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 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.

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].

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.

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.

scholarly journals Constructing analytic solutions on the Tricomi equation

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Emran Khoshrouye Ghiasi ◽  
Reza Saleh
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Tricomi Equation ◽  
Homotopy Analysis ◽  
Optimal Values

AbstractIn this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

scholarly journals Approximate Solutions of Nonlinear Partial Differential Equations by Modifiedq-Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shaheed N. Huseen ◽  
Said R. Grace
Keyword(s):  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
Power Series Solution ◽  
Analysis Method ◽  
Exactly Solvable ◽  
Homotopy Analysis

A modifiedq-homotopy analysis method (mq-HAM) was proposed for solvingnth-order nonlinear differential equations. This method improves the convergence of the series solution in thenHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting thenth-order nonlinear differential equation in the form ofnfirst-order differential equations. The solution of thesendifferential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.

Analytical solutions of linear and nonlinear Klein-Fock-Gordon equation

2015 ◽  
Vol 4 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Sajida Rasheed
Keyword(s):  
Analytical Solutions ◽  
Gordon Equation ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Relativistic Version ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

AbstractIn this paper, we deal with some linear and nonlinear Klein-Fock-Gordon (KFG) equations, which is a relativistic version of the Schrödinger equation. The approximate analytical solutions are obtained by using the homotopy analysis method (HAM). The efficiency of the HAM is that it provides a practical way to control the convergence region of series solutions by introducing an auxiliary parameter }. Analytical results presented are in agreement with the existing results in open literature, which confirm the effectiveness of this method.

scholarly journals Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
O. Abdulaziz ◽  
I. Hashim ◽  
A. Saif
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Coefficient ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Hyperbolic Type ◽  
Series Solutions ◽  
Analysis Method ◽  
Fractional Partial Differential Equations ◽  
Fractional Pdes ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.

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.

Sign in / Sign up

Export Citation Format

Share Document