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Zbl 1197.65096
Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems.
(English)
[J] Numer. Algorithms 54, No. 4, 521-532 (2010). ISSN 1017-1398; ISSN 1572-9265/e

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 $\mathcal{L}$. It can be seen in this paper that the auxiliary parameter $\hbar,$ which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.
MSC 2000:
*65L15 Eigenvalue problems for ODE (numerical methods)
34A08
34L16 Numerical approximation of eigenvalues, etc.
65L20 Stability of numerical methods for ODE
34B24 Sturm-Liouville theory

Keywords: homotopy analysis method; multiple solutions; fractional Sturm-Liouville problems; Caputo's fractional derivative; numerical examples; eigenvalues; convergence

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