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 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 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.

scholarly journals An analytical method for solving the two-phase inverse Stefan problem

2015 ◽  
Vol 63 (3) ◽  
pp. 583-590 ◽  
Author(s):  
E. Hetmaniok ◽  
D. Słota ◽  
R. Wituła ◽  
A. Zielonka
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Stefan Problem ◽  
Homotopy Analysis Method ◽  
Analytical Method ◽  
Original Equation ◽  
Sufficient Condition ◽  
Analysis Method ◽  
Two Phase ◽  
Homotopy Analysis

Abstract In the paper we present an application of the homotopy analysis method for solving the two-phase inverse Stefan problem. In the proposed approach a series is created, having elements which satisfy some differential equation following from the investigated problem. We reveal, in the paper, that if this series is convergent then its sum determines the solution of the original equation. A sufficient condition for this convergence is formulated. Moreover, the estimation of the error of the approximate solution, obtained by taking the partial sum of the considered series, is given. Additionally, we present an example illustrating an application of the described method.

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.

Homotopy analysis method approximate solution for fuzzy pantograph equation

2021 ◽  
Vol 14 (3) ◽  
pp. 286
Author(s):  
A.H. Shather ◽  
A.F. Jameel ◽  
N.R. Anakira ◽  
A.K. Alomari ◽  
Azizan Saaban
Keyword(s):  
Approximate Solution ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Pantograph Equation ◽  
Homotopy Analysis

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 Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

2021 ◽  
Vol 54 (1) ◽  
pp. 11-24
Author(s):  
Atanaska Georgieva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Homotopy Analysis Method ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

Approximate solution of a piecewise linear–nonlinear oscillator using the homotopy analysis method

2017 ◽  
Vol 24 (19) ◽  
pp. 4551-4562 ◽  
Author(s):  
Jixiong Fei ◽  
Bin Lin ◽  
Shuai Yan ◽  
Xiaofeng Zhang
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Homotopy Analysis Method ◽  
Nonlinear Oscillator ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Nonlinear Damping ◽  
Analysis Method ◽  
Bifurcation Diagrams ◽  
Homotopy Analysis

Most of the piecewise oscillators in engineering fields include nonlinear damping or stiffness and the contained damping or stiffness is strongly nonlinear, but to the authors’ knowledge little attention has been paid to those systems. Thus, in the present paper, a sinusoidal excited piecewise linear–nonlinear oscillator is analyzed. The mathematical model of the oscillator is described by a combination of a linear and a nonlinear differential equation which contains strong nonlinear terms of stiffness. An approximate solution for the oscillator is proposed by using the homotopy analysis method and matching method. The validity of the proposed solution is verified by comparing it with the exact solution. It is found that the approximate solution is in good agreement with the exact solution. The influence of some system parameters on the dynamical behavior of the oscillator is also investigated by the bifurcation diagrams of these parameters. From these bifurcation diagrams, one can observe the motion of the oscillator directly.

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