Analytical Solution Research Articles

Estimating Asset Pricing Models in the Presence of Cross-Sectionally Correlated Pricing Errors

In this study, we propose an adversarial learning approach to the asset pricing model estimation problem which aims to find estimates of factors and loadings that capture time-series covariations while minimizing the worst-case cross-sectional pricing errors. The proposed estimator is defined by a novel min-max optimization problem in which finding a solution is known to be difficult. This contrasts with other related estimators that admit a well-defined analytic solution but do not effectively account for correlations among the pricing errors. To this end, we propose an approximate algorithm based on the alternating optimization procedure and empirically demonstrate that our proposed adversarial estimation framework outperforms other existing factor models, especially when the explanatory power of the pricing model is limited.

Analyzing the chaotic and stability behavior of a duffing oscillator excited by a sinusoidal external force

This paper delves into the dynamical, chaotic, and stability aspects of the Duffing oscillator (DO) under sinusoidal external excitation, underscoring its relevance in various scientific and engineering applications. The DO, with its complex behavior, poses a significant challenge to our understanding. A unique perturbation technique, the multiple-scales (MS), is harnessed to tackle this challenge and enrich our knowledge. The analysis entails determining third-order expansions, exploring resonant cases, and factoring in the influence of viscous damping. The accuracy of the analytical solution is cross-validated with numerical results using the Runge–Kutta fourth-order (RK4). The paper also employs effective methods to assess the obtained results qualitatively. As a result, Visual figures, such as bifurcation diagrams and Lyapunov exponents’ spectra (LES), have been presented to illustrate diverse system motions and demonstrate Poincaré diagrams. Moreover, stability analysis has been investigated via resonance curves showing the stable and unstable regions. These aids facilitate our comprehension of the system’s complex behavior and its variations under different conditions. The results are elucidated through displayed curves, offering insights into the dynamics of the DO under sinusoidal excitation and damping effects. The influence of excitation and the quadratic parameter on bifurcation diagrams, LES, and Poincaré maps helps us understand the intricate behavior of nonlinear oscillatory systems, such as DO. The presented oscillator is a compelling example of elucidating the nonlinear behavior observed in numerous engineering and physics phenomena.

Analytical Solutions of the Schrödinger Equation for Two Confined Particles with the van der Waals Interaction

Analytical Solutions of the Schrödinger Equation for Two Confined Particles with the van der Waals Interaction

The Improved tan (\(\frac{\Phi(\xi)}{2}\)) -Expansion Method for The Non-dissipative Double Dispersive Equation in Microstructured Solids

In microstructured solids, the non-dissipative double dispersive equation is a fourth-order non-linear partial differential equation that arises in the study of non-dissipative strain wave propagation. Seeking the exact solution of a nonlinear partial differential equations with a physical background is helpful to understand the motion law of matter and to explain the corresponding physical phenomena scientifically. This equation has been studied widely, and there have been a lot of research methods to find exact solutions to this equation. In this manuscript, we try to study the non-dissipative double dispersive equation by the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. As far as our known, no one has used this method to study the non-dissipative double dispersive equation, partly because of the complicated calculation. We obtain a lot of exact traveling wave solutions, including hyperbolic function solutions, trigonometric function solutions, exponential function solutions, and rational function solutions. Compared with the other methods, some solutions obtained in this manuscript are consistent with existing solutions, which shows the effectiveness of the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. Through this method, Our manuscript obtains more general and new exact solutions. These solutions may play an important role in engineering and physics. What's more, we plot the 3D graphs of some of the solutions obtained in this manuscript, which helps us understand the physical phenomenon of the non-dissipative double dispersive equation. In the future, we will continue to explore its physical significance from the new analytical solution we have obtained.

Theoretical and experimental study of diffraction by a thin cone

A problem of diffraction of ultrasound acoustic waves by an acute-angled hard cone is studied. Within the framework of the parabolic equation method, an analytical solution of the problem for an arbitrarily located point source is built. Specifically, the problem is reduced to the Volterra boundary integral equation, which can be solved using the Fourier transform. An experimental measurement of the diffracted field is carried out. The experiment is based on the MLS method adapted for narrowband sound sources. A comparison of experimental and theoretical results is provided.

Enhanced Attractive Force via Torsion Spring as Rotatable Tip Structure in Beam-Structure Electrostatic Chucks

This research aims to replace ball joints of beam-structure electrostatic chucks (BSESC), which are limited by issues such as structural friction and inconsistent operation, thus enhancing the applicability in industrial settings. Torsion springs are used as rotational degree-of-freedom (RDOF) mechanisms at the tip of BSESCs to reduce friction and improve repeatability, while also highlighting the positive relationship between tip rotatability and the attractive force generated. Following the law of conservation of energy, an analytical calculation was performed using a beam-spring-capacitor model to establish the correlation between tip rotatability and the maximum attractive force generated through energy system analysis. An analytical solution was derived, explicitly defining the relationship between tip rotatability and the achievable attractive force. The results were plotted using MATLAB. Experimental validations were carried out by fabricating devices that differed only in tip rotatability, and measurements of the attractive force were taken. Both analytical and experimental results clearly demonstrate a positive relationship between tip rotatability and attractive force generation in BSESCs. The use of torsion springs effectively reduces the energy concentration at the tip, delaying detachment while simultaneously increasing the attractive force, significantly improving the overall performance of BSESCs. These findings provide a foundation for future industrial applications, particularly for automated large-area manipulation of irregularly shaped targets in industries such as screen production and semiconductors.

Propagation of Hydraulic Fractures and Natural Fractures: The Bypassing Behavior in 3D Space

Summary The interaction between natural fractures (NFs) and hydraulic fractures (HFs) in 3D space poses significant challenges to numerical simulation. The interaction behaviors between HFs and NFs in 3D space include crossing, offset, stopping, and bypassing. Many existing numerical simulators are 2D, which inherently limits their ability to account for the vertical growth of fractures. Consequently, they are unable to model the bypassing behavior effectively. In this paper, a mathematical model for the propagation of HFs and NFs in a naturally fractured reservoir is established. The displacement discontinuity method (DDM) is used to solve the rock deformation, while the finite difference method (FDM) is utilized to solve the fluid flow within fractures. The undirected graph structure is used to represent the complex fracture network, and a dynamic adjustment of grid connectivity method is implemented to describe the process by which HFs bypass NFs. By integrating with graphics processing unit (GPU) computing, an efficient 3D simulator for HFs and NFs propagation is developed. The accuracy is verified against analytical solutions and reference solutions. Subsequently, a series of numerical studies on the bypassing behavior are conducted. The primary findings are as follows: (1) The simulator can accurately capture the interactions between HFs and NFs in 3D space, including crossing, arresting, and bypassing behaviors. (2) The bypassing behavior is characterized by three distinct processes—intersection, height growth of HF, and bypassing. (3) During the intersection stage, both the injection pressure and maximum NF width increase. During the height growth stage, the pressure is relatively stable, while the maximum NF width continues to increase. These two stages together result in the “storage” phenomenon. Once the bypassing behavior occurs, both the pressure and the maximum NF width decrease sharply, leading to a “release” phenomenon. Additionally, the “storage” and “release” phenomena may impact proppant transport. (4) Given the presence of bypassing behavior, it is essential to consider the NFs in 3D fracture simulators. The simulator and its findings can provide valuable insights for field design.

Analytical solution of the Sommerfeld–Page equation

The Sommerfeld–Page equation describes the non-relativistic dynamics of a classical electron modeled by a sphere of finite size with a uniform surface charge density. It is a delay differential equation, and almost no exact solution of this equation was known until recently. However, progress has been made, and an analytical solution was recently found for an almost identical delay differential equation, which arose in the context of the mathematical modeling of the COVID-19 epidemics. Inspired by this research, we offer a pedagogical exposition of how one can find an analytical solution of the Sommerfeld–Page equation.

Large time solution for collisional breakage model: Laplace transformation based accelerated homotopy perturbation method

The behavior of several particulate processes, such as cell interaction, blood clotting, bubble formation, grain breakage, and cheese formation from milk, have been studied using coagulation and fragmentation models (Fogelson and Guy, 2008 [1]; Pazmiño et al., 2022 [2]; Chen et al., [3]). Various studies utilize the linear fragmentation model to simplify the underlying physics. However, in real-life scenarios, particles form due to the collision of two particles, leading to a non-linear collisional breakage model. Unfortunately, the collisional breakage model is less explored due to its complex behavior. While analytical solutions are difficult to compute and are still missing in the literature, this article proposes an approximate solution for the model using the Laplace-based accelerated homotopy perturbation method. Further, coupling with Padé approximant, the accuracy of the solution is extended for the longer time. Considering various physically relevant kernels, the approximate series solutions are compared with the well known finite-volume solutions to measure the accuracy in terms of qualitative and quantitative errors. The article also encompasses theoretical convergence analysis and error estimations to enhance comprehension of the proposed formulation.

Effects of electric and magnetic fields in magnetic mixing in electroosmotic flows

In this paper, mixing efficiency in electroosmotic flow is numerically simulated at the electric double layer region in the presence of a magnetic field. An incompressible and laminar model is adopted to numerically solve the governing equations involving the magnetic field, electric potential fields, concentration distribution for positive and negative ions (Nernst–Planck), and species concentration. To validate the numerical study, an ideal electroosmotic flow with charged walls is simulated and the results are compared with the analytical solution. For the case with a Reynolds number of 0.02, microchannel length of 40 μm, the results show that by applying a magnetic field, the mixing efficiency along the microchannel length increases from 60.5% to 79.67%. It is found that the existence of a magnetic field in the electric double layer has a significant impact on pressure distribution along the microchannel wall, however, its effects on the electric fields (internal and external), the distribution of positive and negative ions, and the net electrical charge density are marginal. In addition, the presence of magnetic field creates two relatively large vortices inside the microchannel. The outcomes of the present study will help to improve mixing efficiency in micro-devices with applications in micro-analysis systems and lab-on-a-chip instruments.

Interactions of localized wave and dynamics analysis in the new generalized stochastic fractional potential-KdV equation.

In this paper, we investigate the new generalized stochastic fractional potential-Korteweg-de Vries equation, which describes nonlinear optical solitons and photon propagation in circuits and multicomponent plasmas. Inspired by Kolmogorov-Arnold network and our earlier work, we enhance the improved bilinear neural network method by using a large number of activation functions instead of neurons. This method incorporates the concept of simulating more complicated activation functions with fewer parameters, with more diverse activation functions to generate more complex and rare analytical solutions. On this basis, constraints are introduced into the method, reducing a significant amount of computational workload. We also construct neural network architectures, such as "2-3-1," "2-2-3-1," "2-3-3-1," and "2-3-2-1" using this method. Maple software is employed to obtain many exact analytical solutions by selecting appropriate parameters, such as the superposition of double-period lump solutions, lump-rogue wave solutions, and three interaction solutions. The results show that these solutions exhibit more complex waveforms than those obtained by conventional methods, which is of great significance for the electrical systems and multicomponent fluids to which the equation is applied. This novel method shows significant advantages when applied to fractional-order equations and is expected to be increasingly widely used in the study of nonlinear partial differential equations.

Periodic orbits of solar sails in the Sun-Earth system

Abstract This paper studies the periodic orbits of solar sails using reflectivity control devices (RCDs) in the circular Sun-Earth system. Significantly, the dynamical equations about the solar sail are given separately in the rotational coordinate system and adjoint coordinate system. Analytical solutions about the periodic orbits of solar sails are given. So as to gain the periodic orbits of solar sails, we discuss the conditions of the attitude angles and reflectivity modulation ratio of solar sails. This paper simulates some periodic orbits of solar sails and analyzes the stability of a periodic orbit of a solar sail by using the Floquet theory. The outcomes show that it is unstable for the periodic orbit of a solar sail. Based on the linear quadratic regulator (LQR), this paper designs a control algorithm to achieve tracking in the target orbit of solar sails.

Influence of plasma density gradient on the tearing mode with the poloidal shear flow

The influence of the plasma density gradient on the m/n = 2/1 and m/n = 4/1 (m is the poloidal mode number and n is the toroidal mode number) tearing mode instability with poloidal flow and flow shear is investigated. Using the analytical solution that we obtained in a previous work and mainly focused on the factors of plasma density and poloidal shear flow, we found that the plasma density has a stabilizing effect on the classical tearing mode, and the poloidal equilibrium flow can intensify this beneficial effect. The density gradient was detrimental to the stability of the tearing mode. The effects of both density and density gradient are slight, but the effect of poloidal flow on the plasma density is significant. Considering that the plasma density changes with the poloidal flow, the values of the tearing mode stability index ∆′ clearly change. Our investigation also found that compared with the negative flow shear, the positive flow shear is beneficial to the stability of the tearing mode, and a larger poloidal flow shear has a better stabilizing effect on the classical tearing mode.

A rapid method for composition tracking in hydrogen-blended pipeline using Fourier neural operator

Blending hydrogen into natural gas for transportation is a crucial approach for achieving the widespread utilization of hydrogen. Tracking the concentration of the hydrogen within the pipeline is important for monitoring gas quality and managing pipeline operations. This study develops a rapid computational model to predict the hydrogen and natural gas concentrations within the pipeline during transportation based on the Fourier Neural Operator (FNO), an operator neural network capable of learning the differential operator in the partial differential equation. In the proposed model, the numerical method is employed to generate datasets, with the spline interpolation used to enhance data smoothness. The initial and boundary conditions are taken as the inputs to accommodate varying transportation scenarios. Comparison results indicate that the proposed model can notably reduce the time needed to predict the hydrogen and natural gas concentrations while maintaining prediction accuracy. The accuracy of the proposed model is validated by comparing its calculated results with the analytical solution and the concentrations of hydrogen and natural gas within the pipeline under two transportation scenarios, with relative errors of 0.49%, 0.31%, and 0.45%, respectively. Notably, the trained model demonstrates strong grid invariance, a type of model generalization. Trained on data generated from a coarse grid of 101 × 41 spatial-temporal resolution, the proposed model can accurately predict results on a fine grid of 401 × 81 spatial-temporal resolution with a relative error of only 0.38%. Regarding the prediction efficiency, the proposed model achieves an average 17.7-fold speedup compared to the numerical method. The positive results indicate that the proposed model can serve as a rapid and accurate solver for the composition transport equation.

Digital rock physics: Calculation of effective elastic properties of heterogeneous materials using graphical processing units (GPUs)

An application based on graphical processing units (GPUs) applied to 3-D digital images is described for computing the linear anisotropic elastic properties of heterogeneous materials. The application can also retrieve the property contribution tensors of individual inclusions of any shape. The code can be executed on professional GPUs as well as on a basic laptop or personal computer Nvidia GPUs. The application is extremely fast: a calculation of the effective elastic properties of volumes consisting of about 7 million voxel elements (1913) takes less than 4 s of computational time using a single A100 GPU; 3 min for 100 million voxel elements (4793) using a single A100 GPU; 14 min for 350 million voxel elements (7033) using a single A100 GPU. Several comparisons against analytical solutions are provided. In addition, an evaluation of the anisotropic effective elastic properties of a 3-D digital image of a cracked Carrara marble sample is presented. The software can be downloaded from a permanent repository Zenodo, the link with a doi is given in the manuscript.

Computer model coupling hemodynamics and oxygen transport in the coronary capillary network: Pulsatile vs. non-pulsatile analysis

Background and Objective:Oxygen transport in the heart is crucial, and its impairment can lead to pathological conditions such as hypoxia, ischemia, and heart failure. However, investigating oxygen transport in the heart using in vivo measurements is difficult due to the small size of the coronary capillaries and their deep embedding within the heart wall. Methods:In this study, we developed a novel computational modeling framework that integrates a 0-D hemodynamic model with a 1-D mass transport model to simulate oxygen transport in/across the coronary capillary network. Results:The model predictions agree with analytical solutions and experimental measurements. The framework is used to simulate the effects of pulsatile vs. non-pulsatile behavior of the capillary hemodynamics on oxygen-related metrics such as the myocardial oxygen consumption (MVO2) and oxygen extraction ratio (OER). Compared to simulations that consider (physiological) pulsatile behaviors of the capillary hemodynamics, the OER is underestimated by less than 9% and the MVO2 is overestimated by less than 5% when the pulsatile behaviors are ignored in the simulations. Statistical analyses show that model predictions of oxygen-related quantities and spatial distribution of oxygen without consideration of the pulsatile behaviors do not significantly differ from those that considered such behaviors (p-values >0.05). Conclusions:This finding provides the basis for reducing the model complexity by ignoring the pulsatility of coronary capillary hemodynamics in the computational framework without a substantial loss of accuracy when predicting oxygen-related metrics.

Analytical properties and solutions for the synchronous pulsating bubble clusters distributed on a spherical surface

The present work concerns with the nonlinear dynamics for the synchronous pulsating bubble clusters uniformly distributed on a spherical surface. First, the governing equation for such clusters with 4/6/8/12/20 coupled bubbles are established. Second, the maximum and minimum radii for the gas-filled bubble clusters are analyzed according to the first integral. Third, by introducing suitable nonlocal transformations, two novel equivalent parametric analytical solutions in the form of Weierstrass elliptic function are constructed for the gas-filled bubble clusters for a specific polytropic exponent κ=3/2 without considering the surface tension, and based on which we immediately derive the parametric analytical solution for the corresponding vapor bubble clusters. Further, to consider the case of arbitrary polytropic exponent and surface tension, we develop a direct approach to construct the parametric analytical solution using Jacobi elliptic function for gas-filled bubble clusters. It is shown that, the behaviors and results for the bubble clusters will degenerate to the corresponding ones for single bubbles as the radius of the bubble cluster approaches infinity. In addition, on the basis of the analytical results, dynamic properties and motion laws of the bubble clusters are also discussed.

Analytical solutions of ray-tracing equations in generalized elliptical anisotropy

Ray-tracing in anisotropic media is pivotal for interpreting observed seismic data and creating high-resolution images of subsurface structures, which are crucial in exploration geophysics. Elliptical anisotropy, a simplified model that approximates a transversely isotropic medium, is particularly relevant for geologic settings such as shale formations or stressed sedimentary layers where directional dependencies of seismic velocities are pronounced. We develop an analytical solution of the ray-tracing equations for a 2D inhomogeneous and anisotropic medium, where velocities depend elliptically on direction and increase linearly with depth — a scenario frequently encountered in stratified geologic formations. Unlike previous studies that assume constant ellipticity throughout the medium, our approach allows for variations in ellipticity, providing a more flexible and realistic representation of subsurface anisotropy. The phase velocities along the x- and z-axes are not necessarily multiples of each other at every point, offering a generalized version of the elliptical anisotropy. This enhancement may enable more accurate predictions and interpretations of observed seismic data, particularly in complex exploration scenarios. The analytical solution yields expressions for the ray paths and the wavefront normals. By setting the normals of the wavefront at the seismic source point and the location of the seismic source as the initial conditions in phase space, we explore the evolution of these wavefront normal curves across different types of the elliptical anisotropy. Our innovative approach includes plotting the evolution of wavefront normal curves on the generalized momentum coordinate plane of the phase space — that commonly overlooked in traditional models focused only on position coordinates.

Calibration and implementation of a design tool for drag anchors in clay

Abstract Drag anchors are considered as one of the most applicable anchor solutions for floating offshore wind as they can be used for a wide range of soil conditions. For feasibility studies of floating offshore wind farms different anchor concepts needs to be compared in order to select the most proper anchor solution. To do so, the level of details in the analysis for the various anchor types must be similar. Currently design charts are adopted as a simple design method for drag anchors. However, design charts are developed for specific soil conditions, and do not account for site specific soil conditions. To have similar precision, as analytical solutions for suction anchors, a simple design methodology that can account for site specific soil conditions for drag anchors must be adopted. This paper presents an example for how a simple analytical solution can be calibrated against design charts, which makes it possible to account for site specific soil conditions, resulting in a better basis for comparing early stage drag anchor design to other anchor solutions.

Slip-mediated Axisymmetric Bending of Multilayered Structures with Different Interfaces: Analytical Solutions and Design Guidelines

Slip-mediated Axisymmetric Bending of Multilayered Structures with Different Interfaces: Analytical Solutions and Design Guidelines