Conformable Fractional Derivative Research Articles

Enhancing the performance of CNN models for pneumonia and skin cancer detection using novel fractional activation function

This paper introduces a novel Riemann–Liouville (RL) conformable fractional derivative based Adaptable-Shifted-Fractional-Rectified-Linear-Unit, briefly called RLASFReLU, and evaluates its efficacy in enhancing the performance of convolutional neural network (CNN) models for pneumonia and skin cancer detection. The study conducts a comprehensive comparative analysis against traditional activation functions and state-of-the-art CNN architectures. The results show that RLASFReLU consistently outperforms other functions, achieving higher accuracy. Comparative evaluations with various neural network architectures reveal that the model equipped with RLASFReLU exhibits superior performance despite its simplicity and fewer trainable parameters, highlighting its efficiency and effectiveness. The findings suggest that RLASFReLU holds promise in improving diagnostic accuracy and efficiency in medical imaging applications, contributing to advancements in healthcare technology and facilitating better patient care. The proposed fractional nonlinear transformation can offer high performance with reduced computational cost, making it practical for deployment in healthcare settings.

Foldy–Wouthuysen transformation of Dirac equation in the context of conformable fractional derivative

Foldy–Wouthuysen transformation of Dirac equation in the context of conformable fractional derivative

Conformable Sumudu Transform Based Adomian Decomposition Method for Linear and Nonlinear Fractional-Order Schrödinger Equations

Fractional-order Schrödinger differential equations extend the classical Schrödinger equation by incorporating fractional calculus to describe more complex physical phenomena. In the literature, the Schr ̈odinger equation is mostly solved using fractional derivatives expressed through the Caputo derivative. However, there is limited research on exact and approximate solutionsinvolving conformable fractional derivatives. This study aims to fill this gap by employing a hybrid approach that combines the Sumudu transform with the decomposition technique to solve the Schrödinger equation with conformable fractional derivatives, considering zero and nonzero trapping potentials. The efficiency of this approach is evaluated through the analysis of relative andabsolute errors, confirming its accuracy. Moreover, the obtained results are compared with other techniques, including the homotopy analysis method (HAM) and the residual power series method (RPSM). The comparison demonstrates strong consistency with these methods, suggesting that our approach is a viable alternative to Caputo derivative-based methods for solving time-fractionalSchr ̈odinger equations. Furthermore, we can conclude that the conformable fractional derivative serves as a suitable substitute for the Caputo derivative in modeling Schrödinger equations.

Solution of Well-known Physical Problems with Fractional Conformable Derivative

This paper presents a new definition of fractional derivatives and integrals through the conformable derivative approach. This innovative framework offers a closer alignment with classical derivative concepts while providing a more practical and intuitive basis for fractional calculus. The new definition is applicable in two primary ranges: 0 ≤ α < 1 and n − 1 ≤ α < n, where n is a positive integer. It is shown that when α = 1, this definition corresponds precisely to the classical first-order derivative. Key benefits of this approach include its improved consistency with traditional calculus and greater computational ease, making it a useful tool for both theoretical research and practical applications. By integrating fractional calculus with conventional derivative ideas, this definition simplifies the analysis and interpretation of fractional differential equations and their solutions. Additionally, we examine the definition’s effects on stability and convergence in numerical methods and provide examples demonstrating its effectiveness and applicability.

A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease.

In this research, the ongoing COVID-19 disease by considering the vaccination strategies into mathematical models is discussed. A modified and comprehensive mathematical model that captures the complex relationships between various population compartments, including susceptible (Sα), exposed (Eα), infected (Uα), quarantined (Qα), vaccinated (Vα), and recovered (Rα) individuals. Using conformable derivatives, a system of equations that precisely captures the complex interconnections inside the COVID-19 transmission. The basic reproduction number (R0), which is an essential indicator of disease transmission, is the subject of investigation calculating using the next-generation matrix approach. We also compute the R0 sensitivity indices, which offer important information about the relative influence of various factors on the overall dynamics. Local stability and global stability of R0 have been proved at a disease-free equilibrium point. By designing the finite difference approach of the conformable fractional derivative using the Taylor series. The present methodology provides us highly accurate convergence of the obtained solution. Present research fills research addresses the understanding gap between conceptual frameworks and real-world implementations, demonstrating the vaccination therapy's significant possibilities in the struggle against the COVID-19 pandemic.

Modeling Pollutant Diffusion in the Ground Using Conformable Fractional Derivative in Spherical Coordinates with Complete Symmetry

The conformal fractional derivative (CFD) has become a hot research topic since it has a physical interpretation and is easier to grasp and apply to problems compared with other fractional derivatives. Its application to heat transfer, diffusion, diffusion-advection, and wave propagation problems can be found in the literature. Fractional diffusion equations have received great attention recently due to their applicability in physical, chemical, and biological processes and engineering. The diffusion of the pollutants within the ground, which is an important environmental problem, can be modeled with a diffusion equation. Diffusion in some porous materials or soil can be modeled more accurately with fractional derivatives or the conformal fractional derivative. In this study, the diffusion problem of a spilled pollutant leaking into the ground modeled with the conformal fractional time derivative in spherical coordinates has been solved analytically using the Fourier series for a constant mass flow rate and complete symmetry under the assumptions of homogeneous and isotropic soil, constant soil temperature, and constant permeability. The solutions have been simulated spatially and in time. A parametric analysis of the problem has been performed for several values of the CFD order. The simulation results are interpreted. It has also been suggested how to find the parameters of the soil to see whether the CFD can be used to model the soil or not. The approach described here can also be used for modeling pollution problems involving different boundary conditions.

Soliton wave profiles and dynamical analysis of fractional Ivancevic option pricing model

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

A new approach to the directional derivative of fractional order

The fractional derivative approximations offer many approaches to understanding real‐world problems. The conformable fractional derivative operator that is one of the fractional derivative operators has recently attracted a lot of interesting. In this paper, a new approach to the directional derivative obtained with the help of the conformal fractional derivative is presented. Considering this approach, the definition of the fractional partial derivative is reformulated. In addition, a new definition of fractional gradient, fractional curl, and fractional divergence is given, and the properties of these new concepts are examined.

Approximate and Exact Solutions of Some Nonlinear Differential Equations Using the Novel Coupling Approach in the Sense of Conformable Fractional Derivative

Several scientific fields utilize fractional nonlinear partial differential equations to model various phenomena. However, most of these equations lack exact solutions. Consequently, techniques for obtaining approximate solutions, which sometimes yield exact solutions, are essential. In this research, we develop a new approach by combining the homotopy perturbation method (HPM) and the conformable natural transform to solve the gas-dynamic equation (GDE), the Fokker-Planck equation (FPE), and the Swift-Hohenberg equation (SHE) in the context of conformable derivatives. The proposed approach is called the conformable natural homotopy perturbation method (CNHPM). This approach has the advantage of not requiring assumptions about significant or minor physical factors. Consequently, it eliminates some of the constraints associated with conventional perturbation methods and can solve both weak and highly nonlinear problems. We consider the absolute, relative, and residual errors numerically and graphically to assess the correctness of our approach. The results show that our approach serves as a suitable alternative to the approximate methods in the literature for solving fractional differential equations.

Exploring solutions to the fractional Boiti–Leon–Manna–Pempinelli equation characterizing wave dynamics in incompressible fluids

In this study, the (3 + 1)-dimensional space-time fractional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated utilizing the Kudryashov method (KM) and the modified Kudryashov method (MKM). These two efficient methods are implemented to acquire exact closed-form solutions to the considered fractional BLMP equation, and novel solitary wave solutions are constructed involving hyperbolic functions, and exponential rational functions with arbitrary constant parameters. With the aid of a unique wave transformation, the nonlinear fractional differential equation is transformed into an ordinary differential equation. In this unique wave transformation, the conformable fractional derivative is considered to which the chain rule is applied. The obtained solutions are indeed beneficial for analyzing the dynamic behavior of the fractional BLMP equation in describing fluid propagation. A few three-dimensional representations of the attained exact analytic solutions are offered by accounting for the proper selection of the appropriate parameters with the help of mathematical software. As a result, novel soliton solutions, such as the anti-kink type and singular kink type waves, are acquired from the attained solutions. This approach is relatively straightforward and efficient, making it possible to utilize it to obtain exact solutions for higher-order nonlinear partial differential equations.

On the positivity and asymptotic stability of the fractional 2D hybrid linear Roesser model

The aim of this paper is to introduce a new 2D fractional hybrid linear system described by the Roesser model with the conformable fractional derivative in the continuous time variables and the fractional difference in the discrete part. The solution of the considered model is derived. The positivity conditions are then extracted and the Cayley–Hamilton theorem is extended. A new necessary and sufficient conditions for the asymptotic stability of the proposed positive fractional 2D hybrid linear model are then established. Finally, some numerical examples are presented to show the applicability and the effectiveness of the presented methods.

Dynamics Behaviours of Kink Solitons in Conformable Kolmogorov–Petrovskii–Piskunov Equation

The current study introduces the generalised New Extended Direct Algebraic Method (gNEDAM) for producing and examining propagation of kink soliton solutions within the framework of the Conformable Kolmogorov–Petrovskii–Piskunov Equation (CKPPE), which entails conformable fractional derivatives into account. The primary justification around employing conformable derivatives in this study is their special ability to comply with the chain rule, allowing for in the solution of aimed nonlinear model. The CKPPE is a crucial model for a number of disciplines, such as mathematical biology, reaction-diffusion mechanisms, and population increase. CKPPE is transformed into a Nonlinear Ordinary Differential Equation by the proposed gNEDAM, and many kink soliton solutions are found by applying the series form solution. These kink soliton solutions shed light on propagation mechanisms within the framework of the CKPPE model. Furthermore, our research offers multiple graphical depictions that facilitate the examination and analysis of the propagation patterns of the identified kink soliton solutions. Through the integration of mathematical biology and reaction-diffusion principles, our research broadens our comprehension of intricate occurrences in various academic domains.

Dynamical investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative

Abstract This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering.

Pioneering the plethora of soliton for the (3+1)-dimensional fractional heisenberg ferromagnetic spin chain equation

Abstract In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions using an improved version of the G ′ G -expansion method, namely the r + G ′ G -expansion method. The constraints for the existence of these solutions are painstakingly explained. Our findings are clearly communicated through a number of 3D and 2D graphical representations displaying periodic, multiple periodic, kink, and shock soliton solutions. These soliton solutions are expressed in several mathematical function forms, such as hyperbolic, trigonometric, and rational functions. Our findings support the suggested method’s efficacy as a powerful symbolic algorithm for discovering innovative soliton solutions within nonlinear evolution systems.

Resonant optical soliton solutions for time-fractional nonlinear Schrödinger equation in optical fibers

In this paper, several new optical soliton solutions of the time-fractional generalized resonant nonlinear Schrödinger equation, relevant to pulse propagation in optical fibers, are constructed using the extended simplest equation approach. The new exact solutions are expressed in terms of hyperbolic functions, trigonometric functions, and rational functions. Further, two-dimensional, three-dimensional, and contour graphs of king-type, wave, dark–bright, bright, and bell-shaped optical soliton solutions are plotted to elucidate the significance of the time-fractional generalized resonant nonlinear Schrödinger equation using appropriate values of physical parameters. The dynamic behavior of the present solutions, incorporating the effect of the conformable fractional derivative, is depicted through two-dimensional graphs. The derived optical solutions are considered novel and have presumably not been previously reported in the literature.

Identification of fractional Hammerstein systems with the conformable fractional derivative

SummaryIn this article, we study parameter estimation problems for fractional commensurate Hammerstein systems utilizing the conformable fractional derivative. Two algorithms are investigated: first, the Poisson moment functions (PMF) method, aiming to transfer the fractional derivative of the measurement signal into PMF using the fractional Laplace transform and convolution; second, a proposed new instrumental variable algorithm, which is based on the conformable fractional derivative. Both algorithms have been analyzed and shown to be consistent. A comprehensive complexity analysis is provided for each algorithm. Furthermore, a kind of special time‐varying systems are discussed under the conformable fractional derivative. Finally, an example is given to illustrate the effectiveness of the proposed algorithms.

Dynamical property of interaction solutions to the Chafee-Infante equation via NMSE method

In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18.

A Comparative Analysis of Conformable, Non-conformable, Riemann-Liouville, and Caputo Fractional Derivatives

This study undertakes a comparative analysis of the non conformable and conformable fractional derivatives alongside the Riemann-Liouville and Caputo fractional derivatives. It examines their efficacy in solving fractional ordinary differential equations and explores their applications in physics through numerical simulations. The findings suggest that the conformable fractional derivative emerges as a promising substitute for the non conformable, Riemann-Liouville and Caputo fractional derivatives within the range of order $\alpha $ where $1/2 < \alpha < 1$.

Pfaffian solutions and nonlinear dynamics of surface waves in two horizontal and one vertical directions with dispersion, dissipation and nonlinearity effects

This study delves into the exploration of the (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt (GFKDKK) system, a crucial nonlinear evolution equation governing wave motion across various physical domains. The core aim is to devise innovative analytical solutions for this system employing the Khater III technique and an enhanced Kudryashov methodology integrated with the conformable fractional derivative.The GFKDKK system holds pivotal significance in modeling and comprehending diverse scientific phenomena, ranging from fluid dynamics to plasma physics and nonlinear optics. This research endeavors to harness the potential of the Khater III technique and the updated Kudryashov approach to offer fresh analytical insights into nonlinear fractional differential equations, thus enriching the existing knowledge base in this domain.The attained solutions furnish invaluable insights into the dynamics of the system and its potential applications, thereby facilitating the development of effective numerical simulations and analytical frameworks. The study’s significance lies in its contribution to the ongoing exploration of nonlinear fractional differential equations and their practical implications across various industries.In conclusion, the research affirms the efficacy of the Khater III method and the enhanced Kudryashov technique, complemented by the conformable fractional derivative, in obtaining analytical solutions for the (3+1)-dimensional GFKDKK system. The novelty lies in the application of these methodologies to this specific system, resulting in the discovery of new analytical solutions that enhance our understanding of its behavior and characteristics.

A novel multi-scale fractional-order Bernoulli grey model and its application on primary school scale prediction

Primary education plays an important role in the whole educational structure. The prediction of the development scale of primary schools is of great significance to the optimal allocation of educational resources by education departments. The paper introduces a novel multi-scale fractional-order Bernoulli grey model by combining a local cumulative operator with a global cumulative operator, abbreviated as MFGM. First, a nonlinear grey Bernoulli model based on conformable fractional derivative is proposed, where a hybrid accumulation is presented. Then, the MFGM parameters are obtained by an PSO optimizer, which are adaptive to data sets. Finally, the proposed model is validated and used to predict the scale of primary school scale. Experimental results show that the MFGM is superior to the other grey models and machine learning models in the fitting and testing primary school scale. The prediction results contribute to the decision-making of education sectors.