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Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications
... Fractional order systems, also known as non-integer order systems, are systems described by fractional-order differential equations, where the order of differentiation is a non-integer value. These systems exhibit more complex behaviors than classical systems, such as long-term memory and non-locality [1]. Because real systems are not exactly integer order, fractional order models have found applications in many fields, including control engineering, signal processing, physics, and finance. ...
... Lastly, ⌅ is made up of the coefficient column vectors ⇠ k 2 R j http://dx.doi.org/10.12785/src/1570966802 that are calculated using a sparse regression algorithm such as the Sequentially Thresholded Least Squares introduced in [15]. The nonzero coefficients of ⇠ k 2 R j s determine which columns in ⇥(X) are active in the dynamics for the kth state equation in (1). That is, after ⌅ is obtained, the kth governing equation is written asẋ ...
... There are several definitions of the fractional derivative, with the Riemann-Liouville (RL), Grünwald-Letnikov (GL), and Caputo definitions being among the most studied. References [1] and [21] present a comprehensive discussion of these definitions and their relationships. ...
This study extends the Sparse Identification of Nonlinear Dynamics (SINDY) algorithm to the discovery of fractional order systems. The SINDY optimization problem was modified to incorporate candidate fractional orders in the Caputo sense, with discretization executed using Gr¨ unwald-Letnikov f inite differences. Identification of the governing equations was done separately for each state. Noise was handled through weak formulation, while sparse regression was performed via sequential thresholding. Discovery of the correct fractional orders was done through parameter sweep. The results showed that the algorithm was able to identify the governing equations of the fractional order Lorenz system, but still suffered from its inherent memory effects.
... Let us mention here some of them, like those proposed by Liouville [7], Riemann [8], Grunwald [9], Letnikov [10] or more recently by Caputo-Fabrizio [11], Lazopoulos [12], Gerasimov-Caputo [13], and Katugampola [14]. This flexibility of FC means that fractional differential equations (integral or differo/integral) (FDEs) do not have, in general, analytical solutions [15,16]-it is therefore necessary to look for robust numerical methods to obtain their solutions (especially for practical cases). ...
... This section is devoted to presenting some required preliminaries of fractional calculus-definitions of Riemann-Liouville integrals and Caputo derivatives [15]. ...
... Substituting Equation (16) into Equation (14) and Equation (17) into Equation (15) and then integrating leads to the expression of the Caputo fractional derivatives as an infinite sum of higher-integer-order derivatives as follows, for the left ...
This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary-even variable-terminals and order. The proposed method involves rearranging the fractional operator as a series of higher-order derivatives considered at a specific point. We demonstrate the effect of the number of terms included in the series expansion on the solution accuracy and error analysis. The advantage of the method is its simplicity and ease of implementation. Additionally, the method allows for a quick estimation of the fractional derivative by using a few first terms of the expansion. The elaborated algorithm is tested against a comprehensive series of illustrative examples, providing very good agreement with the exact/reference solutions. Furthermore, the application of the proposed method to fractional boundary/initial value problems is included.
... The modeling of anomalous diffusion has been actively investigated, especially in the last decade, using the generalized Langevin equation introduced by Kubo [9,10]. Another approach to anomalous diffusion is the use of the generalized diffusion equation [11], the fractional diffusion equation [12,27], the fractional Fokker-Planck equation [13], the Chapman-Kolmogorov equation [14], and the generalized fractional Langevin equation (FGLE) [15,16,17], involving the fractional derivative, where the order of integrals and derivatives is generalized to fractional order [18]. ...
... We present some basic concepts of the Riemann-Liouville (R-L) fractional integral as well as the definition of fractional derivative in the sense of Caputo (see, e.g.,, [18] for more details). The choice of the Caputo derivative is motivated by the fact that it allows for a more intuitive definition of initial conditions and that its derivative for a constant is zero. ...
... Definition 2.5. Let f : [a, b] → R be a continuous real function; its Riemann-Liouville fractional integral, of order α ∈ C, with [α] > 0, is defined by [18]: ...
In this work, we present an extended mathematical study of the fractional anomalous diffusion of a target particle which moves on a curved surface. To do explicit calculations, we assumed that this particle moves on a hypersphere of radius R. We analyzed the particle dynamics using a generalized fractional Langevin equation approach, based on the Caputo fractional derivative and Laplace transform techniques. We derived three physical quantities:the mean square displacement (MSD), the time diffusion coefficient (TDC), and the velocity autocorrelation function (VACF). We performed exact calculations of their temporal evolution by selecting power-law and Dirac delta memory functions. The results obtained are expressed in terms of Mittag-Leffler-type functions. The introduction of the fractional derivative affected the behavior of balistic regime. We plotted the dynamic quantities for different values fractional derivative order and analyzed their influence on diffusion scaling laws.
... The integration and differentiation of non-integer (fractional) order is the mathematical field known as fractional calculus (FC) [1,2]. The concept of FC, which is an extension of classical calculus, was first introduced by Leibniz in the 17th century, but it was not until the late 20th century that it began to be widely studied and applied in various fields such as biology, economics, physics, and engineering [3][4][5][6][7]. ...
... In Sect. 3, we define the mild solution for system (2), employing the Laplace transform, cosine families, and Mainardi's Wright-type function. Section 4 presents two theorems addressing the existence and uniqueness of the mild solution for system (2), establishes a compactness result, and examines the properties of the mild solution operators. ...
... 3, we define the mild solution for system (2), employing the Laplace transform, cosine families, and Mainardi's Wright-type function. Section 4 presents two theorems addressing the existence and uniqueness of the mild solution for system (2), establishes a compactness result, and examines the properties of the mild solution operators. In Sect. ...
Fractional differential equations (FDEs) constitute an essential field of mathematics which has been increasingly studied over recent decades, hence the importance of finding solutions for such systems. In this paper, we prove the existence and the uniqueness of the mild solution for a class of time-fractional evolution systems with an order of differentiation . We make use of different properties of the cosine and sine families as well as the Laplace transform (LT) to obtain a simplified expression of the mild solution, which will be given as an integral formula involving the Mainardi’s wright-type function. Moreover, many useful properties related to the operator appearing in the mild solution are given with proofs. Finally, we provide an illustrative example along with the expression for its mild solution.
... These methods were carried out for classic (integer or rational) systems and are therefore not suited for fractional systems. Indeed the fractional derivative (and consequently fractional systems) has a long-memory property [27,28,32,35]: also, some past of the signals needs to be considered. This is a fundamental property of fractional systems which makes it suitable to model propagation or diffusive phenomena [1,4,23,26,33]. ...
... One of the most well-known fractional derivative definition is the Grünwald-Letnikov one [32]: ...
... It should be noted that R μ in equation (36) is different than R ρ of the coefficient algorithm in (32). To differentiate them, they are denoted as R ρ and R μ . ...
In cardiac surgeries with extracorporeal circulation, the lungs are temporarily disconnected from the body. To minimize the risk of tissue damage, ischemia or pulmonary cell necrosis, the lungs are subjected to mild hypothermia. Incorporating dynamic heat transfer models offer the potential to enhance temperature regulation through a more advanced approach. Thanks to a thermal two-port network formalism, a better comprehension of the lung global heat impedance is obtained which has enabled a frequency analysis of the lungs. This modeling approach can also be adapted to incorporate the blood perfusion, which serves as a natural temperature regulator in the human body. Such a global characterization is too complex to be implemented in real-time application, therefore a model order reduction is developed through several approximations. In addition to the physiological modeling, when the system parameters are unknown, system identification enables estimating and optimizing these parameters. An efficient online algorithm, called LMRPEM, was developed in Victor et al. (Nonlinear Dyn 110:635-648, 2022) but the estimation is carried out with the full input–output data. In an online context, the data acquisition may be long and the computation time may highly increase, and even go beyond the sampling time. Therefore, one of the main contributions of this paper is to propose a truncated-LMRPEM with an analysis of the window length to be set.
... This section introduces fundamental concepts in fractional calculus, which extends classical differentiation and integration to non-integer orders. For a more in-depth discussion, refer to [20,21]. ...
... where m is the smallest integer greater than α (i.e., m − 1 < α < m, with m ∈ Z + ), Γ(·) is the Gamma function [20], t ≥ 0, and α ∈ R + . ...
... where the output signal change is ∆y = K · ∆u and E α represents the one-parameter Mittag-Leffler function [20]. ...
Fractional-order systems capture complex dynamic behaviors more accurately than integer-order models, yet their real-time identification remains challenging, particularly in resource-constrained environments. This work proposes a hybrid framework that combines Particle Swarm Optimization (PSO) with various artificial intelligence (AI) techniques to estimate reduced-order models of fractional systems. First, PSO optimizes model parameters by minimizing the discrepancy between the high-order system response and the reduced model output. These optimized parameters then serve as training data for several AI-based algorithms—including neural networks, support vector regression (SVR), and extreme gradient boosting (XGBoost)—to evaluate their inference speed and accuracy. Experimental validation on a custom-built heating system demonstrates that both PSO and the AI techniques yield precise reduced-order models. While PSO achieves slightly lower error metrics, its iterative nature leads to higher and more variable computation times compared to the deterministic and rapid inference of AI approaches. These findings highlight a trade-off between estimation accuracy and computational efficiency, providing a robust solution for real-time fractional-order system identification on edge devices.
... Most articles dealing with a fractional-order initial value problem begin with the following sentence: consider the Caputo fractional derivative; it has been shown that there is a physical interpretation of its initial conditions, which are the same as the integer-order case. Two monographs have particularly emphasized the use of the Caputo derivative approach, one by Podlubny [1], which is mainly a reference textbook on fractional calculus, and the other by Diethelm [2], related to fractional Cauchy problems. It would be possible to use the Riemann-Liouville derivative, but the interpretation of its initial conditions [3] seems far from physics, whereas the value x(0) (for 0 < n < 1) gives privilege to the Caputo derivative [4] since x(0) seems analogous to the integer-order case. ...
... Knowing this, the following is calculated [1]: ...
... The Riemann-Liouville integral x(t), or fractional integral, of a function v(t) is defined as follows [1]: ...
In this paper, a counter-example based on a realistic initial condition invalidates the usual approach related to the so-called physical initial condition of the Caputo derivative used to solve fractional-order Cauchy problems. Due to Infinite State representation, we prove that the initial condition of the Caputo derivative has to take into account the distributed states of an associated fractional integrator. Then, we prove that the free response of the counter-example requires the knowledge of the associated fractional integrator free response, and a realistic solution is proposed for the convolution problem based on the Mittag–Leffler function. Moreover, a simple and efficient technique based on Infinite State representation is proposed to solve the previous free response problem. Finally, numerical simulations demonstrate that the usual Caputo technique is based on an unrealistic initial condition without any physical meaning.
... A. Problem Formulation Definition 1 [34]: The uniform formula of a fractional integral with ∈ (0, 1) is defined by ...
... Using (18), (33), (34) and Theorem 1, it can be concluded that: ...
In this paper, an observer-based controller design for fractional-order multi-agent systems is discussed. By introducing a novel algorithm and leveraging appropriate lemmas and theoretical frameworks, we propose a stable observer and a distributed consensus protocol tailored for multi-agent systems within the Lipschitz and one-sided Lipschitz classes of nonlinear systems. Lipschitz systems have a bounded rate of change, ensuring proportional output to input differences, while one-sided Lipschitz systems relax this constraint, allowing differential growth in one direction for efficiency. The stability of the observer and the controller in achieving the consensus problem is demonstrated using the Lyapunov's second method. The proposed approach is rigorously developed, ensuring that the designed observer and controller meet the necessary stability criteria. Extensive simulation results validate the theoretical findings, showcasing the method's effectiveness and robustness in practical scenarios. Specifically, the simulations demonstrate that the proposed method achieves global Mittag-Leffler stability, with the estimated states converging to the actual states with minimal deviation. The method's advantages include its ability to handle a broader class of nonlinear systems, including those with large Lipschitz constants, and its robustness to uncertainties and nonlinearities. These simulations confirm the theoretical predictions and illustrate the practical applicability of our approach in real-world multi-agent systems, such as swarm robotics, power grids, and sensor networks.
... In this section, we extend the classical tuberculosis model presented in Equation (5) by incorporating the Caputo fractional derivative operator. Unlike the traditional model described by Equation (5), the fractional-order model provides greater flexibility, allowing its behavior to reflect different dynamic responses [15,20]. This approach enhances the model's capability to capture the memory effects and non-local dependencies inherent in disease transmission, offering a more accurate representation of tuberculosis dynamics. ...
... Given that the term in Eq. (8) on the right is positive, we can now deduce that for In the same way, we also have that since they are positive aspects, the solution will continue to be in for all with positive initial conditions [19,20]. ...
Tuberculosis (TB) remains one of the top infectious disease killers worldwide, with an estimated 10.6 million new cases and 1.3 million deaths reported in 2022 alone (WHO, 2023). The COVID-19 pandemic has further disrupted TB control efforts by limiting access to healthcare services, interrupting treatment regimens, and delaying diagnoses leading to a resurgence in TB transmission It is caused by Mycobacterium tuberculosis and spread through the air, TB poses a serious threat, particularly to vulnerable groups such as individuals with weakened immune systems, including those living with HIV. These challenges emphasize the need for more robust and realistic modeling approaches to inform policy and intervention. In this study, we developed a fractional-order mathematical model to better understand how TB spreads and how it can be controlled. Our model divides the population into six key groups: those susceptible to infection, exposed individuals, people with acute TB, those with chronic TB, individuals undergoing treatment, and those who have recovered. To capture the complexities of TB transmission, we incorporated fractional-order derivatives along with the Adams-Bashforth method, allowing us to account for memory effects and more accurately reflect real-world dynamics. Through sensitivity analysis, we found that increasing treatment rates significantly boosts recovery among infected individuals. Our simulations also explored various intervention strategies, such as improving access to treatment, reducing diagnostic delays, and addressing non-linear transmission patterns. The results highlight the effectiveness of these measures in curbing TB spread and offer insights for improving disease control efforts.
... Fractional calculus represents a highly advanced generalization of traditional calculus of integer orders [1][2][3][4][5][6]. Currently, fractional differential calculus is one of the most extensively used perspectives of fractional calculus. ...
... To facilitate the understanding of this paper, we provide some fundamental definitions of fractional calculus. For more comprehensive and detailed information on the subject, readers are encouraged to refer to [3,4,9]. Definition 1. ...
This research paper investigates a numerical scheme based on cubic B-spline functions for solving the time fractional gas dynamics equation incorporating the Atangana-Baleanu derivative. The Atangana-Baleanu time fractional derivative has been approximated by adopting the typical finite difference formulation while the cubic B-spline functions are employed to accomplish spatial discretization. A conditional bound for stability has been derived and a convergence analysis for cubic B-spline interpolation is conducted to measure the accuracy of the solution. The effectiveness and accuracy of the proposed method are tested through numerical simulations. Graphical and tabular results are exhibited to evaluate the outcomes of the proposed strategy. The major advantage of the suggested scheme is that the algorithm is straightforward to carry out.
... Fractional differential equations (FDEs) involve fractional derivatives or integrals, which extend the concept of integer-order derivatives and integrals. They are useful for modeling processes that exhibit memory or nonlocal behavior [1]. The commutativity property of FDEs allows for rearranging the order of fractional derivatives or integrals without altering the solution. ...
... The Riemann-Liouville fractional integral of order alpha for function f(t) is given by Podlubny [1] ...
In this abstract, we discuss some of the key results and techniques related to the commutativity of fractional-order LTVSs, including the use of linearity, time-varying behavior, fractional-order dynamics, and commutative conditions. These conditions must be satisfied in order for commutativity to hold for these systems. However, the commutativity of fractional-order LTVSs has not been widely explored.
... Many researchers assumed that it is a well-developed topic in mathematics and have been working on it till now. Due to the ideas of German mathematician Leibniz and L-Hospital, the theory of fractional calculus came into existence about 300 years ago [1][2][3][4]. The main benefit of studying FDE is that it gives solutions between intervals, which aids us in examining the results more understandably. ...
... Then, the LT of ( ) f τ in terms of Caputo derivative [2,3] is defined as ...
In this article, we study the time-dependent two-dimensional system of Wu–Zhang equations of fractional order in terms of the Caputo operator, which describes long dispersive waves that minimize and analyze the damaging effects caused by these waves. This article centers on finding soliton solutions of a non-linear ( 2+1 )-dimensional time-fractional Wu–Zhang system, which has become a significant point of interest for its ability to describe the dynamics of long dispersive gravity water waves. The semi-analytical method called the q -homotopy analysis method in amalgamation with the Laplace transform is applied to uncover an analytical solution for this system of equations. The outcomes obtained through the considered method are in the form of a series solution, and they converge swiftly. The results coincide with the exact solution are portrayed through graphs and carried out numerical simulations which shows minimum residual error. This analysis shows that the technique used here is a reliable and well organized, which enhances in analyzing the higher-dimensional non-linear fractional differential equations in various sectors of science and engineering.
... This characteristic coincides with that of fractional calculus. Fractional differentiation offers a more effective representation of a system's biological memory compared to integer differentiation [13]. Although the fractionalorder model was originally introduced in foundational mathematical research, its application remained limited for a considerable time due to the challenges associated with its computation [14]. ...
... LetS,L,B,R be the optimal values corresponding to the variables S, L, B and R, and letλ 1 ,λ 2 ,λ 3 andλ 4 be the solutions to Equation (13). With these clarifications in mind, we are now ready to present the following theorem. ...
The increasing reliance on and remote accessibility of automated industrial systems have shifted SCADA networks from being strictly isolated to becoming highly interconnected systems. The growing interconnectivity among systems enhances operational efficiency and also increases network security threats, especially attacks from industrial viruses. This paper focuses on the stability analysis and optimal control analysis for a fractional-order industrial virus-propagation model based on a SCADA system. Firstly, we prove the existence, uniqueness, non-negativity and boundedness of the solutions for the proposed model. Secondly, the basic reproduction number R0α is determined, which suggests the conditions for ensuring the persistence and elimination of the virus. Moreover, we investigate the local and global asymptotic stability of the derived virus-free and virus-present equilibrium points. As is known to all, there is no unified method to establish a Lyapunov function. In this paper, by constructing an appropriate Lyapunov function and applying the method of undetermined coefficients, we prove the global asymptotic stability for all possible equilibrium points. Thirdly, we formulate our system as an optimal control problem by introducing appropriate control variables and derive the corresponding optimality conditions. Lastly, a set of numerical simulations are conducted to validate the findings, followed by a summary of the overall study.
... And Υ 1 , Υ 2 are the non-negative weight coefficients. The Caputo fractional derivative is define in [18], as follows ...
... with Γ denoting the gamma function. The Riesz fractional derivative on a finite domain [18,21], is given by: ...
This research introduces a spectral element method (SEM) for solving a fractional diffusion model. We propose a discrete-time scheme, using the finite difference method to approach the multi-Caputo fractional derivative on a uniform mesh. In addition, we offer a Galerkin variational formulation to establish the unconditional stability of the scheme. We use the SEM based on Legendre polynomials in the space direction and derive the fully discrete scheme. The error estimation analysis of the fully discrete scheme is proved in the L2 sense. Finally, we demonstrate the method’s effectiveness by numerical experiments and simulations performed in MATLAB.
... Among fractional derivatives, three distinct kinds may be found: Liouville-Caputo, Riemann-Liouville, and Fabrizio-Caputo systems. 25,26 This led to the presentation of several new thermoelastic models based on fractional calculus. By characterizing the material's behavior using the fractional deformation derivative, Magin and Royston pioneered the first model. ...
... where the operator D α t = d α dt α is a fractional order derivative and is given by the Classical Caputo (C-C), Caputo-Fabrizio (C-F), and normal derivative (N-D), respectively, as in the following unified form: 23,25,[43][44][45][46][47][48][49][50] ...
This study introduces a novel mathematical model to investigate heat conduction through a circular micro-plate that is isotropic, homogeneous, and viscothermoelastic. Based on the Kirchhoff–Love plate hypothesis, the governing differential equations have been formulated in the context of the Lord–Shulman theory of generalized thermoelasticity. This model aims to integrate Young’s modulus with fractional operators of the derivative, namely, the Caputo and Caputo–Fabrizio operators, with the conventional derivative. The research illustrates the implementation of scaled viscothermoelasticity on a circular microplate that is simply supported at both extremities. The microplate’s boundary has undergone thermal loading using ramp-type heating. Numerical computations have been conducted to ascertain the inverse of the Laplace transform. The research included graphical comparisons of the definitions of fractional and ordinary derivatives. The effects of the fractional-order and thickness parameters on the thermal wave distribution are deemed minimal; however, the impact of the ramp-time heat parameter is significant. The fractional-order parameter substantially influences the mechanical waves. The ramp heat parameter effectively regulates energy dissipation in ceramic resonators.
... While the Riemann-Liouville fractional derivative (RLFD) approach involves boundary values of fractional derivatives in initial conditions, which can present interpretation challenges, the Caputo fractional derivative (CFD) is favored in physical models. Its advantage lies in its ability to offer a clear interpretation of initial conditions and ensuring measurability, making it preferable in practical applications [1][2][3][4]. ...
... The aim is to determine the initial state of the system (1) from the measurement functional (3). Achieving this, it is necessary to calculate the adjoint of F α,β , which may not always be possible when C is not bounded. ...
In this study, we introduce the concept of observability for a specific category of time-fractional systems characterized by the Hilfer fractional derivative (HFD) of order with type Initially, we provide definitions and discuss key properties of this concept. Subsequently, we outline a technique for determining the system’s state at t=0, which builds upon the principles of the Hilbert uniqueness method (HUM). This methodological extension culminates in an algorithm that yields compelling numerical outcomes. Notably, the reconstruction error associated with the initial state is remarkably low, affirming the efficacy of the approach employed in this investigation.
... Unlike traditional integer-order calculus, fractional calculus generalizes the process of integration to encompass non-integer orders [6,7]. Although this makes the analysis and computation process more complex, the models established will also be more accurate. ...
... Owing to the merits of the Caputo fractional calculus definition, including its minimal dependence on initial values, the facilitation of Laplace transformations, the property that the fractional derivative of constants is zero, and its computational simplicity, Caputo fractional calculus is extensively utilized in engineering research [7,18]. This paper will undertake an analytical study on the modeling of fractional-order synchronous generators employing the Caputo fractional calculus framework. ...
This paper investigates the fractional-order characteristics of the stator and rotor windings of a synchronous generator. Utilizing mechanism-based modeling methodology, it pioneers the derivation of the fractional-order voltage equations for a synchronous generator across both the three-phase stationary coordinate system (A, B, C) and the synchronous rotating coordinate system (d, q, 0). Through simplifying assumptions and rigorous derivations, a 2 + α (α ∈ (0, 2)) order synchronous generator model is formulated. This paper develops a digital simulation model of a fractional-order single-machine infinite bus system and analyzes the impact of the order α on the synchronous generator system’s dynamic performance through disturbance simulation experiments. Experimental results demonstrate that under conventional disturbances, increasing α from 0.8 to 1.2 reduces the system oscillation period and frequency while enhancing mechanical oscillation suppression, whereas decreasing α to 0.8 accelerates the generator terminal voltage response, lowers electromagnetic power overshoot, and improves excitation control effectiveness.
... Fractional calculus [29] plays a prominent role in modern day medical science and medicines [30]. Fractional model is considered in this study because of the fact that cilia is a microscopic structure and its motion, functioning and various other characteristics in the respiratory and reproductive systems of humans/mammals can be better under-stood by using fractional order derivatives i.e., in the range from zero to unity. ...
... In this work, Caputo's definition is used for fractional derivatives. For more details on fractional derivative and Caputo's definition, refer previous studies [38,39]. ...
In the present era of cutting-edge technologies powering electronic devices, energy systems, and modern telecommunication, a mathematical model called the telegraph equation plays a major role. It describes the wave propagation and diffusion processes in a variety of scientific and engineering areas. This work investigates, applies, and compares two semi analytical methods, namely, reduced fractional differential transform method and the Laplace-transformed residual power series method, toward solving the two-dimensional time-fractional telegraph equation. In the recent past, both the methods have been used to solve various linear and non-linear, ordinary, partial, and fractional differential equations. The literature speaks highly about residual power series method being efficient, specially for the non-linear problems. This work puts forth the methodologies and numerical experiments, and it can be observed that both methods result in the same solution for the two-dimensional linear telegraph equation. However, the solutions for the non-linear telegraph equation are better with the differential transform method.
... Recent years have witnessed growing undivided attention being paid to exploring the quest for solitary wave solutions within the domain of nonlinear fractional differential equations (NFDEs) [1,2]. Researchers have devoted significant attention to understanding the formation, propagation, and interaction of solitary waves, which are crucial for modeling a wide range of physical phenomena [3,4]. ...
Citation: Alharthi, N.H.; Kaplan, M.; Alqahtani, R.T. Soliton Dynamics and Modulation Instability in the (3+1)-Dimensional Generalized Fractional Kadomtsev-Petviashvili Equation. Symmetry 2025, 17, 666. Abstract: In this article, novel methods of analysis to solve the (3+1)-dimensional generalized fractional Kadomtsev-Petviashvili equation, which plays a crucial role in the modelling of fluid dynamics, particularly wave propagation in complicated media, are presented. The fractional KP equation, a well-established mathematical model, uses fractional derivatives to more adequately describe more general types of nonlinear wave phenomena, with a richer and improved understanding of the dynamics of fluids with non-classical characteristics, such as anomalous diffusion or long-range interactions. Two efficient methods , the exponential rational function technique (ERFT) and the generalized Kudryashov technique (GKT), have been applied to find exact travelling solutions describing soliton behaviour. Solitons, localized waveforms that do not deform during propagation, are central to the dynamics of waves in fluid systems. The characteristics of the obtained results are explored in depth and presented both by three-dimensional plots and by two-dimensional contour plots. Plots provide an explicit picture of how the solitons evolve in space and time and provide insight into the underlying physical phenomena. We also added modulation instability. Our analysis of modulation instability further underscores the robustness and physical relevance of the obtained solutions, bridging theoretical advancements with observable phenomena.
... Among many models dedicated to such systems like strain-gradient theories [22,23], micropolar theory [24,25] [26,27], couple stress theory [22], dipolar gradient elasticity theory [28] or peridynamic models [29], it appears that the one that bases on fractional analysis are preferable [30,31]. It is because recent achievements in mechanical sciences have shown that the fractional calculus [32,33] enables us to define extremely powerful models for physical phenomena predictions due to its important property that we have infinitely many definitions of fractional derivative operators; let us mention just a few i.e. Riemann (published posthumously) [34], Grunwald [35], Letnikov (1868) [36], Caputo [37], -fractional derivative [38], Atangana-Baleanu [39], Hadamard [40], Tempered-Caputo [41], Gerasimov-Caputo [7], Katugampola [42] and many others. ...
The paper has two primary aims, both related to the description and analysis of scale sensitive truss structures. The first considers the elaboration of the space-fractional (non-local) truss model (s-FTM) based on the principle of virtual work within the framework of space-fractional continuum mechanics (s-FCM)—the model which enables the prediction of strong scale effect. The second encompasses the development of a space-fractional finite element method (s-FFEM) for s-FTM—the numerical method allowing the analysis of arbitrary topological complexity of the truss and arbitrary boundary conditions. It is important that elaborated s-FTM allows to model spatially variable the scale effect, which in phenomenological sense maps non-homogeneous microstructure. Comprehensive numerical study enables understanding of the convergence of s-FFEM, and moreover, it is shown that smooth passage to classical (local) truss theory is obtained as a special case of s-FTM (from both the theoretical and approximated points of view).
... Several formulations of fractional derivatives exist, including Riemann-Liouville, Grunwald-Letnikov, Hadamard, Weyl, Caputo, and conformable derivatives, each providing distinct approaches for modeling and analysis. [5][6][7][8] Among these, the conformable derivative stands out for its simplicity and effectiveness in representing fractional-order systems, offering a clear and straightforward method for extending classical differential equations into the fractional domain. This simplicity makes it especially useful for practical applications, where computational efficiency is often critical. ...
... Surprisingly, it has stimulated astonishing utilization in nuclear reactor dynamics, chemical kinetics (Taghipour and Aminikhah 2022b), advection-diffusion phenomena (Taghipour and Aminikhah 2023), viscoelasticity, biomechanics (Taghipour and Aminikhah 2022a), wave propagation (Taghipour and Aminikhah 2024), optimization and finance (Taghipour and Aminikhah 2022c). An introduction to fractional calculus and its applications can be found in Diethelm and Ford (2002); Oldham and Spanier (1974); Podlubny (1998); Samko (1993); Kilbas et al. (2006). ...
... In fractional calculus (Baleanu et al. 2016;Diethelm 2010;Kilbas et al. 2006;Podlubny 1998), the standard approach when modeling consists in replacing the ordinary derivative to the left-hand side of the equation by a fractional-order derivative. For example, in the well-known logistic model ...
A partial Caputo fractional model mixes, in the same formulation, classical and Riemann–Liouville terms, in the realm of epidemiology and sociology. The Riemann–Liouville part introduces non-Markovian behavior and makes the hazard risk function of transition between the compartments lower as time advances. As a generalization of standard Caputo fractional models, where all of the terms are fractional, here we investigate the mathematical theory on incomplete fractionalization. Essentially, the goal is to prove that partially fractional initial-value problems are well posed, with existence, uniqueness, and continuity of solution with respect to input data. We present fixed-point results, a Cauchy–Kovalevskaya theorem on fractional power series, and Gronwall- and Nagumo-like arguments for uniqueness. Continuity of solutions uses bounds with the Mittag-Leffler function. Linear equations are also addressed, exhibiting global existence, global power-series representation, and certain closed-form solutions with Mikusiński operational calculus, refined bounds, and stability properties. The stochastic counterpart of partial Caputo models is introduced, with several results.
... Its ability to model complex phenomena with greater accuracy and depth has opened up new possibilities for research and development in these areas. More detailed introductions to fractional calculus can be found in [3]. ...
In this paper, we investigate the existence results of solutions for Caputo-type fractional (p,q)-difference equations. Using Banach’s fixed-point theorem, we obtain the existence and uniqueness results. Meanwhile, by applying Krasnoselskii’s fixed-point theorem and Leray-Schauder’s nonlinear alternative, we also obtain the existence results of non-trivial solutions. Finally, we provide examples to verify the correctness of the given results. Moreover, relevant applications are presented through specific examples.
... Fractional calculus (FC), the class of differentials and integrals to non-integer orders development, was the brilliant notion of Gottfried Leibniz at the end of the 17 th century. There are several methods for computing fractional derivations and integrals, including Riemann-Liouville, Caputo, Atangana-Baleanu and Grünwald-Letnikov [42,49,16,29]. The topic has been used primarily in a mathematical framework, but in recent years FC has been recognized as an effective approach for studying and modeling a variety of real-world problems, for example in biology [37], economics [25], finance [32], chemistry [30], physics [11], mechanics [40], epidemiology [8] and engineering [53]. ...
In this research, a novel fractional-order chaotic circuit incorporating a feedback memristor is presented. The structure of the circuit and the associated mathematical model are described in detail. The complex dynamics of the circuit are analyzed using stability analysis, Poincaré maps and numerical simulations, revealing the presence of a saddle fixed point and coexisting attractors. The influence of changing the system variables and initial conditions is studied using bifurcation plots and Lyapunov exponents. The circuit exhibits a variety of nonlinear behaviours including periodic, quasi-periodic, and chaotic dynamics. Furthermore, experimental simulations performed with a chain-ship circuit configuration validate the theoretical results and show strong agreement with the numerical analysis. Finally, a robust sound encryption scheme is presented that exploits the pseudorandom sequences generated by the chaotic memristive circuit.
... We refer to [15][16][17][18] for further study of fractional calculus and fractional differential equations. ...
This paper investigates the existence and nonexistence of positive solutions for a class of nonlinear Riemann–Liouville fractional boundary value problems of order α+2n, where α∈(m−1,m] with m≥3 and m,n∈N. The conjugate fractional boundary conditions are inspired by Lidstone conditions. The nonlinearity depends on a positive parameter on which we identify constraints that determine the existence or nonexistence of positive solutions. Our method involves constructing Green’s function by convolving the Green functions of a lower-order fractional boundary value problem and a conjugate boundary value problem and using properties of this Green function to apply the Guo–Krasnosel’skii fixed-point theorem. Illustrative examples are provided to demonstrate existence and nonexistence intervals.
... The interplay between the mathematical theory and real-world applications of fractional calculus has led to substantial advancements, positioning FDEs as a well-established field with diverse applications across physics, engineering, and technology. Their relevance spans control theory, electrochemistry, electromagnetics, viscoelasticity, and porous media, among others (see [1][2][3]). For further developments in this area, readers may consult [4][5][6]. ...
In this research article, we investigate a three-dimensional dynamical system governed by fractal-fractional-order evolution differential equations subject to terminal boundary conditions. We derive existence and uniqueness results using Schaefer’s and Banach’s fixed-point theorems, respectively. Additionally, the Hyers–Ulam stability approach is employed to analyze the system’s stability. We employ vector terminology for the proposed problem to make the analysis simple. To illustrate the practical relevance of our findings, we apply the derived results to a numerical example and graphically illustrate the solution for different fractal-fractional orders, emphasizing the effect of the derivative’s order on system behavior.
... The mathematical branch known as fractional calculus takes into account the order of differentiation and integration, including arbitrary real numbers or even complex numbers. Fractional Calculus was developed by many mathematicians in the middle of the 20th century, including Riemann, Liouville, Letnikov, Laplace, Weyl, Fourier, Abel, Grunwald, Erdelyi, and Heaviside [1,2]. There has been no remarkable progress in this field for a long time because of the less information about geometrical interpretations and physical applications. ...
The investigation delves into how Lorenz systems behave chaotically when analyzed through both standard and fractional-order derivative approaches. Our research utilizes detailed numerical methods to explore how these systems' dynamic behaviors and stability features change across different parameters. We implement two mathematical frameworks: conventional integer-order derivatives and Caputo-defined fractional-order derivatives to examine the system's evolution. When studying the fractional-order framework, we observed increased complexity in the phase space patterns, specifically as the derivative order shifts within the range of 0 to 1. A significant discovery was the identification of previously undocumented chaotic attractors within fractional-order systems, including cases where the order exceeds 1. This work advances our knowledge of how the application of fractional calculus enhances our grasp of chaotic behavior in nonlinear systems, revealing new layers of complexity in their dynamic patterns
... In chemistry, FDEs are used to model chemical reactions and diffusion processes in heterogeneous media. Furthermore, in fractal and chaos theory, FDEs are instrumental in modeling self-similar structures and complex patterns found in nature (see, e.g., [13,14,21,23,24,31,32]). ...
In this paper, we study generalized (C)-conditions, specifically the Kannan-Suzuki-(C) condition (abbreviated as the (KSC)-condition). We employ the M-iteration process to investigate the convergence behavior of map-pings satisfying the KSC-condition and demonstrate that this approach offers improved convergence speed and computational efficiency compared to other well-known iteration schemes in the literature. To illustrate the advantages of the M-iteration process, we present new numerical examples that highlight its effectiveness. Additionally, we validate our theoretical findings by applying the method to fractional delay differential equations, showcasing its applicability in solving complex mathematical models. Furthermore, we compare the polynomiographs generated by the M-iteration process with those produced by other well-known iteration methods, demonstrating superior vi-sualization properties and computational performance. These results establish the M-iteration process as a powerful tool for studying generalized contraction conditions.
... Specifically, new fractional operators in terms of Mittag-Leffler and exponential kernel types have been defined which offer better adaptability and amenability. Further details on fractional derivatives are given in [15,18,21]. ...
In this paper, the two-dimensional (2-D) fractional cable equation (FCE) with the Caputo variable-order (V-O) derivative was utilized for simulating systems with memory and hereditary characteristics that vary across time and space. This variable-order fractional model is particularly well suited for the description of neuronal dynamics in biological systems. The accurate modeling of dynamic, memory-dependent behaviors that vary over space and time, which are essential for applications such as neuronal dynamics, presents a challenge for conventional numerical methods. Furthermore, there is a lack of stable and effective numerical techniques for 2-D V-O systems, highlighting the need for improved computational approaches. In order to solve the cable equation numerically with high accuracy and computing efficiency, this work primarily focused on using a higher-order finite difference method. The proposed method's robustness was confirmed by stability and convergence analyses, while its efficacy was demonstrated by numerical simulations, which were presented in tabular and graphical formats. These findings demonstrate its precision and efficiency when dealing with the intricate dynamics of V-O fractional equations. The study concludes that the higher-order finite difference method offers an accurate and effective framework for solving fractional partial differential equations (FPDEs), particularly in applications that necessitate precision modeling, such as biological and physical systems. It also creates opportunities for future research, such as the application of the method to multivariate problems, the integration of machine learning techniques, or the adaptation of the method to systems with variable coefficients.
... Specifically, new fractional operators in terms of Mittag-Leffler and exponential kernel types have been defined which offer better adaptability and amenability. Further details on fractional derivatives are given in [15,18,21]. ...
... However, traditional SIR models often simplify disease dynamics by assuming instantaneous interactions and neglecting historical factors, such as the influence of past infection rates on current immunity levels. To address these limitations, fractional-order derivatives, which extend the concept of differentiation to non-integer orders [4][5][6][7], have emerged as a powerful tool. In contrast to integer-order derivatives, fractional derivatives inherently incorporate memory effects and long-range interactions, leading to more accurate representations of biological processes like disease transmission and recovery. ...
This paper introduces a novel fractional Susceptible-Infected-Recovered (SIR) model that incorporates a power Caputo fractional derivative (PCFD) and a density-dependent recovery rate. This enhances the model’s ability to capture memory effects and represent realistic healthcare system dynamics in epidemic modeling. The model’s utility and flexibility are demonstrated through an application using parameters representative of the COVID-19 pandemic. Unlike existing fractional SIR models often limited in representing diverse memory effects adequately, the proposed PCFD framework encompasses and extends well-known cases, such as those using Caputo–Fabrizio and Atangana–Baleanu derivatives. We prove that our model yields bounded and positive solutions, ensuring biological plausibility. A rigorous analysis is conducted to determine the model’s local stability, including the derivation of the basic reproduction number (R0) and sensitivity analysis quantifying the impact of parameters on R0. The uniqueness and existence of solutions are guaranteed via a recursive sequence approach and the Banach fixed-point theorem. Numerical simulations, facilitated by a novel numerical scheme and applied to the COVID-19 parameter set, demonstrate that varying the fractional order significantly alters predicted epidemic peak timing and severity. Comparisons across different fractional approaches highlight the crucial role of memory effects and healthcare capacity in shaping epidemic trajectories. These findings underscore the potential of the generalized PCFD approach to provide more nuanced and potentially accurate predictions for disease outbreaks like COVID-19, thereby informing more effective public health interventions.
... Using Eqs. (13) and (14), we obtain the following inequality ...
This study examines weak sharp solutions for a category of variational inequalities incorporating functionals based on fractional curvilinear integrals, expanding the conventional approach with concepts from fractional calculus. Furthermore, by using the sufficiency property of the minimum principle, the paper establishes and proves results on the weak sharpness characteristic of the solution collection for this type of inequality constraints. An application is provided to illustrate the main theoretical findings.
... These equations are widely utilized in fields such as viscoelasticity, electrochemistry, control systems, porous media, and electromagnetics, among others (see [1][2][3][4][5]). Over the past few years, considerable progress has been made in both the theoretical and practical aspects of fractional differential equations, as highlighted in the monographs by Kilbas et al. [6], Miller and Ross [7], and Podlubny [8], along with several research papers [9][10][11][12][13] and related references. ...
This paper addresses the optimal control problem for a class of nonlinear fractional systems involving Caputo derivatives and nonlocal initial conditions. The system is reformulated as an abstract Hammerstein-type operator equation, enabling the application of operator-theoretic techniques. Sufficient conditions are established to guarantee the existence of mild solutions and optimal control-state pairs. The analysis covers both convex and non-convex scenarios through various sets of assumptions on the involved operators. An optimality system is derived for quadratic cost functionals using the G\^ateaux derivative, and the connection with Pontryagin-type minimum principles is discussed. Illustrative examples demonstrate the effectiveness of the proposed theoretical framework.
In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found that the method produces solutions that are identical to the exact solution and approximate series solutions. The ENDT is its ability to proficiently handle complex systems governed by fractional differential equations while preserving memory and hereditary characteristics. Its simplicity, accuracy, and flexibility render it an effective instrument for replicating real-world phenomena in physics and biology. The ENDT method offers accuracy, stability, and efficiency compared to traditional methods. It effectively handles challenges in complex systems, supports any fractional order, is simple to implement, improves computing efficiency with sophisticated methodologies, and applies it to epidemic predictions and biological simulations.
In this paper the thermoelastic dynamic response of microbeam under laser pulse and the influence by noise interference are investigated for the first time. the governing equations of the microbeam are established by the fractional order two-phase hysteresis model and the Kelvin-Voigt model, and are solved using the shifted Chebyshev polynomials algorithm. Finally, some numerical simulations are provided, which demonstrate the validity the efficiency and robustness by the proposed method.
The protein content of flaxseed (Linum usitatissimum) is a crucial factor influencing its nutritional value and quality. Spectral technology combined with advanced modeling methods offers a fast, accurate, and cost-effective approach for predicting protein content. In this study, visible-near infrared hyperspectral imaging (VNIR-HIS) technology was combined with fractional order ant colony optimization (FOACO) to determine the protein content of flaxseed. Thirty flaxseed varieties commonly cultivated in Northwest China were selected, and hyperspectral data along with protein content measurements were collected. A joint x-y distance algorithm was applied to divide the dataset into calibration and prediction sets after removing outliers. Partial least squares regression (PLSR) models were developed based on both raw and preprocessed spectra, with the Savitzky-Golay (SG) smoothing method found to provide superior performance. The performance of wavelength selection methods based on FOACO, principal component analysis (PCA), and ant colony optimization (ACO) was compared using PLSR and multiple linear regression (MLR) models. The FOACO-MLR model achieved a prediction accuracy of 0.9248, a root mean square error (RMSE) of 0.4346, a relative prediction deviation (RPD) of 3.6458, and a mean absolute error (MAE) of 0.3259. The results show that the FOACO-MLR model provides significant advantages in predicting flaxseed protein content, particularly in terms of prediction accuracy and stability of characteristic bands. By combining VNIR-HIS technology with the FOACO wavelength selection algorithm, this study offers an efficient and rapid method for determining the protein content of flaxseed, providing reliable technical support for the precise detection of nutritional components.
This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders α. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different α show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems.
In this study, a new class of the Benjamin Bona Mahony Burgers equation is introduced, which considers the distributedorder in the time variable and fractional-order space in the Caputo form in the 2D case. The 2D-modified orthonormal normalized shifted Ultraspherical polynomials are derived from 1Dmodified orthonormal normalized shifted Ultraspherical polynomials and 2D-modified orthonormal normalized shifted Ultraspherical polynomials and the orthonormal normalized shifted Ultraspherical polynomials are applied to approximate of the space and time variables, respectively. Moreover, the convergence analysis of these basis functions is investigated. Due to the time variable being in the distributed-order mode and the space variable being in the fractional-order case, to apply the desired numerical algorithm for this type of equation, operational matrices of ordinary, fractional and distributed-order derivatives are computed. In the proposed method, once the unknown function is approximated using the mentioned polynomial, the matrix form of the residual function is derived and then a system of algebraic equations is adopted by applying the collocation approach. An approximate solution is extracted for the original problem by solving constructed equation system. Several examples are examined to demonstrate the accuracy and capability of the method.
This study aims to develop and analyze a model of hepatitis B virus transmission dynamics using integer and fractional derivatives in the Caputo sense. After formulating the models, we conduct an asymptotic stability analysis of the disease-free equilibrium point of both models. The Lyapunov technique demonstrates that under specific conditions, the disease-free equilibrium point in both models remains globally asymptotically stable. The study demonstrates that both models can have at least one endemic equilibrium when , using the vaccination coverage parameter to identify positive equilibrium points. The Banach contraction principle is used to establish the uniqueness and existence of each fractional model’s solutions, followed by demonstrating their global stability using the Ulam-Hyers technique. The model is calibrated using reported hepatitis B cases in Nigeria, allowing for parameter estimations. The study indicates that the disease is endemic in this country, as , indicating a higher level of endemicity. The Adams-Bashforth approach is used to develop a numerical scheme, which is then validated through numerical simulations and evaluated under fractional order parameter variations.
In this paper, we investigate the existence and uniqueness of mild solutions for non-autonomous fractional evolution equations (NFEEs) using the technique of non-compactness measure, focusing on scenarios where the semigroup is non-compact. Furthermore, the optimal control of nonlinear NFEEs with integral index functionals is studied, and the existence of optimal control pairs is proven. Finally, by constructing a corresponding Gramian controllability operator using the solution operator, a sufficient condition is provided for the existence of approximate controllability of the corresponding problem.
This paper discusses the Petrov–Galerkin method's application in solving the time fractional diffusion wave equation (TFDWE). The method is based on using two modified sets of shifted fourth‐kind Chebyshev polynomials (FKCPs) as basis functions. The explicit forms of all spectral matrices were reported. These forms are essential to transforming the TFDWE and its underlying homogeneous conditions into a matrix system. An appropriate algorithm can be used to solve this system to obtain the desired approximate solutions. The error analysis of the method was studied in depth. Four numerical examples were provided that included comparisons with other existing methods in the literature.
This paper is concerned with the relatively exact controllability of a class of fractional stochastic dynamical systems(FSDSs) with distributed delays in control. Firstly, with the aid of unsymmetric Fubini theorem, the solution of this FSDSs are derived. Subsequently, we show that relatively exact controllability of linear systems are equivalent to positive definiteness of Grammian matrix. In addition, Schauder’s fixed point theorem and It isometry are used to prove the relatively exact controllability results for nonlinear case. Finally, an example is offered to illustrated the theory.
This paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series and validates its feasibility through comparison with known exact solutions. The proposed approach introduces a convergence parameter ℏ, which plays a crucial role in adjusting the convergence range of the series solution. By appropriately selecting initial terms, the convergence speed and computational accuracy can be significantly improved. The Jafari transform can be regarded as a generalization of classical transforms such as the Laplace and Elzaki transforms, enhancing the flexibility of the method. Numerical results demonstrate that the proposed technique is computationally efficient and easy to implement. Additionally, when the convergence parameter ℏ=−1, both the homotopy perturbation method and the Adomian decomposition method emerge as special cases of the proposed method. The knowledge gained in this study will be important for model solving in the fields of mathematical economics, analysis of biological population dynamics, engineering optimization, and signal processing.
This paper introduces a collocation approach for treating the ordinary and fractional Newell–Whitehead–Segel equation (NWSE). The integer and fractional derivatives of shifted Schröder polynomials (SPs), as well as some new theoretical results of these polynomials, are presented and used in conjunction with the collocation method to convert the equation with its underlying conditions into a system of equations that can be treated using a suitable numerical solver. A thorough error analysis is performed to evaluate the accuracy and dependability of our suggested method. Some numerical examples show that our suggested strategy is effective and accurate. The numerical results demonstrate that the suggested collocation approach yields accurate solutions using shifted SPs as basis functions.
In this paper, we deal with a semi-infinite variational programming problem (SIVP) involving the Caputo-Fabrizio (CF) fractional derivative operator. Firstly, we formulate the Lagrange dual model for (SIVP) and then by using Slater’s constraint qualification (SCQ) and convexity assumption, we establish the weak and strong duality theorems between primal and dual problems. Later on, the saddle point criteria associated with the Lagrange functional of the corresponding (SIVP) is discussed. Moreover, some numerical examples have been given to support the theoretical results.
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