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A new application of the homotopy analysis method: solving the Sturm-Liouville problems. (English) Zbl 1221.65189
Summary: The homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm-Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. We observe that the auxiliary parameter , which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods.
MSC:
65L99Numerical methods for ODE
References:
[1]Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D thesis, Shanghai Jiao Tong University; 1992.
[2]Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[3]Jafari, H.; Seifi, S.: Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun nonlinear sci numer simulat 14, 1962-1969 (2009) · Zbl 1221.35439 · doi:10.1016/j.cnsns.2008.06.019
[4]Hashim, I.; Abdulaziz, O.; Momani, S.: Homotopy analysis method for fractional ivps, Commun nonlinear sci numer simulat 14, 674-684 (2009) · Zbl 1221.65277 · doi:10.1016/j.cnsns.2007.09.014
[5]Xu, H.; Cang, J.: Analysis of a time fractional wave-like equation with the homotopy analysis method, Phys lett A 372, 1250-1255 (2008) · Zbl 1217.35111 · doi:10.1016/j.physleta.2007.09.039
[6]Abbasbandy, S.; Shirzadi, A.: The series solution of problems in the calculus of variations via the homotopy analysis method, Z naturforsch A 64, 30-36 (2009)
[7]Alomari, A. K.; Noorani, M. S. M.; Nazar, R.: Adaptation of homotopy analysis method for the numeric – analytic solution of Chen system, Commun nonlinear sci numer simulat 4, 2336-2346 (2009) · Zbl 1221.65192 · doi:10.1016/j.cnsns.2008.06.011
[8]Wu, Y.; Wang, C.; Liao, S. J.: Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos solitons fractals 23, 1733-1740 (2005) · Zbl 1069.35060 · doi:10.1016/j.chaos.2004.06.081
[9]Zhang, T. T.; Jia, L.; Wang, Z. C.; Li, X.: The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels, Phys lett A 372, 3223-3227 (2008) · Zbl 1220.76025 · doi:10.1016/j.physleta.2008.01.077
[10]Sajid, M.; Hayat, T.: The application of homotopy analysis method to thin film flows of a third order fluid, Chaos solitons fractals 38, 506-515 (2008) · Zbl 1146.76588 · doi:10.1016/j.chaos.2006.11.034
[11]Zou, L.; Zong, Z.; Dong, G. H.: Generalizing homotopy analysis method to solve lotkavolterra equation, Comput math appl 9, 2289-2293 (2008) · Zbl 1165.34305 · doi:10.1016/j.camwa.2008.03.052
[12]Abbasbandy, S.; Babolian, E.; Ashtiani, M.: Numerical solution of the generalized Zakharov equation by homotopy analysis method, Commun nonlinear sci numer simulat 14, 4114-4121 (2009) · Zbl 1221.65269 · doi:10.1016/j.cnsns.2009.03.001
[13]Molabahrami, A.; Khani, F.: The homotopy analysis method to solve the Burgers – Huxley equation, Nonlinear anal real world appl 10, 589-600 (2009) · Zbl 1167.35483 · doi:10.1016/j.nonrwa.2007.10.014
[14]Cheng, J.; Liao, S. J.; Mohapatra, R. N.; Vajravelu, K.: Series solutions of nano boundary layer flows by means of the homotopy analysis method, J math anal appl 343, 233-245 (2008) · Zbl 1135.76016 · doi:10.1016/j.jmaa.2008.01.050
[15]Van Gorder, R. A.; Vajravelu, K.: 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 (2009) · Zbl 1221.65208 · doi:10.1016/j.cnsns.2009.03.008
[16]Liao, S. J.: Notes on the homotopy analysis method: some definitions and theorems, Commun nonlinear sci numer simulat 14, 983-997 (2009) · Zbl 1221.65126 · doi:10.1016/j.cnsns.2008.04.013
[17]Liao, S. J.: On the relationship between the homotopy analysis method and Euler transform, Commun nonlinear sci numer simulat 15, 1421-1431 (2010) · Zbl 1221.65206 · doi:10.1016/j.cnsns.2009.06.008
[18]Akyildiz, F. Talay; Vajravelu, K.; Liao, S. J.: A new method for homoclinic solutions of ordinary differential equations, Chaos solitons fractals 39, 1073-1082 (2009) · Zbl 1197.65212 · doi:10.1016/j.chaos.2007.04.021
[19]Ghotbi, A. R.; Bararnia, H.; Domairry, G.; Barari, A.: Investigation of a powerful analytical method into natural convection boundary layer flow, Commun nonlinear sci numer simulat 14, 2222-2228 (2009) · Zbl 1221.76145 · doi:10.1016/j.cnsns.2008.07.020
[20]Khan, M.; Abbas, Z.; Hayat, T.: Analytic solution for flow of sisko fluid through a porous medium authors, Trans porous media 71, 23-37 (2008)
[21]Abbasbandy, S.; Hayat, T.: Solution of the MHD Falkner – Skan flow by homotopy analysis method, Commun nonlinear sci numer simulat 14, 3591-3598 (2009) · Zbl 1221.76133 · doi:10.1016/j.cnsns.2009.01.030
[22]Bataineh, A. S.; Noorani, M. S. M.; Hashim, I.: Solutions of time-dependent Emden – Fowler type equations by homotopy analysis method, Phys lett A 371, 72-82 (2007) · Zbl 1209.65104 · doi:10.1016/j.physleta.2007.05.094
[23]Van Gorder, R. A.; Vajravelu, K.: Analytic and numerical solutions to the Lane – Emden equation, Phys lett A 372, 6060-6065 (2008) · Zbl 1223.85004 · doi:10.1016/j.physleta.2008.08.002
[24]Song, L.; Zhang, H.: Application of homotopy analysis method to fractional KdV-Burgers – Kuramoto equation, Phys lett A 367, 88-94 (2007) · Zbl 1209.65115 · doi:10.1016/j.physleta.2007.02.083
[25]Liang, S.; Jeffrey, D. J.: Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation, Commun nonlinear sci numer simulat 14, 4057-4064 (2009) · Zbl 1221.65281 · doi:10.1016/j.cnsns.2009.02.016
[26]Abbasbandy, S.; Magyari, E.; Shivanian, E.: The homotopy analysis method for multiple solutions of nonlinear boundary value problems, Commun nonlinear sci numer simulat 14, 3530-3536 (2009) · Zbl 1221.65170 · doi:10.1016/j.cnsns.2009.02.008
[27]Attili, B. S.: The Adomian decomposition method for computing eigenelements of Sturm – Liouville two point boundary value problems, Appl math comput 168, 1306-1316 (2005) · Zbl 1082.65557 · doi:10.1016/j.amc.2004.10.020
[28]Altintan, D.; U&gbreve, Ö.; Ura: Variational iteration method for Sturm – Liouville differential equations, Comput math appl 58, 322-328 (2009)
[29]Attili, B. S.; Lesnic, D.: An efficient method for computing eigenelements of Sturm – Liouville fourth-order boundary value problems, Appl math comput 182, 1247-1254 (2006) · Zbl 1107.65070 · doi:10.1016/j.amc.2006.05.011
[30]Syam, M. I.; Siyyam, H. I.: An efficient technique for finding the eigenvalues of fourth-order Sturm – Liouville problems, Chaos solitons fractals 39, 659-665 (2009) · Zbl 1197.65039 · doi:10.1016/j.chaos.2007.01.105
[31]Liao, S. J.: Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method, Nonlinear anal real world appl 10, 2455-2470 (2009) · Zbl 1163.35450 · doi:10.1016/j.nonrwa.2008.05.003
[32]Ma, R. Y.; Wang, H. Y.: On the existence of positive solutions of fourth order ordinary differential equation, Appl anal 59, 225-231 (1995) · Zbl 0841.34019 · doi:10.1080/00036819508840401
[33]Schroder, J.: Fourth-order two-point boundary value problems; estimates by two side bounds, Nonlinear anal 8, 107-144 (1984) · Zbl 0533.34019 · doi:10.1016/0362-546X(84)90063-4
[34]Agarwal, R. P.; Chow, M. Y.: Iterative methods for a fourth order boundary value problem, J comput appl math 10, 203-217 (1984) · Zbl 0541.65055 · doi:10.1016/0377-0427(84)90058-X
[35]Agarwal, R. P.: On the fourth-order boundary value problems arising in beam analysis, Diff integral equat 2, 91-110 (1989) · Zbl 0715.34032
[36]O’regan, D.: Solvability of some fourth (and higher) order singular boundary value problems, J math anal appl 161, 78-116 (1991) · Zbl 0795.34018 · doi:10.1016/0022-247X(91)90363-5
[37]Gupta, C. P.: Existence and uniqueness theorems for a bending of an elastic beam equation at resonance, J math anal appl 135, 208-225 (1988) · Zbl 0655.73001 · doi:10.1016/0022-247X(88)90149-7
[38]Liu, L.; Zhang, X.; Wu, Y.: Positive solutions of fourth-order nonlinear singular Sturm – Liouville eigenvalue problems, J math anal appl 326, 1212-1224 (2007) · Zbl 1113.34022 · doi:10.1016/j.jmaa.2006.03.029