Integer Derivatives Research Articles

Numerical discretization of initial–boundary value problems for PDEs with integer and fractional order time derivatives

This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

Fractional order forestry resource conservation model featuring chaos control and simulations for toxin activity and human-caused fire through modified ABC operator

In this work, we proposed a nonlinear mathematical model with fractional-order differential equations employed to illustrate the impacts of depleted forestry resources with the effect of toxin activity and human-caused fire. The numerical and theoretical outcomes are based on the consideration of using a modified ABC-fractional-order depleted forestry resources dynamical system. In the theoretical aspect, we examination of solution positivity, existence, and uniqueness it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. The consecutive recursive sequences are purposefully designed to verify the existence of a solution to the depletion of forestry resources as delineated. To showcase the specificity and stability of the solution within the Hyers–Ulam framework, we employ the concepts and findings of functional analysis. Chaos control will stabilize the system following its equilibrium points by applying the regulate for linear responses technique. Using Lagrange polynomials insight of modified ABC-fractional-order, we conduct simulations and present a comparative analysis in graphical form with classical and integer derivatives. Results also demonstrate the impact of different parameters used in a model that is designed on the system, they provide more understanding and a better approach for real-life problems. Our results demonstrate the significant effects of toxic and fire activities produced by humans on forest ecosystems. More accurate management techniques are made possible by the modified ABC operator’s effectiveness in capturing the long-term effects of these disturbances. The findings highlight how crucial it is to use fractional calculus in ecological modeling to comprehend and manage the intricacies of forest preservation in the face of human pressures. To ensure the sustainable management of forest resources in the face of escalating environmental difficulties, this research offers policymakers and environmental managers a fresh paradigm for creating more robust and adaptive conservation policies.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

Inequalities for strongly s-convex functions via Atangana-Baleanu fractional integral operators

It is more convenient to use fractional derivatives and integrals to express and represent rapid changes than to use integer derivatives and integrals. For this reason, fractional analysis has been found worthy of study in many fields. In recent years, fractional derivatives and integrals have been discussed together with inequality theory and the studies have attracted attention. In this article, we discuss new Hermite-Hadamard type approximations for strongly convex functions with the help of Atangana-Baleanu fractional integral operators. Additionally, new upper bounds have been obtained using various auxiliary inequalities with the help of twice differentiable strongly convex functions.

Comparative study of integer-order and fractional-order artificial neural networks: Application for mathematical function generation

This paper investigates the impact of fractional derivatives on the activation functions of an artificial neural network (ANN). Based on the results and analysis, a three-layer backpropagation neural network model with fractional and integer derivatives employed in the activation function and fractional gradient descent method backpropagation learning algorithm has been proposed. Specifically, three perceptrons have been proposed based on fractional and integer derivatives applied to activation functions and learning algorithms. They are fractional derivative activation function (FDAF) perceptron, integer derivative activation function (IDAF)-fractional derivative learning algorithm (FDLA) perceptron, and fractional derivative learning algorithm (FDLA) perceptron. The Riemann-Liouville (RL) fractional derivative, Grunwald-Letnikov (GL) derivative, Caputo-Fabrizio (CF) fractional derivative, Caputo (C) fractional derivative, and Atangana-Baleanu (AB) fractional derivative have been employed. The impact of these derivatives’ fractional-order (FO) is investigated in a wide range from 0.1−0.9 on testing mean square error (MSE) and time required to train the perceptron. FO-based and IO-based perceptrons are compared with the help of performance metrics such as testing MSE and the time required to train the models. The training and testing simulation results illustrate that Caputo-Fabrizio derivative-based perceptrons outperform other fractional derivatives. Also, the performance of the FO-based perceptron is better in terms of the least MSE.

Understanding COVID-19 propagation: a comprehensive mathematical model with Caputo fractional derivatives for Thailand

This research introduces a sophisticated mathematical model for understanding the transmission dynamics of COVID-19, incorporating both integer and fractional derivatives. The model undergoes a rigorous analysis, examining equilibrium points, the reproduction number, and feasibility. The application of fixed point theory establishes the existence of a unique solution, demonstrating stability in the model. To derive approximate solutions, the generalized Adams-Bashforth-Moulton method is employed, further enhancing the study's analytical depth. Through a numerical simulation based on Thailand's data, the research delves into the intricacies of COVID-19 transmission, encompassing thorough data analysis and parameter estimation. The study advocates for a holistic approach, recommending a combined strategy of precautionary measures and home remedies, showcasing their substantial impact on pandemic mitigation. This comprehensive investigation significantly contributes to the broader understanding and effective management of the COVID-19 crisis, providing valuable insights for shaping public health strategies and guiding individual actions.

Conformal and Non-Minimal Couplings in Fractional Cosmology

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ, to create a non-minimal interaction theory with the coupling, ξRϕ2, between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism.

Adaptive exponential integrate-and-fire model with fractal extension.

The description of neuronal activity has been of great importance in neuroscience. In this field, mathematical models are useful to describe the electrophysical behavior of neurons. One successful model used for this purpose is the Adaptive Exponential Integrate-and-Fire (Adex), which is composed of two ordinary differential equations. Usually, this model is considered in the standard formulation, i.e., with integer order derivatives. In this work, we propose and study the fractal extension of Adex model, which in simple terms corresponds to replacing the integer derivative by non-integer. As non-integer operators, we choose the fractal derivatives. We explore the effects of equal and different orders of fractal derivatives in the firing patterns and mean frequency of the neuron described by the Adex model. Previous results suggest that fractal derivatives can provide a more realistic representation due to the fact that the standard operators are generalized. Our findings show that the fractal order influences the inter-spike intervals and changes the mean firing frequency. In addition, the firing patterns depend not only on the neuronal parameters but also on the order of respective fractal operators. As our main conclusion, the fractal order below the unit value increases the influence of the adaptation mechanism in the spike firing patterns.

Solving Nonlinear Fractional PDEs with Applications to Physics and Engineering Using the Laplace Residual Power Series Method

The Laplace residual power series (LRPS) method uses the Caputo fractional derivative definition to solve nonlinear fractional partial differential equations. This technique has been applied successfully to solve equations such as the fractional Kuramoto–Sivashinsky equation (FKSE) and the fractional generalized regularized long wave equation (GRLWE). By transforming the equation into the Laplace domain and replacing fractional derivatives with integer derivatives, the LRPS method can solve the resulting equation using a power series expansion. The resulting solution is accurate and convergent, as demonstrated in this paper by comparing it with other analytical methods. The LRPS approach offers both computational efficiency and solution accuracy, making it an effective technique for solving nonlinear fractional partial differential equations (NFPDEs). The results are presented in the form of graphs for various values of the order of the fractional derivative and time, and the essential objective is to reduce computation effort.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

Problem of Determining the Density of Sources in a Multidimensional Heat Equation with the Caputo Time Fractional Derivative

In this paper, we propose a new formula for representing the solution of the third initial-boundary value problem for multidimensional fractional heat equation with the Caputo derivative. This formula is obtained by the continuation method used in the theory of partial differential equations with integer derivatives. The Green’s function of the problem is also constructed in terms of the Fox H- function. Involving the results of solving a direct problem and the overdetermination condition, a uniqueness theorem for the definition of the spatial part of the multidimensional source function is proved.

Novel Definitions of α-Fractional Integral and Derivative of the Functions

An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the formulas demonstrate that they are most significant and suitable in fractional integrals and derivatives. The definitions of α-fractional derivative and integral coincide with the existing definitions for the polynomials for 0 ≤ α < 1. Furthermore, if α = 1, the proposed definitions and the usual definition of integer derivative and integral are identical. Some of the properties of the new definitions are discussed and proved, as well, we have introduced some applications in the α- fractional derivatives and integrals. Moreover, α-power series and α–rule of integration by parts have been proposed and implemented in this study.

Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon

The goal of this work is to derive and study a new model which traduces the transmission dynamics of the Buruli ulcer (BU), in which we replace mass action incidences with standard incidences, consider latent period, and by using both integer and fractional derivatives. We first present reported data of Buruli in Cameroon between 01/01/2001 and 31/12/2014, and the statistical model that approximates real data as closely as possible, and thus make a forecasting in the next sixteen years. After that, we formulate a Susceptible humans-Exposed humans-Infectious humans-Recovered humans-Susceptible humans; Susceptible vectors-Infectious vectors (SEIRS-SI) type compartmental model of Buruli Ulcer using integer derivative. We compute the basic reproduction number denoted by R0 and prove the asymptotic stability of the Buruli-free equilibrium whenever R0<1. Then, for R0>1, we prove the existence of an unique endemic equilibrium point and its global stability using the general theory of Lyapunov. We perform parameter estimation to calibrate our model using real data from Cameroon, while sensitivity analysis is conducted to determine important parameters in the BU dynamics. From this model calibration, we obtain R0=2.0843 which is greater than one and implies that BU ulcer is endemic in the country. To determine whether if or not the BU admits one or multiple waves, we compute the strength number denoted by A0. We find that A0≈17>0 which means that Bu admits multiple epidemic waves. We then replace integer derivative with the Caputo fractional derivative. Asymptotic stability of equilibrium points are also performed for the fractional model. This follows by the proof of existence and the uniqueness of solutions of the fractional model. We then construct a numerical scheme based on the Adams–Bashforth–Moulton (ABM) Method. Theoretical results are validated from numerical simulations. These last also permits to evaluate the impact of varying fractional order α in the BU dynamics. This permits to conclude that for α<1, reported cases are closer to model prediction.

Analysis of a non-integer order compartmental model for cholera and COVID-19 incorporating human and environmental transmissions

Fractional differential operators have increasingly gained wider applications in epidemiological modelling due to their ability to capture memory effect in their definitions; an attribute which lacks in the concept of classical integer derivatives. In this paper, employing the Caputo fractional operator with singular kernel, the co-dynamical model for cholera and COVID-19 diseases is proposed and analyzed, incorporating both direct and indirect transmission routes for cholera. The necessary conditions for existence of the unique solution of the proposed model are studied. Using the results from fixed point theory, Ulam-Hyers stability analysis of the system is performed. The model is fitted to real data from Pakistan and the optimized order of the fractional derivative for which the system fits well to data is obtained. Other numerical assessments of the model are also executed. Phase portraits of the infected classes with different initial conditions and various order of the fractional derivative are obtained in the cases when the reproduction number and . It is observed that the trajectories for the infected compartments tend towards the infection-free steady state when and the endemic steady state when irrespective of the initial conditions and the order of the fractional derivative. Increment in the COVID-19 vaccine efficacy, keeping the vaccination rate fixed at , resulted in a decline in the COVID-19 disease. Also, increasing the COVID-19 vaccination rate, keeping the vaccine efficacy for COVID-19 fixed at , led to a decline in the COVID-19 associated reproduction number. The simulations also pointed out the impact of COVID-19 and cholera vaccinations, direct and indirect transmissions of cholera.

A Fractional-Order Memristive Two-Neuron-Based Hopfield Neuron Network: Dynamical Analysis and Application for Image Encryption

This paper presents a study on a memristive two-neuron-based Hopfield neural network with fractional-order derivatives. The equilibrium points of the system are identified, and their stability is analyzed. Bifurcation diagrams are obtained by varying the magnetic induction strength and the fractional-order derivative, revealing significant changes in the system dynamics. It is observed that lower fractional orders result in an extended bistability region. Also, chaos is only observed for larger magnetic strengths and fractional orders. Additionally, the application of the fractional-order model for image encryption is explored. The results demonstrate that the encryption based on the fractional model is efficient with high key sensitivity. It leads to an encrypted image with high entropy, neglectable correlation coefficient, and uniform distribution. Furthermore, the encryption system shows resistance to differential attacks, cropping attacks, and noise pollution. The Peak Signal-to-Noise Ratio (PSNR) calculations indicate that using a fractional derivative yields a higher PSNR compared to an integer derivative.

System of Caputo Fractional Differential Equations with Applications to Predator and Prey Model

In this research article, we provide a methodology to solve the three systems of qth order linear Caputo fractional differential equations, where 0 &lt; q &lt; 1. Since the Caputo derivative is in the convolution form, we can apply the Laplace transform technique. The solution of the two linear system can be used as a tool to study the stability of the equilibrium solution of the Lotka-Volterra predator-prey model. We have referenced three system SIR model in this work. Due to the global nature of the Caputo derivative, the solution obtained is closer to the real data than the integer derivative.

A New Compact Numerical Scheme for Solving Time Fractional Mobile-Immobile Advection-Dispersion Model

This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. The scheme is derived using the effective combination of Euler and Caputo numerical techniques for approximating the integer and fractional time derivatives respectively, and a fourth order exponential compact scheme for spatial derivatives. The Fourier analysis technique is used to prove that the proposed numerical scheme is unconditionally stable and perform convergence analysis. To assess the viability and accuracy of the proposed scheme, some numerical examples are demonstrated with constant as well as variable order time fractional derivatives for this model.

Fractional dynamics of a Chikungunya transmission model

In this paper, we introduce a Caputo-sense fractional derivative to an existing model for the Chikungunya transmission dynamics, replacing integer derivative. After reviewing previous work on the integer order model, we suggest a change whereby the integer derivative is replaced by the Caputo derivative operator, and then we obtain results about the asymptotic stability of equilibrium points in this new fractional model. We prove the existence and uniqueness of the solution as well as the global stability of the fractional system. We then construct a numerical scheme for the fractional model, and use parameter values from the Chikungunya epidemic in Chad to perform numerical simulations. This allows us to validate our theoretical results and compare the dynamics behaviour of both models. We find from numerical simulations, that for the fractional-order parameter η in the range (0.85;1), daily detected cases are closer to those the model predict. Thus in the case of Chikungunya epidemic in Chad, the model with fractional derivative produces better results than which with integer derivative.

Carbon-cement supercapacitors as a scalable bulk energy storage solution.

The large-scale implementation of renewable energy systems necessitates the development of energy storage solutions to effectively manage imbalances between energy supply and demand. Herein, we investigate such a scalable material solution for energy storage in supercapacitors constructed from readily available material precursors that can be locally sourced from virtually anywhere on the planet, namely cement, water, and carbon black. We characterize our carbon-cement electrodes by combining correlative EDS-Raman spectroscopy with capacitance measurements derived from cyclic voltammetry and galvanostatic charge-discharge experiments using integer and fractional derivatives to correct for rate and current intensity effects. Texture analysis reveals that the hydration reactions of cement in the presence of carbon generate a fractal-like electron-conducting carbon network that permeates the load-bearing cement-based matrix. The energy storage capacity of this space-filling carbon black network of the high specific surface area accessible to charge storage is shown to be an intensive quantity, whereas the high-rate capability of the carbon-cement electrodes exhibits self-similarity due to the hydration porosity available for charge transport. This intensive and self-similar nature of energy storage and rate capability represents an opportunity for mass scaling from electrode to structural scales. The availability, versatility, and scalability of these carbon-cement supercapacitors opens a horizon for the design of multifunctional structures that leverage high energy storage capacity, high-rate charge/discharge capabilities, and structural strength for sustainable residential and industrial applications ranging from energy autarkic shelters and self-charging roads for electric vehicles, to intermittent energy storage for wind turbines and tidal power stations.

A case study of monkeypox disease in the United States using mathematical modeling with real data

In this article, we propose the mathematical modeling of monkeypox, a viral zoonotic disease, to study its near outbreaks in the United States. We use integer and fractional derivatives to simulate the transmission model dynamics effectively. The model parameter values are estimated using the Levenberg–Marquardt algorithm with the lsqcurvefit function in MATLAB. We derive the numerical solution of the integer-order model by using a neural network approach. For solving the fractional-order model, we use the finite-difference predictor–corrector method in the sense of the Caputo derivative. We perform several graphical simulations of future disease outbreaks. Implementing the Caputo fractional derivatives is motivated by its nonlocal nature, which includes memory effects in the model. The new data simulation and various solution approaches are the novelty of this research. This work aims to explore the transmission pattern and duration of a monkeypox outbreak in the United States.