The Homotopy Analysis Method for Solving Multi- Fractional Order IntegroDifferential Equations

2011 ◽  
Vol 14 (3) ◽  
pp. 139-143
Author(s):  
Ibtisam K. Hanan ◽  
Keyword(s):  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Integrodifferential Equations ◽  
Analysis Method ◽  
Homotopy Analysis

scholarly journals The Convergence Study of the Homotopy Analysis Method for Solving Nonlinear Volterra-Fredholm Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Behzad Ghanbari
Keyword(s):  
Exact Solution ◽  
Homotopy Analysis Method ◽  
Integrodifferential Equations ◽  
Sufficient Condition ◽  
Analysis Method ◽  
Analytic Treatment ◽  
Homotopy Analysis ◽  
Convergence Study ◽  
Technique Comparison

We aim to study the convergence of the homotopy analysis method (HAM in short) for solving special nonlinear Volterra-Fredholm integrodifferential equations. The sufficient condition for the convergence of the method is briefly addressed. Some illustrative examples are also presented to demonstrate the validity and applicability of the technique. Comparison of the obtained results HAM with exact solution shows that the method is reliable and capable of providing analytic treatment for solving such equations.

scholarly journals On Spectral Homotopy Analysis Method for Solving Linear Volterra and Fredholm Integrodifferential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Z. Pashazadeh Atabakan ◽  
A. Kılıçman ◽  
A. Kazemi Nasab
Keyword(s):  
Exact Solutions ◽  
Homotopy Analysis Method ◽  
Integrodifferential Equations ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Spectral Homotopy Analysis Method

A modification of homotopy analysis method (HAM) known as spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations. Some examples are given in order to test the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to SHAM results and exact solutions.

scholarly journals Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1167
Author(s):  
Said Mesloub ◽  
Saleem Obaidat
Keyword(s):  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
Initial Boundary Value

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.

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 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.

Homotopy analysis method for solving Abel differential equation of fractional order

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Hossein Jafari ◽  
Khosro Sayevand ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Numerical Results ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Abel Differential Equation

AbstractIn this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.

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