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Analysis of one-sided 1-D fractional diffusion operator

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  • This work establishes the parallel between the properties of classic elliptic PDEs and the one-sided 1-D fractional diffusion equation, that includes the characterization of fractional Sobolev spaces in terms of fractional Riemann-Liouville (R-L) derivatives, variational formulation, maximum principle, Hopf's Lemma, spectral analysis, and theory on the principal eigenvalue and its characterization, etc. As an application, the developed results provide a novel perspective to study the distribution of complex roots of a class of Mittag-Leffler functions and, furthermore, prove the existence of real roots.

    Mathematics Subject Classification: Primary: 34B05, 34L10; Secondary: 34B09.

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  • Figure 1.  First real root of Mittag Leffler function $ E_{1+\mu,1+\mu}[z] $ with different values of $ \mu $. The pictures are generated with functions 'mlf' and 'intersections' from Matlab Exchange

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