Multiterm boundary value problem of Caputo fractional differential equations of variable order

Zoubida Bouazza,Mohammed Said Souid,Hatıra Günerhan

doi:10.1186/s13662-021-03553-z

Abstract

In this manuscript, the existence, uniqueness, and stability of solutions to the multiterm boundary value problem of Caputo fractional differential equations of variable order are established. All results in this study are established with the help of the generalized intervals and piece-wise constant functions, we convert the Caputo fractional variable order to an equivalent standard Caputo of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used, the Ulam–Hyers stability of the given Caputo variable order is examined, and finally, we construct an example to illustrate the validity of the observed results. In literature, the existence of solutions to the variable-order problems is rarely discussed. Therefore, investigating this interesting special research topic makes all our results novel and worthy.

Highlights

The main idea of fractional calculus is to constitute the natural numbers in the order of derivation operators with rational ones
A significant number of papers have appeared on this topic; on the contrary, few papers deal with the existence of solutions to problems via variable order, see, e.g., [4, 15, 16, 18, 19]
Some methods are introduced for the approximation of solutions to different FBVPs of variable order

Summary

Introduction

The main idea of fractional calculus is to constitute the natural numbers in the order of derivation operators with rational ones. Lemma 2.2 ([25]) Let u : J → (1, 2] be a continuous function, for f2 ∈ Cδ(J, R) = f2(t) ∈ C(J, R), tδf2(t) ∈ C(J, R), 0 ≤ δ ≤ 1 , the variable-order fractional integral I0u+(t)f2(t) exists for any points on J. Regarding the continuity of function tδf and Lemma 2.1, we deduce that x is the solution of BVP (7). BVP (7) possesses at least one solution in E It follows from the properties of fractional integrals and from the continuity of function tδf that the operator W : E → E defined in (10) is well defined. By Banach’s contraction principle, W has a unique fixed point x ∈ E , which is the unique solution of problem (7), the claim of Theorem 3.1 is proved.

Example

Conclusion