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Optimal solutions for singular linear systems of Caputo fractional differential equations
Author(s) -
Dassios Ioannis,
Baleanu Dumitru
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5410
Subject(s) - mathematics , fractional calculus , smoothing , singular perturbation , derivative (finance) , mathematical analysis , singular solution , mathematical optimization , statistics , financial economics , economics
In this article, we focus on a class of singular linear systems of fractional differential equations with given nonconsistent initial conditions (IC). Because the nonconsistency of the IC can not lead to a unique solution for the singular system, we use two optimization techniques to provide an optimal solution for the system. We use two optimization techniques to provide the optimal solution for the system because a unique solution for the singular system cannot be obtained due to the non‐consistency of the IC. These two optimization techniques involve perturbations to the non‐consistent IC, specifically, an l 2 perturbation (which seeks an optimal solution for the system in terms of least squares), and a second‐order optimization technique at an l 1 minimum perturbation, (which includes an appropriate smoothing). Numerical examples are given to justify our theory. We use the Caputo ( C ) fractional derivative and two recently defined alternative versions of this derivative, the Caputo‐Fabrizio ( C F ) and the Atangana‐Baleanu ( A B ) fractional derivative.