Caputo Fractional Derivative Research Articles

The Influence of the Caputo Fractional Derivative on Time-Fractional Maxwell’s Equations of an Electromagnetic Infinite Body with a Cylindrical Cavity Under Four Different Thermoelastic Theorems

This paper introduces a new mathematical modeling of a thermoelastic and electromagnetic infinite body with a cylindrical cavity in the context of four different thermoelastic theorems; Green–Naghdi type-I, type-III, Lord–Shulman, and Moore–Gibson–Thompson. Due to the convergence of the four theories under study and the simplicity of putting them in a unified equation that includes these theories, the theories were studied together. The bunding plane of the cavity surface is subjected to ramp-type heat and is connected to a rigid foundation to stop the displacement. The novelty of this work is considering Maxwell’s time-fractional equations under the Caputo fractional derivative definition. Laplace transform techniques were utilized to obtain solutions by using a direct approach. The Laplace transform’s inversions were calculated using Tzou’s iteration method. The temperature increment, strain, displacement, stress, induced electric field, and induced magnetic field distributions were obtained numerically and represented in figures. The time-fractional parameter of Maxwell’s equations has a significant impact on all the mechanical studied functions and does not affect the thermal function. The time-fractional parameter of Maxwell’s equations works as a resistance to deformation, displacement, stress, and induced magnetic field distributions, while it acts as a catalyst to the induced electric field through the material.

A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the α-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter α ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with (2−α)-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework.

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.

A fractional-order improved FitzHugh-Nagumo neuron model

Abstract In this paper, we propose a fractional-order improved FitzHugh-Nagumo (FHN) neuron model in terms of a generalised Caputo fractional derivative. Following the existence of a unique solution for the proposed model, we derive the numerical solution using a recently proposed L1 predictor-corrector method. The given method is based on the L1-type discretization algorithm and the spline interpolation scheme. We perform the error and stability analyses for the given method. We perform graphical simulations demonstrating that the proposed FHN neuron model generates rich electrical activities of periodic spiking patterns, chaotic patterns, and quasi-periodic patterns. The motivation behind proposing a fractional-order improved FHN neuron model is that such a system can provide a more nuanced description of the process with better understanding and simulation of the neuronal responses by incorporating memory effects and non-local dynamics, which are inherent to many biological systems.

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

On Riemann-Liouville integrals and Caputo Fractional derivatives via strongly modified (p, h)-convex functions.

The paper introduces a new class of convexity named strongly modified (p, h)-convex functions and establishes various properties of these functions, providing a comprehensive understanding of their behavior and characteristics. Additionally, the paper investigates Schur inequality and Hermite-Hadamard (H-H) inequalities for this new class of convexity. Also, H-H inequalities are proved within context of Riemann-Liouville integrals and Caputo Fractional derivatives. The efficiency and feasibility of Schur inequality and H-H inequalities are supported by incorporating multiple illustrations, that demonstrate the applicability of strongly modified (p, h)-convex functions. The results contribute to the field of mathematical analysis and provide valuable insights into the properties and applications of strongly modified (p, h)-convex functions.

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.

Compartmental Model for Unemployment Dynamics in Ghana using Caputo Fractional Derivative

The study develops a compartmental model for unemployment dynamics in Ghana. The compartmental model is first developed through the method of compartmental designs under certain assumptions such as, all entrants into the labor force qualify for any job, some of the employed become unemployed with time, some of the unemployed go through skill training, among others. The compartmental model is further represented in terms of integral form, and the value of the kernel is thereafter substituted as a power law correlation function to achieve a Caputo fractional derivative model. The Caputo model is analyzed to ascertain its invariant and boundedness, its fixed points and its stability. The unemployment basic reproduction number \(\Re_0\) which determines the threshold of recruitment is analyzed using the Next Generation Method Approach, and the global stability for the unemployment-free equilibrium is also analyzed through the approach of Lyapunov function. The results from the analysis show that, there is attraction of all the solution of the model in \(\mathbb{R}_+^4\) since the region is positively invariant. Again, the results indicates that, the model has two fixed point. Thus, unemployment-free equilibrium and risky-unemployment equilibrium. Notwithstanding, the analysis of both the local and global stability of the model show that, the unemployment free equilibrium is local asymptotically stable (LAS) for \(\Re_0\) &lt;1, and global asymptotically stable (GAS). The results indicate that, the model can examine the unemployment dynamics decision making.

Analysis of Fractional Order-Adaptive Systems Represented by Error Model 1 Using a Fractional-Order Gradient Approach

In adaptive control, error models use system output error and adaptive laws to update controller parameters for control or identification tasks. Fractional-order calculus, involving non-integer-order derivatives and integrals, is increasingly important for modeling, estimation, and control due to its ability to generalize classical methods and offer improved robustness to disturbances. This paper addresses the gap in the literature where fractional-order gradient methods have not yet been extensively applied in identification and adaptive control schemes. We introduce a fractional-order error model with fractional-order gradient (FOEM1-FG), which integrates fractional gradient operators based on the Caputo fractional derivative. By using theoretical analysis and simulations, we confirm that FOEM1-FG maintains stability and ensures bounded output errors across a variety of input signals. Notably, the fractional gradient’s performance improves as the order, β, increases with β&gt;1, leading to faster convergence. Compared to existing integer-order methods, the proposed approach provides a more flexible and efficient solution in adaptive identification and control schemes. Our results show that FOEM1-FG offers superior stability and convergence characteristics, contributing new insights to the field of fractional calculus in adaptive systems.

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.

Determination of two unknown functions of different variables in a time‐fractional differential equation

We study the inverse problem for a differential equation of ‐order with the Caputo fractional derivative over time and Schwartz‐type distribution in its right‐hand side. The generalized solution of the Cauchy problem for such an equation, space‐dependent part of a source, and a time‐dependent reaction coefficient in the equation are unknown. We find sufficient conditions for unique local in time solvability of the inverse problem under time‐ and space‐integral overdetermination conditions.

New modifications of natural transform iterative method and q-homotopy analysis method applied to fractional order KDV-Burger and Sawada–Kotera equations

This manuscript presents enhanced versions of two methods: the natural transform iterative method (NTIM) and the q-homotopy analysis method (q-HAM). These methods harness concepts from Fractional Calculus, particularly leveraging the Caputo fractional derivative operator, to successfully manage the complexities of fractional-order systems. To validate their accuracy and efficiency, we applied the proposed techniques to FPDEs like the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations. Our outcomes, which closely resemble the exact solutions, demonstrate how useful NTIM and q-HAM are for solving difficult FPDEs and improving the study of fractional calculus.

A mathematical modeling of patient-derived lung cancer stem cells with fractional-order derivative

The aim of this article is to help predict the course of lung cancer patients. To make this prediction as close to reality as possible, we used data from lung cancer patients receiving treatment at Erciyes University Hospitals in Kayseri, Turkey. First, we developed a mathematical model considering the cells in the microenvironment of lung cancer tumors with the assistance of Caputo fractional derivatives. Subsequently, we identified the equilibrium points of the proposed mathematical model and examined the coexistence equilibrium point. In addition, we demonstrated the existence and uniqueness of the solutions through the fixed-point theorem. We also investigated the positivity and boundedness of the model’s solutions to show whether they are biologically meaningful. Using laboratory experimental results from cancer stem cells isolated from resected tumor tissues of lung cancer patients, we determined the most biologically realistic parameter values through the least squares curve fitting approach. Then, using these parameter values, we performed numerical simulations with the Adams-Bashforth-Moulton predictor-corrector method to validate the theoretical results. We considered different values of fractional derivatives to investigate how the model is affected by fractional derivatives. As a result, we obtained the dynamics and expectations of lung cancer and made predictions specific to individual patients. In our simulations based on the parameter values obtained from actual patient data, it has been observed that after a certain period, both tumor cells and cancer stem cells have been eliminated. Consequently, an increase in normal tissue cells and immune cells has been observed. This implies that the patient in question, and similar behaving patients, will recover and overcome cancer. The findings from this study provide insights into the dynamics and prognosis of lung cancer, opening up the possibility for more personalized and effective approaches to treatment.

Finite Time Stability Analysis and Feedback Control for Takagi–Sugeno Fuzzy Time Delay Fractional-Order Systems

This study treats the problem of Finite Time Stability Analysis (FTSA) and Finite Time Feedback Control (FTFC), using a Linear Matrix Inequalities Approach (LMIA). It specifically focuses on Takagi–Sugeno fuzzy Time Delay Fractional-Order Systems (TDFOS) that include nonlinear perturbations and interval Time Varying Delays (ITVDs). We consider the case of the Caputo Tempered Fractional Derivative (CTFD), which generalizes the Caputo Fractional Derivative (CFD). Two main results are presented: a two-step procedure is provided, followed by a more relaxed single-step procedure. Two examples are presented to show the reduction in conservatism achieved by the proposed methods. The first is a numerical example, while the second involves the FTFC of an inverted pendulum, which exhibits both symmetrical dynamics for small angular displacements and asymmetrical dynamics for larger deviations.

Subdiffusion of heavy quarks in hot QCD matter by the fractional Langevin equation

The subdiffusion phenomena are studied for heavy quarks dynamics in the hot QCD matter. Our approach aims to provide a more realistic description of heavy quark dynamics through detailed theoretical analyses and numerical simulations, utilizing the fractional Langevin equation framework with the Caputo fractional derivative. I present numerical schemes for the fractional Langevin equation for subdiffusion and calculate the time evolution mean squared displacement and mean squared momentum of the heavy quarks. My results indicate that the mean squared displacements of the heavy quarks for the subdiffusion process deviate from a linear relationship with time. Further, I calculate the normalized momentum correlation function, kinetic energy, and momentum spread. Finally, I show the effect of subdiffusion on experimental observables, the nuclear modification factor. Published by the American Physical Society 2024

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered Ψ–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution.

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

Influence of Fractional Order on the Behavior of a Normalized Time-Fractional SIR Model

In this paper, we propose a novel normalized time-fractional susceptible–infected–removed (SIR) model that incorporates memory effects into epidemiological dynamics. The proposed model is based on a newly developed normalized time-fractional derivative, which is similar to the well-known Caputo fractional derivative but is characterized by the property that the sum of its weight function equals one. This unity property is crucial because it helps with evaluating how the fractional order influences the behavior of time-fractional differential equations over time. The normalized time-fractional derivative, with its unity property, provides an intuitive understanding of how fractional orders influence the SIR model’s dynamics and enables systematic exploration of how changes in the fractional order affect the model’s behavior. We numerically investigate how these variations impact the epidemiological dynamics of our normalized time-fractional SIR model and highlight the role of fractional order in improving the accuracy of infectious disease predictions. The appendix provides the program code for the model.

Fractional derivative of Hermite fractal splines on the fractional-order delayed neural networks synchronization

The purpose of this research is twofold. First, the master–slave synchronization of fractional-order neural networks is explored with time delays using aperiodic intermittent control. Then we present a sufficient condition for master–slave synchronization of delayed fractional-order neural networks via average-width intermittent control technique. A numerical simulation is used to demonstrate the efficacy of the derived results. Second, a novel investigation of the Caputo-fractional derivative of Hermite fractal splines is accomplished. Moreover, its box counting dimension is estimated and related with the Caputo-fractional order. Additionally, we propose an image encryption algorithm utilizing the semi-tensor product (STP). The efficiency of the algorithm is evaluated through the application of statistical measures.

On a non-local problem for a fractional differential equation of the Boussinesq type

In recent years, the fractional partial differential equation of the Boussinesq type has attracted much attention from researchers due to its practical importance. In this paper, we study a non-local problem for the Boussinesq type equation Dtαu(t)+A Dtαu(t)+ν2Au(t) =0, 0&lt;t&lt;T, 1&lt;α&lt;3∕2, where Dtα is the Caputo fractional derivative, and A is an abstract operator. In the classical case, i.e., when α = 2, this problem has been studied previously, and an interesting effect has been discovered: the existence and uniqueness of a solution depend significantly on the length of the time interval and the parameter ν. In this note, we show that in the case of a fractional equation, there is no such effect: a solution of the problem exists and is unique for any T and ν.