Application of the Homotopy Analysis Method for Determining the free Vibrations of Beam
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References
- S. Abbasbandy and E. Shivanian, (2011). A new analytical technique to solve Fredholm’s integral equations, Numer. Algor. 56, 27–43.
- R. Brociek, E. Hetmaniok, J. Matlak, and D. Słota, (2016). Application of the homotopy analysis method for solving the systems of linear and nonlinear integral equations, Math. Model. Anal. 21, 350–370.
- D.S. Chauhan, R. Agrawal, and P. Rastogi, (2012). Magnetohydrodynamic slip flow and heat transfer in a porous medium over a stretching cylinder: homotopy analysis method, Numer. Heat Transfer A 62, 136–157.
- T. Fan and X. You, (2013). Optimal homotopy analysis method for nonlinear differential equations in the boundary leyer, Numer. Algor. 62, 337–354.
- H. Gliński, R. Grzymkowski, A. Kapusta, and D. Słota, (2012) Mathematica 8, WPKJS, Gliwice (in Polish).
- K. Gromysz, (2013). Examination of the Reinforced Concrete Plates Loaded Temporarily, Periodicaly and Kinematically, Monograph no 452, Silesian University of Technology Press, Gliwice (in Polish).
- E. Hetmaniok, I. Nowak, D. Słota, and R. Wituła, (2015). Convergence and error estimation of homotopy analysis method for some type of nonlinear and linear integral equations, J. Numer. Math. 23, 331–344.
- E. Hetmaniok, D. Słota, T. Trawiński, and R. Wituła, (2014). Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind, Numer. Algor. 67, 163–185.
- E. Hetmaniok, D. Słota, R. Wituła, and A. Zielonka, (2015). An analytical method for solving the two-phase inverse Stefan problem, Bull. Pol. Ac.: Tech 63(3), 583–590.
- E. Hetmaniok, D. Słota, R. Wituła, and A. Zielonka, (2015). Solution of the one-phase inverse Stefan problem by using the homotopy analysis method, Appl. Math. Modelling 39, 6793–6805.
- T. Kaczorek, (2016). A new approach to the realization problem for fractional discrete-time linear systems, Bull. Pol. Ac.: Tech 64(1), 9–14.
- Y. Khan, M. Fardi, (2015). A new efficient multi-parametric homotopy approach for two-dimensional Fredholm integral equations of the second kind, Hacettepe J. Math. Stat. 44, 93–99.
- Y. Khan, K. Sayevand, M. Fardi, M. Ghasemi, (2014). A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations, Appl. Math. Comput. 249, 229–236.
- R. Lewandowski, (2006). Dynamics of the Building Structures, Poznań University of Technology Press, Poznań 2006 (in Polish).
- S. Liao, (1998). Homotopy analysis method: a new analytic method for nonlinear problems, Appl. Math. Mech. – Engl. Ed. 19, 957–962.
- S. Liao, (2003). Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall–CRC Press, Boca Raton.
- S. Liao, (2009). Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simulat. 14, 983–997.
- S. Liao, (2010). An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat. 15, 2003–2015.
- S. Liao, (2012). Homotopy Analysis Method in Nonlinear Differential Equations, Springer/Higher Education Press, Berlin/Beijing.
- Z.M. Odibat, (2010). A study on the convergence of homotopy analysis method, Appl. Math. Comput. 217, 782–789.
- P. Ostalczyk, (2015). On simplified forms of the fractional-order backward difference and related fractional-order linear discrete-time system description, Bull. Pol. Ac.: Tech 63(2), 423–433.
- M. Russo and R.A. Van Gorder, (2013). Control of error in the homotopy analysis of nonlinear Klein-Gordon initial value problems, Appl. Math. Comput. 219, 6494–6509.
- D. Słota, (2011). Homotopy Analysis Method and the Examples of Its Applications, Wyd. Pol. Śl., Gliwice (in Polish).
- W. Sumelka, (2016). Fractional calculus for continuum mechanics – anisotropic non-locality, Bull. Pol. Ac.: Tech 64 (2), 361–372.
- M. Turkyilmazoglu, (2010). A note on the homotopy analysis method, Appl. Math. Lett. 23, 1226–1230.
- M. Turkyilmazoglu, (2010). Convergence of the homotopy analysis method, arXiv, 1006.4460v1.
- K. Vajravelu, R.A. Van Gorder, (2012). Nonlinear Flow Phenomena and Homotopy Analysis. Fluid Flow and Heat Transfer, Springer/Higher Education Press, Berlin/Beijing.
- R.A. Van Gorder, (2012). Control of error in the homotopy analysis of semi-linear elliptic boundary value problems, Numer. Algor. 61, 613–629.
- R.A. Van Gorder and K. Vajravelu, (2009). On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Commun. Nonlinear Sci. Numer. Simulat. 14, 4078–4089.
- X. Zhang, B. Tang, and Y. He, (2011). Homotopy analysis method for higher-order fractional integrodifferential equations, Comput. Math. Appl. 62, 3194–3203.
- M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, (2010). The homotopy analysis method for handling systems of fractional differential equations, Appl. Math. Modelling 34, 24–35.
Language: English
Page range: 61 - 71
Submitted on: Oct 24, 2017
Accepted on: Feb 19, 2018
Published on: Apr 1, 2019
Published by: Silesian University of Technology
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year
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© 2019 Krzysztof GROMYSZ, Damian SŁOTA, published by Silesian University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.