Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems

In this paper, 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.

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