Analysis of one-sided 1-D fractional diffusion operator

Abstract

<p style='text-indent:20px;'>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.</p>

A Signed Maximum Principle for Boundary Value Problems for Riemann–Liouville Fractional Differential Equations with Analogues of Neumann or Periodic Boundary Conditions

“…In particular, w 2 ⪰ λ<0 v 2 . Thus, (30) is proved for k = 1. Before applying a straightforward induction to obtain (30), we must show D α 0 w 2 (t) ≥ f (t, w 2 (t)), and…”

“…One can see from this construction that a maximum principle will be valid for λ ∈ (−∞, 0). For the anti-maximum principle, it is shown in ( [30], Corollary 3) that E α,α (−z) has the smallest in modulus root which is a positive root. From the identity,…”

Analysis and Petrov-Galerkin numerical approximation for variable coefficient two-sided fractional diffusion, advection, reaction equations

“…There are fruitful mathematical and numerical results for space-fractional differential equations in the literature [4,5,13,19,21,22,25,26,28,33,34]. The well-posedness of a constant diffusivity coefficient analogue of problem (1.1)-(1.2) and error estimates of its Galerkin finite element approximation was proved in [11].…”