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 Homotopy Approach for Integrodifferential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 904 ◽  
Author(s):  
Damian Słota ◽  
Edyta Hetmaniok ◽  
Roman Wituła ◽  
Krzysztof Gromysz ◽  
Tomasz Trawiński
Keyword(s):  
Approximate Solution ◽  
Reinforced Concrete ◽  
Homotopy Analysis Method ◽  
Concrete Beam ◽  
Reinforced Concrete Beam ◽  
Integrodifferential Equations ◽  
Sufficient Condition ◽  
Analysis Method ◽  
Estimation Of Error ◽  
Homotopy Analysis

In this paper, we present the application of the homotopy analysis method for solving integrodifferential equations. In this method, a series is created, the successive elements of which are determined by calculating the appropriate integral of the previous element. In this elaboration, we prove that, if this series is convergent, then its sum is the solution of the objective equation. We formulate and prove the sufficient condition of this convergence, and we give also the estimation of error of an approximate solution obtained by taking the partial sum of the considered series. Moreover, we present in this paper the example of using the investigated method for determining the vibrations of the freely supported reinforced concrete beam as well as for solving the equation of movement of the electromagnet jumper mechanical system.

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 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 Application of the Homotopy Method for Fractional Inverse Stefan Problem

Energies ◽  
2020 ◽  
Vol 13 (20) ◽  
pp. 5474
Author(s):  
Damian Słota ◽  
Agata Chmielowska ◽  
Rafał Brociek ◽  
Marcin Szczygieł
Keyword(s):  
Heat Flux ◽  
Approximate Solution ◽  
Temperature Distribution ◽  
Continuous Function ◽  
Homotopy Analysis Method ◽  
Input Data ◽  
Sufficient Condition ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
The One

The paper presents an application of the homotopy analysis method for solving the one-phase fractional inverse Stefan design problem. The problem was to determine the temperature distribution in the domain and functions describing the temperature and the heat flux on one of the considered area boundaries. It was demonstrated that if the series constructed for the method is convergent then its sum is a solution of the considered equation. The sufficient condition of this convergence was also presented as well as the error of the approximate solution estimation. The paper also includes the example presenting the application of the described method. The obtained results show the usefulness of the proposed method. The method is stable for the input data disturbances and converges quickly. The big advantage of this method is the fact that it does not require discretization of the area and the solution is a continuous function.

A New Semi-Analytical Solution of the Telegraph Equation with Integral Condition

2011 ◽  
Vol 66 (12) ◽  
pp. 760-768 ◽  
Author(s):  
S. Abbasbandy ◽  
H. Roohani Ghehsarehb
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Iterative Algorithm ◽  
Homotopy Analysis Method ◽  
Integral Condition ◽  
Telegraph Equation ◽  
Current Work ◽  
Analysis Method ◽  
Numerical Examples ◽  
Homotopy Analysis

In the current work, the telegraph equation in its general form and with an integral condition is investigated. Also the well-known homotopy analysis method (HAM) is applied and an interesting iterative algorithm is proposed for solving the problem in general form. Some numerical examples are given and compared with the exact solution to show the effectiveness of the proposed method.

Optimum Path of a Flying Object with Exponentially Decaying Density Medium

2009 ◽  
Vol 64 (7-8) ◽  
pp. 431-438 ◽  
Author(s):  
Said Abbasbandy ◽  
Mehmet Pakdemirli ◽  
Elyas Shivanian
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Dimensionless Form ◽  
Optimum Path ◽  
Homotopy Analysis

AbstractIn this paper, a differential equation describing the optimum path of a flying object is derived. The density of the fluid is assumed to be exponentially decaying with altitude. The equation is cast in to a dimensionless form and the exact solution is given. This equation is then analyzed by homotopy analysis method (HAM). The results showed in the figures reveal that this method is very effective and convenient.

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.

Influence of Wall Properties on the Peristaltic Flow of a Nanofluid in View of the Exact Solutions: Comparisons with Homotopy Analysis Method

2014 ◽  
Vol 69 (5-6) ◽  
pp. 199-206 ◽  
Author(s):  
Abdelhalim Ebaid ◽  
Salem H. Alatawi
Keyword(s):  
Heat Transfer ◽  
Exact Solution ◽  
Heat Transfer Coefficient ◽  
Exact Solutions ◽  
Transfer Coefficient ◽  
Homotopy Analysis Method ◽  
Peristaltic Flow ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
The Heat Transfer Coefficient

No doubt, the exact solution of any physical system is considered optimal when it is available. Such exact solution is of great importance not only in validating the accuracy of the approximate solution obtained for the same problem but also to derive the correct physical interpretation of the involved physical phenomena. In this paper, the system of linear and nonlinear partial differential equations describing the peristaltic flow of a nanofluid in a channel with compliant walls has been solved exactly. These exact solutions have been implemented to explore the exact effects of Prandtl number Pr, thermophoresis parameter NT, Brownian motion parameter NB, and Eckert number Ec on the temperature, the nanoparticle concentration profiles, and the heat transfer coefficient Z(x). In addition, the exact results have been compared with a very recent work via the homotopy analysis method for the same problem. Although these comparisons showed that the published approximate results coincide with the current exact analysis, a few remarkable differences have been detected for the behaviour of the heat transfer coefficient.

Solitary Solution of Nonlinear Equation by Homotopy Analysis Method

2011 ◽  
Vol 130-134 ◽  
pp. 3668-3671
Author(s):  
Xiu Rong Chen ◽  
Wen Shan Cui
Keyword(s):  
Exact Solution ◽  
Nonlinear Equation ◽  
Homotopy Analysis Method ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
Solitary Solution ◽  
Homotopy Analysis

In this paper, we apply homotopy analysis method to solve nonlinear equation and successfully obtain the bell-shaped solitary solution to the nonlinear equation. Comparison between our solution and the exact solution shows that homotopy analysis method is effective and valid for nonlinear problems.

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