Higher Derivatives Research Articles

Semilocal convergence analysis of an eighth order iterative method for solving nonlinear systems

<abstract><p>In this paper, the semilocal convergence of the eighth order iterative method is proved in Banach space by using the recursive relation, and the proof process does not need high order derivative. By selecting the appropriate initial point and applying the Lipschitz condition to the first order Fréchet derivative in the whole region, the existence and uniqueness domain are obtained. In addition, the theoretical results of semilocal convergence are applied to two nonlinear systems, and satisfactory results are obtained.</p></abstract>

Численное решение дифференциально-алгебраических уравнений произвольного индекса с производной Римана-Лиувилля

In the article, linear systems of ordinary differential equations of fractional order α∈(0,1) are investigated. In contrast to previously known results, the authors consider the case when the matrix before the fractional differentiation operation is degenerate. Problems in such a formulation are called differential-algebraic equations of fractional order. The fundamental differences of such systems from the classical problems of fractional differentiation and integration are emphasized, namely, the systems under consideration can have an infinite number of solutions, or a solution of the original problem depends on the high fractional derivative of the right-hand side. Corresponding examples are given. The authors pass to a different, equivalent formulation of the problem, namely, they rewrite it in the form of a system of linear integral equations of the Abel type (with a weak singularity). This technique allows one to use the apparatus of regular matrix bundles to investigate the existence and uniqueness of the original problem. Using this result, the authors give sufficient conditions for the existence of a unique solution to the class of problems under consideration. Further, an algorithm for the numerical solution of such equations is proposed. The method is based on the product integration method and the quadrature formula of right rectangles. Calculations and graphs of the errors of the proposed method for various fractional differentiation exponents and various indices of the initial matrix bundles are presented.

Parametrized error function based Banach space valued univariate neural network approximation

Here we research the univariate quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. We perform also the related Banach space valued ractional approximation. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivaties. Our operators are defined by using a density function induced by a parametrized error function. The approximations are pointwise and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We finish with a convergence analysis.

On the semi-local convergence of a sixth order method in Banach space

High convergence order methods are important in computational mathematics, since they generate sequences converging to a solution of a non-linear equation. The derivation of the order requires Taylor series expansions and the existence of derivatives not appearing on the method. Therefore, these results cannot assure the convergence of the method in those cases when such high order derivatives do not exist. But, the method may converge. In this article, a process is introduced by which the semi-local convergence analysis of a sixth order method is obtained using only information from the operators on the method. Numerical examples are included to complement the theory.

Integrated Chebyshev wavelets for numerical solution of nonlinear one-dimensional and two-dimensional Rosenau equations

In this study, we focus on developing an efficient computational method for one-dimensional and two-dimensional nonlinear Rosenau equations. Suggested computational method in this study uses finite differences for discretization of time variable. For discretization of space variables, firstly unknown function with highest derivative which is appeared in the Rosenau equation is expanded to Chebyshev wavelets and then by successive integrations the unknown function itself is found in terms of truncated Chebyshev wavelets. Also, a linearization technique is applied to handle the nonlinearity. By doing so we obtain a linear system of equations for solving the Rosenau equation. We apply the suggested method to four test problems and compare obtained numerical solutions with exact solutions and with finite element method to assess how accurate are the numerical results. The obtained numerical results endorse the efficiency and practicality of the suggested method for both one-dimensional and two-dimensional nonlinear Rosenau equations.

Spherical Essentially Non-oscillatory (SENO) Interpolation

We develop two new ideas for interpolation on \(\mathbb {S}^2\). In this first part, we will introduce a simple interpolation method named Spherical Interpolation of orDER n (SIDER-n) that gives a \(C^{n}\) interpolant given \(n \ge 2\). The idea generalizes the construction of the Bézier curves developed for \(\mathbb {R}\). The second part incorporates the ENO philosophy and develops a new Spherical Essentially Non-Oscillatory (SENO) interpolation method. When the underlying curve on \(\mathbb {S}^2\) has kinks or sharp discontinuity in the higher derivatives, our proposed approach can reduce spurious oscillations in the high-order reconstruction. We will give multiple examples to demonstrate the accuracy and effectiveness of the proposed approaches.

Resolving Near-Infrared Spectra by Generalized Window Factor Analysis for Understanding Interactions in Aqueous Solution

Near-infrared (NIR) spectroscopy has been demonstrated as a sensitive technique to characterize the structure of water for understanding its interactions in aqueous solutions. Chemometrics, however, is needed to enhance the resolution or to obtain the spectrum of water and analytes. In this work, generalized window factor analysis (GWFA) was developed by defining a virtual window using a high order derivative spectrum. Continuous wavelet transform (CWT) was used for calculating the high order derivative of the determined component. The virtual window was defined as the spectral data that are not zero in the derivative spectrum. The method was tested by separating the water spectrum from aqueous solution. In comparison with the measured spectrum, the calculated water spectrum was more accurate at describing the water structures in the mixture. Therefore, by analyzing the difference spectrum of water between the measured and the calculated one, the structural changes and the interactions may be observed. The spectra of water in the solutions were obtained from which the structure, interactions and the function of water in the systems were determined.

Finite line method for solving high-order partial differential equations in science and engineering

In this paper, a completely new numerical method, named Finite Line Method (FLM), is proposed for solving general linear and non-linear high-order partial differential equations (PDEs) in science as well as engineering problems in heat conduction and mechanics. In this method, the computational domain is discretized into a number of collocation points as in the free element method (Gao et al., 2019; Liu et al., 2019), and at each collocation point a set of straight or curved lines crossing the point is formed, which is represented by a number of nodes distributed over lines. The shape functions for each line are constructed using the Lagrange interpolation formulation and the first and high order spatial partial derivatives of the shape functions with respect to global coordinates are derived through an ingenious technique. The derived partial derivatives can be directly substituted into the governing differential equations and related boundary conditions to set up the discretized system of equations.FLM is a type of collocation method, not needing any integration to establish the solution scheme. And since the Lagrange interpolation formulation is used to construct the shape functions, arbitrarily high order lines can be easily formulated for solving any types of linear and non-linear high order PDEs and engineering problems. A number of numerical examples for PDEs, heat conduction and solid mechanics problems are given to demonstrate the correctness and stability of the proposed method.

On fully-nonlinear symmetry-integrable equations with rational functions in their highest derivative: Recursion operators

We report a class of symmetry-intergable third-order evolution equations in 1+1 dimensions under the condition that the equations admit a second-order recursion operator that contains an adjoint symmetry (integrating factor) of order six. The recursion operators are given explicitly.

Adaptive Interaction Control of Compliant Robots Using Impedance Learning

This paper presents an impedance learning-based adaptive control strategy for series elastic actuator (SEA)-driven compliant robots without the measurement of the robot-environment interaction force. The adaptive controller is designed based on the command filter-based adaptive backstepping approach, where a command filter is used to decrease computational complexity and avoid the requirement of high derivatives of the robot position. In the controller, environmental impedance profiles and robotic parameter uncertainties are estimated using adaptive learning laws. Through a Lyapunov-based theoretical analysis, the tracking error and estimation errors are proven to be semiglobally uniformly ultimately bounded. The control effectiveness is illustrated through simulations on a compliant robot arm.

Brownian axionlike particles

We study the non-equilibrium dynamics of a pseudoscalar axion-like particle (ALP) weakly coupled to degrees of freedom in thermal equilibrium by obtaining its reduced density matrix. Its time evolution is determined by the in-in effective action which we obtain to leading order in the (ALP) coupling but to \emph{all orders} in the couplings of the bath to other fields within or beyond the standard model. The effective equation of motion for the (ALP) is a Langevin equation with noise and friction kernels obeying the fluctuation dissipation relation. A ``misaligned'' initial condition yields damped coherent oscillations, however, the (ALP) population increases towards thermalization with the bath. As a result, the energy density features a mixture of a cold component from misalignment and a hot component from thermalization with proportions that vary in time $(cold)\,e^{-\Gamma t}+(hot)\,(1-e^{-\Gamma t})$, providing a scenario wherein the ``warmth'' of the dark matter evolves in time from colder to hotter. As a specific example we consider the (ALP)-photon coupling $g a \vec{E}\cdot \vec{B}$ to lowest order, valid from recombination onwards. For $T \gg m_a$ the long-wavelength relaxation rate is substantially enhanced $\Gamma_T = \frac{g^2\,m^2_a\,T}{16\pi} $. The ultraviolet divergences of the (ALP) self-energy require higher order derivative terms in the effective action. We find that at high temperature, the finite temperature effective mass of the (ALP) is $m^2_a(T) = m^2_a(0)\Big[ 1-(T/T_c)^4\Big]$, with $T_c \propto \sqrt{m_a(0)/g}$, \emph{suggesting} the possibility of an inverted phase transition, which when combined with higher derivatives may possibly indicate exotic new phases. We discuss possible cosmological consequences on structure formation, the effective number of relativistic species and birefringence of the cosmic microwave background.

Mapping Phonation Types by Clustering of Multiple Metrics

For voice analysis, much work has been undertaken with a multitude of acoustic and electroglottographic metrics. However, few of these have proven to be robustly correlated with physical and physiological phenomena. In particular, all metrics are affected by the fundamental frequency and sound level, making voice assessment sensitive to the recording protocol. It was investigated whether combinations of metrics, acquired over voice maps rather than with individual sustained vowels, can offer a more functional and comprehensive interpretation. For this descriptive, retrospective study, 13 men, 13 women, and 22 children were instructed to phonate on /a/ over their full voice range. Six acoustic and EGG signal features were obtained for every phonatory cycle. An unsupervised voice classification model created feature clusters, which were then displayed on voice maps. It was found that the feature clusters may be readily interpreted in terms of phonation types. For example, the typical intense voice has a high peak EGG derivative, a relatively high contact quotient, low EGG cycle-rate entropy, and a high cepstral peak prominence in the voice signal, all represented by one cluster centroid that is mapped to a given color. In a transition region between the non-contacting and contacting of the vocal folds, the combination of metrics shows a low contact quotient and relatively high entropy, which can be mapped to a different color. Based on this data set, male phonation types could be clustered into up to six categories and female and child types into four. Combining acoustic and EGG metrics resolved more categories than either kind on their own. The inter- and intra-participant distributional features are discussed.

Coefficients at powers of logarithms in the higher-derivatives and minimal-subtractions-of-logarithms renormalization scheme

For renormalizable theories with a single coupling constant regularized by higher derivatives we investigate the coefficients at powers of logarithms present in the renormalization constants assuming that divergences are removed by minimal subtractions of logarithms. According to this higher-derivatives and minimal-subtractions-of-logarithms ($\mathrm{HD}+\mathrm{MSL}$) renormalization prescription, the renormalization constants include only powers of $\mathrm{ln}\mathrm{\ensuremath{\Lambda}}/\ensuremath{\mu}$, where $\mathrm{\ensuremath{\Lambda}}$ and $\ensuremath{\mu}$ are the dimensionful regularization parameter and the renormalization point, respectively. We construct general explicit expressions for arbitrary coefficients at powers of this logarithm present in the coupling constant renormalization and in the field renormalization constant which relate them to the $\ensuremath{\beta}$-function and (in the latter case) to the corresponding anomalous dimension. To check the correctness, we compare the results with the explicit four-loop calculation made earlier for $\mathcal{N}=1$ supersymmetric quantum electrodynamics and (for the supersymmetric case) rederive a relation between the renormalization constants following from the Novikov, Shifman, Vainshtein, and Zakharov equation.

Higher derivative f(R) gravity models for an accelerating universe

In modified gravity, we investigate the universe’s accelerating expansion in a vacuum. We obtain a highly nonlinear differential equation from the variation of Einstein–Hilbert action with the consideration of a Robertson–Walker metric. Because the differential equations have non-analytic solutions, the expansion phase of the Friedmann–Robertson–Walker universe is studied using numerical approaches. We analyzed the acceleration phases of the cosmos for different periods, cosmic jerk and snap parameters, and their dependencies on the initial values and coupling parameters involved in various [Formula: see text] gravity models. Using a parametric technique, we also give the values of the jerk and snap parameters in different accelerating stages of the universe. Finally, the behavior of the dark energy component is addressed for [Formula: see text] models.

General solutions to decay chain equations

We present an explicit compact form of the general solution for linear decay chains with repeated eigenvalues. Comparing with previous solutions of the Bateman equations with identical decay constants, our expression brings new contributions in two ways. First, it is a closed analytical expression valid for arbitrary initial conditions, whereas available solutions typically concern the simplest case of null initial population for all but the first member of the chain. Second, our expression is written in terms of a minimal number of factors and sums, avoiding large nested sums, large products, complicated implicit conditions and high order derivatives. The efficiency of our expression is illustrated with computational implementations of both analytical and numerical applications. To complement our result for linear chains, we present also the solution, for arbitrary initial conditions, of the basic cyclic chain, restricted to the case of negative and distinct eigenvalues.

Richards's curve induced Banach space valued ordinary and fractional neural network approximation.

Here we perform the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivatives. Our operators are defined by using a density function generated by the Richards curve, which is generalized logistic function. The approximations are pointwise and of the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer.

Piezoionic transfer effect in topological borophene-bismuthene derivative micro-leaves for robust supercapacitive electronic skins

To gain high performance in electrochemical device, exquisite accelerating ion-electron transfer is an effective way. Here, a topological idea is realized to design a micro-leave electrode by patterning borophene-bismuthene derivative. The inner bismuthene offers high conductive skeleton, intermediate BiOCl and Bi2O3 derivative as ornament contribute to supercapacitive property, and the outer borophene layer not only boosts supercapacitive property, but also reserves the topological shape. Assisted by a hydrogel electrolyte on a flexible indium-doped tin oxide substrate, a robust pressure sensor is fabricated, in which a piezoionic transfer effect is demonstrated to induce fast ions extraction and migration kinetics among the topological micro-leaves. Then the specific capacitance of electrode increases by 2.483 times comparing with the micro-leaves without borophene. The resultant robust electronic skin sensor presents a comparable sensitivity of 1.60 kPa−1 in the tiny pressure range of ≈ 50–150 Pa, and sensitivity of 1.01 kPa−1 in 150 Pa-8.5 kPa, low detection limit as 0.59 Pa, fast response/recovery time (80/80 ms), wide detection range (∼220 kPa) and reliable capacitance retention of 97.92% even after 10000 cycles. This has been successfully used to monitor the physiological signals and illustrates potential application prospects in human-computer interaction.

A C0 interior penalty method for mth-Laplace equation

In this paper, we propose a C0 interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝd, where m and d can be any positive integers. The standard H1-conforming piecewise r-th order polynomial space is used to approximate the exact solution u, where r can be any integer greater than or equal to m. Unlike the interior penalty method in Gudi and Neilan [IMA J. Numer. Anal. 31 (2011) 1734–1753], we avoid computing Dm of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete Hm-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete Hm-norm. The error estimate under the low regularity assumption of the exact solution is also obtained. Numerical experiments validate our theoretical estimate.

A non-gradient method for solving elliptic partial differential equations with deep neural networks

Deep learning has achieved wide success in solving Partial Differential Equations (PDEs), with particular strength in handling high dimensional problems and parametric problems. Nevertheless, there is still a lack of a clear picture on the designing of network architecture and the training of network parameters. In this work, we developed a non-gradient framework for solving elliptic PDEs based on Neural Tangent Kernel (NTK): 1. ReLU activation function is used to control the compactness of the NTK so that solutions with relatively high frequency components can be well expressed; 2. Numerical discretization is used for differential operators to reduce computational cost; 3. A dissipative evolution dynamics corresponding to the elliptic PDE is used for parameter training instead of the gradient-type descent of a loss function. The dissipative dynamics can guarantee the convergence of the training process while avoiding employment of loss functions with high order derivatives. It is also helpful for both controlling of kernel property and reduction of computational cost. Numerical tests have shown excellent performance of the non-gradient method.

Multi-Fractality, Universality and Singularity in Turbulence

In most geophysical flows, vortices (or eddies) of all sizes are observed. In 1941, Kolmogorov devised a theory to describe the hierarchical organization of such vortices via a homogeneous self-similar process. This theory correctly explains the universal power-law energy spectrum observed in all turbulent flows. Finer observations however prove that this picture is too simplistic, owing to intermittency of energy dissipation and high velocity derivatives. In this review, we discuss how such intermittency can be explained and fitted into a new picture of turbulence. We first discuss how the concept of multi-fractality (invented by Parisi and Frisch in 1982) enables to generalize the concept of self-similarity in a non-homogeneous environment and recover a universality in turbulence. We further review the local extension of this theory, and show how it enables to probe the most irregular locations of the velocity field, in the sense foreseen by Lars Onsager in 1949. Finally, we discuss how the multi-fractal theory connects to possible singularities, in the real or in the complex plane, as first investigated by Frisch and Morf in 1981.