Fractal and Fractional

Research

25 pages, 3204 KiB

Open AccessArticle

Fractional Partial Differential Equation Modeling for Solar Cell Charge Dynamics

by Waleed Mohammed Abdelfattah, Ola Ragb, Mohamed Salah, Mohamed S. Matbuly and Mokhtar Mohamed

Fractal Fract. 2024, 8(12), 729; https://doi.org/10.3390/fractalfract8120729 - 12 Dec 2024

Viewed by 847

This paper presents a groundbreaking numerical approach, the fractional differential quadrature method (FDQM), to simulate the complex dynamics of organic polymer solar cells. The method, which leverages polynomial-based differential quadrature and Cardinal sine functions coupled with the Caputo-type fractional derivative, offers a significant [...] Read more.

This paper presents a groundbreaking numerical approach, the fractional differential quadrature method (FDQM), to simulate the complex dynamics of organic polymer solar cells. The method, which leverages polynomial-based differential quadrature and Cardinal sine functions coupled with the Caputo-type fractional derivative, offers a significant improvement in accuracy and efficiency over traditional methods. By employing a block-marching technique, we effectively address the time-dependent nature of the governing equations. The efficacy of the proposed method is validated through rigorous numerical simulations and comparisons with existing analytical and numerical solutions. Each scheme’s computational characteristics are tailored to achieve high accuracy, ensuring an error margin on the order of

10^{- 8}

or less. Additionally, a comprehensive parametric study is conducted to investigate the impact of key parameters on device performance. These parameters include supporting conditions, time evolution, carrier mobilities, charge carrier densities, geminate pair distances, recombination rate constants, and generation efficiency. The findings of this research offer valuable insights for optimizing and enhancing the performance of organic polymer solar cell devices. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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25 pages, 14310 KiB

Open AccessArticle

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

by Waleed Mohammed Abdelfattah, Ola Ragb, Mohamed Salah and Mokhtar Mohamed

Fractal Fract. 2024, 8(11), 647; https://doi.org/10.3390/fractalfract8110647 - 6 Nov 2024

Cited by 2 | Viewed by 843

Abstract

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work [...] Read more.

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10⁻⁸. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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22 pages, 2980 KiB

Open AccessArticle

Approximate Solutions of Fractional Differential Equations Using Optimal q-Homotopy Analysis Method: A Case Study of Abel Differential Equations

by Süleyman Şengül, Zafer Bekiryazici and Mehmet Merdan

Fractal Fract. 2024, 8(9), 533; https://doi.org/10.3390/fractalfract8090533 - 11 Sep 2024

Viewed by 1167

Abstract

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives [...] Read more.

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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19 pages, 346 KiB

Open AccessArticle

Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay

by Aeshah Abdullah Muhammad Al-Dosari

Fractal Fract. 2024, 8(4), 235; https://doi.org/10.3390/fractalfract8040235 - 17 Apr 2024

Cited by 2 | Viewed by 1296

Abstract

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area [...] Read more.

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (

E_{α, β}

), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

19 pages, 4410 KiB

Open AccessArticle

Modeling and Analysis of Caputo–Fabrizio Definition-Based Fractional-Order Boost Converter with Inductive Loads

by Donghui Yu, Xiaozhong Liao and Yong Wang

Fractal Fract. 2024, 8(2), 81; https://doi.org/10.3390/fractalfract8020081 - 26 Jan 2024

Cited by 8 | Viewed by 1708

Abstract

This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model [...] Read more.

This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model of a fractional-order Boost converter by taking advantage of the state space averaging method, providing accurate analytical solutions for the quiescent operating point and the ripple parameters of the circuit with three state variables. Secondly, this paper constructs a small signal model of the C-F definition-based fractional-order Boost converter by small signal linearization, providing the transfer function of the fractional-order system with three state variables. Finally, this paper conducts circuit-oriented simulation experiments where the steady-state parameters and the transfer function of the circuit are obtained, and then the effect of the order of capacitor, induced inductor, and load inductor on the quiescent operating point and ripple parameters is analyzed. The experimental results show that the simulation results are consistent with those obtained by the proposed mathematical model and that the three fractional orders in the fractional model with three state variables have a significant impact on the DC component and steady-state characteristics of the fractional-order Boost converter. In conclusion, the proposed mathematical model can more comprehensively analyze the system characteristics of the C-F definition-based fractional-order Boost converter with fractional-order inductive loads, benefiting the circuit design of Boost converters. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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12 pages, 2379 KiB

Open AccessArticle

Fractional Photoconduction and Nonlinear Optical Behavior in ZnO Micro and Nanostructures

by Victor Manuel Garcia-de-los-Rios, Jose Alberto Arano-Martínez, Martin Trejo-Valdez, Martha Leticia Hernández-Pichardo, Mónica Araceli Vidales-Hurtado and Carlos Torres-Torres

Fractal Fract. 2023, 7(12), 885; https://doi.org/10.3390/fractalfract7120885 - 15 Dec 2023

Cited by 4 | Viewed by 1817

Abstract

A fractional description for the optically induced mechanisms responsible for conductivity and multiphotonic effects in ZnO nanomaterials is studied here. Photoconductive, electrical, and nonlinear optical phenomena exhibited by pure micro and nanostructured ZnO samples were analyzed. A hydrothermal approach was used to synthetize [...] Read more.

A fractional description for the optically induced mechanisms responsible for conductivity and multiphotonic effects in ZnO nanomaterials is studied here. Photoconductive, electrical, and nonlinear optical phenomena exhibited by pure micro and nanostructured ZnO samples were analyzed. A hydrothermal approach was used to synthetize ZnO micro-sized crystals, while a spray pyrolysis technique was employed to prepare ZnO nanostructures. A contrast in the fractional electrical behavior and photoconductivity was identified for the samples studied. A positive nonlinear refractive index was measured on the nanoscale sample using the z-scan technique, which endows it with a dominant real part for the third-order optical nonlinearity. The absence of nonlinear optical absorption, along with a strong optical Kerr effect in the ZnO nanostructures, shows favorable perspectives for their potential use in the development of all-optical switching devices. Fractional models for predicting electronic and nonlinear interactions in nanosystems could pave the way for the development of optoelectronic circuits and ultrafast functions controlled by ZnO photo technology. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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11 pages, 533 KiB

Open AccessArticle

Application of Riemann–Liouville Derivatives on Second-Order Fractional Differential Equations: The Exact Solution

by Abdulrahman B. Albidah

Fractal Fract. 2023, 7(12), 843; https://doi.org/10.3390/fractalfract7120843 - 28 Nov 2023

Cited by 3 | Viewed by 1712

Abstract

This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional [...] Read more.

This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fractional differential equations under appropriate initial conditions. It is shown that the zeroth lower bound implies implicit solutions in terms of the Mittag–Leffler functions while explicit solutions are derived when negative infinity is taken as a lower bound. Such explicit solutions are obtained for the current two classes in terms of trigonometric and hyperbolic functions. Some theoretical results are introduced to facilitate the solutions procedures. Moreover, the characteristics of the obtained solutions are discussed and interpreted. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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23 pages, 1284 KiB

Open AccessArticle

Fractional-Order Zener Model with Temperature-Order Equivalence for Viscoelastic Dampers

by Kang Xu, Liping Chen, António M. Lopes, Mingwu Wang, Ranchao Wu and Min Zhu

Fractal Fract. 2023, 7(10), 714; https://doi.org/10.3390/fractalfract7100714 - 28 Sep 2023

Cited by 5 | Viewed by 1505

Abstract

Viscoelastic (VE) dampers show good performance in dissipating energy, being widely used for reducing vibration in engineering structures caused by earthquakes and winds. Experimental studies have shown that ambient temperature has great influence on the mechanical behavior of VE dampers. Therefore, it is [...] Read more.

Viscoelastic (VE) dampers show good performance in dissipating energy, being widely used for reducing vibration in engineering structures caused by earthquakes and winds. Experimental studies have shown that ambient temperature has great influence on the mechanical behavior of VE dampers. Therefore, it is important to accurately model VE dampers considering the effect of temperature. In this paper, a new fractional-order Zener (AEF-Zener) model of VE dampers is proposed. Firstly, the important influence of fractional orders on the energy dissipation ability of materials is analyzed. Secondly, an equivalent AEF-Zener model is developed that incorporates the ambient temperature and fractional-order equivalence principle. Finally, the chaotic fractional-order particle swarm optimization (CFOPSO) algorithm is used to determine the model’s parameters. The accuracy of the AEF-Zener model is verified by comparing model simulations with experimental results. This study is helpful for designing and analyzing vibration reduction techniques for civil structures with VE dampers under the influence of temperature. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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