On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

Abstract

In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.

Existence results of self-similar solutions of the space-fractional diffusion equation involving the generalized Riesz-Caputo fractional derivative

“…Now, we define the R-CGFD. Definition 2.6 (Riesz-CGFD (see [1])) Let α, ρ ∈ C with Re(α), Re(ρ) > 0 and g(η) ∈ X p c (0, µ) for 0 ≤ η ≤ µ. Then the R-CGFD is defined as…”

“…where C D α,ρ 0 + and C D α,ρ µ − are left and right sided of CGFDs which defined in ( 5) and ( 6), respectively. Lemma 2.7 (see [1])…”

“…Lemma 2.13 (The generalized Gronwall inequality (see [1])) Let α > 0, 0 < η < µ and assume that g(η), u 1 (η) and u 2 (η) are locally integrable, nonnegative and non-decreasing functions. Also, assume that v 1 (η) and v 2 (η) are a non-decreasing continuous functions such that 0 ≤ v 1 (η), v 2 (η) ≤ L, where L is a constant.…”

“…Definition 2.5 (Riesz-generalized fractional integral. (see [1])) Let g(η) ∈ X p c (0, µ) and α, ρ > 0. Then, for 0 ≤ η ≤ µ, the generalized Riesz type integral is defined as…”

“…It represents a generalization of the Caputo and Caputo-Hadamard FDs. Aleem et al in [1], presented a generalisation of the Riesz fractional operator, where this operator covers as particular cases the classical Riesz fractional derivative. In the same paper, the authors have also proposed a Caputo-type modification of this operator.…”