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1

The homotopy analysis Rangaig transform method for nonlinear partial differential equations

Ziane Djelloul, Cherif Mountassir Hamdi

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 2

111--122

EN

The idea suggested in this article is to combine the Rangaig transform with the homotopy analysis method in order to facilitate the solution of nonlinear partial differentia equations. This method may be called the homotopy analysis Rangaig transform method (HARTM). The proposed example results showed that HARTM is an effective method for solving nonlinear partial differential equations.

2

Numerical scheme methods for solving nonlinear pseudo-hyperbolic partial differential equations

Abdulazeez Sadeq Taha, Modanli Mahmut, Husien Ahmad Muhamad

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 4

5--15

EN

The numerical solutions to the nonlinear pseudo-hyperbolic partial differentia equation with nonlocal conditions are presented in this study. This equation is solved using the homotopy analysis technique (HAM) and the variational iteration method (VIM). Both strategies are compared and contrasted in terms of approximate and accurate solutions. The results show that the HAM technique is more appropriate, effective, and close to the exact solution than the VIM method. Finally, the graphical representations of the obtained results are given.

3

Nonlinear vibration of a beam resting on a nonlinear viscoelastic foundation traversed by a moving mass: a homotopy analysis

Pourseifi Mehdi, Monfared Mojtaba Mahmoudi

Engineering Transactions

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2022

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Vol. 70, iss. 4

355--371

EN

In this study, the dynamic response of an Euler-Bernoulli beam resting on the nonlinear viscoelastic foundation under the action of a moving mass by considering the stretching effect of the beam’s neutral axis is investigated. A Dirac-delta function is applied to model the location of the moving mass along the beam as well as its inertial effects. The Galerkin decomposition method is used to transform a partial dimensionless nonlinear differential equation of dynamic motion into an ordinary nonlinear differential equation. Subsequently, the well-known homotopy analysis method (HAM) is employed to obtain an approximate analytical solution of this equation. The validity and accuracy of the solution are examined numerically using the fourth-order Runge-Kutta method. Finally, several examples are provided to show the effects of parameters such as linear and nonlinear stiffness coefficients of a viscoelastic foundation, velocity of the moving mass as well as Coriolis force, centrifugal force and inertia force of the moving mass on the dynamic deflection of the beam.

4

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

Georgieva Atanaska

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

11--24

EN

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.

5

Existence and convergence results for Caputo fractional Volterra integro-differential equations

Hamoud A. A., Bani Issa M. Sh., Ghadle K. P., Abdulghani M.

Journal of Mathematics and Applications

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2018

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Vol. 41

109--121

EN

In this article, homotopy analysis method is successfully applied to find the approximate solution of Caputo fractional Volterra integro-differential equation. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximate. Moreover, we proved the existence and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

6

Unsteady hydromagnetic flow of Oldroyd-B fluid over an oscillatory stretching surface: a mathematical model

Khan S. U., Ali N.

Technical Sciences / University of Warmia and Mazury in Olsztyn

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2017

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nr 20(1)

87--100

EN

In the present work, we have studied an unsteady, two-dimensional boundary layer flow of a magnetohydrodynamics (MHD) Oldroyd-B fluid over an oscillatory stretching surface. The problem is modeled by using constitutive equations. The number of independent variables in the governing equations are reduced by using appropriate dimensionless variables. The analytical solution is computed by using homotopy analysis method. The influences of various physical parameters such as Deborah numbers, ratio of angular frequency to stretching rate parameter and Hartmann number on time-series of velocity and transverse velocity profiles at different time instants are investigated and discussed quantitatively with the help of various graphs. It is observed that amplitude of velocity increases by increasing ratio of oscillating frequency to stretching rate parameter while decreases by increasing Hartmann number. It is further observed that the magnitude of velocity decreases by increasing Hartmann number and Deborah numbers in the terms of relaxation time parameter.

7

Stagnation point flow of Eyring Powell fluid in a vertical cylinder with heat transfer

Rehman A., Achakzai S., Nadeem S., Iqbal S.

Journal of Power Technologies

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2016

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Vol. 96, nr 1

57--62

EN

In this paper an analysis is carried out to examine the effects of natural convection heat transfer for steady boundary layer flow of an Eyring Powell fluid flowing through a vertical circular cylinder. The governing partial differential equations along with the boundary conditions are reduced to dimensionless form by using the boundary layer approximation and applying suitable similarity transformations. The resulting nonlinear coupled system of ordinary differential equations subject to the appropriate boundary conditions is solved using the analytic technique homotopy analysis method (HAM). The effects of the physical parameters on the flow and heat transfer characteristics are presented. The behavior of skinfriction coefficient and Nusselt numbers are also studied for different parameters.

8

Homotopy approach for solving two-dimensional integral equations of the second kind

Hetmaniok E., Nowak I., Słota D., Zielonka A.

Computer Assisted Methods in Engineering and Science

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2016

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Vol. 23, no. 1

19--41

EN

In this paper, the two-dimensional linear and nonlinear integral equations of the second kind is analyzed. The homotopy analysis method (HAM) is used for determining the solution of the investigated equation. In this method, a solution is sought in the series form. It is shown that if this series is convergent, its sum gives the solution of the considered equation. The sufficient condition for the convergence of the series is also presented. Additionally, the error of approximate solution, obtained as partial sum of the series, is estimated. Application of the HAM is illustrated by examples.

9

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

Hetmaniok E., Słota D., Wituła R., Zielonka A.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2015

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Vol. 63, nr 3

583--590

EN

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.

10

Solving of the two-dimensional unsteady heat transfer problem by using the homotopy analysis method

Brociek R., Hetmaniok E., Matlak J., Słota D.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

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2014

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z. 4

89--102

PL

W artykule opisano rozwiązanie dwuwymiarowego niestacjonarnego zagadnienia przewodzenia ciepła przy wykorzystaniu homotopijnej metody analizy. W metodzie tej tworzony jest szereg funkcyjny. Podano warunek wystarczający zbieżności tego szeregu, a także oszacowanie błędu rozwiązania przybliżonego, które uzyskujemy, biorąc sumę częściową szeregu.

EN

In this paper a solution of the two-dimensional unsteady heat transfer problem by using the homotopy analysis method is described. In presented method the functional series is generated. This paper contains the sufficient condition for convergence of this series. We also give the estimation of error of the approximate solution obtained by taking the partial sum of received series.

11

Usage of the homotopy analysis method for determining the temperature in the casting-mould system

Hetmaniok E., Pleszczyński M., Słota D., Zielonka A.

Hutnik, Wiadomości Hutnicze

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2014

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Vol. 81, nr 1

50--54

EN

In the paper we present an application of the homotopy analysis method for solving the heat conduction equation in the non-homogeneous casting-mould system with the perfect contact condition between casting and mould. In the described method the series is created, elements of which satisfy some differential equation resulting from the considered problem. If this series is convergent, then its sum gives the solution of initial problem. In the paper we give the sufficient condition for this convergence and the estimation of error of approximate solution which we obtain by taking only the partial sum of considered series. An example illustrating the application of investigated method is presented as well.

PL

W pracy przedstawiono zastosowanie homotopijnej metody analizy do rozwiązania równania przewodnictwa ciepła w niejednorodnym układzie odlew-forma, przy założeniu idealnego kontaktu na styku odlewu i formy. W opisywanej metodzie tworzony jest szereg, którego elementy spełniają pewne równanie różniczkowe wynikające z rozważanego zagadnienia. Jeśli szereg ten jest zbieżny, to jego suma jest rozwiązaniem wyjściowego równania. W pracy podano warunek wystarczający tej zbieżności oraz oszacowanie błędu rozwiązania przybliżonego, które uzyskujemy ograniczając się do sumy częściowej rozważanego szeregu. Przedstawiono także przykład ilustrujący zastosowanie omawianej metody.

12

Modelling of flow and heat transfer in a generalized second grade fluid

Hayat T., Ellahi R., Asghar S.

International Journal of Applied Mechanics and Engineering

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2008

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Vol. 13, no 1

101-121

EN

The flow of a second grade fluid past a porous plate has been studied using a modified model of second grade fluid that has shear dependent viscosity and can predict the normal stress difference. The boundary value problem subject to two different sets of boundary conditions is investigated. In the first instance, we consider that the plate is at temperature higher than the fluid. The second case deals with the analysis of an insulated plate. The differential equations governing the flow are solved using the homotopy analysis method (HAM). Expressions for velocity and temperature profiles are plotted and discussed.