Fractional Derivative Research Articles

Controllability Analysis of Neutral Stochastic Differential Equation Using $$\psi $$-Hilfer Fractional Derivative with Rosenblatt Process

Controllability Analysis of Neutral Stochastic Differential Equation Using $$\psi $$-Hilfer Fractional Derivative with Rosenblatt Process

Investigating the fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation: an iterative framework for nonlinear wave dynamics

Abstract This paper addresses the solution of the fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation using the Conformable Laplace Decomposition Method (CLDM). The CDGSK equation, a fundamental model in wave dynamics and fluid mechanics, is explored for its applications in quantum mechanics and nonlinear optics. By employing fractional calculus, we demonstrate how fractional derivatives influence the physical characteristics of wave propagation in both optical and quantum systems. The exact solutions obtained provide insight into soliton behavior, essential for understanding wave-particle interactions in quantum fields and light–matter interactions in optics. The fractional nature of the equation allows for more accurate modeling of non-integer order dynamics commonly found in optical fibers and quantum waveguides. The CLDM method proves to be highly effective, providing approximate solutions with minimal computational effort. These findings offer significant contributions to the fields of quantum mechanics and nonlinear optics, where the fractional CDGSK equation can be applied to solve complex wave equations with great accuracy.

Effects of Memory on Inventory Control and Pricing Policy for Imperfect Production with Rework Process

Fractional calculus is a pertinent way to study the memory effect in an inventory model for investigating its dynamical behavior. Since inventory management is a memory-dependent process, fractional calculus approach may be employed to discover some fruitful insights and can help to gain more profit. In realistic scenarios, the manufacturing process cannot be perfect, and it delivers some faulty units due to many inevitable reasons. In literature, an imperfect production inventory problem under the memory effect has not been studied. Our study aims to investigate the memory effect on a production inventory system. In this article, a fractional order inventory control problem is formulated by considering an imperfect manufacturing process and price-sensitive demand. The faulty units are repaired through a rework process. Caputo fractional derivatives and integrals are used to consider the memory effect. Due to the nonlinear cost elements in the formulated problem, optimal pricing and production policies are investigated by using a quasi-Newton optimization algorithm and particle swarm optimization approach. The managerial implications of the proposed study are discussed with the help of numerical illustrations. The numerical outcomes suggest that consideration of memory in the inventory system boosts the profitability of the firm.

Existence of solutions for time fractional semilinear parabolic equations in Besov–Morrey spaces

AbstractWe consider the Cauchy problem for a time fractional semilinear heat equation "Equation missing"where $$0<\alpha <1,\, \gamma >1,\, N\in \mathbb {Z}_{\geqslant 1}$$ 0 < α < 1 , γ > 1 , N ∈ Z ⩾ 1 and $$\mu (x)$$ μ ( x ) belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative $$^{C}\partial ^{\alpha }_{t}$$ C ∂ t α is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data.

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Normalized acceleration based online tuning of variable-order fractional derivatives: a case study on quadcopter position control

In recent years, the use of Variable-Order (VO) fractional operators in control system design has been gaining popularity due to their adaptive nature, enabled by their dynamically adjustable fractional derivative and integral orders. This paper presents an online tuning method for adjusting the VO fractional derivatives in fractional controllers. The method is formulated to strategically accelerate or decelerate the system response's rate of change to enhance reference tracking and disturbance rejection performance while preserving closed-loop stability. It uses the normalised acceleration of the system response, a metric that provides insights into the ‘fastness’ or ‘slowness’ of the system response. The effectiveness of the proposed online tuning method is validated through a case study on quadcopter position control. Our research includes both simulation and real-time testing of Variable-Order Fractional PD (VOFPD) controllers, which utilise our online tuning method to adjust their fractional derivatives in real-time. Stability analysis via the D-decomposition method confirms that the quadcopter's closed-loop stability is preserved. Comparative results show significant improvements in reference tracking and disturbance rejection in terms of time-domain criteria.

Compact ADI Difference Scheme for the 2D Time Fractional Nonlinear Schrödinger Equation

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the L1−2−3 formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of O(τ1+α+hx4+hy4)(α∈(0,1)) is obtained by adding a small term, where τ, hx, hy are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme.

Reliable Computational Method for Systems of Fractional Differential Equations Endowed with ψ-Caputo Fractional Derivative

This study develops a highly convergent computational method, the ψ-Laplace Adomian Decomposition Method (ψ-LADM), for solving coupled systems ofψ-Caputo Fractional Differential Equations (FDEs). The effectiveness of the proposed method has been assessed using various numerical test examples, including a real-world application for atmospheric convection models utilizing Lorenz chaotic dynamical systems. Notably, the method consistently produced solutions that matched the true solutions of the governing models. In the case of the Lorenz chaotic system, the obtained solutions accurately portrayed the characteristic phase portraits of a true chaotic system.

On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel

Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures.

Novel Fractional Order Differential and Integral Models for Wind Turbine Power–Velocity Characteristics

This work presents an improved modelling approach for wind turbine power curves (WTPCs) using fractional differential equations (FDE). Nine novel FDE-based models are presented for mathematically modelling commercial wind turbine modules’ power–velocity (P-V) characteristics. These models utilize Weibull and Gamma probability density functions to estimate the capacity factor (CF), where accuracy is measured using relative error (RE). Comparative analysis is performed for the WTPC mathematical models with a varying order of differentiation (α) from 0.5 to 1.5, utilizing the manufacturer data for 36 wind turbines with capacities ranging from 150 to 3400 kW. The shortcomings of conventional mathematical models in various meteorological scenarios can be overcome by applying the Riemann–Liouville fractional integral instead of the classical integer-order integrals. By altering the sequence of differentiation and comparing accuracy, the suggested model uses fractional derivatives to increase flexibility. By contrasting the model output with actual data obtained from the wind turbine datasheet and the historical data of a specific location, the models are validated. Their accuracy is assessed using the correlation coefficient (R) and the Mean Absolute Percentage Error (MAPE). The results demonstrate that the exponential model at α=0.9 gives the best accuracy of WTPCs, while the original linear model was the least accurate.

Fractional Derivative-based Burger Creep Model for Soft Rocks and its Verification Using Tunnel Monitoring Results and Experimental Data

AbstractConsidering the creep behavior of soft and weak rocks is critical for analyzing the long-term stability of underground constructions. This paper introduces a novel creep constitutive model to characterize the creep behavior of rocks under uniaxial and triaxial stress states. The fractional derivative Abel dashpot was used to improve the Burger model, and a viscoplastic component was added in series with the modified Burgers model to replicate the tertiary phase of rock creep. The effectiveness of the model was verified using creep test data from various soft rocks and monitoring measurements from a tunnel excavated in heavily jointed weak rock masses. Furthermore, a sensitivity analysis was carried out to assess the impact of the model parameters on creep deformation, and a comparative study was performed to evaluate the efficacy of the suggested model in modeling the accelerated stage of rock creep compared with some existing models. The strong agreement observed between the calculated results and both the creep test data and tunnel monitoring measurements underscores the accuracy and validity of the proposed model. The comparative analysis further revealed that the proposed model offers the highest fitting efficiency for describing the tertiary stage of rock creep. These findings suggest that the model effectively captures the creep behavior of rocks and precisely represents the entire creep process.

Controllability of Impulsive Damped Fractional Order Systems Involving State Dependent Delay

In this article, the concept of controllability on fractional order impulsive systems involving state dependent delay and damping behavior is analysed by utilizing Caputo fractional derivative. The main motivation is to derive the sufficient conditions for the controllability of the considered systems. Based on the Laplace transform and inverse Laplace transform, the solution of fractional-order dynamical systems are obtained. The results are established by utilizing basic ideas of fractional calculus, Mittag-Leffler function and Banach fixed point theorem. Finally, an application is provided to illustrate the derived result.

Nonlinear dynamics of the complex periodic coupled system via a proportional generalized fractional derivative

Nonlinear dynamics of the complex periodic coupled system via a proportional generalized fractional derivative

A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equation

In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations (2D-TF-RASDE) characterized by the tempered parameter λ and a tempered fractional derivative of order α, (where 0<α<1). The novel fast tempered scheme is based on the sum of exponents (SOE) technique. Additionally, we develop a novel high order compact finite difference (CFD) scheme using an alternating direction implicit (ADI) formulation for the 2D-TF-RASDE. We investigate the stability and convergence of this FLλ2−1σ-ADI-CD scheme. Numerical simulations reveal a convergence order of O(Δt1+α+hϰ4+hy4+ϵ) under robust regularity assumptions underlining the scheme's superior computational efficiency. Furthermore, our computational results align well with theoretical analysis, demonstrating both accuracy and reduced computational complexity and storage requirements as compared to the standard tempered Lλ2−1σ-ADI-CD scheme. Notably, the fast tempered FLλ2−1σ-ADI-CD scheme exhibits competitive performance and reduced CPU time relative to the standard Lλ2−1σ-ADI-CD scheme, thereby offering a distinct advantage for efficiently solving complex fractional differential equations.

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.

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

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

On the Solutions of Nonlinear Implicit ω-Caputo Fractional Order Ordinary Differential Equations with Two-Point Fractional Derivatives and Integral Boundary Conditions in Banach Algebra

This article delves into the analysis of nonlinear implicit ω-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives and integral boundary conditions within the context of Banach algebra. The primary focus is on demonstrating the existence and uniqueness of solutions for these complex fractional differential equations by utilizing Banach's and Krasnoselskii's fixed point theorems. Furthermore, the study explores the stability of these solutions through the Ulam-Hyers and Ulam-Hyers-Rassias stability criteria, thereby assessing the robustness of the proposed model. To illustrate the versatility of the generalized model, several special cases are examined, showcasing its ability to encompass various classical models. The practical applicability of the theoretical findings is underscored through a numerical example, which demonstrates the feasibility and relevance of the proposed methodology. This thorough investigation advances the comprehension of nonlinear fractional differential equations with integral boundary conditions, highlighting the intricate relationship between fractional derivatives, nonlinearities, and integral terms. The results offer significant insights into the behavior and stability of solutions within this demanding mathematical framework.

A Special Note According to Possible Applications of Fractional-Order Calculus for Various Special Functions

The main aim of this special study is to recall certain information about fractional (arbitrary) order calculus, which has wide and fruitful applications in science and engineering. Then, it aims to consider various essential definitions related to fractional order integrals and derivatives for stating and proving some results, as well as to present some of their possible applications to the attention of related researchers.

On the Global Existence of Solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.