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Journal articles on the topic 'Homotopy analysis method'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

He, Ji-Huan. "Comparison of homotopy perturbation method and homotopy analysis method." Applied Mathematics and Computation 156, no. 2 (September 2004): 527–39. http://dx.doi.org/10.1016/j.amc.2003.08.008.

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2

Ahmad Soltani, L., E. Shivanian, and Reza Ezzati. "Shooting homotopy analysis method." Engineering Computations 34, no. 2 (April 18, 2017): 471–98. http://dx.doi.org/10.1108/ec-10-2015-0329.

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Purpose The purpose of this paper is to present a new method based on the homotopy analysis method (HAM) with the aim of fast searching and calculating multiple solutions of nonlinear boundary value problems (NBVPs). Design/methodology/approach A major problem with the previously modified HAM, namely, predictor homotopy analysis method, which is used to predict multiplicity of solutions of NBVPs, is a time-consuming computation of high-order HAM-approximate solutions due to a symbolic variable namely “prescribed parameter”. The proposed new technique which is based on traditional shooting method, and the HAM cuts the dependency on the prescribed parameter. Findings To demonstrate the computational efficiency, the mentioned method is implemented on three important nonlinear exactly solvable differential equations, namely, the nonlinear MHD Jeffery–Hamel flow problem, the nonlinear boundary value problem arising in heat transfer and the strongly nonlinear Bratu problem. Originality/value The more high-order approximate solutions are computable, multiple solutions are easily searched and discovered and the more accurate solutions can be obtained depending on how nonhomogeneous boundary conditions are transcribed to the homogeneous boundary conditions.

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3

Liao, Shijun. "Comparison between the homotopy analysis method and homotopy perturbation method." Applied Mathematics and Computation 169, no. 2 (October 2005): 1186–94. http://dx.doi.org/10.1016/j.amc.2004.10.058.

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4

He, Ji-Huan. "Homotopy Perturbation Method with an Auxiliary Term." Abstract and Applied Analysis 2012 (2012): 1–7. http://dx.doi.org/10.1155/2012/857612.

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The two most important steps in application of the homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be such constructed that when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach to construction of the homotopy equation with an auxiliary term; Dufing equation is used as an example to illustrate the solution procedure.

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5

Nave, Ophir, Shlomo Hareli, and Vladimir Gol’dshtein. "Singularly perturbed homotopy analysis method." Applied Mathematical Modelling 38, no. 19-20 (October 2014): 4614–24. http://dx.doi.org/10.1016/j.apm.2014.03.013.

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6

Huseen, Shaheed N. "A Comparative Study of q-Homotopy Analysis Method and Liao’s Optimal Homotopy Analysis Method." Advances in Computer and Communication 1, no. 1 (December 15, 2020): 36–45. http://dx.doi.org/10.26855/acc.2020.12.004.

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7

Liang, Songxin, and David J. Jeffrey. "Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation." Communications in Nonlinear Science and Numerical Simulation 14, no. 12 (December 2009): 4057–64. http://dx.doi.org/10.1016/j.cnsns.2009.02.016.

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8

LIU, CHENG-SHI, and YANG LIU. "COMPARISON OF A GENERAL SERIES EXPANSION METHOD AND THE HOMOTOPY ANALYSIS METHOD." Modern Physics Letters B 24, no. 15 (June 20, 2010): 1699–706. http://dx.doi.org/10.1142/s0217984910024079.

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A simple analytic tool, namely the general series expansion method, is proposed to find the solutions for nonlinear differential equations. A set of suitable basis functions [Formula: see text] is chosen such that the solution to the equation can be expressed by [Formula: see text]. In general, t0 can control and adjust the convergence region of the series solution such that our method has the same effect as the homotopy analysis method proposed by Liao, but our method is simpler and clearer. As a result, we show that the secret parameter h in the homotopy analysis methods can be explained by using our parameter t0. Therefore, our method reveals a key secret in the homotopy analysis method. For the purpose of comparison with the homotopy analysis method, a typical example is studied in detail.

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9

Das, S., and P. K. Gupta. "Application of homotopy perturbation method and homotopy analysis method to fractional vibration equation." International Journal of Computer Mathematics 88, no. 2 (November 28, 2010): 430–41. http://dx.doi.org/10.1080/00207160903474214.

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10

Motsa, Sandile Sydney, Precious Sibanda, Gerald T. Marewo, and Stanford Shateyi. "A Note on Improved Homotopy Analysis Method for Solving the Jeffery-Hamel Flow." Mathematical Problems in Engineering 2010 (2010): 1–11. http://dx.doi.org/10.1155/2010/359297.

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This paper presents the solution of the nonlinear equation that governs the flow of a viscous, incompressible fluid between two converging-diverging rigid walls using an improved homotopy analysis method. The results obtained by this new technique show that the improved homotopy analysis method converges much faster than both the homotopy analysis method and the optimal homotopy asymptotic method. This improved technique is observed to be much more accurate than these traditional homotopy methods.

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11

Hashim, I., O. Abdulaziz, and S. Momani. "Homotopy analysis method for fractional IVPs." Communications in Nonlinear Science and Numerical Simulation 14, no. 3 (March 2009): 674–84. http://dx.doi.org/10.1016/j.cnsns.2007.09.014.

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12

Chu, Moody T., T. Y. Li, and Tim Sauer. "Homotopy Method for General $\lambda $-Matrix Problems." SIAM Journal on Matrix Analysis and Applications 9, no. 4 (October 1988): 528–36. http://dx.doi.org/10.1137/0609043.

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13

Hon, Benny Y. C., Engui Fan, and Qi Wang. "Homotopy Analysis Method for Ablowitz–Ladik Lattice." Zeitschrift für Naturforschung A 66, no. 10-11 (November 1, 2011): 599–605. http://dx.doi.org/10.5560/zna.2011-0022.

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In this paper, the homotopy analysis method is successfully applied to solve the systems of differential-difference equations. The Ablowitz-Ladik lattice system are chosen to illustrate the method. Comparisons between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective and simple in solving systems of differential-difference equations.

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14

Nargund, Achala, R. Madhusudhan, and S. B. Sathyanarayana. "HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS." JOURNAL OF ADVANCES IN PHYSICS 10, no. 3 (October 6, 2015): 2825–33. http://dx.doi.org/10.24297/jap.v10i3.1322.

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In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysisÂ method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-SykesÂ plot for the region of convergence. We have applied Pade for the HAM series to identify theÂ singularity and reflect it in the graph. The HAM is a analytical technique which is used toÂ solve non-linear problems to generate a convergent series. HAM gives complete freedom toÂ choose the initial approximation of the solution, it is the auxiliary parameter h which gives us aÂ convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

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15

Fernández, Francisco M. "On two new applications of the homotopy analysis method and the homotopy perturbation method." Physica Scripta 81, no. 3 (February 17, 2010): 037002. http://dx.doi.org/10.1088/0031-8949/81/03/037002.

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16

Chowdhury, M. S. H., I. Hashim, and O. Abdulaziz. "Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems." Communications in Nonlinear Science and Numerical Simulation 14, no. 2 (February 2009): 371–78. http://dx.doi.org/10.1016/j.cnsns.2007.09.005.

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17

Huseen, Shaheed N., Said R.Grace, and Magdy A. El-Tawil. "The Optimal q-Homotopy Analysis Method (Oq-HAM)." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 11, no. 8 (November 20, 2013): 2859–66. http://dx.doi.org/10.24297/ijct.v11i8.3003.

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In this paper, an optimal q-homotopy analysis method (Oq-HAM) is proposed. We present some examples to show the reliability and efficiency of the method. It is compared with the one-step optimal homotopy analysis method. The results reveal that the Oq-HAM has more accuracy to determine the convergence-control parameter than the one-step optimal HAM.

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18

Chen, Xiu Rong, and Wen Shan Cui. "Solitary Solution of Nonlinear Equation by Homotopy Analysis Method." Applied Mechanics and Materials 130-134 (October 2011): 3668–71. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.3668.

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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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19

Biazar, Jafar, Zainab Ayati, and Mohammad Reza Yaghouti. "Homotopy perturbation method for homogeneous Smoluchowsk's equation." Numerical Methods for Partial Differential Equations 26, no. 5 (July 2, 2009): 1146–53. http://dx.doi.org/10.1002/num.20480.

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20

Olvera, Daniel, and Alex Elías-Zúñiga. "Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/486509.

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We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameterhby following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

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21

Abbasbandy, S. "Homotopy analysis method for the Kawahara equation." Nonlinear Analysis: Real World Applications 11, no. 1 (February 2010): 307–12. http://dx.doi.org/10.1016/j.nonrwa.2008.11.005.

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22

Elwakil, Elsayed Abd Elaty, and Mohamed Aly Abdou. "New Applications of the Homotopy Analysis Method." Zeitschrift für Naturforschung A 63, no. 7-8 (August 1, 2008): 385–92. http://dx.doi.org/10.1515/zna-2008-7-801.

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An analytical technique, namely the homotopy analysis method (HAM), is applied using a computerized symbolic computation to find the approximate and exact solutions of nonlinear evolution equations arising in mathematical physics. The HAM is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The validity and reliability of the method is tested by application on three nonlinear problems, namely theWhitham-Broer-Kaup equations, coupled Korteweg-de Vries equation and coupled Burger’s equations. Comparisons are made between the results of the HAM with the exact solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics.

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23

Sakuma, Takayuki. "Homotopy Analysis Method Applied SABR and XVA." Wilmott 2019, no. 99 (January 2019): 62–69. http://dx.doi.org/10.1002/wilm.10738.

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24

Liu, YanBin, and YuShu Chen. "KBM method based on the homotopy analysis." Science China Physics, Mechanics and Astronomy 54, no. 6 (February 19, 2011): 1137–40. http://dx.doi.org/10.1007/s11433-011-4250-z.

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25

El-Tawil, Magdy A., and Shaheed N. Huseen. "On convergence of q-homotopy analysis method." International Journal of Contemporary Mathematical Sciences 8 (2013): 481–97. http://dx.doi.org/10.12988/ijcms.2013.13048.

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26

Xuan, Liu, and Haidong Qu. "Homotopy analysis method for Cauchy Riemann equations." International Journal of Mathematical Analysis 8 (2014): 1099–103. http://dx.doi.org/10.12988/ijma.2014.44108.

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27

Liu, Cheng-shi. "The essence of the homotopy analysis method." Applied Mathematics and Computation 216, no. 4 (April 2010): 1299–303. http://dx.doi.org/10.1016/j.amc.2010.02.022.

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28

Turkyilmazoglu, M. "A note on the homotopy analysis method." Applied Mathematics Letters 23, no. 10 (October 2010): 1226–30. http://dx.doi.org/10.1016/j.aml.2010.06.003.

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29

Abbasbandy, S. "Homotopy analysis method for heat radiation equations." International Communications in Heat and Mass Transfer 34, no. 3 (March 2007): 380–87. http://dx.doi.org/10.1016/j.icheatmasstransfer.2006.12.001.

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30

Mastroberardino, Antonio. "Homotopy analysis method applied to electrohydrodynamic flow." Communications in Nonlinear Science and Numerical Simulation 16, no. 7 (July 2011): 2730–36. http://dx.doi.org/10.1016/j.cnsns.2010.10.004.

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31

Abbasbandy, S., Y. Tan, and S. J. Liao. "Newton-homotopy analysis method for nonlinear equations." Applied Mathematics and Computation 188, no. 2 (May 2007): 1794–800. http://dx.doi.org/10.1016/j.amc.2006.11.136.

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32

Sadaf, Maasoomah, and Ghazala Akram. "An improved adaptation of homotopy analysis method." Mathematical Sciences 11, no. 1 (January 4, 2017): 55–62. http://dx.doi.org/10.1007/s40096-016-0204-y.

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33

Fallahzadeh, Amir, and Mohammad Ali Fariborzi Araghi. "Homotopy analysis method for fuzzy Boussinesq equation." Mathematical Sciences 9, no. 3 (July 8, 2015): 145–52. http://dx.doi.org/10.1007/s40096-015-0161-x.

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34

Sadigh Behzadi, Sh. "Convergence of Iterative Methods Applied to Generalized Fisher Equation." International Journal of Differential Equations 2010 (2010): 1–16. http://dx.doi.org/10.1155/2010/254675.

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A generalized Fisher's equation is solved by using the modified Adomian decomposition method (MADM), variational iteration method (VIM), homotopy analysis method (HAM), and modified homotopy perturbation method (MHPM). The approximation solution of this equation is calculated in the form of series whose components are computed easily. The existence, uniqueness, and convergence of the proposed methods are proved. Numerical example is studied to demonstrate the accuracy of the present methods.

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35

Domairry, G., and N. Nadim. "Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation." International Communications in Heat and Mass Transfer 35, no. 1 (January 2008): 93–102. http://dx.doi.org/10.1016/j.icheatmasstransfer.2007.06.007.

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36

Nasabzadeh, H., and F. Toutounian. "Convergent Homotopy Analysis Method for Solving Linear Systems." Advances in Numerical Analysis 2013 (October 8, 2013): 1–6. http://dx.doi.org/10.1155/2013/732032.

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By using homotopy analysis method (HAM), we introduce an iterative method for solving linear systems. This method (HAM) can be used to accelerate the convergence of the basic iterative methods. We also show that by applying HAM to a divergent iterative scheme, it is possible to construct a convergent homotopy-series solution when the iteration matrix G of the iterative scheme has particular properties such as being symmetric, having real eigenvalues. Numerical experiments are given to show the efficiency of the new method.

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37

Jafari, H., and M. Alipour. "Solution of the Davey–Stewardson equation using homotopuy analysis method." Nonlinear Analysis: Modelling and Control 15, no. 4 (October 25, 2010): 423–33. http://dx.doi.org/10.15388/na.15.4.14313.

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In this paper, the homotopy analysis method (HAM) proposed by Liao is adopted for solving Davey–Stewartson (DS) equations which arise as higher dimensional generalizations of the nonlinear Schrödinger (NLS) equation. The results obtained by HAM have been compared with the exact solutions and homotopy perturbation method (HPM) to show the accuracy of the method. Comparisons indicate that there is a very good agreement between the HAM solutions and the exact solutions in terms of accuracy.

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38

Zedan, Hassan A., and Eman El Adrous. "The Application of the Homotopy Perturbation Method and the Homotopy Analysis Method to the Generalized Zakharov Equations." Abstract and Applied Analysis 2012 (2012): 1–19. http://dx.doi.org/10.1155/2012/561252.

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We introduce two powerful methods to solve the generalized Zakharov equations; one is the homotopy perturbation method and the other is the homotopy analysis method. The homotopy perturbation method is proposed for solving the generalized Zakharov equations. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions; the homotopy analysis method is applied to solve the generalized Zakharov equations. HAM is a strong and easy-to-use analytic tool for nonlinear problems. Computation of the absolute errors between the exact solutions of the GZE equations and the approximate solutions, comparison of the HPM results with those of Adomian’s decomposition method and the HAM results, and computation the absolute errors between the exact solutions of the GZE equations with the HPM solutions and HAM solutions are presented.

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39

Liao, Shi-Jun, and A. T. Chwang. "Application of Homotopy Analysis Method in Nonlinear Oscillations." Journal of Applied Mechanics 65, no. 4 (December 1, 1998): 914–22. http://dx.doi.org/10.1115/1.2791935.

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In this paper, we apply a new analytical technique for nonlinear problems, namely the Homotopy Analysis Method (Liao 1992a), to give two-period formulas for oscillations of conservative single-degree-of-freedom systems with odd nonlinearity. These two formulas are uniformly valid for any possible amplitudes of oscillation. Four examples are given to illustrate the validity of the two formulas. This paper also demonstrates the general validity and the great potential of the Homotopy Analysis Method.

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40

Zhao, X., S. G. Zhang, Y. T. Yang, and Q. H. Liu. "Homotopy Method for a General Multiobjective Programming Problem under Generalized Quasinormal Cone Condition." Abstract and Applied Analysis 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/591612.

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A combined interior point homotopy continuation method is proposed for solving general multiobjective programming problem. We prove the existence and convergence of a smooth homotopy path from almost any interior initial interior point to a solution of the KKT system under some basic assumptions.

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41

Kumar, Devendra, Jagdev Singh, and A. Kılıçman. "An Efficient Approach for Fractional Harry Dym Equation by Using Sumudu Transform." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/608943.

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An efficient approach based on homotopy perturbation method by using sumudu transform is proposed to solve nonlinear fractional Harry Dym equation. This method is called homotopy perturbation sumudu transform (HPSTM). Furthermore, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement, and, hence, this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the HPSTM show that the approach is easy to implement and computationally very attractive.

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42

Jafari, Hossein, Khadijeh Bagherian, and Seithuti P. Moshokoa. "Homotopy Perturbation Method to Obtain Positive Solutions of Nonlinear Boundary Value Problems of Fractional Order." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/919052.

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We use the homotopy perturbation method for solving the fractional nonlinear two-point boundary value problem. The obtained results by the homotopy perturbation method are then compared with the Adomian decomposition method. We solve the fractional Bratu-type problem as an illustrative example.

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43

Inc, Mustafa, and Ali Akgül. "Numerical Solution of Seventh-Order Boundary Value Problems by a Novel Method." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/745287.

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We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventh-order boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), variation of parameters method (VPM), and homotopy analysis method (HAM). Obtained results show that our method is very effective.

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44

Liu, Yin-Ping, and Zhi-Bin Li. "Homotopy Analysis Method for Nonlinear Differential Equations with Fractional Orders." Zeitschrift für Naturforschung A 63, no. 5-6 (June 1, 2008): 241–47. http://dx.doi.org/10.1515/zna-2008-5-602.

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The aim of this paper is to solve nonlinear differential equations with fractional derivatives by the homotopy analysis method. The fractional derivative is described in Caputo’s sense. It shows that the homotopy analysis method not only is efficient for classical differential equations, but also is a powerful tool for dealing with nonlinear differential equations with fractional derivatives.

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45

Mehmet Baskonus, Haci, Hasan Bulut, and Yusuf Pandir. "On the Solution of Nonlinear Time-Fractional Generalized Burgers Equation by Homotopy Analysis Method and Modified Trial Equation Method." International Journal of Modeling and Optimization 4, no. 4 (August 2014): 305–9. http://dx.doi.org/10.7763/ijmo.2014.v4.390.

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46

Hussain, Khawlah. "New reliable modifications of HPM and HAM." Indonesian Journal of Electrical Engineering and Computer Science 19, no. 1 (July 1, 2020): 371. http://dx.doi.org/10.11591/ijeecs.v19.i1.pp371-379.

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<p>In this article, a new modification of the homotopy perturbation method (HPM) and homotopy analysis method (HAM) is presented and applied to non-homogeneous fractional Volterra integro-differential equations with boundary conditions. A comparative study between the new modified homotopy perturbation method (MHPM) and the new modified homotopy analysis method (MHAM). Several illustrative examples are given to demonstrate the effectiveness and reliability of the methods.</p>

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47

Ghoreishi, M., A. I. B. MD Ismail, and A. K. Alomari. "Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth-order integro-differential equation." Mathematical Methods in the Applied Sciences 34, no. 15 (September 8, 2011): 1833–42. http://dx.doi.org/10.1002/mma.1483.

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48

Alomari, A. K., M. S. M. Noorani, and R. Nazar. "Comparison between the homotopy analysis method and homotopy perturbation method to solve coupled Schrodinger-KdV equation." Journal of Applied Mathematics and Computing 31, no. 1-2 (October 22, 2008): 1–12. http://dx.doi.org/10.1007/s12190-008-0187-4.

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49

Arshad, Sarmad, Abdul M. Siddiqui, Ayesha Sohail, Khadija Maqbool, and ZhiWu Li. "Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons." Advances in Mechanical Engineering 9, no. 3 (March 2017): 168781401769294. http://dx.doi.org/10.1177/1687814017692946.

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50

Liao, Shijun. "General boundary element method: an application of homotopy analysis method." Communications in Nonlinear Science and Numerical Simulation 3, no. 3 (September 1998): 159–63. http://dx.doi.org/10.1016/s1007-5704(98)90007-7.

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