key secret in the homotopy analysis method. For the purpose of comparison with the homotopy analysis method, a typical example is studied in detail.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

A Mathematical Study on Non-Linear MHD Boundary Layer Past a Porous Shrinking Sheet with Suction

2019 ◽  
Vol 2 (2) ◽  
pp. 32-48
Author(s):  
V. Ananthaswamy ◽  
K. Renganathan
Keyword(s):  
Boundary Layer ◽  
Homotopy Analysis Method ◽  
Dimensionless Temperature ◽  
Shrinking Sheet ◽  
Analysis Method ◽  
Approximate Analytical Expression ◽  
Boundary Layer Equations ◽  
Mhd Boundary Layer ◽  
Homotopy Analysis ◽  
Good Agreement

In this paper we discuss with magneto hydrodynamic viscous flow due to a shrinking sheet in the presence of suction. We also discuss two dimensional and axisymmetric shrinking for various cases. Using similarity transformation the governing boundary layer equations are converted into its dimensionless form. The transformed simultaneous ordinary differential equations are solved analytically by using Homotopy analysis method. The approximate analytical expression of the dimensionless velocity, dimensionless temperature and dimensionless concentration are derived using the Homotopy analysis method through the guessing solutions. Our analytical results are compared with the previous work and a good agreement is observed.

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