Sturm Liouville Equations in the frame of fractional operators with exponential kernels and their discrete versions

A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators

“…Let λ be an eigenvalue of (17)- (19) and let ξ be the corresponding eigenfunction. Then ξ and its complex conjugate ξ satisfy…”

“…Proof. Assume λ 1 and λ 2 are distinct eigenvalues of (17)- (19) and let ξ λ1 and ξ λ2 be the corresponding eigenfunctions. Then we have…”

“…The SLEs, which were investigated a long time ago, have many applications in various areas of science, engineering, and mathematics [34,53]. However, its fractional formulation and numerical methods for the solutions have started very recently [18,19,21,22,33,43,48,51,52]. The classical Sturm-Liouville problem (SLP ) for a linear second order differential equation is a boundary value problem (BV P ) of the form:…”

Integral Balance Methods for Stokes’ First Equation Described by the Left Generalized Fractional Derivative

“…There exist many other fractional derivatives such as the Riemann-Liouville fractional derivative [6,20], the Atangana-Baleanu fractional derivative [9,21,22], the Caputo-Fabrizio fractional derivative [23], the Caputo fractional derivative [6,20], and the conformable fractional derivative [24]. In differentiation and integration to non-integer order, we distinguish the fractional derivatives and the fractal derivatives.…”

“…In differentiation and integration to non-integer order, we distinguish the fractional derivatives and the fractal derivatives. For recent advancements of the fractional derivatives, see [21,23,24] and, for the fractal derivatives, see [25][26][27][28].…”