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29 pages, 464 KiB

Open AccessArticle

On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

by Sina Etemad, Sotiris K. Ntouyas, Ivanka Stamova and Jessada Tariboon

Fractal Fract. 2024, 8(4), 236; https://doi.org/10.3390/fractalfract8040236 - 17 Apr 2024

Cited by 9 | Viewed by 1629

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned

(p; q)

-difference Navier problems in two single-valued and set-valued versions. We use the main properties of the

(p; q)

-operators in this regard. Application of the fixed points of the

ρ

θ

-contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential

(p; q)

-difference Navier problems. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

14 pages, 317 KiB

Open AccessArticle

On Solvability of Fractional (p,q)-Difference Equations with (p,q)-Difference Anti-Periodic Boundary Conditions

by Ravi P. Agarwal, Hana Al-Hutami and Bashir Ahmad

Mathematics 2022, 10(23), 4419; https://doi.org/10.3390/math10234419 - 23 Nov 2022

Cited by 6 | Viewed by 1442

Abstract

We discuss the solvability of a

(p, q)

-difference equation of fractional order

α \in (1, 2]

, equipped with anti-periodic boundary conditions involving the first-order

(p, q)

-difference operator. The desired results are [...] Read more.

We discuss the solvability of a

(p, q)

-difference equation of fractional order

α \in (1, 2]

, equipped with anti-periodic boundary conditions involving the first-order

(p, q)

-difference operator. The desired results are accomplished with the aid of standard fixed point theorems. Examples are presented for illustrating the obtained results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

12 pages, 296 KiB

Open AccessArticle

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

by Piyachat Borisut, Poom Kumam, Idris Ahmed and Kanokwan Sitthithakerngkiet

Symmetry 2019, 11(6), 829; https://doi.org/10.3390/sym11060829 - 22 Jun 2019

Cited by 41 | Viewed by 4462

Abstract

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): ^c [...] Read more.

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): ^c

D_{0^{+}}^{q} u (t) = f (t, u (t)), t \in [0, T]

u^{(k)} (0) = ξ_{k}, u (T) = \sum_{i = 1}^{m} {β_{i}}_{R L} I_{0^{+}}^{p_{i}} u (η_{i}),

where

n - 1 < q < n, n \geq 2, m, n \in N

ξ_{k}, β_{i} \in R

k = 0, 1, \dots, n - 2

i = 1, 2, \dots, m

, and ^c

D_{0^{+}}^{q}

is the Caputo fractional derivatives,

f : [0, T] \times C ([0, T], E) \to E

, where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be

R

(with the absolute value) or

C ([0, T], R)

with the supremum-norm. _RL

I_{0^{+}}^{p_{i}}

is the Riemann–Liouville fractional integral of order

p_{i} > 0

η_{i} \in (0, T)

, and

\sum_{i = 1}^{m} β_{i} η_{i}^{p_{i} + n - 1} \frac{Γ (n)}{Γ (n + p_{i})} \neq T^{n - 1}

. Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

12 pages, 1732 KiB

Open AccessArticle

Atangana–Baleanu and Caputo Fabrizio Analysis of Fractional Derivatives for Heat and Mass Transfer of Second Grade Fluids over a Vertical Plate: A Comparative Study

by Arshad Khan, Kashif Ali Abro, Asifa Tassaddiq and Ilyas Khan

Entropy 2017, 19(8), 279; https://doi.org/10.3390/e19080279 - 18 Aug 2017

Cited by 96 | Viewed by 8407

Abstract

This communication addresses a comparison of newly presented non-integer order derivatives with and without singular kernel, namely Michele Caputo–Mauro Fabrizio (CF)

^{C F} (\partial^{β} / \partial t^{β})

and Atangana–Baleanu (AB) [...] Read more.

This communication addresses a comparison of newly presented non-integer order derivatives with and without singular kernel, namely Michele Caputo–Mauro Fabrizio (CF)

^{C F} (\partial^{β} / \partial t^{β})

and Atangana–Baleanu (AB)

^{A B} (\partial^{α} / \partial t^{α})

fractional derivatives. For this purpose, second grade fluids flow with combined gradients of mass concentration and temperature distribution over a vertical flat plate is considered. The problem is first written in non-dimensional form and then based on AB and CF fractional derivatives, it is developed in fractional form, and then using the Laplace transform technique, exact solutions are established for both cases of AB and CF derivatives. They are then expressed in terms of newly defined M-function

M_{q}^{p} (z)

and generalized Hyper-geometric function

_{p} Ψ_{q} (z)

. The obtained exact solutions are plotted graphically for several pertinent parameters and an interesting comparison is made between AB and CF derivatives results with various similarities and differences. Full article

(This article belongs to the Special Issue Non-Equilibrium Thermodynamics of Micro Technologies)

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