Order Derivatives Research Articles

Exploring Intrinsic Bond Properties with the Fukui Matrix from Conceptual Density Matrix Functional Theory.

We extend the traditional conceptual density functional theory (CDFT) to conceptual density matrix functional theory (CDMFT) by replacing the external potential v(r) by the one-electron integral hrs in the energy functional. This approach provides a new path for investigating intrinsic bond properties such as bond reactivity. The derivation of the Fukui matrix, i.e., derivative of the density matrix P with respect to the number of electrons N, is elucidated, and the result is illustrated in a case study on H2O. The matrix is shown to play a crucial role in quantifying changes of bond strength for electron removal or addition processes via the bond order derivative . Using the Mayer bond order and different atoms-in-molecules partitioning methods, we show that as a first-order response quantity, the bond order derivative agrees well with the finite difference bond order changes. The bond order derivative (bond Fukui function) is a bond reactivity descriptor. We demonstrate this by predicting the regioselectivity of a classical electrophilic addition reaction (the bromination of alkenes) and predicting the initial electron-driven bond cleavage in mass spectrometry. Specifically, the bond order derivative captures all of the major signals from the experimental mass spectra for a series of small molecules with a variety of functional groups.

Testing and simulation of the hysteresis characteristics of magnetostrictive materials under harmonic excitation

As various power electronic devices are widely used in the power grid, the current flowing through the drive winding of the magnetostrictive actuator inevitably contains harmonic components. The flux density within the hysteresis material is non-sinusoidal with harmonic components, which distorts the hysteresis characteristics of the material and affects the accurate calculation of losses. This paper builds a harmonic magnetic performance testing platform for magnetostrictive materials to test and analyze the magnetic properties under different excitation conditions (magnetic density amplitude, harmonic order, and harmonic content). To complete the mathematical description of the static hysteresis model containing closed minor hysteresis loops, two pinning coefficients k1(Bn,kTHD) and k2(Bn,kTHD), which are related to the degree of harmonic distortion, are used to express the rate of change of the irreversible magnetization components. Fractional order derivative and harmonic frequency-dependent terms are used to characterize the eddy losses and excess losses mathematically. Finally, the dynamic hysteresis model under harmonic excitation is developed. Comparative analysis of model calculation results and experimental test data shows that the model can not only accurately simulate the distorted hysteresis characteristics but also predict the loss of magnetostrictive materials. This research is of significance to the optimal design and temperature rise analysis of magnetostrictive devices.

A new epidemic model of sexually transmittable diseases: a fractional numerical approach

This study aims at investigating the dynamics of sexually transmitted infectious disease (STID), which is serious health concern. In so doing, the integer order STID model is progressed in to the time-delayed non-integer order STID model by introducing the Caputo fractional derivatives in place of integer order derivatives and including the delay factors in the susceptible and infectious compartments. Moreover, unique existence of the solution for the underlying model is ensured by establishing some benchmark results. Likewise, the positivity and boundedness of the solutions for the projected model is explored. The basic reproduction number is is found out for the model. The time-delayed non-integer order STID model holds two steady states, namely, the STID free and endemic steady state. The model stability is carried out at the steady states. The non-standard finite difference (NSFD) technique is hybridized with the Grunwald Letnikov (GL) approximation for finding the numerical solutions of the time-delayed non-integer order STID model. The boundedness and non-negativity of the numerical scheme is confirmed. The simulated graphs are presented with the help of an appropriate test example. These graphs show that the proposed numerical algorithm provides the positive bounded solutions. The article is ended with productive outcomes of the study.

Approximation properties relative to continuous scale space for hybrid discretisations of Gaussian derivative operators

This paper presents an analysis of properties of two hybrid discretisation methods for Gaussian derivatives, based on convolutions with either the normalised sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretisation methods is that in situations when multiple spatial derivatives of different orders are needed at the same scale level, they can be computed significantly more efficiently, compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian derivative kernels or integrated Gaussian derivative kernels. We characterise the properties of these hybrid discretisation methods in terms of quantitative performance measures, concerning the amount of spatial smoothing that they imply, as well as the relative consistency of the scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretisation methods. The presented results are intended as a guide, when designing as well as interpreting the experimental results of scale-space algorithms that operate at very fine scale levels.

Elite paradox

The paradoxical nature of the existence and functioning of power elites is not a new fact. However, it has been the object of few studies. This is due to the ambiguity of the term and of the phenomenon itself,which is described under the name “elite”. This applies equally to the “paradox”. The author, based on the principles of institutional analysis and starting from the theory of structuration and the theory of paradox in its managerial version, studies both concepts and offers an approach to the analysis of the elite paradox. Firstly, the contradictory position of the power elites in the societal system is revealed: being a subsystem, they regulate and direct the functioning of the entire system. Secondly, the multidirectional nature of the system requirements for the elites is studied. Thirdly, the discrepancy perceived by the observer between the expected characteristics and functions (based on ideas about the role of elites in the social system) and those that are actually recorded are considered. These variants represent the modes of the basic structural paradox of the part and the whole. Elites are derivatives of the social order, and they are also active agents in its formation and consolidation of the principles of its functioning. At the same time, elites are forced to go beyond the framework of the normative system they themselves protect. Otherwise, their main function - the stabilization of the system - will not be realized. The poly-role nature of elite individuals and their control of resources can lead to the privatization of power institutions and the realization of group and personal interests to the detriment of public ones. In modern society, the forced openness of the elite leads to a paradox of public intimacy. Elites are doomed to a transborder existence and ambivalent characteristics, primarily moral ones.

Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM

This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integrodifferential equations (FracVF-IDEs). Other than that, the existance and uniqueness of solution as well as the norm convergence with respect to ND-HAM, were proven in a Hilbert space. In addition, three numerical examples (including multi-term fractional IDEs) are presented and compared with the HAM, modified HAM and ”Generalized block pulse operational differentiation matrices method” developed in previous works by illustrating the accuracy as well as validity with respect to ND-HAM. Empirical investigations reveal that ND-HAM and the modified HAM yields the same results when control parameter ℏ is chosen as ℏ = −1 and is comparable to the standard HAM. The findings discovered that the ND-HAM is highly convenient, effective, as well as in line with theoretical results.

New results about controllability and stability of ψ-Caputo-type stochastic fractional integro-differential systems with control delay

Abstract This study explores the controllability and stability criteria for the fractional integro-differential stochastical systems with control delay employing the Ψ-Caputo-type fractional derivative (Ψ-CTFD) of order σ ∈ (0, 1). The necessary and sufficient conditions for the controllability of linear stochastical systems are obtained by utilizing the positive definiteness of the Grammian matrix. The sufficient conditions for the controllability criteria of semilinear and integro-differential stochastical systems are derived by employing Banach’s contraction principle. The sufficient conditions for the stability of integro-differential stochastical systems are derived from the Hyer’s-Ulam stability criteria. Additionally, a few examples and appropriate graphical representations are included in the work to help understand the theoretical conclusions.

Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations

In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations.

Solvability of Boundary Value Problems for Differential Equations Combining Ordinary and Fractional Derivatives of Non-Autonomous Variable Order

This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecewise constant functions, we propose a new fractional operator better suited for this problem. We analyze the existence and uniqueness of solutions, establishing the conditions necessary for these properties to hold using the Krasnoselskii fixed-point theorem and Banach’s contraction principle. Our study also addresses the Ulam–Hyers stability of the proposed problems, examining how variations in boundary conditions influence the solution dynamics. To support our theoretical framework, we provide numerical examples that not only validate our findings but also demonstrate the practical applicability of these mixed derivative equations across various scientific domains. Additionally, concepts such as symmetry may offer further insights into the behavior of solutions. This research contributes to a deeper understanding of complex differential equations and their implications in real-world scenarios.

Construction of the Closed Form Wave Solutions for TFSMCH and (1 1) Dimensional TFDMBBM Equations via the EMSE Technique+

The purpose of this study is to investigate a series of novel exact closed form traveling wave solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE technique. The considered FONLEEs are used to delineate the characteristic of diffusion in the creation of shapes in liquid beads arising in plasma physics and fluid flow and to estimate the external long waves in nonlinear dispersive media. These equations are also used to characterize various types of waves, such as hydromagnetic waves, acoustic waves, and acoustic gravity waves. Here, we utilize the Caputo-type fractional order derivative to fractionalize the considered FONLEEs. Some trigonometric and hyperbolic trigonometric functions have been used to represent the obtained closed form traveling wave solutions. Furthermore, here, we reveal that the EMSE technique is a suitable, significant, and dominant mathematical tool for finding the exact traveling wave solutions for various FONLEEs in mathematical physics. We draw some 3D, 2D, and contour plots to describe the various wave behaviors and analyze the physical consequence of the attained solutions. Finally, we make a numerical comparison of our obtained solutions and other analogous solutions obtained using various techniques.

Sub-pico-second chirped optical solitons in birefringent fibers for space--time fractional Kaup-Newell equation

The present work is devoted to investigate the chirped bright and dark optical solitons of fractional Kaup-Newell equation (KNE) in birefringent fibers. The study is carried out analytically by the traveling wave hypothesis with the conformable derivative which reduces the governing model to an ordinary differential equation (ODE). The obtained equation is handled with the aid of an exotic integration scheme that utilizes the Jacobi elliptic equation in the form of a first-order nonlinear ODE with three-degree terms. Taking the modulus of Jacobi elliptic function to unity, distinct types of bright and dark optical solitons are derived with their corresponding chirping. The fractional order derivative is noted to have a significant influence on the pulse propagation. Additionally, the nonlinearity amount causes also marked variations in the amplitude and width of solitons. The modulation instability of the KNE is reported by implementing the linear stability analysis which confirms that all solutions are stable. The revealed results can be capitalized in improving the relevant physical and engineering applications in the field of birefringent fiber.

Regularity Results for Hybrid Proportional Operators on Hölder Spaces

Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems.

Stability analysis of nonlinear Riemann-Liouville fractional differential equations

In this paper, we give sufficient conditions to guarantee the asymptotic stability of the zero solution to a kind of nonlinear fractional differential equations with the Riemann Liouville fractional derivative of order α∈(n-1,n) by using Krasnoselskii's fixed point theorem and the Banach contraction mapping principle in a weighted Banach space. The results obtained here extend the work of Li and Kou [6].

Estimation of Radon Flux Density Changes in Temporal Vicinity of the Shipunskoe Earthquake with Mw = 7.0, 17 August 2024 with the Use of the Hereditary Mathematical Model

Using the data of radon accumulation in a chamber with excess volume at one of the points of the Kamchatka subsurface gas-monitoring network, the change in radon flux density due to seismic waves and post-seismic relaxation of the medium is shown. A linear fractional equation is considered to be a model equation. The change of radon-transport intensity due to changes in the state of the geo-environment is described by a fractional Gerasimov–Caputo derivative of constant order. Presumably, the order of the fractional derivative is related to the radon-transport intensity in the geosphere. Using the Levenberg–Marquardt method, the optimal values of the model parameters were determined based on experimental data: air exchange coefficient and order of fractional derivative, which allowed the solving of the problems of radon flux density determination. Data in the temporal neighborhood of a strong earthquake with Mw=7.0, which occurred in the northern part of Avacha Bay on 17 August 2024, were used. As a result of the modeling, it is shown that the strong seismic impact and subsequent processes led to changes in the radon flux in the accumulation chamber. The obtained model curves agree well with the real data, and the obtained estimates of radon flux density agree with the theory.

The Efficacy of Haar Wavelets in Addressing Discontinuities of McKean Equations with Heaviside Functions

This study explores the application of Haar wavelets to solve the McKean equation, a reaction-diffusion equation with discontinuous Heaviside step function. Haar wavelets, with their compact support and orthogonality, offer straightforward but yet powerful tool for addressing the equation’s nonlinear dynamics. We focus on the time-independent solution of the McKean equation, which is crucial for understanding the threshold phenomenon that determines the system’s behavior. Despite the existence of analytical time-independent solution to the McKean equation, achieving such solutions in closed form is uncommon for more complicated systems, highlighting the utility of the Haar wavelet approach. The proposed method integrates the Haar series expansion of the highest order derivative, enabling systematic solution derivation. Through a detailed comparison with analytical solution, we validate the Haar wavelet approach as a robust and computationally feasible tool for solving complex reaction-diffusion system. The results also demonstrate the method’s accuracy and efficiency, offering insights into its broader applicability to more complex reaction-diffusion system, especially those with discontinuity and sharp transitions.

New approaches to numerical differentiation for second and third order

In this article, new numerical methods for calculation of second and third order derivatives are designed by using basic finite difference methods; forward, central and backward finite difference approaches. Those approaches are originally derived from the well-known Taylor series. Main advantage of new numerical formulas (named as Improved Backward Finite Difference Method, Improved Forward Finite Difference Method) is that they produce more accurate numerical results with smaller step size than the well-known backward and forward finite difference methods. For this purpose, some numerical examples are given to compare these new formulas with the traditional finite difference methods; backward and forward. The performance of the new methods in terms of error analysis and elapsed time for both second and third order derivative computations is also presented.

Asymptotic stability of fractional order switching nonlinear system based on short memory principle

AbstractFractional order derivatives have memory effects and are widely used in real world applications. However, they require large storage space and lead to low computational efficiency. Therefore, fractional order systems based on the short memory principle have gradually attracted the scholars' attention. In this paper, the asymptotic stability of Caputo fractional order switching nonlinear systems is investigated based on the Markov process and short memory principle. Firstly, a model of Caputo fractional order Markovian switching nonlinear systems (CFMNSs) based on the short memory principle is constructed so that the lower bound initial time and the corresponding initial state values are updated synchronously with switching. Secondly, the stability of the system is investigated based on the probabilistic analysis method and stochastic multi‐Lyapunov functions and the sufficient conditions for the asymptotic stability of the system are given. Using a similar method, we also study the asymptotic stability of CFMNs with variable fractional order. Finally, the simulation results show that the proposed stability scheme is effective and reasonable.

PARALLEL PERFORMANCE OF THE EXPLICIT FDTD METHODS APPLIED ON 3D ACOUSTIC WAVE EQUATION

The 3D acoustic wave equation is a second-order linear hyperbolic partial differential equation. It is widely studied in acoustics, fluid dynamics, electromagnetism, and seismology within both science and engineering fields. Among the various numerical methods, the finite difference method is considered to be the simpler to understand and easy to implement. Based on the discretize schemes finite difference time domain method described by the explicit scheme in which spatial second order derivatives are evaluated at the previous time step. The solution of 3D acoustic wave equation may be required on the high-resolution mesh points consequently more computational resources are required. In this study parallel algorithm for the numerical solution of 3D acoustic wave equation is proposed and designed by using data parallel approach and message passing schemes. The algorithm is implemented on shared memory parallel system using MATLAB parallel computing mode. The parallel performance of the designed algorithm is analyzed on different mesh sizes and time steps. It is revealed that the proposed algorithm may reduce computational time up to 3 times as compared to sequential solution algorithm. The proposed parallel algorithm remains more efficient on workers while for the efficiency of the algorithm drops because of the high communication time. The results of the proposed research could be utilized in the large-scale simulations of 3D acoustic wave equations and may enable to simulate the acoustic wave pressure at more refined meshes on high performance computing systems.

Expectation formulas for $ q $-probability distributions: a new extension via Andrews-Askey integral

<p>In this paper, we utilize the $ q $-Chu-Vandermonde formula to derive a novel expectation formula for the $ q $-probability distribution $ W(x, y; q) $, extending previously known results. Several applications are presented, including a broader generalization of the Andrews-Askey integral. Although fractional $ q $-calculus is not directly employed in this work, its potential for future extensions is discussed, as non-integer order derivatives and integrals could offer deeper insights into $ q $-series and probability distributions.</p>

Stability analysis and optimal control of tumour-immune interaction problem using fractional order derivative

Stability analysis and optimal control of tumour-immune interaction problem using fractional order derivative