## 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 $$\mathcal{L}$$. 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:

 65L15 Numerical solution of eigenvalue problems involving ordinary differential equations 34A08 Fractional ordinary differential equations 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators 65L20 Stability and convergence of numerical methods for ordinary differential equations 34B24 Sturm-Liouville theory

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### References:

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