Computation of eigenvalues of fractional Sturm–Liouville problems
We consider the eigenvalues of the fractional-order Sturm--Liouville equation of the form \begin{equation*} -{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0<\alpha\leq 1,\quad t\in[0,1], \end{equation*} with Dirichlet boundary conditions $$I_{0^+}^{1-\alpha}y(t)\v...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Ferdowsi University of Mashhad
2021-03-01
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| Series: | Iranian Journal of Numerical Analysis and Optimization |
| Subjects: | |
| Online Access: | https://ijnao.um.ac.ir/article_39622_986a1d19d921e33a18083f766e24c2b7.pdf |
| Summary: | We consider the eigenvalues of the fractional-order Sturm--Liouville equation of the form \begin{equation*} -{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0<\alpha\leq 1,\quad t\in[0,1], \end{equation*} with Dirichlet boundary conditions $$I_{0^+}^{1-\alpha}y(t)\vert_{t=0}=0\quad\mbox{and}\quad I_{0^+}^{1-\alpha}y(t)\vert_{t=1}=0,$$ where $q\in L^2(0,1)$ is a real-valued potential function. The method is used based on a Picard's iterative procedure. We show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives. |
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| ISSN: | 2423-6977 2423-6969 |