TOTAL VIEWS: 7443

Published: December 15,2020

In this paper, comparative study of q-homotopy analysis method (q-HAM) with the Liao’s optimal homotopy analysis method (OHAM) is proposed. We solved two examples, first example is a system of Volterra integro-differential equations and the second one is a nonlinear integro-differential equation. The results show that the q-HAM was more accuracy than the OHAM.

[1] Liao, S. J. (1992). The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. thesis, Shanghai Jiao Tong University.

[2] Liao, S. J. (2003). Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman &Hall/CRC, Boca Raton, Fla, USA.

[3] Liao, S. J. (2004). On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computa-tion, Vol. 147, No. 2, pp. 499-513.

[4] Liao, S. J. (2005). Comparison between the homotopy analysis method and homotopy perturbation method. Applied Mathematics and Computation, Vol. 169, No. 2, pp. 1186-1194.

[5] Liao, S. J. (1997). Homotopy analysis method: a new analytical technique for nonlinear problems. Communications in Nonlinear Science and Numerical Simulation, Vol. 2, No. 2, pp. 95-100.

[6] Abbasbandy, S. (2008). Soliton solutions for the 5th-order KdV equation with the homotopy analysis Abbasbandy S., Soliton solutions for the 5th-order KdV equation with the homotopy analysis method. Nonlinear Dynamics, Vol. 51, No. 1-2, pp. 83-87.

[7] Arafa, A. A. M., Rida, S. Z., and Mohamed, H. (2014). An Application of the Homotopy Analysis Method to the Transient Behavior of a Biochemical Reaction Model. Inf. Sci. Lett.3, No. 1, pp. 29-33.

[8] Jafari, H., Saeidy, M., and Firoozjaee, M. A. (2010). The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficients. Numerical Methods for Partial Differential Equations, Vol. 26, No. 5, pp. 1021-1032.

[9] Onyejek, O. N. (2014). Solutions of some parabolic inverse problems by homotopy analysis method. International Journal of Applied Mathematical Research, 3(1).

[10] Zhu, S. P. (2006). An exact and explicit solution for the valuation of American put options. Quantitative Finance, Vol. 6, No. 3, pp. 229-242.

[11] Yabushita, K., Yamashita, M., and Tsuboi, K. (2007). An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A-Math. Theor., 40, 8403-8416.

[12] Akyildiz, F. T. and Vajravelu, K. (2008). Magneto hydrodynamic flow of a viscoelastic fluid. Phys. Lett. A., 372, 3380-3384.

[13] Niu, Z. and Wang, C. (2010). A one-step optimal homotopy analysis method for nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 15, 2026-2036.

[14] Liao, S. J. (2010). An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat., 15, 2003-2016.

[15] El-Tawil, M. A. and Huseen, S. N. (2012). The q-Homotopy Analysis Method (q-HAM). International Journal of Applied Mathematics and Mechanics, 8(15): 51-75.

[16] El-Tawil, M. A. and Huseen, S. N. (2013). On Convergence of The q-Homotopy Analysis Method. Int. J. Contemp. Math. Sciences, Vol. 8, No. 10, 481-497.

[17] Huseen, S. N. and Grace, S. R. (2013). Approximate Solutions of Nonlinear Partial Differential Equations by Mod-ified q-Homotopy Analysis Method (mq-HAM). Hindawi Publishing Corporation, Journal of Applied Mathematics, Article ID 569674, 9 pages.

[18] Huseen, S. N., Grace, S. R., and El-Tawil, M. A. (2013). The Optimal q-Homotopy Analysis Method (Oq-HAM). International Journal of Computers & Technology, Vol 11, No. 8.

[19] Huseen, S. N. (2015). Application of optimal q-homotopy analysis method to second order initial and boundary value problems. Int J Sci Innovative Math Res (IJSIMR), 3(1):18–24.

[20] Huseen, S. N. (2015). Solving the K (2, 2) Equation by Means of the q-Homotopy Analysis Method (q-HAM). In-ternational Journal of Innovative Science, Engineering & Technology, Vol. 2, Issue 8.

[21] Huseen, S. N. (2016). Series Solutions of Fractional Initial-Value Problems by q-Homotopy Analysis Method. In-ternational Journal of Innovative Science, Engineering & Technology, Vol. 3, Issue 1.

[22] Huseen, S. N. (2017). A Numerical Study of One-Dimensional Hyperbolic Telegraph Equation. Journal of Mathe-matics and System Science, 7, 62-72.

[23] Huseen, S. N. and Nada M. Ayay. (2018). A New Technique of The q-Homotopy Analysis Method for Solving Non-Linear Initial Value Problems. Journal of Progressive Research in Mathematics, Volume 14, Issue 1, 2292-2307.

[24] Huseen, S. N. and Rawan A. shlaka. (2019). The Regularization q-Homotopy Analysis Method for (1 and 2)-Dimensional Non-linear First Kind Fredholm Integral Equations. Journal of Progressive Research in Mathemat-ics, Volume 15, Issue 3, 2721-2743.

[25] Akinyemi, L. (2019). q-Homotopy analysis method for solving the seventh-order time-fractional Lax’s Korte-weg-deVries and Sawada-Kotera equations. Comp. Appl. Math., 38(4), 191.

[26] L. Akinyemi, S. N. Huseen (2020). A powerful approach to study the new modified coupled Korteweg-de Vries system. Mathematics and Computers in Simulation, 177, 556-567.

[27] Huseen, S. N. (2020). On Analytical Solution of Time-Fractional Type Model of the Fisher’s Equation. Iraqi Jour-nal of Science, Vol. 61, No. 6, pp. 1419-1425.

[28] Huseen, S. N., El-Tawil, M. A., Grace S. R., and Ismail, G. A. (2020). Solving High-Order Nonlinear Partial Diffe-rential Equations by Modified q-Homotopy Analysis Method. AJMS/Apr-Jun, Vol. 4, Issue 2, pp. 25-46.

[29] Jafar S. Nadjafi and Hossein S. Jafari. (2011). Comparison of Liao’s optimal HAM and Niu’s one-step optimal HAM for solving integro-differential equations. Journal of Applied Mathematics and Bioinformatics, Vol. 1, No. 2, 85-98.

A Comparative Study of q-Homotopy Analysis Method and Liao’s Optimal Homotopy Analysis Method

**How to cite this paper:** Shaheed N. Huseen. (2020) A Comparative Study of q-Homotopy Analysis Method and Liao’s Optimal Homotopy Analysis Method. *Advances in Computer and Communication*, **1**(**1**), 36-45.

DOI: http://dx.doi.org/10.26855/acc.2020.12.004

- Evaluation of Student Academic Performance Using Naïve Bayes Classifier
- A Comparative Study of q-Homotopy Analysis Method and Liao’s Optimal Homotopy Analysis Method
- Autopoietic Computing Systems and Triadic Automata: The Theory and Practice
- Wavelength-Adaptation Underwater Optical Wireless Sensor Network Using Visible Light Communications
- URL Security Detection on the Basis of a New Optimization Algorithm