A fractional approach to the Sturm-Liouville problem
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Cites background from "A fractional approach to the Sturm-..."
...Bas and Metin [6], Klimek and Agrawal [13], Zayernouri and Karniadakis [30], and Rivero, Trujillo, and Velasco [25] defined different classes of fractional Sturm–Liouville operators and investigated the properties of the corresponding eigenfunctions and the eigenvalues....
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...[25] M. Rivero, J. J. Trujillo, and M. P. Velasco, A fractional approach to the Sturm–Liouville problem, Central European J. Phys., 11 (2013), pp. 1246–1254....
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73 citations
Cites background or methods from "A fractional approach to the Sturm-..."
...Moreover, in [68], a general approach of (4) not only with both the left and right sided Riemann-Liouville and Caputo fractional derivatives on a compact interval [a, b], but the Liouville fractional derivatives on R have been examined and some spectral properties, such as orthogonality of the eigenfunctions and the fact that the eigenvalues are all real valued, have been proved, but in these papers, the authors did not obtain any explicit form of eigenfunctions for (4)....
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...Due to the orthogonality of the eigenfunctions of (4) [68, 44], the explicit form of the eigenfunctions has very important role to establish the theories of the spectral methods such as Galerkin, Tau, Petrov-Galerkin, collocation and pseudo-spectral methods based on the eigenfunctions obtained from equation (4)....
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...It is worth observing that in [68], the authors have used some other fractional operators to construct FSLPs....
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73 citations
Cites background from "A fractional approach to the Sturm-..."
..., [1,2,9,10,11,12,13,15,16,18,20]) which are integrals and derivatives of arbitrary real or complex order....
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Additional excerpts
...Let λ > 0 and <(α) ≥ 0, then: Dα −e = λαe−λx (27) Dα +eλx = λαeλx (28) The following rules for fractional integration by parts for the Riemann-Liouville, Caputo and Liouville fractional derivatives hold: Lemma 1....
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..., n (25) Dα b−(b− x)α−j = 0, j = 1, 2, ....
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"A fractional approach to the Sturm-..." refers background in this paper
...In this section, we introduce the fractional derivatives and integrals used in this work and some their properties (see also [11], [17], [23], [25], [26], [28], [30])....
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