Blowing-up solutions of multi-order fractional differential equations with the periodic boundary condition

Abstract

In this paper, we analyze the boundary value problem of a class of multi-order fractional differential equations involving the standard Caputo fractional derivative with the general periodic boundary conditions:where, a i ∈ R, a n = 0, and f : [0, T] × R → R is a continuous operation. We get the Green's function in terms of the Laplace transform. We obtain the existence and uniqueness of solution for the class of multi-order fractional differential equations. We investigate the blowing-up solutions to the special case f (t, u(t)) = |u(t)| p , a i ≥ 0, and give an upper bound on the blow-up time T max .

Stability of Ulam–Hyers and Ulam–Hyers–Rassias for a class of fractional differential equations

“…Fractional differential equations and fractional integral equations appeared in various fields such as polymer rheology, blood flow phenomena, electrodynamics of complex medium, modeling and control theory, signal processing, and so on; see [1,2]. In recent years, many researchers proved the existence and uniqueness of solutions to fractional differential equations [3][4][5][6][7][8]. Moreover, integral boundary problems had a variety of applications in real-life problems such as blood flow, underground water flow, population dynamics, thermoplasticity, chemical engineering, and so on; see [9][10][11].…”