Discussing a Solution to Nonlinear Duffing Oscillator with Fractional Derivatives Using Homotopy Analysis Method (HAM)

2021 ◽  
pp. 57-81
Author(s):  
C. L. Ejikeme ◽  
M. O. Oyesanya ◽  
D. F. Agbebaku ◽  
M. B. Okofu
Keyword(s):  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Duffing Oscillator ◽  
Analysis Method ◽  
Homotopy Analysis

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.

scholarly journals On the Application of Homotopy Analysis Method to Fractional Differential Equations

2021 ◽  
pp. 1-18
Author(s):  
Khalid Suliman Aboodh ◽  
Abu baker Ahmed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
Fractional Differential ◽  
Convergence Control Parameter

In this paper, an attempt has been made to obtain the solution of linear and nonlinear fractional differential equations by applying an analytic technique, namely the homotopy analysis method (HAM). The fractional derivatives are described by Caputo’s sense. By this method, the solution considered as the sum of an infinite series, which converges rapidly to exact solution with the help of the nonzero convergence control parameter ℏ. Some examples are given to show the efficiently and accurate of this method. The solutions obtained by this method has been compared with exact solution. Also our graphical represented of the solutions have been given by using MATLAB software.

scholarly journals Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Asmat Ara
Keyword(s):  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Higher Dimensions ◽  
Analysis Method ◽  
Transform Method ◽  
Coupled Equations ◽  
Fractional Schrödinger Equations ◽  
Differential Transform ◽  
Homotopy Analysis

We construct the approximate solutions of the time-fractional Schrödinger equations, with zero and nonzero trapping potential, by homotopy analysis method (HAM). The fractional derivatives, in the Caputo sense, are used. The method is capable of reducing the size of calculations and handles nonlinear-coupled equations in a direct manner. The results show that HAM is more promising, convenient, efficient and less computational than differential transform method (DTM), and easy to apply in spaces of higher dimensions as well.

scholarly journals Homotopy Analysis Method for Solving Foam Drainage Equation with Space- and Time-Fractional Derivatives

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Hadi Hosseini Fadravi ◽  
Hassan Saberi Nik ◽  
Reza Buzhabadi
Keyword(s):  
Analytical Solution ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Foam Drainage Equation ◽  
Foam Drainage

The analytical solution of the foam drainage equation with time- and space-fractional derivatives was derived by means of the homotopy analysis method (HAM). The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons show that the homotopy analysis method is very effective and convenient. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained.

Homotopy Analysis Method for Nonlinear Differential Equations with Fractional Orders

2008 ◽  
Vol 63 (5-6) ◽  
pp. 241-247 ◽  
Author(s):  
Yin-Ping Liu ◽  
Zhi-Bin Li
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Nonlinear Differential Equations ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fractional Orders

The aim of this paper is to solve nonlinear differential equations with fractional derivatives by the homotopy analysis method. The fractional derivative is described in Caputo’s sense. It shows that the homotopy analysis method not only is efficient for classical differential equations, but also is a powerful tool for dealing with nonlinear differential equations with fractional derivatives.

scholarly journals Application of Extended Homotopy Analysis Method to the Two-Degree-of-Freedom Coupled van der Pol-Duffing Oscillator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Y. H. Qian ◽  
S. M. Chen ◽  
L. Shen
Keyword(s):  
Homotopy Analysis Method ◽  
Duffing Oscillator ◽  
Approximate Solutions ◽  
Analytical Approximation ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Different Types ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

The extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing oscillator. Meanwhile, the comparisons between the results of the EHAM and standard Runge-Kutta numerical method are also presented. The results demonstrate that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions. For EHAM as an analytical approximation method, we are not sure whether it can apply to all of the nonlinear systems; we can only verify its effectiveness through specific cases. As a result of the existence of nonlinear terms, we must study different types of systems, no matter from the complication of calculation and physical significance.

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 Davey-Stewartson Equation with Fractional Coordinate Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
H. Jafari ◽  
K. Sayevand ◽  
Yasir Khan ◽  
M. Nazari
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Good Agreement

We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under consideration are discussed. Furthermore, comparisons indicate that there is a very good agreement between the solutions of homotopy analysis method and the exact solutions in terms of accuracy.

The Series Solution of Fractional Partial Differential Equations via Homotopy Analysis Method

2014 ◽  
Vol 530-531 ◽  
pp. 613-616
Author(s):  
Jia Ju Yu
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Analysis Method ◽  
Fractional Partial Differential Equations ◽  
Homotopy Analysis ◽  
Fractional Equation

In this letter, we apply the homotopy analysis method (HAM) to obtain analytical solution of the fractional equation where the fractional derivatives are Caputo sense. The example is given to show the efficiency of the method.

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