Fourth-order Derivative Research Articles

Experimental identification of the bending stiffness and damping of plates using the frequency-adapted virtual fields Method

Inverse vibroacoustic methods can be used to identify the complex bending stiffness of a plate from its vibratory response. This work focuses on the Virtual Fields Method (VFM) and the Force Analysis Technique (FAT). The VFM uses functions called virtual fields to solve the Principle of Virtual Work, a weak form of equilibrium, and identifies complex bending stiffness. Here, the virtual fields are defined as piecewise functions over a surface smaller than the plate (virtual window). FAT uses a finite-difference scheme to discretize the fourth-order spatial derivatives of the displacement in the local equilibrium of the plate and thus estimates the bending stiffness. The application of a finite-difference scheme creates a bias in the identified stiffness, which increases with frequency. The Corrected Force Analysis Technique (CFAT) corrects this bias in the high-frequency domain. In this study, it is proposed to apply the CFAT principles to the VFM to adapt the method, using the virtual window size, so it can be applied it in the high-frequency domain. FAT, CFAT, the VFM and the Frequency-Adapted VFM will be presented. The method was tested on experimental data to identify the complex bending stiffness of an aluminium plate partially covered with a damping material.

Combined methods for solving degenerate unconstrained optimization problems

UDC 519.853.6 : 519.613.2 We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems. The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives. Our approach is based on the decomposition of ℝ n into the direct sum of the kernel of a Hessian matrix and its orthogonal complement. The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement. This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix. In order to get the second-order method, Newton's method is combined with the steepest-descent method. We study the efficiency of these methods and analyze their convergence rates. We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case. The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions.

Derivative Spectrophotometric Methods for Simultaneous Determination of Quercetin and Gentisic acid in Capparis spinosa L.

الشفلح هو أحدی النباتات الطبية الذي یستخدم في الطب التقليدي و یتضمن المواد الكيميائية النباتية العديدة مثل المركبات بوليفينولیة. تعتبر كلتا المركبتين، كویرستین وحامض الجینتسیك، کمركب فينولية مهمة اللتان توجدان في النباتات ولهما الخصائص الطبية الکثیرة مثل مضاد الالتهابات، مضاد الميكروبات، مضاد الأكسدة ومضاد للسرطان. لم يتم التحقق بعد من تحديد كلا المركبين معًا في مزیج ثنائي باستخدام طرق القياس الطيفي. في هذه الدراسة، تم تطوير طريقتين طيفيتين مشتقتين بسيطتين سريعتين و دقيقتين لاستخدامهما للتقدير الكمي الآنی لمادة كویرستین وحامض الجینتسیك في المزيج الثنائية وخلاصة أوراق الشفلح. الطريقة الأولی تعتمد على استخدام طریق العبور الصفري (المشتق الأول والرابع) ، بينما الطريقة الثانية هي المشتق الأول لأطياف النسبة. الخطية لمنحنيات المعايرة لكلتا الطريقتين الطيفية المشتقة يتراوح تركيزهما بين 2.0-30 ميكروغرام / مل و 4.0-80 ميكروغرام / مل لكویرستین وحامض الجینتسیك، على التوالي، في حين تراوحت نسب الاسترداد من 94.06٪ الی 105.98٪ (كویرستین) و 94.29٪ الی 113.37٪ (حمض الجینتسیك). تم استخدام الطرق المطورة بشكل فعال في التقدیر الكمي لكل من المركبات الفينولية في أوراق الشفلح..

Validation of spectrophotometric methods for the simultaneous determination of fluconazole and riparin B in the development of lipid nanoparticles modified by β-cyclodextrin: Application for in vitro characterization and ex vivo studies of nail retention

Fluconazole (FLC) has been used in the clinic for almost thirty years as a treatment for fungal infections, but its widespread use has led to the emergence of resistant strains. In this context, synthetic analogues of riparins, phytocompounds isolated from Aniba riparia (Nees) Mez (Lauraceae), such as Riparin B (RIP), stand out as an important agent for modulating antimicrobial resistance, potentially useful for treating superficial mycoses resistant to conventional antifungals. Onychomycosis is one of the main fungal infections affecting the nails. Its local treatment is desirable but has limited efficacy due to poor drug permeation. The use of nanoparticles and permeation promoters can increase permeability and improve treatment. Therefore, nanostructured lipid carriers coated with β-cyclodextrin (MNLC) were developed for the topical application of FLC and RIP on nails. To support the formulation, four spectrophotometric methods (amplitude factor, amplitude subtraction, modified amplitude subtraction, and fourth-order derivative), based on derivative spectroscopy and simple mathematical manipulations, were validated, in accordance with the International Conference on Harmonization (ICH), for simultaneous determination of FLC and RIP in the MNLC and in in vitro characterization studies of the formulation. In addition, the methods were applied to quantify drug retention in porcine hooves using an optimized extraction process. All the methods developed were found to be linear (r > 0.999), precise (RSD < 5 %), and accurate (92.33 to 108.86 %) without being affected by the components of the formulation or the biological matrix. The methods developed made it possible to determine the encapsulation efficiency and sustained release profile of the drugs. In addition, successive extractions revealed the amount of FLC and RIP retained in the superficial and deeper regions of porcine hooves. Thus, the results indicated that FLC and RIP can be determined simultaneously, quickly, simply, and economically, using any of the proposed methods, in complex MNLC samples and the biological matrix of porcine hooves.

Numerical Accuracy Matters: Applications of Machine Learned Potential Energy Surfaces.

The role of numerical accuracy in training and evaluating neural network-based potential energy surfaces is examined for different experimental observables. For observables that require third- and fourth-order derivatives of the potential energy with respect to Cartesian coordinates single-precision arithmetics as is typically used in ML-based approaches is insufficient and leads to roughness of the underlying PES as is explicitly demonstrated. Increasing the numerical accuracy to double-precision gives a smooth PES with higher-order derivatives that are numerically stable and yield meaningful anharmonic frequencies and tunneling splitting as is demonstrated for H2CO and malonaldehyde. For molecular dynamics simulations, which only require first-order derivatives, single-precision arithmetics appears to be sufficient, though.

Free-Vibration Analysis for Truncated Uflyand–Mindlin Plate Models: An Alternative Theoretical Formulation

Plates are flat structural elements whose thickness is small in relation to the size of the surface. Their use may include engine foundations, reinforced concrete bridge elements or parts of various floating structures. Consequently, knowledge of their mechanical behavior under static and dynamic loads is of primary importance in engineering applications and of interest from a structural point of view. As a result, numerous works existing in the literature have investigated the mechanical properties of plates using various plate models, such as Reissner’s theory, Levinson’s theory, Kirchhoff’s theory and Mindlin’s theory, and their static and dynamic behavior has been examined. In the present paper the truncated Uflyand–Mindlin plate equation is proposed. According to Uflyand–Mindlin theory, an alternative theoretical formulation is presented for the free-vibration analysis of plates, and the equations of motion and the general corresponding boundary conditions are derived. This paper develops the truncated Uflyand–Mindlin plate equation, i.e., without the fourth-order derivative, by means of the direct method and variational formulation. The first-order shear deformable plate theory developed by Elishakoff, which takes into account rotational inertia and shear deformation and does not include a fourth-order time derivative, is variationally derived here. This derivation complements that performed by Mindlin some 70 years ago. The innovative aspect of the suggested strategy is that variational and direct methods for studying plate dynamics are analogous. Finding the third equation of the reduced Uflyand–Mindlin equations, the accompanying boundary conditions and their mathematical resemblance are the goals of the presented formulations. In order to solve the dynamic equilibrium problem of a truncated Uflyand–Mindlin equation via a variational formulation, it is demonstrated that the differential equations and the corresponding boundary conditions have the same form as those found using the direct technique. This paper successfully completes this task. Finally, in order to validate the effectiveness and correctness of the proposed procedure, a numerical example of the case of a plate simply supported at all four ends is proposed.

Nonlinear stability of shock-fronted travelling waves in reaction-nonlinear diffusion equations

Reaction-nonlinear diffusion PDEs can be derived as continuum limits of stochastic models for biological and ecological invasion. We numerically investigate the nonlinear stability of shock-fronted travelling waves arising in these RND PDEs, in the presence of a fourth-order spatial derivative multiplied by a small parameter ɛ that models high-order regularization. Once we have verified sectoriality of our linear operator, our task is reduced to checking spectral stability of our family of travelling waves. Motivated by the authors’ recent stability analysis of shock-fronted travelling waves under viscous relaxation, our numerical analysis suggests that near the singular limit, the associated eigenvalue problem for the linearized operator admits a fast–slow decomposition similar to that constructed by Alexander, Gardner, and Jones in the early 90s. In particular, our numerical results suggest a reduction of the complex four-dimensional eigenvalue problem into a real one-dimensional problem defined along the slow manifolds; i.e. slow eigenvalues defined near the tails of the shock-fronted wave for ɛ=0 govern the point spectrum of the linearized operator when 0<ɛ≪1.

A novel reduced-order method using mixed nonlinear kinematics for geometrically nonlinear analysis of thin-walled structures

Reduced-order methods based on the Koiter perturbation theory are applicable only to structural buckling problems, and high-order strain energy derivatives influence the computational efficiency of the methods. In this paper, a novel reduced-order method is proposed for the geometrically nonlinear analysis of thin-walled structures with large deflections. Mixed nonlinear kinematics are developed for the reduced-order method using co-rotational and updated von Kármán formulations. The co-rotational kinematics are applied to determine the internal force and tangent stiffness, whereas the third- and fourth-order strain energy derivatives are calculated using the updated von Kármán kinematics. Reduced-order models with one degree of freedom is constructed based on the perturbation theory, the solutions of which are nonlinear predictors of the geometrically nonlinear response. The use of mixed nonlinear kinematics significantly reduces the computational cost of constructing a reduced-order model. Nonlinear predictors are corrected using internal force-based residuals to ensure the accuracy of the geometrically nonlinear analysis. A geometrically nonlinear response with favorable smoothness is efficiently achieved using large step sizes in the path-following analysis. Various numerical examples, including the stiffened plate and wing structure, are provided to validate the accuracy and efficiency of the proposed method.

A dual number formulation to efficiently compute higher order directional derivatives

This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular cases of symmetric multilinear forms; therefore, using their symmetric properties and their coordinate representation, we implement functions to calculate mixed partial derivatives. Moreover, with directional derivatives, we deduce concise formulas for the velocity, acceleration, jerk, and jounce/snap vectors. The utility of our formulation is proved with three examples. The first example presents a comparison against the forward mode of finite differences to compute the fourth-order directional derivative of a scalar function. To this end, we have coded the finite differences method to calculate partial derivatives until the fourth order, to any order of approximation. The second example presents efficient computations of the velocity, acceleration, jerk, and jounce/snap. Finally, the third example is related to the computation of some partial derivatives. The implemented code of the proposed formulation and the finite differences method is proportioned as additional material to this article.

Investigating Unitarity Violation of Lee–Wick’s Complex Ghost with Quantum Field Theory

Theories with fourth-order derivatives like Lee–Wick’s quantum electrodynamics model or quadratic gravity result in complex ghosts above a definite energy threshold that violate unitarity.

On the convergence of a new fourth-order method for finding a zero of a derivative

&lt;abstract&gt;&lt;p&gt;On the basis of Wang's method, a new fourth-order method for finding a zero of a derivative was presented. Under the hypotheses that the third and fourth order derivatives of nonlinear function were bounded, the local convergence of a new fourth-order method was studied. The error estimate, the order of convergence, and uniqueness of the solution were also discussed. In particular, Herzberger's matrix method was used to obtain the convergence order of the new method to four. By comparing the new method with Wang's method and the same order method, numerical illustrations showed that the new method has a higher order of convergence and accuracy.&lt;/p&gt;&lt;/abstract&gt;

Об интегральном подходе при использовании метода коллокации

The article describes a matrix method of polynomial Chebyshev approximation using an integral approach to construct a solution to a nonhomogeneous fourth-order differential equation with mixed boundary conditions of the first kind. The proposed method is based on the expansion of the fourth-order derivative of the desired function into a series in terms of Chebyshev polynomials of the first kind and the representation of the partial sum of this series as a product of matrices whose elements are, respectively, the Chebyshev polynomials and the coefficients in this expansion. In this paper, using analytical formulas for calculating integrals of Chebyshev polynomials, we obtain a representation of the desired function in terms of the product of the matrices defined above. The use of points of extrema and zeros of Chebyshev polynomials of the first kind as nodes, as well as the properties of the sums of products of Chebyshev polynomials at these points, made it possible to reduce the boundary value problem by the collocation method to a system of inhomogeneous linear algebraic equations with a sparse matrix of this system. It is shown that the solution constructed in this way satisfies the differential equation at all nodes, including the boundary ones, in contrast to the approximate solution obtained by approximating the exact solution in the form of a finite sum of the Chebyshev series. The effectiveness of the proposed method is demonstrated by considering a boundary value problem with a known analytical solution. The convergence analysis of the constructed solution is carried out.

Non-perturbative Lee-Wick gauge theory: Towards Confinement & RGE with strong couplings

We consider a non-Abelian Lee–Wick gauge theory and discuss Becchi-Rouet-Stora-Tyutin invariance. It contains fourth-order derivative as extensions of the kinetic term, leading to massive ghosts in the theory upon quantization. We particularly provide essential clues towards confinement conditions in strongly-coupled regimes, using the Kugo–Ojima approach, and obtain the functions in the non-perturbative regimes. This is achieved using a set of exact solutions of the corresponding local theory in terms of Jacobi elliptical functions. We obtain a similar function just as for the ordinary Yang-Mills theory but the main differences are that now, the cut-off arises naturally from the Lee–Wick heavy mass scales (M). We show that the fate of the ghosts are fixed in these regimes: they are no more the propagating degrees of freedom in the infrared-limit. As it also happens for the ordinary case, confinement is due to the non-Abelian nature of the theory. In the limit , one recovers the standard results for the local non-Abelian Yang-Mills theory.

Unitarity violation in field theories of Lee–Wick’s complex ghost

Abstract Theories with fourth-order derivatives, including the Lee–Wick finite QED model and quadratic gravity, have a better UV behavior, but the presence of negative metric ghost modes endangers unitarity. Noticing that the ghost acquires a complex mass by radiative corrections, Lee and Wick, in particular, claimed that such complex ghosts would never be created by collisions of physical particles because of energy conservation, so that the physical S-matrix unitarity must hold. We investigate the unitarity problem faithfully, working in the operator formalism of quantum field theory. When complex ghosts participate, a complex delta function (a generalization of the Dirac delta function) appears at each interaction vertex, which enforces a specific conservation law of complex energy. Its particular property implies that the naive Feynman rule is wrong if the four-momenta are assigned to the internal lines after taking account of the conservation law in advance. We show that complex ghosts are actually created and unitarity is violated in such fourth-order derivative theories. We also find a definite energy threshold below which ghosts cannot be created: The theories are unitary and renormalizable below the threshold.

Exploring the couple stress flow of sodium alginate-based casson tetra hybrid nanofluid between inclined parallel microplates using Fourier analysis and Fick's laws

Compared to a regular fluid like water, the Couple Stress Fluid (CSF) has an additional material constant that manages both couple stress and lubricant viscosity. However, because this material constant creates a fourth-order derivative term in the momentum equation, there has been little research done on this type of fluid, even for traditional fluid problems. In this article, we analyze the flow of a tetra hybrid nanofluid consisting of Casson (sodium alginate-based) couple stress between two parallel microplates that are inclined infinitely. Fourier and Fick's laws are used to investigate the flow, where one plate remains stationary and the other oscillates at a constant speed. The flow is driven by the buoyant force generated by heat transfer and free convection, which results in the uniform dispersion of all spherical dust particles throughout the fluid. The system of partial differential equations (PDEs) is expanded using the Caputo-Fabrizio (CF) fractional derivative to simulate fluid flow. The problem is tackled by using finite sine Fourier and Laplace transforms to obtain closed-form solutions for temperature and velocity profiles. CF time fractional derivatives are used to provide CSF (Couette flow with suction or injection) results for various stimulating fluid parameters and examine their impact on the system. Graphical representations of the CSF, dust particle, and temperature profiles are presented, and skin friction and Nusselt number are calculated using Mathcad-15. It is found that an increase in the couple stress parameter leads to a retardation of both velocity profiles.

Impacts of Arrhenius energy and viscous dissipation on variable properties of viscoelastic nanofluid flow with slip velocity

ABSTRACT In this article, the viscoelastic nanofluid external flow using the model that considers the Brownian motion and thermophores is examined. A non-Newtonian liquid has variable properties, namely dynamic viscosity and thermal conductivity, and these depended on temperature distributions. Third order mathematical formulations include some important impacts such as Lorentz force, non-linear radiation, exponential heat generation, viscous dissipation, and Arrhenius activation energy. Also, the flow on the outer edge has slip conditions, variable nanoparticle distributions, and convective boundary conditions. The solution technique is based on reducing the fourth order derivatives of ODEs (Ordinary Differential Equations) and shooting method. From the major results, it is noted that the variable property case gives a higher rate of heat transfer compared to the constant case. The slip-velocity condition in the case of the viscoelastic nanofluids causes higher velocity features for the non-Newtonian suspension compared to the case of Newtonian nanofluids. Minimization of the viscoelastic parameter, viscosity parameter, and mixed convection parameter is the best to obtain the higher values of skin friction coefficient.

An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation

The stability and accuracy of explicit high-order finite difference (HOFD) algorithms have been research hotspots in different fields. To improve the stability and accuracy of the HOFD algorithms in thermal simulations, we present a Lax-Wendroff high-order finite difference (LHOFD) algorithm to solve the 2D transient heat transfer equation in this paper and develop an improved LHOFD (IHOFD) algorithm to improve the stability of the LHOFD algorithm. The formulas of the general high-order central FD (HOCFD) coefficients and the truncation error coefficient as well as the high-order non-central FD (HONFD) coefficients and the truncation error coefficient of the fourth-order spatial derivative are derived concisely in a different way. Furthermore, a unified analytical formula of the general HOCFD and HONFD coefficients, which can calculate the spatial derivative of any integer order, is derived. A new strategy of combination with the HOCFD and HONFD approximations under the same high-order accuracy as the internal computational domain is proposed to calculate the mixed derivatives of the boundary domains with high accuracy, no additional computational cost, and easy implementation. Then, the accuracy analysis, stability analysis, and comparative analysis of numerical simulation results obtained by the LHOFD and IHOFD algorithms with the exact solution show the correctness and validity of the proposed algorithms and their stability formulas, and the advantages of the proposed algorithms. The proposed algorithms are valid under both symmetric and asymmetric boundary conditions. The stability factor of the LHOFD algorithm is slightly higher than that of the conventional algorithm. The stability factor of the IHOFD algorithm is twice that of the conventional algorithm, and the maximum absolute error of the thermal simulation is within 0.015 (°C).

Semiclassical dynamics of Hawking radiation

We consider gravity in 3+1 spacetime dimensions coupled to N scalar matter fields in a semiclassical limit where . The dynamical evolution of a black hole including the back-reaction of the Hawking radiation on the metric is formulated as an initial-value problem. The quantum stress-energy tensor is evaluated using a point-splitting regularization along spacelike geodesics. To account for the quantum entanglement of the matter fields, they are treated as a set of bilocal collective fields defined on spacelike hypersurfaces. The resulting semiclassical field equations include terms up to fourth order in derivatives that can be treated in a perturbative expansion. The formulation we arrive at should be amenable to numerical simulation of time-dependent semiclassical spacetime.

Path integral quantization of generalized Stueckelberg electrodynamics: A Faddeev-Jackiw approach

We build a setup for path integral quantization through the Faddeev-Jackiw approach, extending it to include Grassmannian degrees of freedom, to be later implemented in a model of generalized electrodynamics that involves fourth-order derivatives in the components of a massive vector field being endowed with gauge freedom, due to an additional scalar field. Namely, the generalized Stueckelberg electrodynamics. In the first instance, we work on the free case to gain some familiarity with the program and, subsequently, we add the interaction with fermionic matter fields to complete our aim. In addition to deriving the correct classical brackets for such a model, we get the full expression for the associated generating functional and its associated integration measure.

A highly efficient semi-implicit corrective SPH scheme for 2D/3D tumor growth model

In this work, a semi-implicit corrective SPH method is proposed to solve the multi-component Cahn–Hilliard equation with fourth-order derivatives, and it is further used to predict the high-dimensional tumor growth model. The scheme is motivated by: (a) the fourth-order spatial derivative is discretized continuously by a corrective SPH formula for approximating second-order derivative twice, and the Neumann boundary is imposed by a ghost technique; (b) the temporal direction is approximated by the implicit scheme, and an iterative concept is employed to handle the above implicit form; (c) the multi-CPUs MPI parallelization is adopted to reduce the computing cost. Firstly, the second-order convergence rate of the proposed method for 2D/3D equation is shown and discussed by two analytical examples, and the mass conservation and energy properties are also demonstrated. Secondly, the efficiency of the proposed approach for multi-phase separation phenomenon is illustrated in an irregular domain. Finally, the 2D/3D tumor growth evolution at a short time is predicted by the proposed method and qualitatively compared with other numerical results. The numerical experiments show that the proposed scheme for phase-separation phenomenon or tumor growth model is highly efficient and reliable.