Bernstein Polynomial Method Research Articles

Dymnikova black hole from an infinite tower of higher-curvature corrections

Recently, in [1], it was demonstrated that various regular black hole metrics can be derived within a theory featuring an infinite number of higher curvature corrections to General Relativity. Moreover, truncating this infinite series at the first few orders already yields a reliable approximation of the observable characteristics of such black holes [2]. Here, we further establish the existence of another regular black hole solution, particularly the D-dimensional extension of the Dymnikova black hole, within the equations of motion incorporating an infinite tower of higher-curvature corrections. This solution is essentially nonperturbative in the coupling parameter, rendering the action, if it exists, incapable of being approximated by a finite number of powers of the curvature. In addition, we compute the dominant quasinormal frequencies of such black holes using both the Bernstein polynomial method and the 13th order WKB method with Padé approximants, obtaining a high degree of agreement between them.

Quasinormal Spectrum of (2+1)$(2+1)$‐Dimensional Asymptotically Flat, dS and AdS Black Holes

AbstractWhile ‐dimensional black holes in the Einstein theory allow for only the anti‐de Sitter (AdS) asymptotic, when the higher curvature correction is tuned on, the asymptotically flat, de Sitter, and AdS cases are included. Here, a detailed study of the stability and quasinormal spectra of the scalar field perturbations around such black holes with all three asymptotics is proposed. Calculations of the frequencies are fulfilled with the help of the 6th order Wentzel–Kramers–Brillouin (WKB) method with Pade approximants, Bernstein polynomial method, and time‐domain integration. Results obtained by all three methods are in a very good agreement in their common range of applicability. When the multipole moment is equal to zero, the purely imaginary, i.e., non‐oscillatory, modes dominate in the spectrum for all types of the asymptotic behavior, while the spectrum at higher resembles that in four‐dimensional spacetime with the corresponding asymptotic.

A continuous-time voltage control method based on hierarchical coordination for high PV-penetrated distribution networks

The photovoltaic (PV) integration brings both fast photovoltaic generation (PVG) variations and a large number of PV inverters to the distribution networks (DNs). The management of the PVG variation and the PV inverter is an urgent problem. The discrete-time model can’t describe PVG variations during the schedule interval and the regulation potential of the PV inverter is limited by a fixed dispatching value. This paper proposes a continuous-time voltage control method based on hierarchical coordination for high PV-penetrated DNs. In the central schedule, based on the Bernstein polynomial method, continuous-time models of PVGs are established and reference values of local inverter control curves are dispatched as continuous-time functions in a sub-hourly timescale. In the local control, reactive power and active power curtailment of PV inverters are set as the reference values, i.e., the corresponding values of the continuous-time functions, every 30 s to optimize the central dispatching objective under the voltage constrains, and deviate from the reference values according to both the local control curve and voltage measurement only when the real-time bus voltage violation occurs. In addition, the cooperation between inverter and mechanical equipment in the continuous-time framework is also considered, and the Bernstein polynomial method is developed to the continuous-time modeling of voltage control in DNs. The simulation shows that the proposed method has lower maximum voltage magnitude and lower energy loss compared with discrete-time central schedule method.

Gohar Fractional Derivative: Theory and Applications

The local fractional derivatives marked the beginning of a new era in fractional calculus. Due to their that have never been observed before in the field, they are able to fill in the gaps left by the nonlocal fractional derivatives and substantially increase the field’s theoretical and applied potential. In this article, we introduce a new local fractional derivative that possesses some classical properties of the integer-order calculus, such as the product rule, the quotient rule, the linearity, and the chain rule. It meets the fractional extensions of Rolle’s theorem and the mean value theorem and has more properties beyond those of previously defined local fractional derivatives. We reveal its geometric interpretation and physical meaning. We prove that a function can be differentiable in its sense without being classically differentiable. Moreover, we apply it to solve the Riccati fractional differential equations to demonstrate that it provides more accurate results with less error in comparison with the previously defined local fractional derivatives when applied to solve fractional differential equations. The numerical results obtained in this work by our local fractional derivative are shown to be in excellent agreement with those produced by other analytical and numerical methods such as the enhanced homotopy perturbation method (EHPM), the improved Adams-Bashforth-Moulton method(IABMM), the modified homotopy perturbation method (MHPM), the Bernstein polynomial method (BPM), the fractional Taylor basis method (FTBM), and the reproducing kernel method (RKM).

Statistical inference for the two-sample problem under likelihood ratio ordering, with application to the ROC curve estimation.

The receiver operating characteristic (ROC) curve is a powerful statistical tool and has been widely applied in medical research. In the ROC curve estimation, a commonly used assumption is that larger the biomarker value, greater severity the disease. In this article, we mathematically interpret "greater severity of the disease" as "larger probability of being diseased." This in turn is equivalent to assume the likelihood ratio ordering of the biomarker between the diseased and healthy individuals. With this assumption, we first propose a Bernstein polynomial method to model the distributions of both samples; we then estimate the distributions by the maximum empirical likelihood principle. The ROC curve estimate and the associated summary statistics are obtained subsequently. Theoretically, we establish the asymptotic consistency of our estimators. Via extensive numerical studies, we compare the performance of our method with competitive methods. The application of our method is illustrated by a real-data example.

Rogosinsky–Bernstein Polynomial Method of Summation of Trigonometric Fourier Series

Rogosinsky–Bernstein Polynomial Method of Summation of Trigonometric Fourier Series

Approximate solutions of nonlinear two‐dimensional Volterra integral equations

The present work is concerned with examining the Optimal Homotopy Asymptotic Method (OHAM) for linear and nonlinear two‐dimensional Volterra integral equations (2D‐VIEs). The result obtained by the suggested method for linear 2D‐VIEs is compared with the differential transform method, Bernstein polynomial method, and piecewise block‐plus method and result of the proposed method for nonlinear 2D‐VIEs is compared with 2D differential transform method. The proposed method provides us with efficient and more accurate solutions compared to the other existing methods in the literature.

Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate

In the present paper, we introduce the fractional Bernstein series solution (FBSS) to solve the fractional diffusion equation, which is a generalization of the classical diffusion equation. The Bernstein polynomial method is a promising one and can be generalized to more complicated problems in fractional partial differential equations. To get the FBSS, we first convert all terms in the problem to matrix forms. Then, the fundamental matrix equation is obtained and thus, the solution is obtained. Two error estimation methods based on a residual correction procedure and the consecutive approximations are incorporated to find the estimate and bound of the absolute error. The perturbation and stability analysis of the method is given. We apply the method to some illustrative examples. The numerical results are compared with the exact solutions and known second-order methods. The outcomes of the numerical examples are very encouraging and show that the FBSS is highly useful in solving fractional partial problems. The results show the accuracy and effectiveness of the method.

An Efficient Operational Matrix Method for a Few Nonlinear Differential Equations Using Wavelets

In this paper, a theoretical model of biofiltration of mixture of hydrophilic (methanol) and hydrophobic (α-pinene) volatile organic compounds is discussed. An efficient wavelet-based operational matrix method is developed for the analytical expressions pertaining to the concentration profiles of methanol and α-pinene in the air stream and bio-film stream. The convergence and supporting analysis of the proposed method is discussed. The operational matrix of derivatives contains many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with the Bernstein polynomial method and Adomian decomposition method. The efficiency of the proposed method is confirmed by using computational CPU runtime.

An efficient method for flutter stability analysis of aeroelastic systems considering uncertainties in aerodynamic and structural parameters

Abstract The aim of this work is to propose a novel numerical method for analyzing the flutter stability of aeroelastic systems considering uncertain parameters. Unlike the probabilistic method which is dependent on large quantities of statistical information, interval models are utilized to quantify the uncertain aerodynamic and structural parameters. An interval-form of the aeroelastic equation is established by substituting interval variables into a deterministic aeroelastic model. The flutter stability of an aeroelastic system is analyzed as a generalized interval eigenvalue problem via eigenvalue-based approach. By using the sensitivity analysis based vertex method (SAVM), interval bounds on eigenvalues can be calculated through solving two additional deterministic eigenvalue problems. A novel uncertain propagation method, called Bernstein polynomial method (BPM), is proposed to calculate the interval bounds on eigenvalues. The range of critical flutter speed can be predicted based on the upper and lower bounds on the maximum real part of all eigenvalues. For comparison, the perturbation-based probabilistic uncertainty propagation method (PPUPM) is given to determine the bounds on eigenvalues and the critical flutter speed. Numerical results demonstrate that BPM is consistent with Monte-Carlo simulation method within significantly less computation time and verify the compatibility between BPM and PPUPM.

Two-photon transitions in confined hydrogenic atoms

Two-photon transitions from ground state to excited and ionized states are studied. The energy levels and radial matrix elements of animpenetrable spherically confined hydrogenic atom embedded in plasma environment are evaluated using accurate Bernstein-polynomial(B-polynomial) method. Transition probability amplitudes, transparency frequencies and resonance enhancement frequencies for varioustransitions, namely, $1s-2s$, $1s-3s$ and $1s-3d$ are evaluated for various values of confining Debye potential parameter. The effect ofspherical confinement is studied and explained.

Two Photon Processes in an Atom Confined in Gaussian Potential

Transitions of an atom under the effect of a Gaussian potential and loose spherical confinement are studied. An accurate Bernstein-polynomial (B-polynomial) method has been applied for the calculation of the energy levels and radial matrix elements. The transition probability amplitudes, transparency frequencies, and resonance enhancement frequencies for transitions to various excited states have been evaluated. The effect of the shape of confining potential on these spectral properties is studied.

Intense field induced excitation and ionization of an atom confined in a dense quantum plasma

Exponential cosine screened Coulomb potential (ECSCP) has been widely used in various branches of physics e.g., solid-state physics, nuclear physics and plasma physics. The atomic photoionization processes under plasma shielding can serve as an efficient tool for study of plasma properties in various environments ranging from nano-scale devices to astrophysical objects. In the present study, ECSCP has been used to characterize a dense quantum plasma and its effect on the spectrum of an atom encaged in a spherical box has been investigated. The work has further been extended to study the response of such a system to a periodic laser field. Photoexcitation and ionization probabilities of the system have been studied as a function of applied laser field parameters using the non-perturbative Floquet technique. As the Floquet method requires exact energy values and oscillator strengths, the spectrum of confined system has been calculated using Bernstein-polynomial method. The variation of energy spectrum and oscillator strengths with screening as well as confinement parameters has also been explored.

Photoexcitation and ionization of hydrogen atom confined in Debye environment

The dynamics of a hydrogen atom confined in an impenetrable spherical box and under the effect of Debye screening, in the presence of intense short laser pulses of few femtosecond (fs) is studied in detail. The energy spectra and wave functions have been calculated using Bernstein polynomial (B-polynomial) method. Variation of transition probabilities for various transitions due to changes in Debye screening length, confinement radius as well as the parameters characterizing applied laser pulse is studied and explained.

Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model

In this paper, a computational technique is introduced for the solution of the first order multi-pantograph differential equation (MPDE) through some well-known optimization algorithms like sequential quadratic programming (SQP) and Active Set Technique (AST). Furthermore, artificial neural network (ANN) is used for networking of the first order multi-pantograph differential equation in used to provide mathematical model based on unsupervised error for equation. Moreover, mathematical modeling has been performed perfectly through multi-runs for simulation to justify the better convergence of the solutions. Also, two examples are presented to exhibit the aptitude of the method SQP and AST. The comparative study will be made with reported techniques such as variational iteration technique (VIT) [6] and collocation based on Bernstein polynomial method (BCM) [6].

Laser-induced excitation and ionization of a confined hydrogen atom in an exponential-cosine-screened Coulomb potential

The energy spectra of spherically confined hydrogen atom embedded in an exponential-cosine-screened Coulomb potential is worked out by using the Bernstein-polynomial method. The interaction of short laser pulses in the femtosecond range with the system is studied in detail. The effect of shape of laser pulse, confinement radius, Debye screening length as well as different laser parameters on the dynamics of the system has been explored and analyzed.

The Role of Logarithmic Expansions for Nonsinglet QCD Analysis of [formula omitted

In this paper the analysis of certain logarithmic expansions, which developed for precision studies of the evolution of the QCD parton distributions (pdf) at the Large Hadron Collider, is used. The x F 3 data of the CCFR collaboration is used to parameterize the non-singlet parton distributions in this order. In the fitting procedure, Bernstein polynomial method is employed. The results of valence quark distributions in the NLO are compered with exact and truncated solutions for non-singlet case.

Improved analysis of moments of [formula omitted] in neutrino–nucleon scattering using the Bernstein polynomial method

We use recently calculated next-to-next-to-leading order (NNLO) anomalous dimension coefficients for the moments of the x F 3 structure function in νN scattering, together with the corresponding three-loop Wilson coefficients, to obtain improved QCD predictions for both odd and even moments of F 3 . To investigate the issue of renormalization scheme dependence, the Complete Renormalization Group Improvement (CORGI) approach is used, in which all dependence on renormalization and factorization scales is avoided by a complete resummation of RG-predictable scale logarithms. We also consider predictions using the method of effective charges, and compare with the standard ‘physical scale’ choice. The Bernstein polynomial method is used to construct experimental moments (from the x F 3 data of the CCFR Collaboration) that are insensitive to the value of x F 3 in the region of x which is inaccessible experimentally. Direct fits for Λ MS ¯ ( 5 ) ( α s ( M Z ) ) are then performed. The CORGI fits including target mass corrections give a value α s ( M Z ) = 0.1189 −0.0019 +0.0019 , consistent with the world average. The effective charge and physical scale fits give slightly smaller values, which are still consistent within the errors.

The NNLO non-singlet QCD analysis of parton distributions based on Bernstein polynomials

A non-singlet QCD analysis of the structure function $xF_3$ up to NNLO is performed based on the Bernstein polynomials approach. We use recently calculated NNLO anomalous dimension coefficients for the moments of the $xF_3$ structure function in $\nu N$ scattering. In the fitting procedure, Bernstein polynomial method is used to construct experimental moments from the $xF_3$ data of the CCFR collaboration in the region of $x$ which is inaccessible experimentally. We also consider Bernstein averages to obtain some unknown parameters which exist in the valence quark densities in a wide range of $x$ and $Q^2$. The results of valence quark distributions up to NNLO are in good agreement with the available theoretical models. In the analysis we determined the QCD-scale $\Lambda^ {\bar{MS}}_{QCD, N_{f}=4}=211$ MeV (LO), 259 MeV (NLO) and 230 MeV (NNLO), corresponding to $\alpha_s(M_Z^2)=0.1291$ LO, $\alpha_s(M_Z^2)=0.1150$ NLO and $\alpha_s(M_Z^2)=0.1142$ NNLO. We compare our results for the QCD scale and the $\alpha_s(M_Z^2)$ with those obtained from deep inelastic scattering processes.

Non-singlet structure function in the NNLO approximation

We use recently the next-to-next-to-leading order (NNLO) contributions to anomalous dimension which is governing the evolution of non-singlet quark distributions. The x F 3 data of the CCFR collaboration is used to parameterize the non-singlet parton distributions in higher order. In the fitting procedure, Bernstein polynomial method is employed. The results of valence quark distributions in the LO, NLO and NNLO are compered with different theoretical models.