Derivatives Of Arbitrary Order Research Articles

Finite Element Method applied to the Fractional Partial Derivative Equation of Anomalous Diffusion

Diffusion, a fundamental phenomenon in nature, operates on both microscopic and macroscopic scales. At the microscopic level, it manifests itself as a stochastic process, while at the macroscopic level it represents a uniform drift toward equilibrium. The Fokker-Planck equation is widely used to model diffusion phenomena in various disciplines, including physics, chemistry, biology, and others. The classical Fick’s law cannot correctly represent the movement of species through inhomogeneous materials, such as porous media. This phenomenon is known as non-Fickian or anomalous diffusion and presents behavior that is difficult to analyze and require computationally intensive simulations. To address this, recent studies have explored modeling anomalous diffusion using fractional derivatives in time or space. Fractional calculus, a branch of classical calculus, provides a framework for handling integrals and derivatives of arbitrary order. Its application has proven to be particularly effective in systems that exhibit hysteresis, allowing the computation of associated memory effects. One example is the modeling of anomalous diffusion in polymeric coatings used to protect flexible pipelines in subsea oil exploration. Under such adverse conditions, extreme depths and temperatures, the properties of the polymer change over time, affecting the ingress of corrosive ions into the metal structure. It left unchecked, this threatens operational safety and environmental integrity. In this work, the Finite Element Method (FEM) is applied to anomalous diffusion described by Fractional Partial Derivative Equations (FPDEs), seeking to predict these intricate transport dynamics.

On general tempered fractional calculus with Luchko kernels

In this paper, we construct the n-fold ψ-fractional integrals and derivatives and study their properties. This construction is purely based on the approach proposed by Luchko (2021). The fundamental theorems of fractional calculus are formulated and proved for the proposed n-fold ψ-fractional integrals and derivatives. On the other hand, a suitable generalization of the Luchko condition is presented to discuss the ψ-tempered fractional calculus of arbitrary order. We introduce an important class of kernels that satisfy this condition. For the ψ-tempered fractional integrals and derivatives of arbitrary order, two fundamental theorems are proven, along with a relation between Riemann–Liouville and Caputo derivatives. Finally, Cauchy problems for the fractional differential equations with the ψ-tempered fractional derivatives are solved.

Automatic Differentiation for Explicitly Correlated MP2.

Automatic differentiation (AD) offers a route to achieve arbitrary-order derivatives of challenging wave function methods without the use of analytic gradients or response theory. Currently, AD has been predominantly used in methods where first- and/or second-order derivatives are available, but it has not been applied to methods lacking available derivatives. The most robust approximation of explicitly correlated MP2, MP2-F12/3C(FIX)+CABS, is one such method. By comparing the results of MP2-F12 computed with AD versus finite-differences, it is shown that (a) optimized geometries match to about 10-3 Å for bond lengths and a 10-6 degree for angles, and (b) dipole moments match to about 10-6 D. Hessians were observed to have poorer agreement with numerical results (10-5), which is attributed to deficiencies in AD implementations currently. However, it is notable that vibrational frequencies match within 10-2 cm-1. The use of AD also allowed the prediction of MP2-F12/3C(FIX)+CABS IR intensities for the first time.

Complete topological asymptotic expansion for L2 and H1 tracking-type cost functionals in dimension two and three

In this paper, we study the topological asymptotic expansion of a topology optimisation problem that is constrained by the Poisson equation with the design/shape variable entering through the right hand side. Using an averaged adjoint approach, we give explicit formulas for topological derivatives of arbitrary order for both an L2 and H1 tracking-type cost function in both dimension two and three and thereby derive the complete asymptotic expansion. As the asymptotic behaviour of the fundamental solution of the Laplacian differs in dimension two and three, also the derivation of the topological expansion significantly differs in dimension two and three. The complete expansion for the H1 cost functional directly follows from the analysis of the variation of the state equation. However, the proof of the asymptotics of the L2 tracking-type cost functional is significantly more involved and, surprisingly, the asymptotic behaviour of the bi-harmonic equation plays a crucial role in our proof.

Dxtb-An efficient and fully differentiable framework for extended tight-binding.

Automatic differentiation (AD) emerged as an integral part of machine learning, accelerating model development by enabling gradient-based optimization without explicit analytical derivatives. Recently, the benefits of AD and computing arbitrary-order derivatives with respect to any variable were also recognized in the field of quantum chemistry. In this work, we present dxtb-an open-source, fully differentiable framework for semiempirical extended tight-binding (xTB) methods. Developed entirely in Python and leveraging PyTorch for array operations, dxtb facilitates extensibility and rapid prototyping while maintaining computational efficiency. Through comprehensive code vectorization and optimization, we essentially reach the speed of compiled xTB programs for high-throughput calculations of small molecules. The excellent performance also scales to large systems, and batch operability yields additional benefits for execution on parallel hardware. In particular, energy evaluations are on par with existing programs, whereas the speed of automatically differentiated nuclear derivatives is only 2 to 5 times slower compared to their analytical counterparts. We showcase the utility of AD in dxtb by calculating various molecular and spectroscopic properties, highlighting its capacity to enhance and simplify such evaluations. Furthermore, the framework streamlines optimization tasks and offers seamless integration of semiempirical quantum chemistry in machine learning, paving the way for physics-inspired end-to-end differentiable models. Ultimately, dxtb aims to further advance the capabilities of semiempirical methods, providing an extensible foundation for future developments and hybrid machine learning applications. The framework is accessible at https://github.com/grimme-lab/dxtb.

GBasis: A Python library for evaluating functions, functionals, and integrals expressed with Gaussian basis functions.

GBasis is a free and open-source Python library for molecular property computations based on Gaussian basis functions in quantum chemistry. Specifically, GBasis allows one to evaluate functions expanded in Gaussian basis functions (including molecular orbitals, electron density, and reduced density matrices) and to compute functionals of Gaussian basis functions (overlap integrals, one-electron integrals, and two-electron integrals). Unique features of GBasis include supporting evaluation and analytical integration of arbitrary-order derivatives of the density (matrices), computation of a broad range of (screened) Coulomb interactions, and evaluation of overlap integrals of arbitrary numbers of Gaussians in arbitrarily high dimensions. For circumstances where the flexibility of GBasis is less important than high performance, a seamless Python interface to the Libcint C package is provided. GBasis is designed to be easy to use, maintain, and extend following many standards of sustainable software development, including code-quality assurance through continuous integration protocols, extensive testing, comprehensive documentation, up-to-date package management, and continuous delivery. This article marks the official release of the GBasis library, outlining its features, examples, and development.

Analysis of COVID-19 epidemic with intervention impacts by a fractional operator

This study introduces an innovative fractional methodology for analyzing the dynamics of COVID-19 outbreak, examining the impact of intervention strategies like lockdown, quarantine, and isolation on disease transmission. The analysis incorporates the Caputo fractional derivative to grasp long-term memory effects and non-local behavior in the advancement of the infection. Emphasis is placed on assessing the boundedness and non-negativity of the solutions. Additionally, the Lipschitz and Banach contraction theorem are utilized to validate the existence and uniqueness of the solution. We determine the basic reproduction number associated with the model utilizing the next generation matrix technique. Subsequently, by employing the normalized sensitivity index, we perform a sensitivity analysis of the basic reproduction number to effectively identify the controlling parameters of the model. To validate our theoretical findings, numerical simulations are conducted for various fractional order values, utilizing a two-step Lagrange interpolation technique. Furthermore, the numerical algorithms of the model are represented graphically to illustrate the effectiveness of the proposed methodology and to analyze the effect of arbitrary order derivatives on disease dynamics.

Decay of higher order derivatives for Lp solutions to the compressible fluid model of Korteweg type

As a follow-up study of [31], we present a derivation of upper bounds for the decay of arbitrary higher order derivatives of solutions to the compressible Navier-Stokes-Korteweg system in the Lp critical Besov space. The approach, based on Gevrey regularizing estimates, is to establish uniform bounds on the growth of the radius of analyticity of the solution in negative Besov norms, which may be applicable to a wide range of dissipative systems.

A Novel Attitude Control Strategy for a Quadrotor Drone with Actuator Dynamics Based on a High-Order Sliding Mode Disturbance Observer

In the attitude control of quadrotor drones, it is necessary to cope with matched and unmatched disturbances caused by nonlinear couplings, model uncertainties, and external disturbances, as well as to consider the effects caused by actuator dynamics. Aiming to accurately track desired trajectories under the above factors, a novel control strategy is proposed by combining a state feedback control with a high-order sliding mode disturbance observer (HOSMDO). The HOSMDO is motivated by the higher-order sliding mode (HOSM) differentiator and extended state observer (ESO) technique, allowing for the exact robust estimation of disturbances and their arbitrary order derivatives in finite time. Unlike the control schemes based on back-stepping methods, the proposed controller is designed with a holistic mindset. Specifically, a baseline feedback framework is constructed firstly, and the disturbances and relevant derivatives required for the baseline framework are then generated by the HOSMDOs to obtain the overall control scheme. The stability conditions of the controllers designed with and without considering the actuator dynamics are analyzed separately. In the latter case, the actuator dynamics imposed additional constraints on the control parameters. Numerical simulations validate the effectiveness of the proposed control strategy.

A RELIABLE NUMERICAL ALGORITHM FOR TREATMENT OF FRACTIONAL MODEL OF CONVECTIVE STRAIGHT FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY

Fins serve the purpose of enhancing heat transfer between their surroundings and the underlying surface, resembling the function of heat exchangers in various scientific domains. This research undertakes an investigation into the temperature distribution within convective straight fins, considering the impact of both constant and thermally conductive temperature-dependent properties. The chosen approach utilizes a numerical technique known as the Laguerre operational matrix collocation method, which relies on the Laguerre operational matrix and Laguerre collocation method for Caputo fractional derivatives of arbitrary orders. This methodology effectively converts the issue into a set of nonlinear algebraic equations (NLAEs). To evaluate this arbitrary order model, we employed the collocation method to obtain an approximate solution. The numerical results obtained through this approach demonstrate its ease of application and computational efficiency. In addition, we performed a comparative analysis with the Homotopy Perturbation Sumudu Transform Method (HPSTM) and the Variational Iteration Method (VIM). The numerical and graphical presentation of the results underscores the efficacy and precision of the methodologies employed within the scope of this investigation.

General Fractional Calculus Operators with the Sonin kernels and Some of Their Applications

In this survey paper, the general fractional integrals and derivatives with the Sonin kernels, the regularized general fractional derivatives, the sequential generalized fractional derivatives as well as some of their applications and the fractional differential equations with these derivatives are discussed. We start with a short summary of different kinds of the general Fractional Calculus operators with the Sonin kernels and then proceed with a presentation of some recent results including the fundamental theorems of Fractional Calculus for the general fractional derivatives of arbitrary order, for the regularized general fractional derivatives of arbitrary order, and for the sequential general fractional derivatives. As an application of these results, the generalized convolution Taylor formulas and the generalized convolution Taylor series with the coefficients and remainders in terms of the general fractional derivatives and the regularized general fractional derivatives are presented. Finally, we discuss a method for construction of solutions to the fractional differential equations with the general fractional derivatives in terms of the so called convolution series.

Computational analysis of fractional Michaelis-Menten enzymatic reaction model

<abstract><p>In this study for examining the fractional Michaelis-Menten enzymatic reaction (FMMER) model, we suggested a computational method by using an operational matrix of Jacobi polynomials (JPs) as its foundation. We obtain an operational matrix for the arbitrary order derivative in the Caputo sense. The fractional differential equations (FDEs) are then reduced to a set of algebraic equations by using attained operational matrix and the collocation method. The approach which utilized in this study is quicker and more effective compared to other schemes. We also compared the suggested method with the Vieta-Lukas collocation technique (VLCM) and we obtain excellent results. A comparison between numerical outcomes is shown by figures and tables. Error analysis of the recommended methods is also presented.</p></abstract>

Novel solitary wave solutions to the fractional new (3+1)-dimensional Mikhailov–Novikov–Wang equation

This paper addresses the new [Formula: see text]-dimensional Mikhailov–Novikov–Wang (MNW) equation with arbitrary order derivative and presents novel exact solutions of it by implementing [Formula: see text]-expansion, modified Kudryashov, generalized [Formula: see text]-expansion, and modified extended tanh-function methods. This equation emphasizes significant connection between the integrability and water waves’ phenomena. Employing the conformable derivative definition, a variety of soliton (bright, dark, anti-kink) solutions of the model are obtained. Therefore, it would appear that these approaches might yield noteworthy results in producing the exact solutions to the fractional differential equations in a wide range. In addition, 2D, 3D, and contour plots of the solutions are drawn for specific values to demonstrate the physical behaviors of the solutions.

Sharp unifying generalizations of Opial’s inequality

Opial’s inequality and its ramifications play an important role in the theory of differential and difference equations. A sharp unifying generalization of Opial’s inequality is presented that contains both its continuous and discrete versions. This generalization, based on distribution functions, is extended to the case of derivatives of arbitrary order. This extension optimizes and improves the constant given in the literature. The special case of derivatives of second order is studied in more detail. Two closely related Opial inequalities with a weight function are presented as well. The associated Wirtinger inequality is studied briefly.

Fractional order mathematical modeling of lumpy skin disease

In this article, we study the fractional-order SEIR mathematical model of Lumpy Skin Disease (LSD) in the sense of Caputo. The existence, uniqueness, non-negativity and boundedness of the solutions are established using fixed point theory. Using a next-generation matrix, the reproduction number $R_{0}$ is determined for the disease’s prognosis and durability. Using the fractional Routh-Hurwitz stability criterion, the evolving behaviour of the equilibria is investigated. Generalized Adams–Bashforth–Moulton approach is applied to arrive at the solution of the proposed model. Furthermore, to visualise the efficiency of our theoretical conclusions and to track the impact of arbitrary-order derivative, numerical simulations of the model and their graphical presentations are carried out using MATLAB(R2021a).

ADER discontinuous Galerkin Material Point Method

AbstractThe first‐order accurate discontinuous Galerkin Material Point Method (DGMPM), initially introduced by Renaud et al. (J Comput Phys. 2018;369:80–102.), considers a solid body discretized by a collection of material points carrying the history of the matter, embedded in an arbitrary grid on which a nodal discontinuous Galerkin approximation is defined, and that serves to solve balance equations. This method has been shown to be promising, especially for solving hyperbolic problems in finite deforming solids (Renaud et al. Int J Numer Methods Eng. 2020;121(4):664–689. Renaud et al. Comput Methods Appl Mech Eng. 2020;365:112987.). The main goal of this research is to extend the first‐order DGMPM to arbitrary high‐order accurate approximations. This is performed by adapting the ADER (Arbitrary high order DErivative Riemann problem) approach (Busto et al. Front Phys. 2020;8:32.) to the particular spatial discretization of the DGMPM. First, the predictor step permits to design a particle‐to‐grid projection of arbitrary high order of accuracy, consistent with that of the nodal discontinuous Galerkin approximation defined on the arbitrary grid. This is performed using a moving least square approximation for the ADER predictor field. Second, since the degrees of freedom of the predictor field are now defined at material points, the computation of the constitutive response of the material is ensured to be always performed at these material points. This is of crucial importance for history‐dependent constitutive models because it avoids any diffusive transfer of internal variables on a new computational grid. Finally, a total Lagrangian formulation of equations is kept, which allows to precompute once and for all both the nodal discontinuous Galerkin approximation and that of the ADER predictor field, until the arbitrary grid is discarded if required. The method is illustrated on a few two‐dimensional numerical examples, on which comparisons are shown with the ADER‐DGFEM and Runge–Kutta‐DGFEM.

Optimally analyzed fractional Coronavirus model with Atangana–Baleanu derivative

In this article, we develop a nonlinear SEIQHR fractional model with Atangana–Baleanu (ABC) derivative for the Corona virus disease (Covid-19). It is significant to mention that using a fractional-order derivative can provide more intricate insights into the complex dynamics of underlying models. We provide two unique equilibrium states of the model in order to analyze the problem. To explore the long-term dynamics of a disease, a threshold parameter for the model utilizing next-generation approach is computed. Local and global asymptotic behaviors of the proposed model at both the equilibrium states are established by imposing some necessary conditions on threshold parameter. To validate our obtained analytical results and to examine the importance of arbitrary order derivative, we implement a recently proposed Toufik–Atangana numerical technique. To reach our conclusions, we investigate a thorough quantitative analysis of the model through the adjustment of quarantine and hospitalization rates as a first constant control technique. Through numerical experiments, it is asserted that the Covid-19 pandemic may be eliminated more quickly if a human community selfishly adopted both of the necessary control measures at various coverage levels with appropriate awareness. The developed model is subjected to sensitivity analysis and the most sensitive parameters are identified. In addition, the bifurcation nature of the Covid-19 model is examined. Furthermore, we develop an optimal control problem along with the associated optimality conditions of Pontryagin type to discover the most effective controls for the both strategies, one for exposed and another for infected individuals. The goal is to reduce both the financial burden of executing these strategies as well as the number of exposed and infected people. The extremals are obtained numerically. The effectiveness and efficiency of the optimal control strategy are finally demonstrated by numerical simulations before and after the optimization. The importance of the current research work is the use of developed structure preserving Toufik-Atangana numerical scheme, backward in time, for the first time to analyze optimally the epidemic models, for example the proposed SEQIHR Covid-19 model.

General fractional integrals and derivatives and their applications

In this survey paper, some recent results obtained for the general fractional integrals and derivatives and for the regularized general fractional derivatives are discussed. We start with a short historical survey of the general Fractional Calculus operators with the Sonin kernels and continue with a presentation of the recent developments regarding the general fractional derivatives and their applications. In particular, we introduce the first and the second fundamental theorems of Fractional Calculus for the general fractional derivatives of arbitrary order, for the regularized general fractional derivatives of arbitrary order, and for the sequential general fractional derivatives. As an application of these results, the generalized convolution Taylor formulas and the generalized convolution Taylor series with the coefficients and remainders in terms of the general fractional derivatives and the regularized general fractional derivatives are presented. Finally, we consider some Cauchy problems for the fractional differential equations with the general fractional derivatives.

Computational Algorithm for Fractional Fredholm Integro-Differential Equations

Fractional calculus is a fascinating field of mathematics that focuses on the study of integrals and derivatives of arbitrary orders, extending the principles of basic calculus. Its applications span across various scientific, engineering, and other disciplines. In this study, the collocation method, in conjunction with the utilization of the fourth kind Chebyshev polynomials, is employed to explore solutions for fractional integro-differential equations of Fredholm type. By applying the collocation method, the problem at hand is transformed into a system of linear algebraic equations. These equations are subsequently solved by employing matrix inversion techniques to determine the unknown constants. To provide a comprehensive understanding and visualization of the results, the research incorporates tables and figures, which present numerical examples and comparisons. These comparisons serve to highlight the superior performance of the proposed method in terms of efficiency and convenience when compared to traditional methods. By showcasing the advantages of the collocation method and the utilization of fourth kind Chebyshev polynomials, the research underscores the potential of these approaches in solving fractional integro-differential equations.

Approximation on hexagonal domains by Taylor-Abel-Poisson means

Approximative properties of the Taylor-Abel-Poisson linear summation method of Fourier series are considered for functions of several variables, periodic with respect to the hexagonal domain, in the integral metric. In particular, direct and inverse theorems are proved in terms of approximations of functions by the Taylor-Abel-Poisson means and K-functionals generated by radial derivatives. Bernstein type inequalities for L1-norm of radial derivatives of arbitrary order of the Poisson kernel are also obtained.