Abstract
In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach’s fixed-point criterion. We validate our study by implementing two numerical schemes to handle some Riemann–Liouville fractional boundary value problems and obtain approximate series solutions that converge to the exact ones. In particular, we present several examples that illustrate the closeness of the approximate solutions to the exact solutions.
Highlights
We first derive an integral equation corresponding to the given multiterm multi-order RLFBVP (1) and establish required conditions to confirm the existence of solutions for (1)
We study the existence of solutions for a multi-term multi-order RLFBVP with integral boundary conditions in the first step
We apply two numerical methods (i.e., Dafterdar-Gejji and Jafari method (DGJIM) and Adomian decomposition method (ADM) algorithms) for solving the suggested multi-term fractional differential equation based on the decomposition technique
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Of fractional differential equations usually yield other operational mathematical models for the description of special chemical, physical, and biological processes, which one can find in recently published works [12–19] Along with these real models describing the phenomena, many mathematicians conduct research on the existence theory of solutions for different abstract structures of FBVPs with general boundary conditions including three-point, multi-point, multi-order, multi-strip, and nonlocal integral ones [20–29]. We must emphasize that the novelty and motivation of our work is that, other papers use the ADM and DGJIM methods for solving IVPs, we here intend to compute approximate solutions for a complicated multi-order multi-term RLFBVP with boundary conditions including double-order RL-fractional integrals.
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