A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions

Abstract

We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results. MSC: Primary 34A08; secondary 34A12Keywords: Caputo fractional derivative; Hybrid fractional differential equation and inclusion; Thermostat modelingh(t,z(t)) ] + g(t, z(t)) = 0 (t ∈ [0, 1], p ∈ (1, 2]), z(0) = z(1) = 0.

General conformable estimators with finite-time stability

“…Consider system (7) with pair (A, C) observable. IfĀ = A -K 1 C is stable and M i = P i-1 K i C is positive semidefinite for 2 ≤ i ≤ m, then the LQR-based estimator (9) is a finite-time general conformable exponentially stable estimator for system (6).…”

“…The high-gain observer (17) is a finite-time general conformable exponentially stable estimator for system (6).…”

“…Fractional calculus, the generalization of calculus to noninteger orders, besides looking to extend the classical mathematical results, has had many applications to physical systems since the 1970s of the twentieth century. The first results in this area involved the Riemann-Liouville integral and the Riemann-Liouville and Caputo derivatives, which are still widely studied and used [1][2][3][4][5][6][7][8]. Nevertheless, other definitions of noninteger operators have been developed; some recent classifications of the main operators and their properties appear in [9,10].…”

On Hyers–Ulam stability of a multi-order boundary value problems via Riemann–Liouville derivatives and integrals

“…Fractional differential problems have drawn much interest in recent years owing to their extensive utilization in different branches of science such as engineering, mechanics, potential theory, biology, chemistry, etc. (see [1][2][3][4][5][6][7][8][9][10][11]). Many researchers helped in developments on the existence and uniqueness results of fractional differential equations [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Fractional hybrid inclusion version of the Sturm–Liouville equation

“…During the last decades, many researchers have been studying some well-known problems involving differential equations such as Sturm-Lioville boundary value problems from different views (see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16]). It is important that researchers try to investigate distinct versions of famous and applicable differential equations (see, for example, [17][18][19][20]). On the other hand, some interesting integro-differential equations have been investigated by researchers.…”