Order Derivatives Research Articles

Qualitative analysis of fourth-order hyperbolic equations

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

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.

Optical solitons, qualitative analysis, and chaotic behaviors to the highly dispersive nonlinear perturbation Schrödinger equation

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic‐power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell‐shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

Complete Euler deconvolution

Euler deconvolution is widely used as a semi-automatic interpretation technique for potential field data. In its original form it uses derivatives of orders 0 and 1, while second order Euler deconvolution uses derivatives of orders 0 and 2. Complete Euler deconvolution is introduced here, and this combines both methods to use derivatives of orders 0, 1, and 2. The use of the additional information in the three orders of derivatives provides increased accuracy compared to the original method, and when combined with other Euler equations it can also determine the structural index of the source. The methods are applied to both synthetic data and to two aeromagnetic data profiles from South Africa.

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.

Fractional-Order Modeling and Stochastic Dynamics Analysis of a Nonlinear Rubbing Overhung Rotor System

To study the nonlinear dynamic behavior and system stability of a rubbing overhung rotor with viscoelastic and memory-effect damping and random uncertain parameters, this paper introduces a fractional-order modeling and stochastic dynamic analysis method for the nonlinear overhung rotor system with frictional impact faults. Firstly, the dynamic equations of the overhung rotor considering friction effect and fractional damping effect are established based on the transfer matrix method and fractional order derivative. Then, the time-domain response of the fractional-order dynamic equations is solved by combining the Runge–Kutta method and the continuous fractional expansion, and the steady-state response characteristics of different fractional damping are analyzed in the deterministic case. Finally, to analyze the response of the system under the effect of stochastic parameters, the sparse grid-based PCE metamodel of the system response is developed. Statistical moments, probability distributions, and sensitivity indices of the response of stochastic systems are revealed. The results of this paper provide a theoretical basis for efficient and accurate prediction of the stochastic response of nonlinear rubbing overhung rotor systems.

Cauchy discretization method for solve fractional linear optimal control problems

This paper presents a numerical scheme designed for solving fractional linear optimal control problems (FLOCPs) governed by Caputo fractional derivatives (CFDs) of order 0<α≤1. The method transforms the original fractional control problem into an equivalent linear programming problem (LPP), enabling a more straightforward solution process. To evaluate the performance and accuracy of this approach, several numerical experiments were conducted using MATLAB. These experiments highlight the method’s computational efficiency and its simplicity in terms of implementation. The proposed approach offers accurate results, particularly in scenarios where traditional methods may fall short due to the complexities introduced by fractional derivatives. A detailed numerical example is presented to further illustrate the method’s effectiveness in addressing FLOCPs. The outcomes suggest that this approach is not only practical but also provides valuable insights for researchers dealing with fractional calculus in optimal control theory. This contribution is expected to be beneficial for applications in engineering and the physical sciences, where fractional-order systems play a significant role.

Faedo-Galerkin method for the time-fractional reaction-diffusion model

This paper investigates a time-fractional reaction-diffusion equation characterized by a Caputo derivative of fractional order , incorporates non-local and memory effects that are essential for accurately modeling complex diffusion processes in such environments. These effects arise from the fractal nature of the media, which leads to anomalous diffusion behavior that cannot be adequately described by classical diffusion models. We start by giving a definition of a weak solution for this problem, ensuring that the solution satisfies the equation in a generalized sense. Subsequently, the Faedo-Galerkin method combined with compactness arguments is employed to establish the existence and uniqueness of the weak solution. This approach involves approximating the solution using a sequence of finite-dimensional functions and then passing to the limit to obtain the desired solution . Moreover, we establish the stability of these solutions, which is fundamental for understanding their behavior under varying conditions and ensuring the reliability of the model . Our findings contribute to a deeper understanding of anomalous diffusion phenomena in fractal media and provide a solid foundation for further research in this area.

Novel method in mice to evaluate left atrial (LA) systolic contribution to left ventricular volume (LV) using pressure-volume (PV) loops

Abstract Background LA myopathy is a critically important non-LV contributor to the progression of HFpEF (heart failure with preserved ejection fraction) in patients and is associated with a severe phenotype, leading to impaired cardiac output reserve and reduced exercise capacity. In normal humans LA systole contributes ~20% of total LV filling, but with HFpEF, this contribution decreases to <16% with a greater reliance on this contribution to LV filling. Purpose We sought to investigate LA function as a critical component of cardiac function analysis by utilizing the LV pressure signal to identify and quantify the contribution of volume during LA systole to the total LV volume in mice with HFpEF. Methods Hypertension (HTN)-associated HFpEF C57BL/6J mice were subjected to chronic aldosterone infusion and 1% NaCl for 4 weeks (SAUNA model). Obesity after a high fat diet (HFD: 60% fat diet) for 16 weeks followed by the SAUNA protocol, induced HTN and obesity associated HFpEF. Blood pressure was monitored weekly. Mice were anesthetized with isoflurane, intubated, chest opened to expose the heart and allow placement of a PV catheter (Transonic) which was introduced into the LV via apical stab with a 27G needle. Calibration of the PV catheter was performed according to the manufacturer’s instructions. Atrial systole was identified using 2nd order derivative of the LV pressure signal. Contribution of LA systole was calculated using volume at the beginning of LA systole identified by the 2nd order derivative of the LV pressure signal and subtracting from the LV end diastolic volume. The % of LA systole contribution was calculated as a % of the total LV stroke volume. Data was analyzed with LabChart pro software (AD Instruments). Results SAUNA mice had normal LVEF, moderate HTN and lung congestion. PV relationships demonstrated Tau was impaired in HFpEF vs. control mice (14±1 vs. 10±0.5 ms) with a shift of the PV loop to the left (Fig. A.) In control mice, the atrial systole contributed 39± 4% to LV volume and was impaired in HFpEF mice 20± 3% a 1.95-fold reduction (P<0.01). Importantly atrial systole contribution was increased in obese mice 53±5 %. However, the combination of obesity and HTN, both common comorbidities in HFpEF, resulted in a reduced LA systole contribution by 1.43-fold to 37± 9%, (P<0.005; Fig B). Conclusion A novel method of % atrial systole from LV PV loop using 2nd order derivative to quantify the volume contribution of atrial systole to total LV stroke volume was identified in mice. LA dysfunction was present in murine HTN-associated HFpEF with/without obesity.

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

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

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

Retraction Note: A study on the reflection of quasi Rayleigh waves in a magneto self reinforced thermoelastic medium with fractional order derivative

Retraction Note: A study on the reflection of quasi Rayleigh waves in a magneto self reinforced thermoelastic medium with fractional order derivative

The Orthogonal Riesz Fractional Derivative

The aim of this paper is to extend the concept of the orthogonal derivative to provide a new integral representation of the fractional Riesz derivative. Specifically, we investigate the orthogonal derivative associated with Gegenbauer polynomials Cn(ν)(x), where ν&gt;−12. Building on the work of Diekema and Koornwinder, the n-th derivative is obtained as the limit of an integral involving Gegenbauer polynomials as the kernel. When this limit is omitted, it results in the approximate Gegenbauer orthogonal derivative, which serves as an effective approximation of the n-th order derivative. Using this operator, we introduce a novel extension of the fractional Riesz derivative, denoted as Dαx, providing an alternative framework for fractional calculus.

On a fractional Cauchy problem with singular initial data

Abstract This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: D α x ( t ) = F ( t , x ( t ) ) x ( 0 ) = x 0 , \left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where x 0 {x}_{0} is the singular generalized function and F satisfies L ∞ {L}^{\infty } logarithmic type, D α {D}^{\alpha } is the Caputo derivative of order m − 1 &lt; α &lt; m m-1\lt \alpha \lt m with m ∈ N * m\in {{\mathbb{N}}}^{* } , which we will confirm to be present in Colombeau algebra. The Gronwall lemma is used in Colombeau’s algebra to establish the main results. To illustrate our theoretical analysis, we ended our work with an example.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

Comprehending symmetry in epidemiology: A review of analytical methods and insights from models of COVID-19, Ebola, Dengue, and Monkeypox.

The challenge of developing comprehensive mathematical models for guiding public health initiatives in disease control is varied. Creating complex models is essential to understanding the mechanics of the spread of infectious diseases. We reviewed papers that synthesized various mathematical models and analytical methods applied in epidemiological studies with a focus on infectious diseases such as Severe Acute Respiratory Syndrome Coronavirus-2, Ebola, Dengue, and Monkeypox. We address past shortcomings, including difficulties in simulating population growth, treatment efficacy and data collection dependability. We recently came up with highly specific and cost-effective diagnostic techniques for early virus detection. This research includes stability analysis, geographical modeling, fractional calculus, new techniques, and validated solvers such as validating solver for parametric ordinary differential equation. The study examines the consequences of different models, equilibrium points, and stability through a thorough qualitative analysis, highlighting the reliability of fractional order derivatives in representing the dynamics of infectious diseases. Unlike standard integer-order approaches, fractional calculus captures the memory and hereditary aspects of disease processes, resulting in a more complex and realistic representation of disease dynamics. This study underlines the impact of public health measures and the critical importance of spatial modeling in detecting transmission zones and informing targeted interventions. The results highlight the need for ongoing financing for research, especially beyond the coronavirus, and address the difficulties in converting analytically complicated findings into practical public health recommendations. Overall, this review emphasizes that further research and innovation in these areas are crucial for addressing ongoing and future public health challenges.

Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations

In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

A Fractional Order Derivative Newton-Raphson Method for the Computation of the Power Flow Problem Solution in Energy Systems

A Fractional Order Derivative Newton-Raphson Method for the Computation of the Power Flow Problem Solution in Energy Systems

Calculation of machine precision second order derivatives using dual-complex numbers

Calculation of machine precision second order derivatives using dual-complex numbers

Acoustic and spectroscopic characterization of glycol ethers in vitamin B7 over the range of temperatures (288.15, 293.15, 298.15, 303.15) K

Density (ρ) and ultrasonic speed (u) of butoxyethanol and phenoxyethanol in aqueous solution of biotin concentrations (0.0012, 0.0017, 0.0022) mol·kg−1 at varying temperature range (288.15, 293.15, 298.15 and 303.15) K at 0.1 MPa were measured using Anton Paar DSA 5000 M. The volumetric and acoustical properties such as apparent molar properties (V∅), partial molar volume (V∅0),partial molar volume of transfer (ΔV∅0),apparent molar isentropic compression (K∅),partial molar isentropic compression (K∅0),partial molar isentropic compression of transfer are computed using the experimental data of density (ρ) and ultrasonic speed (u). This investigation is useful to understand the solvation behavior of biotin. Additionally, partial molar expansibilities (E∅0) along with their first order derivatives (∂E∅0/∂T)pare determined using the obtained experimental data. These results are effective in understanding the nature of interaction occurring inside the solution. Further, pair and triplet coefficients are evaluated which determines the structure making and breaking ability of glycol ethers in aqueous solution of biotin. Also, spectroscopic study of solutions is mentioned in current research which reveals the nature and strength of bond formation and various interactions in the mixture. The present study suggests the molecular interaction between the solute and solvent present in aqueous solution of Biotin (Vitamin B7) as solvent, butoxyethanol (BE) and phenoxyethanol (PE) as solute.