Second-order Terms Research Articles

Chelyshkov polynomials method for distributed-order time fractional nonlinear diffusion-wave equations

This work deals with the distributed-order time fractional nonlinear diffusion-wave equations. These equations are generated by replacing the first- and second-order time derivative terms with the distributed-order fractional derivative terms. The distributed-order fractional derivatives used in these problems are in the Caputo sense. The Chelyshkov polynomials as a well-known family of basis functions are used to develop a spectral collocation method for these problems. Through the way, some operational matrices regarding the classical and distribute-order fractional derivatives for these polynomials are extracted. In the proposed method, after approximating the solution of the problem in terms of the Chelyshkov polynomials and employing the expressed matrices, solving the primary equations transforms into solving algebraic systems which can be easily solved. Some numerical examples are studied to show the adequacy of the approach.

Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

Nonlinear acoustic propagation in low-density media

Nonlinear acoustic propagation in the atmosphere is usually modeled using an augmented Burgers equation accounting for atmospheric absorption and weak nonlinearity. However, the weak-nonlinearity assumption may not apply to acoustic signals propagating from the lower atmosphere that are subsequently refracted downward from the upper atmosphere (e.g., stratosphere and thermosphere) due to the decreasing air density with increasing altitude [Lonzaga, et al., Geophys. J. Int. 200(3), 1347–1361]. Consequently, this paper discusses the effects of a strong nonlinearity that lead to an amplitude-dependent increase in signal propagation speed. This propagation speed is obtained using a perturbation expansion where the leading-order term is simply the small-amplitude sound speed while the first-order term is the existing expression that gives rise to waveform steepening and stretching. Furthermore, the second-order term is proportional to the spectral density of the acoustic signal and is inversely proportional to the local density of the medium. Consequently, for an impulsive signal such as a sonic boom or an infrasound, the second-order effect causes a dispersion of the signal similar to the observed dispersion of acoustic signals from supersonic Concorde as well as from large explosions. This paper also discusses the extension of these results in general low-density propagation media.

Nekhoroshev estimates for the orbital stability of Earth’s satellites

We provide stability estimates, obtained by implementing the Nekhoroshev theorem, in reference to the orbital motion of a small body (satellite or space debris) around the Earth. We consider a Hamiltonian model, averaged over fast angles, including the J_2 geopotential term as well as third-body perturbations due to Sun and Moon. We discuss how to bring the Hamiltonian into a form suitable for the implementation of the Nekhoroshev theorem in the version given by Pöschel, (Math Z 213(1):187–216, 1993) for the ‘non-resonant’ regime. The manipulation of the Hamiltonian includes (i) averaging over fast angles, (ii) a suitable expansion around reference values for the orbit’s eccentricity and inclination, and (iii) a preliminary normalization allowing to eliminate particular terms whose existence is due to the nonzero inclination of the invariant plane of secular motions known as the ‘Laplace plane’. After bringing the Hamiltonian to a suitable form, we examine the domain of applicability of the theorem in the action space, translating the result in the space of physical elements. We find that the necessary conditions for the theorem to hold are fulfilled in some nonzero measure domains in the eccentricity and inclination plane (e, i) for a body’s orbital altitude (semimajor axis) up to about 20 000 km. For altitudes around 11 000 km, we obtain stability times of the order of several thousands of years in domains covering nearly all eccentricities and inclinations of interest in applications of the satellite problem, except for narrow zones around some so-called inclination-dependent resonances. On the other hand, the domains of Nekhoroshev stability recovered by the present method shrink in size as the semimajor axis a increases (and the corresponding Nekhoroshev times reduce to hundreds of years), while the stability domains practically all vanish for a>20{,}000 km. We finally examine the effect on Nekhoroshev stability by adding more geopotential terms (J_3 and J_4) as well as the second-order terms in J_2 in the Hamiltonian. We find that these terms have only a minimal effect on the domains of applicability of Nekhoroshev theorem and a moderate effect on the stability times.

Hyperspectral subpixel target detection based on interaction subspace model

In this paper, we examine the problem of detecting subpixel targets in hyperspectral images. The so-called subpixel target refers to a target that only occupies a part of a pixel due to the low spatial resolution of hyperspectral sensors. Considering that the subpixel target spectrum is not always reliable (e.g., due to spectral variability), an interaction subspace model is designed to deal with this problem. In this subspace model, the second-order interaction terms are introduced to better describe the spectral variability, thereby improving the robustness. Specifically, the subspace model uses a hyperplane in a high-dimensional space to model spectral variability, while traditional models (e.g., the additive model and the replacement model) use a line in the high-dimensional space to model spectral variability. Obviously, the stronger description ability of the hyperplane makes the subspace model more tolerant to the mismatch of the target spectrum. Based on this interaction subspace model, we derive adaptive detectors according to the one-step generalized likelihood ratio test and its two-step variant. Experiments conducted on hyperspectral data demonstrate that the proposed two-step detector exhibits the strongest robustness in cases where the target spectrum is not very reliable.

A Design of All-Optical Integrated Linearized Modulator Based on Asymmetric Mach-Zehnder Modulator

We proposed an all-optical linearized modulator model suitable for an integration platform. The linearized modulator is formed by an asymmetric Mach–Zehnder modulator and a carrier-separated waveguide. We focus on the suppression of both intermodulation distortion (IMD3) and second-order harmonic distortions (SHD). As a result, the third-order nonlinear terms of IMD3 and second-order nonlinear terms of SHD are entirely suppressed, leading to a spurious free dynamic range of IMD3 (SFDRIMD3) improvement of ~14 dB and SFDRSHD improvement of ~3 dB compared with a standard MZM model. The influence of large RF input signals, losses, and fabrication deviations are also discussed to prove the feasibility of the model.

Scanning gate microscopy in graphene nanostructures

The conductance of graphene nanoribbons and nanoconstrictions under the effect of a scanning gate microscopy tip is systematically studied. Using a scattering approach for noninvasive probes, the first- and second-order conductance corrections caused by the tip potential disturbance are expressed explicitly in terms of the scattering states of the unperturbed structure. Numerical calculations confirm the perturbative results, showing that the second-order term prevails in the conductance plateaus, exhibiting a universal scaling law for armchair graphene strips. For stronger tips, at specific probe potential widths and strengths beyond the perturbative regime, the conductance corrections reveal the appearance of resonances originated from states trapped below the tip. The zero-transverse-energy mode of an armchair metallic strip is shown to be insensitive to the long-range electrostatic potential of the probe. For nanoconstrictions defined on a strip, scanning gate microscopy allows to get insight into the breakdown of conductance quantization. The first-order correction generically dominates at low tip strength, while for Fermi energies associated with faint conductance plateaus, the second-order correction becomes dominant for relatively small potential tip strengths. In accordance with the spatial dependence of the partial local density of states, the largest tip effect occurs in the central part of the constriction, close to the edges. Nanoribbons and nanoconstrictions with zigzag edges exhibit a similar response as in the case of armchair nanostructures, except when the intervalley coupling induced by the tip potential destroys the chiral edge states.

Bifurcation Control for a Ship Maneuvering Model with Nonsmooth Nonlinearities

We consider a widely used form of models for ship maneuvering, whose nonlinearities entail continuous but nonsmooth second-order modulus terms. For such models bifurcations of straight motion are not amenable to standard center manifold reduction and normal forms. Based on a recently developed analytical approach, we nevertheless determine the character of local bifurcations when stabilizing the straight motion course with standard proportional control. For a specific model class we perform a detailed analysis of the linearization to determine the location of these bifurcations in the control parameter space and its dependence on selected design parameters. By computing the analytically derived characteristic parameters, we find that “safe” supercritical Andronov–Hopf bifurcations are typical. Through numerical continuation we provide a more global bifurcation analysis, which identifies the arrangement and relative location of stable and unstable equilibria and periodic orbits.

Thin-shell wormhole under non-commutative geometry inspired Einstein–Gauss–Bonnet gravity

Einstein–Gauss–Bonnet gravity is a generalization of the general relativity to higher dimensions in which the first- and second-order terms correspond to general relativity and Einstein–Gauss–Bonnet gravity, respectively. We construct a new class of five-dimensional (5D) thin-shell wormholes by the ‘Cut–Paste’ technique from black holes in Einstein–Gauss–Bonnet gravity inspired by non-commutative geometry starting with a static spherically symmetric, Gaussian mass distribution as a source and for this structural form of the thin-shell wormhole we have explored several salient features of the solution, viz., pressure–density profile, equation of state, the nature of wormhole, total amount of exotic matter content at the shell. We have also analyzed the linearized stability of the constructed wormhole. From our study we can assert that our model is found to be plausible with reference to the other model of thin-shell wormhole available in literature.

Local Adaptiveness of Mixed Higher Order Partial Differential Equations and Its Application in Image Denoising

Background: Image denoising methods based on partial differential equations have attracted much attention due to their "infinite" local adaptation capabilities, high flexibility, and strong mathematical theoretical support. Methods: This paper proposes a mixed higher order partial differential equation denoising model for the step effect caused by the second-order denoising model and the edge blur caused by the fourth-order denoising model. The model combines the second-order and fourth-order terms based on the relationship between the variational energy minimization and the partial differential equations. The fourth-order term is used to remove noise in the uniform area of the image to avoid the step effect, and the second-order term is used at the edge to avoid boundary blur. Results: Theoretical analysis and numerical experiment results show that the proposed model has weak solutions and can effectively avoid the step effect and maintain the edge. Conclusion: The image denoising results of the model are better than those of other improved denoising models in subjective effect, and objective evaluation indicators, such as SNR, PSNR, and MSSIM.

Model position sensing method for low fineness ratio models in a magnetic suspension and balance system.

A new model-position-sensing method for the levitation of models with a low fineness ratio (ratio of the longitudinal length to the diameter) in a magnetic suspension and balance system (MSBS) is proposed. MSBS is an ideal model-support device for wind-tunnel testing, which enables the study of flow fields around blunt bodies without flow disturbances introduced by mechanical support devices, with the aerodynamic forces determined from the magnetic forces using a pre-calibrated relationship. The new method allows wind tunnel experiments without mechanical supports with a low fineness ratio model. This method adopts two line sensors placed parallel to the central axis of the model image and measures the position with a resolution finer than 0.06mm or degeven for thin model geometries. In addition, measurement errors were reduced by correcting a second-order term in the depth direction of the camera. A low fineness ratio circular cylinder model was levitated following sensor calibration. The model was supported in conditions with and without freestream flow. This position measurement method was also applied to a reentry capsule model. The model was levitated while keeping its position and attitude stabilized near the origin.

Towards Tropically Counting Binodal Surfaces

We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor plans to recover the count of multinodal curves. We then prove that for $$\delta =2$$ or 3 nodes, tropical surfaces with unseparated nodes contribute asymptotically to the second-order term of the polynomial giving the degree of the family of complex projective surfaces in $${\mathbb {P}}^3$$ of degree d with $$\delta $$ nodes. We classify when two nodes in a surface tropicalize to a vertex dual to a polytope with 6 lattice points, and prove that this only happens for projective degree d surfaces satisfying point conditions in Mikhalkin position when $$d>4$$ .

Particle-size segregation in self-channelized granular flows

Geophysical mass flows such as debris flows, dense pyroclastic flows and snow avalanches can self-channelize on shallow slopes. The confinement afforded by formed levees helps to maintain the flow depth, and hence mobility, allowing self-channelized flows to run out significantly farther than unconfined, spreading flows. Levee formation and self-channelization are strongly associated with particle-size segregation, but can also occur in monodisperse flows. This paper uses the monodisperse depth-averaged theory of Rochaet al.(J. Fluid Mech., vol. 876, 2019, pp. 591–641), which incorporates a hysteretic friction law and second-order depth-averaged viscous terms. Both of these are vital for the formation of a travelling wave that progressively deposits a pair of levees just behind the front. The three-dimensional velocity field is reconstructed in a frame moving with the front assuming Bagnold flow. This enables a bidisperse particle-size segregation theory to be used to solve for the large and small particle concentrations and particle paths in three-dimensions, for the first time. The model shows that the large particles tend to segregate to the surface of the flow, forming a carapace that extends over the centre of the channel, as well as along the external sides and base of the levee walls. The small particles segregate downwards, and are concentrated in the main channel and in the inner levee walls. This supports the contention that a low-friction channel lining provides a secondary mechanism for run-out enhancement. It is also shown that the entire theory scales with particle diameter, so experiments with millimetre-sized particles provide important insights into geophysical-scale flows with boulders and smaller rock fragments. The model shows that self-channelization does not need particle-size segregation to occur, but supports the hypothesis that particle-size segregation and the associated frictional feedback can significantly enhance both the flow mobility and the levee strength.

Improving T2*-weighted human cortico-spinal acquisitions with a dedicated algorithm for region-wise shimming

Cortico-spinal fMRI acquisitions aim to investigate direct interactions between brain and spinal cord, e.g. during motor output or pain processing, by covering both regions in a single measurement. Due to their large distance and location in the body, a dynamic shim update of constant and linear shim terms is required when using echo-planar imaging (EPI) to achieve reasonable image quality in both target regions. A previously presented approach with region-wise shim settings is based on a standard single-region shim algorithm and suffers from (i) non-optimal shim settings because it combines linear and second-order shim terms optimized for different volumes, and (ii) significant user interactions making it rather cumbersome, time consuming, and error-prone. Here, a dedicated (“CoSpi”) shim algorithm for cortico-spinal fMRI is presented that performs joint optimization of static second-order shim terms and one set of linear and constant shim terms for each region in a single run and with minimal user interaction. Field map and T2*-weighted EPI measurements were performed on a clinical 3 T whole-body MR system in water phantoms and five healthy volunteers using the conventional region-wise and CoSpi shim settings as well as “gold standard” shim settings optimized for one of the target regions only. With CoSpi shim settings, (i) overall field inhomogeneity was reduced by about 65% / 75% (brain / spinal cord volume) compared to the conventional region-wise approach and in vivo was within 5% of the values obtained with the single-volume shim settings, (ii) geometric distortions derived from voxel displacement maps were reduced on average by about 35% / 70%, (iii) the temporal SNR determined from an EPI time series that may reflect the impact of through-slice dephasing, was increased by about 17% / 10%, and (iv) the variation of the mean field between slices, a measure targeting the predisposition to insufficient fat saturation and GRAPPA-related ghosting artifacts, was reduced by about 90% / 45%. Thus, the presented algorithm not only speeds up and simplifies the shim procedure considerably, but also provides a better field homogeneity and image quality, which both could help to significantly improve the applicability of cortico-spinal fMRI.

A Bias-Reduced Solution for Multistatic Localization Using Differential Delays and Doppler Shifts

In this paper, the multistatic localization problem with unknown propagation speed is investigated using differential delays and Doppler shifts between the signals from direct and indirect paths. A series of pseudo-linear equations are formulated via the transformation of measurement models. A weighted least squares (WLS) formulation is then proposed after ignoring the second-order error terms, which can be rewritten as a non-convex optimization problem with the relationships among variables included as constraints. To deal with the non-convexity of the problem, semidefinite relaxation is applied, resulting in a convex semidefinite program (SDP). Several reasonable second-order cone constraints constructed via basic inequality and Cauchy-Schwarz inequality are added to tighten the relaxed SDP problem. By preserving the second-order error terms in equations, the bias of the estimate from the WLS formulation is also derived and then subtracted to nearly eliminate the bias and reach a bias-reduced solution. Simulation results show that the mean square error (MSE) of the proposed method approaches the Cramer-Rao lower bound (CRLB) and the bias is reduced significantly.

A Linearized Alternating Direction Method of Multipliers Algorithm for Prestack Seismic Inversion in VTI Media Using the Quadratic PP-Reflectivity Approximation

Periodic horizontal thin interlayers, or formations developing horizontally layered fractures in a homogeneous background, are commonly studied as a transverse isotropy with a vertical axis of symmetry (VTI) media. Anisotropy needs to be considered in predicting such special reservoirs, and can be inferred from the amplitude variation with offset (AVO) by the exact or linear reflectivity equations. However, the linear approximation is usually derived based on the assumptions of weak anisotropy and weak contrast, the accuracy suffers when the anisotropy or the contrast of two adjacent media increases. In addition, due to the high nonlinearity of the Graebner equation, the stability and efficiency of the exact inversion are limited. To address these issues, we propose a robust pre-stack seismic inversion for VTI media using quadratic PP-reflectivity approximation. Firstly, by deriving quadratic approximations for phase velocity and polarization, we analyze the advantage of the second-order terms at a higher-contrast interface. Then, based on perturbation theory, we derive a quadratic PP-reflectivity approximation from the exact equation. It consists of first- and second-order terms with isotropic and anisotropic components, and the accuracy is significantly improved even with enhanced anisotropy. Finally, we introduce the Hadamard product operator to linearly characterize the objective function, and develop an improved Alternating Direction Method of Multipliers (ADMM) algorithm to handle the nonlinearity of the quadratic equations. The feasibility and stability of the proposed method are tested on two datasets, that is, the logging curves used as numerical experiments and a field dataset from a shale reservoir.

Implementation of Effective Scale Factor Matching and Axial Centripetal Error Compensation for a Rotating Accelerometer Gravity Gradiometer

Rotating accelerometer gravity gradiometer (RAGG) is the only gravity gradient measuring device in the world to achieve commercial applications. When the parameter consistency among multiple accelerometers of the RAGG is not sufficient, its combined output cannot effectively suppress common-mode noise in the environment. A method for suppressing RAGG’s linear and angular motion noise is presented in this article by projecting the parameters, including scale factor, misalignment angle, second-order terms, and cross-coupling terms, onto the Cartesian coordinate system of the turntable plane. Proposing the algorithm implementation scheme of three effective scale factor solver loops, and for further canceling the motion residual error, high-precision gyroscope compensation loops are introduced. The added error compensation loop obtains the angular acceleration by differencing the angular velocity measured by the gyroscope, which in turn adjusts the loop parameters. Experimental results show that with the proposed method, the noise suppression of linear motion common-mode acceleration reaches 6.5 orders of magnitude, that is, the noise level <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.7 \times {10^{ - 8}} {\mathrm{ g/}}\sqrt {{\mathrm{Hz}}} $ </tex-math></inline-formula> of the accelerometer at the rotation frequency, and the angular motion common-mode acceleration noise is suppressed by six orders of magnitude, and the noise level <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6.8 \times {10^{ - 8}} {\mathrm{ g/}}\sqrt {{\mathrm{Hz}}} $ </tex-math></inline-formula> of the accelerometer at the swing frequency is also obtained.

Accurate non-covalent interaction energies on noisy intermediate-scale quantum computers via second-order symmetry-adapted perturbation theory.

The calculation of non-covalent interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, the use of the standard supermolecular method with the variational quantum eigensolver (VQE) would require extremely precise resolution of the total energies of the fragments to provide for accurate subtraction to the interaction energy. Here we present a symmetry-adapted perturbation theory (SAPT) method that may provide interaction energies with high quantum resource efficiency. Of particular note, we present a quantum extended random-phase approximation (ERPA) treatment of the SAPT second-order induction and dispersion terms, including exchange counterparts. Together with previous work on first-order terms (Chem. Sci., 2022, 13, 3094), this provides a recipe for complete SAPT(VQE) interaction energies up to second order, which is a well established truncation. The SAPT interaction energy terms are computed as first-level observables with no subtraction of monomer energies invoked, and the only quantum observations needed are the VQE one- and two-particle density matrices. We find empirically that SAPT(VQE) can provide accurate interaction energies even with coarsely optimized, low circuit depth wavefunctions from a quantum computer, simulated through ideal statevectors. The errors of the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions. In addition, we present heme-nitrosyl model complexes as a system class for near term quantum computing simulations. They are strongly correlated, biologically relevant and difficult to simulate with classical quantum chemical methods. This is illustrated with density functional theory (DFT) as the predicted interaction energies exhibit a strong sensitivity with respect to the choice of functional. Thus, this work paves the way to obtain accurate interaction energies on a NISQ-era quantum computer with few quantum resources. It is the first step in alleviating one of the major challenges in quantum chemistry, where in-depth knowledge of both the method and system is required a priori to reliably generate accurate interaction energies.

Bounding the photon mass with the dedispersed pulses of the Crab pulsar and FRB 180916B

Tight limits on the photon mass have been set through analyzing the arrival time differences of photons with differentfrequencies originating from the same astrophysical source. However, all these constraints have relied on usingthe first-order Taylor expansion of the dispersion due to a nonzero photon mass. In this work, we present an analysisof the nonzero photon mass dispersion with the second-order derivative of Taylor series. If the arrival time delay correctedfor all known effects (including the first-order delay time due to the plasma and photon mass effects) is assumed to bedominated by the second-order term of the nonzero photon mass dispersion, a conservative upper limit on the photon mass can be estimated. Here we show that the dedispersed pulses with the second-order time delays from the Crab pulsar andthe fast radio burst FRB 180916B pose strict limits on the photon mass, i.e.,mγ,2 ≤ 5.7 × 10-46 kg ≃ 3.2 × 10-10 eV/c 2and mγ,2 ≤ 6.0 × 10-47 kg ≃ 3.4 × 10-11 eV/c 2, respectively.This is the first time to study the possible second-order photon mass effect.

Wavelength scaling of strong-field Rydberg-state excitation: Toward an effective S -matrix theory

$S$-matrix theory and the Born expansion, the lowest-order term of which is known as the ``strong-field approximation'' (SFA), play an indispensable role in our understanding of atomic and molecular processes in intense laser fields. Most phenomena in this field are reproduced by the first two terms of the Born series. However, for a long-range potential such as the Coulomb potential, the second-order term may be larger than the SFA term, raising the problem of the convergence of the Born series. By simultaneously measuring and simulating ionization and Rydberg-state excitation of an argon atom subject to a strong laser field for various wavelengths, we demonstrate that the wavelength scaling law of the measured ratio of ${\mathrm{Ar}}^{*}$ over ${\mathrm{Ar}}^{+}$ and the period of its oscillation with respect to the laser intensity can be well reproduced by the second-order term of the $S$-matrix expansion in terms of the Coulomb potential, but not by the lowest-order (SFA) term. We conjecture that the second-order term of the $S$-matrix expansion may provide an effective theory for the intense-laser--atom interaction.