Application Of Fractional Calculus Research Articles

Applications of Fractional Calculus and Borel Distribution Series for Multivalent Functions on Complex Hilbert Space

In this paper, we introduce applications of fractional calculus techniques for a family of multivalent analytic functions defined by the Borel distribution on Hilbert space. We derive several interesting properties, including coefficient estimates, extreme points, and convex combinations.

Application of Fractional Modified Taylor Wavelets in the Dynamical Analysis of Fractional Electrical Circuits under Generalized Caputo Fractional Derivative

Abstract This study examines the application of fractional calculus in the analysis and modeling of electrical circuits of fractional order, highlighting its potential to explain the behaviour of complex electrical circuits accurately. In the domain of electrical circuits, fractional differential equations are employed in the analysis and simulation of systems that consist of resistors, capacitors and inductors. In the present paper, a novel approach utilizing fractional order modified Taylor wavelets is implemented to solve the fractional model of RL, LC, RC and RLC electrical circuits under generalized Caputo fractional derivative which offers precise and flexible modeling of non-locality and hereditary characteristics in complex systems. Furthermore, an additional parameter $\sigma$ (time scale parameter) is incorporated in fractional circuit dynamics to maintain the physical dimensionality. The considered wavelets with the collocation technique offer an efficient solution by converting the fractional model of electrical circuits into a set of algebraic equations which are further solved by using the Newton iteration method. Moreover, this study discusses the significance of Ulam-Hyers stability, emphasizing its role in ensuring stable and reliable circuit performance. The impact of fractional order on the dynamics of the electric circuit model is presented by tables and graphs. The approximate solutions obtained by the proposed method are well comparable with exact solutions and some other existing wavelet-based techniques. The residual errors are also evaluated under various model parameters for fractional orders. Furthermore, the graphs illustrate that the error progressively decreases as the number of wavelets basis increases.

Chaos-driven detection of methylene blue in wastewater using fractional calculus and laser systems

Abstract Methylene blue (MB) concentrations in residual water were detected using fractional calculus, the Rössler chaotic attractor and laser systems. A Nd:YVO4 nanosecond pulsed laser at 532 nm, with pulse energies ranging from 2 µJ to 7 µJ, was applied to irradiate different water samples containing MB concentrations from 20 µl to 100 µl. Fractional calculus was employed with the purpose of modeling the temperature distribution in the samples, with the Caputo fractional derivative describing photothermal effects induced by laser irradiation. Different MB concentrations were detected by using the Rössler chaotic attractor, it monitored variation on concentrations, associating attractor shapes with MB concentrations. Lower concentrations showed a weaker attractor response, whereas higher concentrations manifest stronger attractor shapes in magnitude. Raman spectroscopy confirmed the detection of MB in residual water from the Requena dam, located in Tepeji del Río de Ocampo, Hidalgo, Mexico. The application of fractional calculus improved the prediction of heat distribution in the samples, by incorporating numerical simulation. The results suggest that this approach is suitable for real-time monitoring, as it associates MB concentrations with distinct chaotic attractor shapes. This technique shows promise for the detection of other contaminants as well. Future research should focus on refining this method and expanding its application to develop innovative monitoring solutions.

Hybrid finite element and laplace transform method for efficient numerical solutions of fractional PDEs on graphics processing units

Fractional Partial Differential equations (FPDEs) are essential for modeling complex systems across various scientific and engineering areas. However, efficiently solving FPDEs presents significant computational challenges due to their inherent memory effects, often leading to increased execution times for numerical solutions. This study proposes a highly parallelizable hybrid computational approach that combines the Finite Element Method (FEM) for spatial discretization with Numerical Inversion of the Laplace Transform (NILT) for time-domain solutions, optimized for execution on Graphics Processing Units (GPUs). The NILT method’s high parallelizability, stemming from the independence of its series terms, combined with the robust spatial discretization provided by FEM, enables the efficient and accurate solution of FPDEs on GPUs, demonstrating substantial performance improvements over traditional CPU-based implementations. We observe a generalized pattern in execution time behavior that accounts for both the number of nodes and the number of NILT terms. Specifically, execution time scales quadratically with the number of nodes, while showing only a logarithmic marginal increase with the number of NILT terms These behaviors not only enables consistent performance assessment but also highlights potential areas for algorithm optimization. Validation against exact solutions of fractional diffusion and wave equations, employing Caputo, modified Caputo-Fabrizio, and modified Atangana-Baleanu derivatives, demonstrates the accuracy and convergence of the hybrid FEM-NILT method. Notably, the exact solutions of wave equation are novel in literature. The results highlight the method’s potential for enabling high-precision, large-scale simulations in fractional calculus applications, thereby advancing computational capabilities and efficiency in the field.

Hyers–Ulam stability of Nipah virus model using Atangana–Baleanu–Caputo fractional derivative with fixed point method

In this study, we present a novel investigation into the dynamics of the Nipah virus through the lens of fractional differential equations (FDEs), employing the Atangana–Baleanu–Caputo fractional derivative (ABCFD) and the fixed-point approach (FPA). The core contribution of this work lies in establishing the existence and uniqueness of solutions to the proposed FDEs, a critical step for validating the model. Furthermore, we explore the Hyers–Ulam (HU) stability of these generalized FDEs, providing a rigorous mathematical foundation for the stability analysis within the context of viral dynamics. By leveraging the ABCFD, our work extends the classical stability criteria, offering new insights into the role of memory effects in disease modeling. Additionally, we present approximate solutions across various compartments and fractional orders, highlighting the sensitivity of the system to key parameters. Numerical simulations, conducted using the Cullis method, illustrate the impact of fractional orders and validate the theoretical findings, positioning this work as a significant advancement in the application of fractional calculus to epidemiological models.

Series form solutions of time–space fractional Black–Scholes model via extended He-Aboodh algorithm

The objective of the current study is analyze linear and nonlinear time–space fractional Black–Scholes models via modified homotopy perturbation method (m-HPM). In current investigation, memory effects in financial markets are explored through fractional derivative in Caputo sense. The effectiveness of proposed methodology is checked numerically by finding residual errors and presented in tables. These tables also provide a benchmark for the comparison with already existing results in literature. Furthermore, solutions are graphically analyzed via 3D and contour plots across a range of parameters under varying market conditions. Analysis confirms the efficiency of m-HPM for predicting solutions of time–space fractional Black–Scholes models. The current study can contribute in understanding the applications of fractional calculus in finance, and can be a valuable computational tool for pricing financial derivatives in fractional environments.

A novel methodology for anomaly detection in smart home networks via Fractional Stochastic Gradient Descent

With the rising integration of IoT devices within smart home environments, securing these interconnected systems against unauthorized access and cyber threats has become increasingly critical. This study introduces a novel methodology utilizing an advanced Artificial Neural Network (ANN) frame-work to enhance attack and anomaly detection capabilities within smart home networks. The objective is to develop a robust model that outperforms traditional detection methods and provides high accuracy and low false positive rates in identifying potential security threats. The research employs a pioneering approach to train the ANN by incorporating a Fractional Stochastic Gradient Descent optimizer grounded in Grunwald–Letnikov fractional calculus. This method was chosen for its potential to refine learning processes and improve detection accuracy over standard optimizers. The evaluation of the model was performed using the DS2OS traffic traces dataset, applying precision, sensitivity (recall), and specificity metrics to assess performance. The proposed model demonstrated exceptional performance with an accuracy of 0.9951. It significantly surpassed traditional methods like Logistic Regression and Support Vector Machines. The precision achieved was high, indicating a low rate of false positives, while the sensitivity and specificity values underscore the model’s ability to identify both typical and unconventional behaviours within the network accurately. This study introduces a robust and efficient ANN-based methodology for enhancing security in smart home IoT networks. Using a fractional stochastic gradient descent optimizer has proven effective in improving the model’s accuracy and reliability in detecting anomalies and attacks. The findings suggest significant implications for the future of IoT security, highlighting the potential for broader applications of fractional calculus in machine learning to enhance cybersecurity measures in various domains.

An extension of Schweitzer's inequality to Riemann-Liouville fractional integral

Abstract This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.

Modeling the Dynamics of Supercapacitors by Means of Riemann–Liouville Integral Definition

The application of fractional calculus to obtain dynamic models for supercapacitors represents an alternative approach to obtaining simpler and more accurate models. This paper presents a model for the supercapacitor in the time domain, based on the use of the fractional or non-integer order integral. This fractional model is compared with the conventional simple model, which is typically used in industrial applications. This fractional integral-based model provides satisfactory fits in relation to the number of parameters used in the model. Furthermore, an interpretation of the effect of the application of fractional integration is presented for constant current charging and discharging processes at constant current, using the Riemann–Liouville definition for the non-integer order integral. Supercapacitors are devices that exhibit non-linear behavior, with a distinct charging and discharging operation. There are several methods of dynamic analysis for the characterization of supercapacitors. The information extracted from these methods is essential for understanding the behavior of supercapacitors and, thus, ensuring that processes involving supercapacitors are as efficient as possible. This paper presents a dynamic analysis based on charge and discharge operations with constant currents. The conclusion is that the fractional model provides fairly accurate fits.

Analysis of some dynamical systems by combination of two different methods

In this study, we introduce a novel iterative method combined with the Elzaki transformation to address a system of partial differential equations involving the Caputo derivative. The Elzaki transformation, known for its effectiveness in solving differential equations, is incorporated into the proposed iterative approach to enhance its efficiency. The system of partial differential equations under consideration is characterized by the presence of Caputo derivatives, which capture fractional order dynamics. The developed method aims to provide accurate and efficient solutions to this complex mathematical system, contributing to the broader understanding of fractional calculus applications in the context of partial differential equations. Through numerical experiments and comparisons, we demonstrate the efficacy of the proposed Elzaki-transform-based iterative method in handling the intricate dynamics inherent in the given system. The study not only showcases the versatility of the Elzaki transformation but also highlights the potential of the developed iterative technique for addressing similar problems in various scientific and engineering domains.

The application of fractional calculus in economic growth modelling: An approach based on regression analysis

This paper establishes a fractional-order economic growth model to model the gross domestic product (GDP). The fractional-order model consists of a differential equation of integer and fractional orders, where the GDP is a function of several exploratory variables. An empirical application is adopted using Malaysia's GDP data from 1956 to 2018, incorporating exploratory variables such as total population, crude death rate, production of logs, gross fixed capital formation, exports of goods and services, general government final consumption expenditure, private final consumption expenditure, and the impact of investment. Extensive comparisons were carried out to evaluate the modelling performance of the full and reduced fractional-order multiple linear regression models with the benchmark models, namely full and reduced integer-order multiple linear regression models. Results indicate that the reduced fractional-order model with six exploratory variables, excluding the crude death rate and production of logs, predominates other models for the in-sample model fitting based on the Akaike information criterion, coefficient of determination and other criteria. Furthermore, the fractional-order model offers the best-of-sample forecasts evaluated based on the root mean square forecast error and mean absolute forecast error. The application of the Diebold–Mariano test also serves to confirm the superior performance of the suggested fractional-order model, revealing a significant difference in forecasting ability between the fractional-order and integer-order models.

Fuzzy discrete fractional calculus and fuzzy fractional discrete equations

This paper aims to highlight certain limitations in the study of fuzzy fractional discrete equations (FFDEs) based on the generalized Hukuhara difference (gH-difference) in the previous papers. In general, the equivalence between FFDEs and the associated fuzzy discrete fractional sum equations (FDFSEs) is not achieved, requiring the introduction of an appropriate hypothesis to establish this equivalence. Furthermore, this paper introduces the fundamental theory of fuzzy fractional discrete calculus through granular arithmetic operations between fuzzy intervals to address restrictions in the formerly mentioned approaches involving the generalized Hukuhara difference. These operations are constructed based on the concept of the horizontal membership function (HMF) utilized in multidimensional fuzzy arithmetic (MFA). Additionally, the paper proposes the application of fractional discrete calculus to two types of time-discretization diffusion equations with non-zero right-hand sides. Finally, several numerical examples are provided to validate the main results.

Erroneous Applications of Fractional Calculus: The Catenary as a Prototype

In this work, we study the equation of the catenary curve in the context of the Caputo derivative. We solve this equation and compare the solution with real physical models. From the experiments, we find that the best approximation is achieved in the classical case. Therefore, introducing a fractional parameter arbitrarily can be detrimental. However, we observe that, when adding a certain weight to the chain, fractional calculus produces better results than classical calculus for modeling the minimum height.

Solving a Novel System of Time-Dependent Nuclear Reactor Equations of Fractional Order

Building upon the previous research that solved neutron diffusion equations in simplified slab geometry, this study advances the field by addressing the more complex cylindrical geometry, focusing on neutron diffusion equations that are coupled with delayed neutrons in cylindrical reactors of fractional order. The method of solving used integrates the technique of residual power series (RPS) with the Laplace transform (LT) method. Anomalous neutron behavior is explained by examining the non-Gaussian scenario with various fractional parameters α. The LRPSM Laplace transform and residual power series method employed in this approach eliminates the complex difficulties. This simplicity makes the method particularly coherent with different fractional calculus applications. To validate the proposed method, numerical simulations are conducted with two different initial conditions representing distinct scenarios. The obtained results are presented in suitable tables and figures. It should be emphasized that this system is solved for the first time utilizing fractional calculus techniques. The outcomes are consistent with those achieved using the Adomian decomposition method.

Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

An efficient computational scheme for solving coupled time-fractional Schrödinger equation via cubic B-spline functions.

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schrödinger equation using B-spline functions and the Atangana-Baleanu fractional derivative. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while a θ-weighted scheme is used to discretize the space directions. The efficiency of the method is demonstrated through numerical results, and error norms are examined at various values of the non-integer parameter, temporal directions, and spatial directions.

Applying Laplace Transformation on Epidemiological Models as Caputo Derivatives

This paper delves into the application of fractional calculus, with a focus on Caputo derivatives, in epidemiological models using ordinary differential equations. It highlights the critical role Caputo derivatives play in modeling intricate systems with memory effects and assesses various epidemiological models, including SIR variants, demonstrating how Caputo derivatives capture fractional-order dynamics and memory phenomena found in real epidemics. The study showcases the utility of Laplace transformations for analyzing systems described by ordinary differential equations with Caputo derivatives. This approach facilitates both analytical and numerical methods for system analysis and parameter estimation. Additionally, the paper introduces a tabular representation for epidemiological models, enabling a visual and analytical exploration of variable relationships and dynamics. This matrix-based framework permits the application of linear algebra techniques to assess stability and equilibrium points, yielding valuable insights into long-term behavior and control strategies. In summary, this research underscores the significance of Caputo derivatives, Laplace transformations, and matrix representation in epidemiological modeling. We assume that by using this type of methodology we can get analytic solutions by hand when considering a function as constant in certain cases and it will not be necessary to search for numerical methods.

Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative

Abstract This study demonstrates the use of fractional calculus in the field of epidemiology, specifically in relation to dengue illness. Using noninteger order integrals and derivatives, a novel model is created to examine the impact of temperature on the transmission of the vector–host disease, dengue. A comprehensive strategy is proposed and illustrated, drawing inspiration from the first dengue epidemic recorded in 2009 in Cape Verde. The model utilizes a fractional-order derivative, which has recently acquired popularity for its adaptability in addressing a wide variety of applicable problems and exponential kernel. A fixed point method of Krasnoselskii and Banach is used to determine the main findings. The semi-analytical results are then investigated using iterative techniques such as Laplace-Adomian decomposition method. Computational models are utilized to support analytical experiments and enhance the credibility of the results. These models are useful for simulating and validating the effect of temperature on the complex dynamics of the vector–host interaction during dengue outbreaks. It is essential to note that the research draws on dengue outbreak studies conducted in various geographic regions, thereby providing a broader perspective and validating the findings generally. This study not only demonstrates a novel application of fractional calculus in epidemiology but also casts light on the complex relationship between temperature and the dynamics of dengue transmission. The obtained results serve as a foundation for enhancing our understanding of the complex interaction between environmental factors and infectious diseases, leading the way for enhanced prevention and control strategies to combat global dengue outbreaks.

Caputo Fractional Derivative Inequalities for Refined Modified ( h , m ) -Convex Functions

This paper introduces a novel type of convex function known as the refined modified (h,m)-convex function, which is a generalization of the traditional (h,m)-convex function. We establish Hadamard-type inequalities for this new definition by utilizing the Caputo k-fractional derivative. Specifically, we derive two integral identities that involve the nth order derivatives of given functions and use them to prove the estimation of Hadamard-type inequalities for the Caputo k-fractional derivatives of refined modified (h,m)-convex functions. The results obtained in this research demonstrate the versatility of the refined modified (h,m)-convex function and the usefulness of Caputo k-fractional derivatives in establishing important inequalities. Our work contributes to the existing body of knowledge on convex functions and offers insights into the applications of fractional calculus in mathematical analysis. The research findings have the potential to pave the way for future studies in the area of convex functions and fractional calculus, as well as in other areas of mathematical research.

Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method

<p>We investigated the dynamics of highly dispersive nonlinear gap solitons in optical fibers with dispersive reflectivity, utilizing a conformable fractional derivative model. The modified extended direct algebraic method was employed to obtain various soliton solutions, including bright solitons and singular solitons, as well as hyperbolic and trigonometric solutions. The key findings demonstrated that the fractional derivative parameter ($ \alpha $) can effectively control the wave propagation, causing a shift in the wave signal while maintaining the same amplitude. This is a novel contribution, as the ability to control soliton properties through the conformable derivative is explored for the first time in this work. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. This research provides insights into the potential applications of fractional calculus in the design and optimization of photonic devices for optical communication systems.</p>