Application of the Homotopy Method for Fractional Inverse Stefan Problem

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.

Download Full-text

The One Step Optimal Homotopy Analysis Method to Circular Porous Slider

Mathematical Problems in Engineering ◽

10.1155/2012/135472 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Mohammad Ghoreishi ◽

Ahmad Izani B. Md. Ismail ◽

Abdur Rashid

Keyword(s):

Reynolds Number ◽

Approximate Solution ◽

Homotopy Analysis Method ◽

Momentum Equation ◽

Analysis Method ◽

Lateral Velocity ◽

Optimal Homotopy Analysis Method ◽

Homotopy Analysis ◽

One Step ◽

The One

An incompressible Newtonian fluid is forced through the porous of a circular slider which is moving laterally on a horizontal plan. In this paper, we introduce and apply the one step Optimal Homotopy Analysis Method (one step OHAM) to the problem of the circular porous slider where a fluid is injected through the porous bottom. The effects of mass injection and lateral velocity on the heat generated by viscous dissipation are investigated by solving the governing boundary layer equations using one step optimal homotopy technique. The approximate solution for the coupled nonlinear ordinary differential equations resulting from the momentum equation is obtained and discussed for different values of the Reynolds number of the velocity field. The solution obtained is also displayed graphically for various values of the Reynolds number and it is shown that the one step OHAM is capable of finding the approximate solution of circular porous slider.

Download Full-text

Homotopy Approach for Integrodifferential Equations

Mathematics ◽

10.3390/math7100904 ◽

2019 ◽

Vol 7 (10) ◽

pp. 904 ◽

Cited By ~ 4

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.

Download Full-text

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

Bulletin of the Polish Academy of Sciences Technical Sciences ◽

10.1515/bpasts-2015-0068 ◽

2015 ◽

Vol 63 (3) ◽

pp. 583-590 ◽

Cited By ~ 2

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.

Download Full-text

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

The Scientific World JOURNAL ◽

10.1155/2014/465951 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 2

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.

Download Full-text

Optimal Homotopic Exploration of features of Cattaneo-Christov model in Second Grade Nanofluid flow via Darcy-Forchheimer medium subject to Viscous Dissipation and Thermal Radiation

Combinatorial Chemistry & High Throughput Screening ◽

10.2174/1386207324666210903144447 ◽

2021 ◽

Vol 24 ◽

Author(s):

Ghulam Rasool ◽

Anum Shafiq ◽

Yu-Ming Chu ◽

Muhammad Shoaib Bhutta ◽

Amjad Ali

Keyword(s):

Temperature Distribution ◽

Differential Equations ◽

Thermal Radiation ◽

Skin Friction ◽

Homotopy Analysis Method ◽

Stream Velocity ◽

Analysis Method ◽

Nanofluid Flow ◽

Optimal Homotopy Analysis Method ◽

Homotopy Analysis

Introduction: In this article Optimal Homotopy analysis method (oHAM) is used for exploration of the features of Cattaneo-Christov model in viscous and chemically reactive nanofluid flow through a porous medium with stretching velocity at the solid/sheet surface and free stream velocity at the free surface. Methods: The two important aspects, Brownian motion and Thermophoresis are considered. Thermal radiation is also included in present model. Based on the heat and mass flux, the Cattaneo-Christov model is implemented on the Temperature and Concentration distributions. The governing Partial Differential Equations (PDEs) are converted into Ordinary Differential Equations (ODEs) using similarity transformations. The results are achieved using the optimal homotopy analysis method (oHAM). The optimal convergence and residual errors have been calculated to preserve the validity of the model. Results: The results are plotted graphically to see the variations in three main profiles i.e. momentum, temperature and concentration profile. Conclusion: The outcomes indicate that skin friction enhances due to implementation of Darcy medium. It is also noted that the relaxation time parameter results in enhancement of the temperature distribution. Thermal radiation enhances the temperature distribution and so is the case with skin friction.

Download Full-text

One Dimensional Model of Martensitic Transformation Solved by Homotopy Analysis Method

Zeitschrift für Naturforschung A ◽

10.5560/zna.2012-0016 ◽

2012 ◽

Vol 67 (5) ◽

pp. 230-238 ◽

Cited By ~ 1

Author(s):

Chen Xuan ◽

Cheng Peng ◽

Yongzhong Huo

Keyword(s):

Phase Transition ◽

Martensitic Transformation ◽

Analytical Solutions ◽

Homotopy Analysis Method ◽

Lagrange Equation ◽

Nonlinear Ordinary Differential Equation ◽

Physical Parameters ◽

Analysis Method ◽

Homotopy Analysis ◽

The One

The homotopy analysis method (HAM) is applied to solve a nonlinear ordinary differential equation describing certain phase transition problem in solids. Both bifurcation conditions and analytical solutions are obtained simultaneously for the Euler-Lagrange equation of the martensitic transformation. HAM is capable of providing an analytical expression for the bifurcation condition to judge the occurrence of the phase transition, while other numerical techniques have difficulties in bifurcation analysis. The convergence of the analytical solutions on the one hand can be adjusted by the auxiliary parameter and on the other hand is always obtainable for all relevant physical parameters satisfying the bifurcation condition.

Download Full-text