# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems. (English) Zbl 1197.65096
Summary: The homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator $ℒ$. It can be seen in this paper that the auxiliary parameter $\hslash ,$ which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.
##### MSC:
 65L15 Eigenvalue problems for ODE (numerical methods) 34A08 Fractional differential equations 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum 65L20 Stability and convergence of numerical methods for ODE 34B24 Sturm-Liouville theory
BVPh
##### References:
 [1] Liao, S.J.: 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. Chapman and Hall/CRC Press, Boca Raton (2003) [3] Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. In: Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 983–997 (2009) [4] Xu, H., Liao, S.J., You, X.C.: Analysis of nonlinear fractional partial differential equations with the homotopy analysis method. In: Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 1152–1156 (2009) [5] Cang, J., Tan, Y., Xu, H., Liao, S.J.: Series solutions of non-linear Riccati differential equations with fractional order. Chaos, Solitons Fractals 40, 1–9 (2009) · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018 [6] 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 [7] Jafari, H., Seifi, S.: Solving a system of nonlinear fractional partial differential equations using homotopy analysis method. In: Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 1962–1969 (2009) [8] Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional IVPs. In: Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 674–684 (2009) [9] 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 [10] Zurigat, M., Momani, S., Odibat, Z., Alawneh, A.: The homotopy analysis method for handling systems of fractional differential equations. Appl. Math. Model. 34, 24–35 (2010) · Zbl 1185.65140 · doi:10.1016/j.apm.2009.03.024 [11] Talay Akyildiz, F., 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 [12] 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 [13] Ghotbi, A.R., Bararnia, 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 [14] Khan, M., Abbas, Z., Hayat, T.L: Analytic solution for flow of Sisko fluid through a porous medium Authors. Trans. Porous Media 71, 23–37 (2008) · doi:10.1007/s11242-007-9109-4 [15] Abbasbandy, S., Shirzadi, A.: The series solution of problems in calculus of variations via homotopy analysis method. Z. Naturforsch. A 64(a), 30–36 (2009) [16] 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 [17] 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 [18] 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 [19] 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 [20] Abbasbandy, S., Hayat, T., Ellahi, R., Asghar, S.: Numerical results of flow in a third grade fluid between two porous walls. Z. Naturforsch. A 64(a), 59–64 (2009) [21] López, J.L., Abbasbandy, S., López-Ruiz, R.: Formulas for the amplitude of the van der Pol limit cycle through the homotopy analysis method. In: Scholarly Research Exchange, vol. 2009, article ID 854060 (2009) [22] Abbasbandy, S., Magyari, E., Shivanian, E.: The homotopy analysis method for multiple solutions of nonlinear boundary value problems. In: Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 3530–3536 (2009) [23] Al-Mdallal, Q.M.: An efficient method for solving fractional Sturm-Liouville problems. Chaos, Solitons Fractals 40, 183–189 (2009) · Zbl 1197.65097 · doi:10.1016/j.chaos.2007.07.041 [24] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) [25] Podlubny, I.: Fractional Differential Equations. Academic, New York (1999) [26] Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, New York (1974)