Some notes concerning the convergence control parameter in homotopy analysis method
الموضوعات :
1 - Department of Mathematics, Islamic Azad University, Hamedan Branch , Hamedan, 6518118413,
Iran.
2 - Department of Mathematics and Computer Science, Tarbiat Moallem University, 599 Taleghani
avenue, Tehran 1561836314, Iran.
الکلمات المفتاحية: Homotopy Analysis Method, Convergence-control parameter, Zeroth order deformation equation,
ملخص المقالة :
omotopy analysis method (HAM) is a promising method for handling func-tional equations. Recent publications proved the e ectiveness of HAM in solvingwide variety of problems in di erent elds. HAM has a unique property whichmakes it superior to other analytic methods, this property is its ability to con-trol the convergence region of the solution series. In this work, we clari ed theadvantages and e ects of convergence-control parameter through an example.
[1] S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int.
Commun. Heat Mass Transf. 34 (2007), 380{387.
[2] S. Abbasbandy, The application of the homotopy analysis method to nonlinear
equations arising in heat transfer, Phys. Lett. A 360 (2006) 109-113.
[3] S. Abbasbandy, F. Samadian Zakaria, Soliton solutions for the 5th-order KdV
equation with the homotopy analysis method, Nonlinear Dyn. 51 (2008) 83{87.
[4] S. Abbasbandy, Y. Tan and S. J. Liao, Newton-homotopy analysis method for
nonlinear equations, Appl. Math. Comput. 188 (2007), 1794{1800.
[5] T. Hayat, T. Javed, M. Sajid, Analytic solution for rotating ow and heat transfer
analysis of a third-grade uid, Acta Mech 191 (2007), 219{229.
[6] T. Hayat, M. Sajid, Homotopy analysis of MHD boundary layer ow of an upper-
convected Maxwell uid, Int. J. Eng. Sci. 45 (2007), 393{401.
[7] T. Hayat, M. Sajid, Analytic solution for axisymmetric ow and heat transfer
of a second grade uid past a stretching sheet, Int. J. Heat Mass Transfer, 50
(2007), 75{84.
[8] M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann
boundary conditions by the homotopy analysis method, Phys. Lett. A 365 (2007),
412{415.
[9] S. J. Liao, Proposed homotopy analysis technique for the solution of nonlinear
problems, Ph.D. dissertation, Shanghai Jiao Tong University, 1992.
[10] S. J. Liao, A kind of approximate solution technique which does not depend upon
small parameters: a special example, Int. J. of Non-Linear Mech. 30 (1995),
371{380.
[11] S. J. Liao, A kind of approximate solution technique which does not depend upon
small parameters (II): an application in uid mechanics, Int. J. of Non-Linear
Mech. 32 (1997) 815{822.
[12] S. J. Liao, A. T. Chwang, Application of homotopy analysis method in nonlinear
oscillations, ASME. J. of Appl. Mech. 65 (1998), 914{922.
[13] S. J. Liao, Beyond perturbation: An introduction to homotopy analysis method,
Chapman Hall/CRC Press, Boca Raton, 2003.
[14] S. J. Liao, On the relationship between the homotopy analysis method and Euler
transform, Commun. Nonlin. Sci. Num. Simul. 18 (2010), 1421-1431.
[15] S. J. Liao, E. Magyari, Exponentially decaying boundary layers as limiting cases
of families of algebraically decaying ones, ZAMP 57 (2006), 777{792.
[16] S. J. Liao, A new branch of solutions of boundary-layer ows over a permeable
stretching plate, Int. J. Non-Linear Mech. 42 (2007), 819{830.
[17] S. J. Liao, Notes on the homotopy analysis method: Some de nitions and theo-
rems, Commun. Nonlin. Sci. Num. Simul. 14 (2009) 983{997.
[18] S. J. Liao, An optimal homotopy analysis approach for strongly nonlinear di er-
ential equations, Commun. Nonlin. Sci. Num. Simul. 15 (2010), 2003{2016.
[19] Y. P. Liu, Z. B. Li, The homotopy analysis method for approximating the solution
of the modi ed Korteweg-de Vries equation, Chaos, Solitons and Fractals, 39
(2009), 1{8.
[20] M. Paripour, E. Babolian, J. Saeidian, Analytic solutions to di usion equations,
Math. Comput. Mod. 51 (2010), 649{657.
[21] M. Sajid, T. Hayat, S. Asghar, Comparison between the HAM and HPM solutions
of thin lm ows of non-Newtonian uids on a moving belt, Nonlinear Dyn. 50
(2007), 27{35.
[22] Z. Wang, L. Zou, H. Zhang, Applying homotopy analysis method for solving
di erential-di erence equation, Phys. Lett. A 369 (2007), 77{84.
[23] S. P. Zhu, An exact and explicit solution for the valuation of American put
options, Quantitative Finance 6 (2006), 229{242.
[24] S. P. Zhu, A closed-form analytical solution for the valuation of convertible bonds
with constant dividend yield, ANZIAM J. 47 (2006), 477{494.
[25] L. Zou, Z. Zong, Z. Wang, L. He, Solving the discrete KdV equation with homo-
topy analysis method, Phys. Lett. A 370 (2007), 287{294.
