Caputo Definition Research Articles

Analytic Approach Solution to Time-Fractional Phi-4 Equation with Two-Parameter Fractional Derivative

This paper is devoted to building a general framework for constructing a solution to fractional Phi-4 differential equations using a Caputo definition with two parameters. We briefly introduce some definitions and properties of fractional calculus in two parameters and the Phi-4 equation. By investigating the homotopy analysis method, we built the solution algorithm. The two parameters of the fractional derivative gain vary the behavior of the solution, which allows the researchers to fit their data with the proper parameter. To evaluate the effectiveness and accuracy of the proposed algorithm, we compare the results with those obtained using various numerical methods in a comprehensive comparative study.

Finite‐time consensus for variable‐order fractional non‐linear multi‐agent systems under actuator faults and external disturbances

AbstractThe present investigation aims to address the leader‐following consensus issue for variable‐order fractional multi‐agent systems (VOFMASs) under actuator faults and unknown external disturbances. The Caputo definition for the variable‐order fractional (VOF) derivative is used to model the non‐linear dynamics of the leader and follower agents. Consequently, two lemmas are developed for the Caputo VOF derivative of the Lyapunov function. In the first case, it is assumed that the multi‐agent system (MAS) operates without actuator faults and an adaptive controller is proposed. With the aid of the developed lemmas, assurance is provided for the finite‐time bounded cooperative tracking of the VOFMAS despite the presence of unknown external disturbances. In the second case, a novel fault‐tolerant controller is designed for the finite‐time consensus of the MAS under two common kinds of actuator faults: loss of effectiveness fault and bias fault. Finally, the efficacy of the proposed controller is demonstrated through the presentation of results from three simulation examples.

Enhancing Medical Image Classification with an Advanced Feature Selection Algorithm: A Novel Approach to Improving the Cuckoo Search Algorithm by Incorporating Caputo Fractional Order.

Glaucoma is a chronic eye condition that seriously impairs vision and requires early diagnosis and treatment. Automated detection techniques are essential for obtaining a timely diagnosis. In this paper, we propose a novel method for feature selection that integrates the cuckoo search algorithm with Caputo fractional order (CFO-CS) to enhance the performance of glaucoma classification. However, when using the infinite series, the Caputo definition has memory length truncation issues. Therefore, we suggest a fixed memory step and an adjustable term count for optimization. We conducted experiments integrating various feature extraction techniques, including histograms of oriented gradients (HOGs), local binary patterns (LBPs), and deep features from MobileNet and VGG19, to create a unified vector. We evaluate the informative features selected from the proposed method using the k-nearest neighbor. Furthermore, we use data augmentation to enhance the diversity and quantity of the training set. The proposed method enhances convergence speed and the attainment of optimal solutions during training. The results demonstrate superior performance on the test set, achieving 92.62% accuracy, 94.70% precision, 93.52% F1-Score, 92.98% specificity, 92.36% sensitivity, and 85.00% Matthew's correlation coefficient. The results confirm the efficiency of the proposed method, rendering it a generalizable and applicable technique in ophthalmology.

Analysis to the dynamic response of a functionally graded spherical microshell in consideration of fractional-order dual-phase-lag heat conduction and nonlocal effect

Functionally graded materials (FGMs) stand as one of the most representative composite materials due to the superior thermal resistance in non-isothermal environment. On another front, the applicability of integer-order heat conduction models is increasingly questionable because the heat transfer shares the history-dependent property at time microscale. It is noted that the small-scale effect of elastic deformation has become significant due to the development of micro-devices. To capture the small-scale effect of elastic deformation and the memory-dependent effect of heat conduction in the FGM microstructures, as a first attempt, the present study focuses on developing a thermoelastic model by incorporating the fractional-order dual-phase-lag (FODPL) heat conduction model and the Eringen’s nonlocal model. Moreover, the extension of the Caputo definition, i.e., the Tempered-Caputo (TC) definition of fractional derivative ruling out singular kernel is adopted to depict the memory-dependent effect of the heat conduction. Then, the modified model is used to investigating the dynamic response of a FGM spherical microshell subjected to a sinusoidal thermal-mechanical loading. The corresponding governing equations are formulated and then solved by Laplace transform techniques. Some parametric results are demonstrated to display the influences of the fractional-order parameter, the nonlocal parameter and the power-law index on the considered physical quantities.

Galerkin approximation for multi-term time-fractional differential equations

Fractional differential equations (FDEs) are utilized as a precise model for describing a wide range of biological and physical processes, benefiting from the inherent symmetry feature in natural phenomena. The introduction of multi-term time-fractional order equations serves to tackle intricate phenomena spanning various physical domains. This study solves multi-term time-fractional DEs using the quadratic B-spline Galerkin method. In Galerkin approach, the quadratic B-spline is utilized as both a test and a basis function. The fractional portion of time is evaluated using the Caputo definition. In addition, the Gauss quadrature formula is used to evaluate the integration of complex function which enhance the accuracy. Lax-Richtmyer's stability criterion is employed to evaluate the stability of the proposed scheme. The E2 and E∞ norms are computed and displayed in the tables to illustrate the robustness and efficiency of the method. We compute simulation for various nodal points as well as various mesh sizes to analyze the solution behavior. The obtained results are compared with the exact and available results, indicating that the proposed scheme is more efficient. The graphical solution of each example is also demonstrated which verifies that exact and approximate solutions graphs closed with each other.

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

Efficient Numerical Solutions for Fuzzy Time Fractional Diffusion Equations Using Two Explicit Compact Finite Difference Methods

This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev’s scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev’s is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes.

Differences between Riemann–Liouville and Caputo calculus definitions in analyzing fractional boost converter with discontinuous‐conduction‐mode operation

SummaryFractional‐order calculus is an extension of the integer‐order calculus in that its order is not limited to integers but can be noninteger as well, thus making the application of fractional‐order calculus more flexible. It has been shown that actual reactive elements such as capacitors and inductors have fractional‐order properties and that the theory of fractional‐order calculus can be utilized to more accurately characterize circuits and systems containing reactive elements. However, there is no uniform definition of fractional order calculus. Different definitions will yield different results in solving the same fractional‐order circuits and systems, and it is worth exploring whether these differences have an impact on the accuracy of solving fractional‐order systems. Taking a fractional‐order boost converter running in discontinuous conduction mode (DCM) as an example, this paper focuses on the differences between the Riemann–Liouville (R–L) and the Caputo fractional calculus definitions. Effects of the two definitions on the division of the converter's CCM (continuous conduction mode) and DCM operating regions are analyzed, as well as their impact on the characterization of a fractional‐order boost converter in DCM operation, including the DC (direct current) operating point, current ripple, and voltage gain. Simulations and experiments show that the analytical results based on the R–L definition can more accurately describe the operation of a real fractional‐order converter than those based on the Caputo definition.

Analysis of fractional electrical circuit with rectangular input signal using Caputo and conformable derivative definitions

Analysis of fractional electrical circuit with rectangular input signal using Caputo and conformable derivative definitions

Research on the higher-order logic formalization of fractance element

Fractance element reflects the fractional order behavior of circuits, which can show the characteristics of the actual circuits. Higher-order logic theorem proving is based on the rigorous and correct mathematical theories. It becomes more and more important in the verifications of high-reliability systems. Fractance element is formalized using the proof of higher-order logic theorem in this paper. Firstly, the formalized model of fractional calculus which is based on Caputo definition is established in higher-order logic theorem proof tool. Then some properties of fractional calculus are proved, including the zero order property, the fractional differential of a constant and the consistency of fractional calculus and integer order calculus. Finally, fractance element and fractional differential circuit constituted by fractance element are formally analyzed. These formalizations demonstrate the effectiveness of the formal method in the analysis of fractance element.

Fractional perspective evaluation of chikungunya infection with saturated incidence functions

It is significant to investigate the transmission dynamics of vector-borne infection because it has a global impact, can help predict and prevent future outbreaks, is important for understanding the impact of climate change on public health, can lead to more effective control strategies, and can improve our comprehension of the dynamics of these infections. Our paper presents a new model for chikungunya virus infection, which considers treatment and vaccination, using the Atangana-Baleanu derivative within the framework of the Caputo definition. First of all, we examine the positivity and uniqueness of the solution for the model. Then, we find out the fundamental results of the suggested model, such as the steady-state without infection and the R0 value that indicates endemicity pertaining to the system. Furthermore, we verify the local asymptotic stability of the infection-free steady-state. Through the application of fixed-point theory, we demonstrate the existence of the solution and propose a numerical methodology to investigate the dynamic behavior of the model. Finally, we performed simulations to demonstrate the effects of vaccination, transmission probability, treatment, and index of memory on the system's solution pathways. The findings of this research suggest the most sensitive parameters of the system responsible for the control and prevention of the infection. Emphasizing the significance of the memory index as an attractive parameter, we propose its consideration to the policy makers as a control parameter for the prevention of the infection.

Barycentric Interpolation Collocation Method for Solving Fractional Linear Fredholm-Volterra Integro-Differential Equation

In this article, barycentric interpolation collocation method (BICM) is presented to solve the fractional linear Fredholm-Volterra integro-differential equation (FVIDE). Firstly, the fractional order term of equation is transformed into the Riemann integral with Caputo definition, and this integral term is approximated by the Gauss quadrature formula. Secondly, the barycentric interpolation basis function is used to approximate the unknown function, and the matrix equation of BICM is obtained. Finally, several numerical examples are given to solve one-dimensional differential equation.

Fractional melting process in inclined containers using (NePCM) and hybrid nanoparticles

Using substances of high latent heat such as phase change materials is a perfect technique in the energy storage units. Additionally, examining the heat transport and melting process for these materials is beneficial for a wide variety of solar-related applications. Therefore, this study aims to examine the time-dependent fractional melting process within inclined containers filled with Nanoparticles-enhanced Phase-Change Materials (NePCM) via the model of the enthalpy-porosity. The used NePCM is octadecane and the fractional derivatives are considered for all the time-dependent variables, namely, velocities, temperature and liquid fraction. The Caputo definition is applied to estimate the non-integer derivatives and the fractional order takes the values between 0.75 and 0.95. The solution methodology is depending on the Finite Volume technique with SIMPLE approach. The range of the Fourier number is between 0.05 and 0.4 and the resulting data is presented in terms of melting interface, liquid fraction, streamlines, isotherms and heat transfer rate. The main findings revealed that the influences of order of the fractional derivatives more significant at the higher values of the Fourier number and the melting interface points move towards the heated wall as order of the fractional derivatives is reduced. Also, at higher values of the fractional derivative's order (0.9), the maximizing of inclination angle causes a diminishing in the rate of the heat transfer.

Assessing Impedance Analyzer Data Quality by Fractional Order Calculus: A QCM Sensor Case Study

The paper presents the theoretical, simulation, and experimental results on the QCM sensor based on the Butterworth van Dyke (BVD) model with lumped reactive motional circuit elements of fractional order. The equation of the fractional order BVD model of the QCM sensor has been derived based on Caputo definitions and its behavior around the resonant frequencies has been simulated. The simulations confirm the ability of fractional order calculus to cover a wide range of behaviors beyond those found in experimental practice. The fractional order BVD model of the QCM sensor is considered from the perspective of impedance spectroscopy to give an idea of the advantages that fractional order calculus brings to its modeling. For the true values of the electrical parameters of the QCM sensor based on the standard BVD model, the experimental investigations confirm the equivalence of the measurements after the standard compensation of the virtual impedance analyzer (VIA) and the measurements without compensation by fitting with the fractional order BVD model. From an experimental point of view, using fractional order calculus brings a new dimension to impedance analyzer compensation procedures, as well as a new method for validating the compensation.

Stabilizability of Riccati Matrix Fractional Delay Differential Equation

In this article, the backstepping control scheme is proposed to stabilize the fractional order Riccati matrix differential equation with retarded arguments in which the fractional derivative is presented using Caputo's definition of fractional derivative. The results are established using Mittag-Leffler stability. The fractional Lyapunov function is defined at each stage and the negativity of an overall fractional Lyapunov function is ensured by the proper selection of the control law. Numerical simulation has been used to demonstrate the effectiveness of the proposed control scheme for stabilizing such type of Riccati matrix differential equations.

Continuous Adaptive Finite-Time Sliding Mode Control for Fractional-Order Buck Converter Based on Riemann-Liouville Definition.

This study proposes a continuous adaptive finite-time fractional-order sliding mode control method for fractional-order Buck converters. In order to establish a more accurate model, a fractional-order model based on the Riemann-Liouville (R-L) definition of the Buck converter is developed, which takes into account the non-integer order characteristics of electronic components. The R-L definition is found to be more effective in describing the Buck converter than the Caputo definition. To deal with parameter uncertainties and external disturbances, the proposed approach combines these factors as lumped matched disturbances and mismatched disturbances. Unlike previous literature that assumes a known upper bound of disturbances, adaptive algorithms are developed to estimate and compensate for unknown bounded disturbances in this paper. A continuous finite-time sliding mode controller is then developed using a backstepping method to achieve a chattering-free response and ensure a finite-time convergence. The convergence time for the sliding mode reaching phase and sliding mode phase is estimated, and the fractional-order Lyapunov theory is utilized to prove the finite-time stability of the system. Finally, simulation results demonstrate the robustness and effectiveness of the proposed controller.

Computational Algorithm for Approximating Fractional Derivatives of Functions

This paper presents an algorithmic approach for numerically solving Caputo fractional differentiation. The trapezoidal rule was modified, the new modification was used to derive an algorithm to approximate fractional derivatives of order α > 0, the fractional derivative used was based on Caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The trapezoidal rule was used in conjunction with the finite difference scheme which is the forward, backward and central difference to derive the computational algorithm for the numerical approximation of Caputo fractional derivative for evaluating functions of fractional order. The study was conducted through some illustrative examples and analysis of error.

Barycentric interpolation collocation algorithm to solve fractional differential equations

Fractional equations have been paid much attention in recent years. Barycentric interpolation collocation algorithm (BICA) is proposed to solve the fractional differential equations in this manuscript. In order to calculate fractional derivatives in fractional differential equations, the Gauss quadrature formula with weights ρ(τ)=(t−τ)ξ−α is constructed and the error estimation of this quadrature formula is proved. The fractional differential term of equation is transformed into Riemann integral under Caputo definition. Barycentric interpolation is used to approximate the unknown function and matrix equation of transformed fractional differential equation is obtained by BICA. The error results show exponential convergence rate of Gauss quadrature formula with weights ρ(τ)=(t−τ)ξ−α by comparing with the Gauss–Legendre quadrature formula. In addition, several examples are given to solve the fractional differential equation.

Dual‐phase‐lag thermoelastic diffusion analysis of a size‐dependent microplate based on modified fractional‐order heat conduction model

AbstractWith the rapid evolution of small‐scale devices, the effect of size dependence on elastic deformation becomes more and more obvious. Furthermore, the physical processes usually involve memory and inheritance for some anomalous diffusion that the constitutive relation does not obey the standard gradient rate. Owing to capturing the memory‐dependent effect and the size‐dependent effect, the existing thermoelastic diffusion theories cannot be applicable. To further refine the thermoelastic diffusion model for micro/nano structures, a refined nonlocal thermoelastic diffusion model is developed by combining the fractional‐order dual‐phase‐lag (DPL) heat conduction model and fractional‐order diffusion model in this paper. In addition, the Tempered‐Caputo (TC) definition is an extension of the Caputo definition without the fractional derivative of the singular kernel, which is adopted for the first time to reflect the memory‐dependent effects of heat conduction and stress–strain relationship. Then, this new model is applied to investigating the transient response of an elastic microplate subjected to a sinusoidal thermal loading. The corresponding governing equations are formulated and solved by the Laplace transform method and its numerical inversion. In calculation, the influences of the fractional‐order parameter, the tempered parameter and the nonlocal parameter on the variations of the considered quantities are presented and discussed in detail. It is expected to provide new insights into thermoelastic diffusion behaviors of structures at the micro/nanoscale.

Numerical treatments for the optimal control of two types variable-order COVID-19 model

In this paper, a novel variable-order COVID-19 model with modified parameters is presented. The variable-order fractional derivatives are defined in the Caputo sense. Two types of variable order Caputo definitions are presented here. The basic reproduction number of the model is derived. Properties of the proposed model are studied analytically and numerically. The suggested optimal control model is studied using two numerical methods. These methods are non-standard generalized fourth-order Runge–Kutta method and the non-standard generalized fifth-order Runge–Kutta technique. Furthermore, the stability of the proposed methods are studied. To demonstrate the methodologies’ simplicity and effectiveness, numerical test examples and comparisons with real data for Egypt and Italy are shown.