Abstract
Abstract In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the aforementioned problems and present the Ulam-Hyers stability of the fractional differential equation involving two different fractional derivatives. An example is given to illustrate our theoretical result.
Highlights
The fractional differential equations can model some engineering and scientific disciplines in the fields of physics, chemistry, electrodynamics of complex medium, polymer rheology, etc
The forward and backward fractional derivatives provide an excellent tool for the description of some physical phenomena such as the fractional oscillator equations and the fractional Euler-Lagrange equations [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
A linear boundary value problem (BVP) involving both the right Caputo and the left RiemannLiouville fractional derivatives has been studied by some authors [10,11,12,13,14]
Summary
The fractional differential equations can model some engineering and scientific disciplines in the fields of physics, chemistry, electrodynamics of complex medium, polymer rheology, etc. A linear boundary value problem (BVP) involving both the right Caputo and the left RiemannLiouville fractional derivatives has been studied by some authors [10,11,12,13,14]. Studying the stability of Ulam-Hyers for fractional differential equations is gaining much importance and attention [18,19]. The Ulam-Hyers stabilities of differential equations involving with the forward and backward fractional derivatives have not yet been investigated. We investigate the following BVP of the fractional differential equation with two different fractional derivatives: cD1β−(LD0α+ + λ)u(t) = f (t, u(t)), t ∈ J ≔
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