Approximation Theory Research Articles

The Peano–Sard theorem for Caputo fractional derivatives and applications

The classical Peano–Sard theorem is a very useful result in approximation theory, bounding the errors of approximations that are exact on sets of polynomials. A fractional version was developed by Diethelm for fractional derivatives of Riemann–Liouville type, which we here extend to fractional derivatives of Caputo type. We indicate some applications to quadrature and interpolation formulae. These results will be useful in the approximate solution of fractional differential equations involving Caputo-type operators, which are often said to be more natural for applications.

Controlling factors of coating thickness of Sisko fluid in blade coating process

Blade coating is a technique for coating the moving substrate with a protective fluid layer through the blade as a common smoothing device. The heat transfer and slip effects of the rheological Sisko fluid model for plane coater are considered under blade coating process. This model combines the power-law and Newtonian fluids and taken as the limiting behavior of a blade coating in the stiff blade limit. The flow of greases is the most significant example of this type of fluid, which is easily found in nature and has many practical uses. To solve the non-linear expressions, LAT (Lubrication Approximation Theory) is applied, and a shooting technique is used for numerical solutions. The impacts of several parameters on flow characteristics, particularly the velocity ratio K, the Sisko fluid parameter b, the power-law index n, and the slip parameter α are investigated. The effects of both shear-thinning and shear-thickening behaviors are also investigated. The lubrication pressure and temperature increase with the Sisko fluid parameter b and decrease with the slip parameter α. As material parameter b increases from (0.001–1), 20.7% reduction in coating thickness and a 53.2% pressure reduction is noted as compared to the Newtonian case.

Error approximation for backwards and simple continued fractions

In this paper, we provide a new framework for studying continued fractions (CFs) by means of the backwards continued fraction (BCF). We develop an approximation theory for BCFs based on taking expansions of a fixed length, show the correspondence between continued fractions and their BCFs counterpart, and illustrate a rich approximation theory for continued fractions based off the methods of the approximation theory for the backwards case. In particular, we construct explicit functions that are sharp bounds for the BCF or CF error infinitely often over any BCF or CF cylinder set, and work out the details to pass seamlessly between the BCF and CF expansion of any real number.

Viscosity and thermal conductivity on magneto-hydrodynamic chemically reacting nanofluid over a vertical cone embedded in porous medium

The purpose of current investigations is to explore the impacts of variable fluid properties on the vertical cone embedded in a porous medium for reduced skin friction rates and enhanced mass transfer rates. The flow governing systems of highly nonlinear partial differential equations is obtained by using the Rivilin Erickson tensor along with the theory of boundary layer approximation. Buongiorno model is analyzed using the impacts of variable properties of the fluid based on the nanofluids characteristics. The present model has been numerically tackled using the boundary value problem technique in MATLAB and the criterion for convergence or tolerance is taken as 10−6. The outcome of present study shows the variable thermal conductivity, which is improved by using the heat transfer coefficient and reduces the skin friction based on the nanofluid. The Sherwood number is also increased under the Thermophoresis diffusion and the Brownian motion. Momentum boundary layer has expanded under the adjustable thermal conductivity along with the viscidity parameter. It is stated that these investigations have not been found by using the boundary value algorithm to solve the utilizing Buongiorno nanofluid model over the vertical cone based on the permeable medium.

Optimized Self-Similar Borel Summation

The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to infinity, is described. The method is based on the combination of optimized perturbation theory, self-similar approximation theory, and Borel-type transformations. General Borel Fractional transformation of the original series is employed. The transformed series is resummed in order to adhere to the asymptotic power laws. The starting point is the formulation of dynamics in the approximations space by employing the notion of self-similarity. The flow in the approximation space is controlled, and “deep” control is incorporated into the definitions of the self-similar approximants. The class of self-similar approximations, satisfying, by design, the power law behavior, such as the use of self-similar factor approximants, is chosen for the reasons of transparency, explicitness, and convenience. A detailed comparison of different methods is performed on a rather large set of examples, employing self-similar factor approximants, self-similar iterated root approximants, as well as the approximation technique of self-similarly modified Padé–Borel approximations.

Beyond superoscillation: general theory of approximation with bandlimited functions

We give a general strategy to construct superoscillating/supergrowing functions using an orthogonal polynomial expansion of a bandlimited function. The degree of superoscillation/supergrowth is controlled by an anomalous expectation value of a pseudodistribution that exceeds the band limit. The function is specified via the rest of its cumulants of the pseudodistribution. We give an explicit construction using Legendre polynomials in the Fourier space, which leads to an expansion in terms of spherical Bessel functions in the real space. The other expansion coefficients may be chosen to optimize other desirable features, such as the range of super behavior. We provide a prescription to generate bandlimited functions that mimic an arbitrary behavior in a finite interval. As target behaviors, we give examples of a superoscillating function, a supergrowing function, and even a discontinuous step function. We also look at the energy content in a superoscillating/supergrowing region and provide a bound that depends on the minimum value of the logarithmic derivative in that interval. Our work offers a new approach to analyzing superoscillations/supergrowth and is relevant to the optical field spot generation endeavors for far-field superresolution imaging.

Surface Motion and Topographic Effects on Ice Thickness Inversion for High Mountain Asia Glaciers: A Comparison Study from Three Numerical Models

Several studies have evaluated the accuracy of ice thickness model inversions, but the effects of surface velocity and topographic factors have not been fully studied. Because of the important potential of models for estimating regional and global ice thickness, the influence of the characteristics of the glaciers themselves on the models is unknown. Therefore, we estimated the thickness of 10 glaciers in HMA based on remote sensing data using the GlabTop2, GV22, and VOLTA models, and we analyzed the factors affecting accuracy. The first two models are representative of the widely used shallow ice approximation and laminar flow theory, and the latter is a new model based on shallow ice approximation. The results show that the RMSEs for GlabTop2, VOLTA, and GV22 are 36.6 m, 56.8 m, and 63.2 m, respectively. The accuracy of the three models differed obviously under the influence of different slopes, aspects, and surface velocities. In contrast to glaciers with higher surface velocities, GV22 is more suitable for glaciers with smaller surface velocities, and the accuracy of the model is significantly reduced when surface velocities are greater than 50 m/yr. Aspect has an effect on the accuracy of the GV22 ice thickness inversion but not on the other models. Ice thicknesses estimated using the three models is more reliable at surface slopes of 16–24°. These three models do not apply to gentle glaciers such as icecaps. Our study provides an important reference for choosing a suitable model to invert the glacier thickness in HMA.

Benchmark Data Set of Crystalline Organic Semiconductors.

This work reports a Benchmark Data set of Crystalline Organic Semiconductors to test calculations of the structural and electronic properties of these materials in the solid state. The data set contains 67 crystals consisting of mostly rigid molecules with a single dominant conformer, covering the majority of known structural types. The experimental crystal structure is available for the entire data set, whereas zero-temperature unit cell volume can be reliably estimated for a subset of 28 crystals. Using this subset, we benchmark r2SCAN-D3 and PBE-D3 density functionals. Then, for the entire data set, we benchmark approximate density functional theory (DFT) methods, including GFN1-xTB and DFTB3(3ob-3-1), with various dispersion corrections against r2SCAN-D3. Our results show that r2SCAN-D3 geometries are accurate within a few percent, which is comparable to the statistical uncertainty of experimental data at a fixed temperature, but the unit cell volume is systematically underestimated by 2% on average. The several times faster PBE-D3 provides an unbiased estimate of the volume for all systems except for molecules with highly polar bonds, for which the volume is substantially overestimated in correlation with the underestimation of atomic charges. Considered approximate DFT methods are orders of magnitude faster and provide qualitatively correct but overcompressed crystal structures unless the dispersion corrections are fitted by unit cell volume.

Nonmonotonic enhancement of diversity-induced resonance in systems of mobile oscillators.

Diversity is omnipresent in natural and synthetic extended systems, the phenomenon of diversity-induced resonance (DIR), wherein a moderate degree of the diversity can provoke an optimal collective response, provides researchers a brand-new strategy to amplify and utilize the weak signal. As yet the relevant advances focus mostly on the ideal situations where the interactions among elements are uncorrelated with the physical proximity of agents. Such a consideration overlooks interactions mediated by the motion of agents in space. Here, we investigate the signal response of an ensemble of spatial mobile heterogeneous bistable oscillators with two canonical interacting modes: dynamic and preset. The oscillators are considered as mass points and perform random walks in a two-dimensional square plane. Under the dynamic scheme, the oscillators can only interact with other oscillators within a fixed vision radius. For the preset circumstance, the interaction among oscillators occurs only when all of them are in a predefined region at the same moment. We find that the DIR can be obtained in both situations. Additionally, the strength of resonance nonmonotonically rises with respect to the increase of moving speed, and the optimal resonance is acquired by an intermediate magnitude of speed. Finally, we propose reduced equations to guarantee the occurrence of such mobility-optimized DIR on the basis of the fast switching approximation theory and also examine the robustness of such phenomenon through the excitable FitzHugh-Nagumo model and a different spatial motion mechanism. Our results reveal for the first time that the DIR can be optimized by the spatial mobility and thus has promising potential application in the communication of mobile agents.

Calculation of specific loss power of magnetic fluids by mean spherical approximation (MSA) model

According to various theoretical models the heating performance, i.e. the volumetric power dissipation of magnetic fluids under alternating magnetic field is proportional to the imaginary part of the complex susceptibility. However, the magnetic field strength dependence of the susceptibility is usually taken into account only by the Langevin theory. The main goal of our work was to propose a theoretical model based on the series expansion of the mean spherical approximation (MSA) theory to calculate the magnetic properties, and derive the power dissipation by including the dipole-dipole interactions. The predictions of the theory for the field strength dependence of the power dissipation was verified by comparison with experimental data. It was shown that compared to the MSA theory the generally used Langevin limiting case underestimates the power dissipation. The contribution of the higher-order terms in the approximated dynamic susceptibility at different field strengths was also investigated.

Acoustic Performance of a Metamaterial Coating with Uncertainty in Design Parameters

The effect of uncertainty in design parameters on the acoustic performance of a viscoelastic metamaterial coating is investigated. The coating is composed of a layer of hard inclusions embedded in a soft host medium. The layer of inclusions in the soft medium is modelled as a homogenised layer using effective medium approximation theory. Stochastic models based on the polynomial chaos expansion (PCE) method and Monte Carlo (MC) simulations are developed. The effect of uncertainty in geometric and material parameters are quantified by examining the mean and envelopes of the absorption coefficient. Uncertainty in geometric and material parameters produce similar levels of variation in the absorption coefficient of the coating, particularly around the peaks of the absorption coefficient spectrum.

Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials

Approximate solutions for a family of nonlinear fractional-order differential equations are introduced in this work. The fractional-order operator of the derivative are provided in the Caputo sense. The third-kind Chebyshev polynomials are discussed briefly, then operational matrices of fractional and integer-order derivatives for third-kind Chebyshev polynomials are constructed. These obtained matrices are a critical component of the proposed strategy. The created matrices are used in the context of approximation theory to solve the stated problem. The fundamental advantage of this method is that it converts the nonlinear fractional-order problem into a system of algebraic equations that can be numerically solved. The error bound for the suggested technique is computed, and numerical experiments are presented to verify and support the accuracy and efficiency of the proposed method for solving the class of nonlinear multi-term fractional-order differential equations.

Research on basketball game tactics based on multivariate statistical methods

Abstract This paper analyzes and studies the tactics in basketball games using the main branches of cluster analysis using multivariate statistical methods. A brief overview of multivariate statistical analysis is summarized by analyzing the application of multivariate statistical methods. The main method for classifying basketball tactics is cluster analysis, and nonlinear programming functions with constraints are constructed using mean square approximation theory. The classification results are reasonable when combined with the least square error sum. According to the results, man-to-man defensive tactics were most frequently used in basketball, with 72 instances accounting for 75.19% of all defensive strategies. The most frequent offensive tactic in the basketball game was dribbling with a fast break, and 15 times succeeded in scoring during 20 dribbling with a fast break in the first half, with a success rate of 75%.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

On analytical theories for conductivity and self-diffusion in concentrated electrolytes.

Describing analytically the transport properties of electrolytes, such as their conductivity or the self-diffusion of the ions, has been a central challenge of chemical physics for almost a century. In recent years, this question has regained some interest in light of Stochastic Density Field Theory (SDFT) - an analytical framework that allows the approximate determination of density correlations in fluctuating systems. In spite of the success of this theory to describe dilute electrolytes, its extension to concentrated solutions raises a number of technical difficulties, and requires simplified descriptions of the short-range repulsion between the ions. In this article, we discuss recent approximations that were proposed to compute the conductivity of electrolytes, in particular truncations of Coulomb interactions at short distances. We extend them to another observable (the self-diffusion coefficient of the ions) and compare them to earlier analytical approaches, such as the mean spherical approximation and mode-coupling theory. We show how the treatment of hydrodynamic effects in SDFT can be improved, that the choice of the modified Coulomb interactions significantly affects the determination of the properties of the electrolytes, and that comparison with other theories provides a guide to extend SDFT approaches in this context.

Analytical solution to incident angle quasi-phase-matching engineering for second harmonic generation in a periodic-poled lithium niobate crystal

Phase matching or quasi-phase matching (QPM) is of significant importance to the conversion efficiency of second harmonic generation (SHG) in artificial nonlinear crystals like lithium niobate (LN) crystal or microstructured nonlinear crystals like periodic-poled lithium niobate (PPLN) crystals. In this paper, we propose and show that the incident angle of pump laser light can be harnessed as an alternative versatile tool to engineer QPM for high-efficiency SHG in a PPLN crystal, in addition to conventional means of period adjusting or temperature tuning. A rigorous model is established and analytical solution of the nonlinear conversion efficiency under the small and large signal approximation theory is obtained at different incident angles. The variation of phase mismatching and walk-off length with incident angle or incident wavelength are also explored. Numerical simulations for a PPLN crystal with first order QPM structure are used to confirm our theoretical predictions based on the exact analytical solution of the general large-signal theory. The results show that the narrow-band tunable SHG output covers a range of 532 nm–552.8 nm at the ideal incident angle from 0° to 90°. This theoretical scheme, fully considering the reflection and transmission at the air-crystal interface, would offer an efficient theoretical system to evaluate the nonlinear frequency conversion and help to obtain the maximum SHG conversion efficiency by selecting an optimum incident wavelength and incident angle in a specially designed PPLN crystal, which would be very helpful for the design of tunable narrow-band pulse nanosecond, picosecond, and femtosecond laser devices via PPLN and other microstructured LN crystals.

Properties of focused Laguerre–Gaussian beam propagating in anisotropic ocean turbulence

We analyze the properties of a focused Laguerre–Gaussian (LG) beam propagating through anisotropic ocean turbulence based on the Huygens–Fresnel principle. Under the Rytov approximation theory, we derive the analytical formula of the channel capacity of the focused LG beam in the anisotropic ocean turbulence, and analyze the relationship between the capacity and the light source parameters as well as the turbulent ocean parameters. It is found that the focusing mirror can greatly enhance the channel capacity of the system at the geometric focal plane in oceanic turbulence. The results also demonstrate that the communication link can obtain high channel capacity by adopting longer beam wavelength, greater initial beam waist radius, and larger number of transmission channels. Further, the capacity of the system increases with the decrease of the mean squared temperature dissipation rate, temperature-salinity contribution ratio and turbulence outer scale factor, and with the increase of the kinetic energy dissipation rate per unit mass of fluid, turbulence inner scale factor and anisotropy factor. Compared to a Hankel–Bessel beam with diffraction-free characteristics and unfocused LG beam, the focused LG beam shows superior anti-turbulence interference properties, which provide a theoretical reference for research and development of underwater optical communication links using focused LG beams.

Phase separation in binary Bose mixtures at finite temperature

We investigate the magnetic behavior of finite-temperature repulsive two-component Bose mixtures by means of exact path-integral Monte-Carlo simulations. Novel algorithms are implemented for the free energy and the chemical potential of the two components. Results on the magnetic susceptibility suggest that the conditions for phase separation are not modified from the zero temperature case. This contradicts previous predictions based on approximate theories. We also determine the temperature dependence of the chemical potential and the contact parameters for experimentally relevant balanced mixtures.

Mechanisms of like-charge attraction in many-body systems

The physics of like-charge attraction in a collinear binary system of N particles has been investigated via a polarizable-ion model. It was found that like-charge attraction between an array of (N−1) type-A particles on the left and a single type-B particle on the right is conditional upon a sufficient degree of dissimilarities between the two types of particles. By assuming that (1) the particles of one type are strongly polarizing (but weakly polarizable) such that the Coulomb fields they produce are dominant all over other electric fields, and that (2) the particles of the other type are strongly polarizable (but weakly polarizing) such that their Coulomb-field-induced dipoles are the only polarization effects worth considering, a good agreement between our approximate analytic theory and exact numerical solutions has been obtained. Our analytic results show that the attractive components of the electrostatic force between the two types of particles could be thought of as arising from imaginary charges induced in the polarized particle(s) by the polarizing ones, resulting in a screening of the real charge(s) in the polarized particle(s). Like-charge attraction occurs if the field contributions from the imaginary charges collectively outweigh the field contribution(s) from the real charge(s).

Tunneling Current Variations in Small-Sized Devices Based on a Compact Threshold Voltage Model

Accurate modeling of threshold voltage is necessary in the integrated circuit design of strained silicon devices. Thoroughly researching the factors that affect threshold voltage and establishing a more precise threshold voltage model, can provide essential theoretical support for integrated circuit design. By solving a Poisson equation, in this paper, we demonstrate a comprehensive physical model for the threshold voltage of strained Si NMOSFETs using the gradual channel approximation theory and a quasi-two-dimensional analysis. The model investigates the physical effects such as short-channel, narrow-channel, non-uniform doping, and drain-induced barrier lowering effects on the threshold voltage. After substituting the extracted parameters into the model, a comparison was made with experimental results to validate the accuracy and correctness of the established model. Additionally, variations in the tunneling current of small-sized devices were studied. The two models provide essential references for the analysis and design of strained Si large-scale integrated circuits.