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27 pages, 383 KiB

Open AccessArticle

Qualitative Analysis of Stochastic Caputo–Katugampola Fractional Differential Equations

by Zareen A. Khan, Muhammad Imran Liaqat, Ali Akgül and J. Alberto Conejero

Axioms 2024, 13(11), 808; https://doi.org/10.3390/axioms13110808 - 20 Nov 2024

Cited by 3 | Viewed by 967

Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we [...] Read more.

Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we achieve significant results for these equations within the context of Caputo–Katugampola derivatives. First, we establish the existence and uniqueness of solutions by employing the contraction mapping principle with a suitably weighted norm and demonstrate that the solutions continuously depend on both the initial values and the fractional exponent. The second part examines the regularity concerning time. Third, we illustrate the results of the averaging principle using techniques involving inequalities and interval translations. We generalize these results in two ways: first, by establishing them in the sense of the Caputo–Katugampola derivative. Applying condition

β = 1

, we derive the results within the framework of the Caputo derivative, while condition

β \to 0^{+}

yields them in the context of the Caputo–Hadamard derivative. Second, we establish them in

L^{p}

space, thereby generalizing the case for

p = 2

. Full article

(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)

27 pages, 368 KiB

Open AccessArticle

Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat

Axioms 2024, 13(7), 438; https://doi.org/10.3390/axioms13070438 - 28 Jun 2024

Cited by 3 | Viewed by 1224

Abstract

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable [...] Read more.

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in

L^{p}

spaces with

p \geq 2

under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end. Full article

(This article belongs to the Special Issue Advances in Partial Differential Equations: Qualitative Analysis and Numerical Methods)

14 pages, 380 KiB

Open AccessArticle

On Pantograph Problems Involving Weighted Caputo Fractional Operators with Respect to Another Function

by Saeed M. Ali

Fractal Fract. 2023, 7(7), 559; https://doi.org/10.3390/fractalfract7070559 - 19 Jul 2023

Cited by 3 | Viewed by 1532

Abstract

In this investigation, weighted

p s i

-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes [...] Read more.

In this investigation, weighted

p s i

-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes into account the situation where

ψ (x) = x

. With the aid of Banach’s and Krasnoselskii’s classic fixed point results, we have established a the qualitative study. Different values of

ψ (x)

and

w (x)

are discussed as special cases that are relevant to our current results. Additionally, in light of our findings, we provide a more general fractional system with the weighted

ψ

-Caputo-type that takes into account both the new problems and certain previously existing, related problems. Finally, we give two illustrations to support and validate the outcomes. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

17 pages, 358 KiB

Open AccessArticle

Nonlinear Piecewise Caputo Fractional Pantograph System with Respect to Another Function

by Mohammed S. Abdo, Wafa Shammakh, Hadeel Z. Alzumi, Najla Alghamd and M. Daher Albalwi

Fractal Fract. 2023, 7(2), 162; https://doi.org/10.3390/fractalfract7020162 - 6 Feb 2023

Cited by 10 | Viewed by 1779

Abstract

The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel

p s i

-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the [...] Read more.

The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel

p s i

-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the literature. The required results are proven using Banach’s contraction mapping and Krasnoselskii’s fixed-point theorem. Additionally, results pertaining to UH stability are obtained using traditional procedures of nonlinear functional analysis. Additionally, in light of our current findings, a more general challenge for the pantograph system is presented that includes problems similar to the one considered. We provide a pertinent example as an application to support the theoretical findings. Full article

(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)

21 pages, 839 KiB

Open AccessArticle

Controllability and Observability Analysis of a Fractional-Order Neutral Pantograph System

by Irshad Ahmad, Saeed Ahmad, Ghaus ur Rahman, Shabir Ahmad and Wajaree Weera

Symmetry 2023, 15(1), 125; https://doi.org/10.3390/sym15010125 - 1 Jan 2023

Cited by 4 | Viewed by 1740

Abstract

In the recent past, a number of research articles have explored the stability, existence, and uniqueness of the solutions and controllability of dynamical systems with a fractional order (FO). Nevertheless, aside from the controllability and other dynamical aspects, very little attention has been [...] Read more.

In the recent past, a number of research articles have explored the stability, existence, and uniqueness of the solutions and controllability of dynamical systems with a fractional order (FO). Nevertheless, aside from the controllability and other dynamical aspects, very little attention has been given to the observability of FO dynamical systems. This paper formulates a novel type of FO delay system of the Pantograph type in the Caputo sense and explores its controllability and observability results. This research endeavor begins with the conversion of the proposed dynamical system into a fixed-point problem by utilizing Laplace transforms, the convolution of Laplace functions, and the Mittag–Leffler function (MLF). We then set out Gramian matrices for both the controllability and observability of the linear parts of our proposed dynamical system and prove that both the Gramian matrices are invertible, thus confirming the controllability and observability in a given domain. Considering the controllability and observability results of the linear part along with some other assumptions, we investigate the controllability and observability results related to the nonlinear system. The Banach contraction result, the fixed-point result of Schaefer, the MLF, and the Caputo FO derivative are used as the main tools for establishing these results. To establish the authenticity of the established results, we add two examples at the end of the manuscript. Full article

9 pages, 286 KiB

Open AccessArticle

On Averaging Principle for Caputo–Hadamard Fractional Stochastic Differential Pantograph Equation

by Mounia Mouy, Hamid Boulares, Saleh Alshammari, Mohammad Alshammari, Yamina Laskri and Wael W. Mohammed

Fractal Fract. 2023, 7(1), 31; https://doi.org/10.3390/fractalfract7010031 - 28 Dec 2022

Cited by 26 | Viewed by 1996

Abstract

In this paper, we studied an averaging principle for Caputo–Hadamard fractional stochastic differential pantograph equation (FSDPEs) driven by Brownian motion. In light of some suggestions, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean [...] Read more.

In this paper, we studied an averaging principle for Caputo–Hadamard fractional stochastic differential pantograph equation (FSDPEs) driven by Brownian motion. In light of some suggestions, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean square. We expand the classical Khasminskii approach to Caputo–Hadamard fractional stochastic equations by analyzing systems solutions before and after applying averaging principle. We provided an applied example that explains the desired results to us. Full article

(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)

18 pages, 14064 KiB

Open AccessArticle

Stability Analysis of Fractional-Order Mathieu Equation with Forced Excitation

by Ruihong Mu, Shaofang Wen, Yongjun Shen and Chundi Si

Fractal Fract. 2022, 6(11), 633; https://doi.org/10.3390/fractalfract6110633 - 31 Oct 2022

Cited by 2 | Viewed by 2496

Abstract

The advantage of fractional-order derivative has attracted extensive attention in the field of dynamics. In this paper, we investigated the stability of the fractional-order Mathieu equation under forced excitation, which is based on a model of the pantograph–catenary system. First, we obtained the [...] Read more.

The advantage of fractional-order derivative has attracted extensive attention in the field of dynamics. In this paper, we investigated the stability of the fractional-order Mathieu equation under forced excitation, which is based on a model of the pantograph–catenary system. First, we obtained the approximate analytical expressions and periodic solutions of the stability boundaries by the multi-scale method and the perturbation method, and the correctness of these results were verified through numerical analysis by Matlab. In addition, by analyzing the stability of the k’T-periodic solutions in the system, we verified the existence of the unstable k’T-resonance lines through numerical simulation, and visually investigated the effect of the system parameters. The results show that forced excitation with a finite period does not change the position of the stability boundaries, but it can affect the expressions of the periodic solutions. Moreover, by analyzing the properties of the resonant lines, we found that when the points with k’T-periodic solutions were perturbed by the same frequency of forced excitation, these points became unstable due to resonance. Finally, we found that both the damping coefficient and the fractional-order parameters in the system have important influences on the stability boundaries and the resonance lines. Full article

(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)

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15 pages, 323 KiB

Open AccessArticle

Exact Solvability Conditions for the Non-Local Initial Value Problem for Systems of Linear Fractional Functional Differential Equations

by Natalia Dilna and Michal Fečkan

Mathematics 2022, 10(10), 1759; https://doi.org/10.3390/math10101759 - 21 May 2022

Cited by 4 | Viewed by 1652

Abstract

The exact conditions sufficient for the unique solvability of the initial value problem for a system of linear fractional functional differential equations determined by isotone operators are established. In a sense, the conditions obtained are optimal. The method of the test elements intended [...] Read more.

The exact conditions sufficient for the unique solvability of the initial value problem for a system of linear fractional functional differential equations determined by isotone operators are established. In a sense, the conditions obtained are optimal. The method of the test elements intended for the estimation of the spectral radius of a linear operator is used. The unique solution is presented by the Neumann’s series. All theoretical investigations are shown in the examples. A pantograph-type model from electrodynamics is studied. Full article

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

17 pages, 1952 KiB

Open AccessArticle

Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System

by Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Juan L. G. Guirao and Tareq Saeed

Fractal Fract. 2021, 5(4), 277; https://doi.org/10.3390/fractalfract5040277 - 17 Dec 2021

Cited by 8 | Viewed by 2495

Abstract

The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation [...] Read more.

The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation and the detailed discussions of the singularity, shape factor terms along with the fractional order forms. The numerical discussions of the FOPSS are described based on the fractional Meyer wavelets (FMWs) as a neural network (NN) with the optimization procedures of global/local search procedures of particle swarm optimization (PSO) and interior-point algorithm (IPA), i.e., FMWs-NN-PSOIPA. The FMWs-NN strength is pragmatic and forms a merit function based on the differential system and the initial conditions of the FOPSS. The merit function is optimized, using the integrated capability of PSOIPA. The perfection, verification and substantiation of the FOPSS using the FMWs is pragmatic for three cases through relative investigations from the true results in terms of stability and convergence. Additionally, the statics’ descriptions further authorize the presentation of the FMWs-NN-PSOIPA in terms of reliability and accuracy. Full article

(This article belongs to the Special Issue Numerical Methods and Simulations in Fractal and Fractional Problems)

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