Browsing by Subject "lower and upper solutions"

Now showing 1 - 2 of 2

Access status: Open Access ,
Monotone method for Riemann-Liouville multi-order fractional differential systems
(2016) Denton, Zachary
In this paper we develop the monotone method for nonlinear multi-order $N$-systems of Riemann-Liouville fractional differential equations. That is, a hybrid system of nonlinear equations of orders $q_i$ where $0 \lt q_i \lt 1$. In the development of this method we recall any needed existence results along with any necessary changes. Through the method's development we construct a generalized multi-order Mittag-Leffler function that fulfills exponential-like properties for multi-order systems. Further we prove a comparison result paramount for the discussion of fractional multi-order inequalities that utilizes lower and upper solutions of the system. The monotone method is then developed via the construction of sequences of linear systems based on the upper and lower solutions, and are used to approximate the solution of the original nonlinear multi-order system.
Access status: Open Access ,
Quasilinearization method for finite systems of nonlinear RL fractional differential equations
(Wydawnictwa AGH, 2020) Denton, Zachary; Ramírez, Juan Diego
In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order $0\lt q\lt 1$. Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.