Gradient Of Solution Research Articles

Gradient Higher Integrability for Degenerate Parabolic Double-Phase Systems

We prove a local higher integrability result for the gradient of a weak solution to degenerate parabolic double-phase systems of p-Laplace type. This result comes with reverse Hölder type estimates. The proof is based on a careful phase analysis, estimates in the intrinsic geometries and stopping time arguments.

The inverse relationship between income and the prevalence of dementia in high-income nations

Dementia continues to be a serious problem worldwide. A negative relationship between the prevalence of dementia and the level of socioeconomic status (SES), such as income can be observed in most countries. The aim of the study in this paper is to explore the possible pathways that contribute to this negative relationship. This article analyses and presents three possible causes of health inequalities between SES gradients in dementia and possible solutions to them, through a large collection of articles. Differences in health literacy, housing levels and social support caused by SES such as different incomes lead to social inequalities in health through behavioral/cultural pathway material pathway and psychosocial pathway respectively.

Prediction and analysis of Q-markers of Elephantopus scaber based on its UPLC fingerprint, content determination of components, and in vitro a nti-tumor activity

This study aimed to provide scientific evidence for predicting quality markers(Q-markers) of Elephantopus scaber by establishing UPLC fingerprint of E. scaber from different geographical origins and determining the content of 13 major components, as well as conducting in vitro anti-cancer activity investigation of the main components. The chromatographic column used was Waters CORTECS UPLC C_(18)(2.1 mm×150 mm, 1.6 μm), and the mobile phase consisted of acetonitrile and 0.1% formic acid solution(gradient elution). The column temperature was set at 30 ℃, and the flow rate was 0.2 mL·min~(-1). The injection volume was 1 μL, and the detection wavelength was 240 nm. The UPLC fingerprint of E. scaber was fitted using the Similarity Evaluation System for Chromatographic Fingerprint of Traditional Chinese Medicine(2012 edition) to determine common peaks, evaluate similarity, identify and determine the content of major components. The CCK-8 assay was used to explore the inhibitory effect of the main components on the proliferation of lung cancer cells. The results showed that in the established UPLC fingerprint of E. scaber, 35 common peaks were identified. Thirteen major components, including neochlorogenic acid(peak 1), chlorogenic acid(peak 2), cryptochlorogenic acid(peak 3), caffeic acid(peak 4), schaftoside(peak 6), galuteolin(peak 9), isochlorogenic acid B(peak 10), isochlorogenic acid A(peak 12), isochlorogenic acid C(peak 18), deoxyelephantopin(peak 28), isodeoxyelephantopin(peak 29), isoscabertopin(peak 31), and scabertopin(peak 32) were identified and quantified, and a quantitative analysis method was established. The results of the in vitro anti-cancer activity study showed that deoxyelephantopin, isodeoxyelephantopin, isoscabertopin, and scabertopin in E. scaber exhibited inhibition rates of lung cancer cell proliferation exceeding 80% at a concentration of 10 μmol·L~(-1), higher than the positive drug paclitaxel. These results indicate that the fingerprint of E. scaber is highly characteristic, and the quantitative analysis method is accurate and stable, providing references for the research on quality standards of E. scaber. Four sesquiterpene lactones in E. scaber show significant anti-cancer activity and can serve as Q-markers for E. scaber.

On Higher Integrability of the Gradient of Solutions to the Zaremba Problem for p-Laplace Equation

On Higher Integrability of the Gradient of Solutions to the Zaremba Problem for p-Laplace Equation

Evolution Analysis of Strain Waves for the Fractal Nonlinear Propagation Equation of Longitudinal Waves in a Rod

Based on the layered and porous characteristics of functionally graded materials and the finite deformation assumption of solids, the fractal nonlinear propagation equation of longitudinal waves in a functionally graded rod is derived. A large number of exact displacement gradient traveling wave solutions of the fractal equation are obtained by using an equivalent simplified extended (G′/G) expansion method. Three sets of existing and different displacement gradient solutions are obtained by analyzing these exact solutions, and then three corresponding fractal dimension strain waves are derived. The results of numerical simulation of the evolution of these three strain waves with fractal dimension show that when the strain wave propagates in the rod, the smaller the fractal dimension or, the larger the radius of the rod, the higher the tensile strength of the material.

On the reduction in accuracy of finite difference schemes on manifolds without boundary

Abstract We investigate error bounds for numerical solutions of divergence structure linear elliptic partial differential equations (PDEs) on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of stability that guarantees the existence of a bounded solution. In many settings including the Dirichlet problem, it is easy to show that the resulting solution error is proportional to the formal consistency error of the scheme. We make the surprising observation that this need not be true for PDEs posed on compact manifolds without boundary. We propose a particular class of approximation schemes built around an underlying monotone scheme with consistency error $O(h^{\alpha })$. By carefully constructing barrier functions, we prove that the solution error is bounded by $O(h^{\alpha /(d+1)})$ in dimension $d$. We also provide a specific example where this predicted convergence rate is observed numerically. Using these error bounds, we further design a family of provably convergent approximations to the solution gradient.

A Liouville theorem of the 2-Hessian equation in half-space

In this paper, we establish the gradient and second derivative estimates for solutions to the 2-Hessian equation in half-space under certain boundary data and growth condition. We also use such estimates to obtain the Liouville theorem.

An efficient quadratic interpolation scheme for a third-order cell-centered finite-volume method on tetrahedral grids

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of the source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.

A Discontinuous Petrov-Galerkin Method for One-Dimensional Hyperbolic Conservation Law Equations Based on HWENO Limiter

The discontinuous Petrov-Galerkin method (DPG) is a new type of numerical solution to the hyperbolic conservation law equations, which distinguishes itself from the DG method by its high accuracy based on compact stencils. However, in order to overcome the unphysical oscillations of the higher order linear schemes near the large gradient solution, the DPG method often needs to incorporate limiter functions to obtain a high-resolution image of the numerical solution. This paper attempts to combine HWENO as limiter function with DPG to solve the discontinuous initial value problems for the hyperbolic conservation law equations. The single-step high-accuracy SSP Runge-Kutta method is used for time discretization, and a new HWENO-based process is used as the limiter of the RKDPG methods, which requires only one reconstruction to complete the update of the higher-order moments without calculating the linear weight coefficients. . Since the accuracy does not meet the design requirements, the HWENO limiter above is partially improved for the identification of the problem cells in the HWENO limiter above, with the original numerical solution used at the smooth solution. This paper only gives the calculation results of P1 ~P3, and the limiter is also applicable to the DPG method for higher elements. Numerical examples show that the HWENO limiter can effectively suppress non-physical oscillations in the problem cells and keep the original accuracy at the non-problem cells. The high accuracy and compactness characteristics of the DPG method are maintained. The numerical calculation solutions are efficient and accurate.

Convection in a Compressible Rotating Viscoelastic Fluid

The aim of the present work was to study the thermosolutal convection in a compressible rotatory Rivlin-Ericksen viscoelastic fluid in permeable media. Following linear stability theory and normal mode analysis method, the dispersion relation is obtained. For the case of stationary convection, the Rivlin-Ericksen viscoelastic fluid behaves like an ordinary Newtonian fluid. The compressibility, stable solute gradient and rotation are found to postpone the onset of convection, whereas medium permeability hastens or postpones the onset of convection for the case of stationary convection. Also it is found that the system is stable for and under the condition , the system becomes unstable. The case of overstability has also been considered wherein sufficient conditions for the non-existence of overstability are obtained. The stable solute gradient and rotation are found to introduce oscillatory modes in the system.

Conservative semi-Lagrangian method for continuity equation with various time steps in different parts of computational domain

The system of Navier Stokes differential equations describes gas flow near rocket nozzle. To find solution of this system in general case, scientists and engineers use numerical methods. Modern numerical methods do not allow engineers to model all features of gas flow. It happened because of sophisticated physical processes and limitations of computational hardware. So, at least where are two ways to improve it: to enlarge hardware or reduce computational complicacy. The global aim of many science investigations is to develop numerical approach with reduced computational complicacy and at the same time without loss of computational accuracy. In 1959, Aksel C. Wiin-Nielsen proposed a new and effective trajectories method for problem of numerical forecasting. In 1966, K. M. Magomedov developed similar approach (method of characteristics) for numerical modelling of space (three dimensional on space) gas flow. In 1982, O. Pironneau showed a new and effective approach for two dimensional approximation of the Navier Stokes problem. It was based on method of characteristics also. Nowadays, these methods are called semi-Lagrangian or Eulerian Lagrangian methods. They use Lagrangian nature of the transport process. To apply this advantage, scientists decompose each equation of Navier Stokes system into three parts: convective part (hyperbolic type), elliptic part and part of right-hand side. Scientists use semi-Lagrangian approach to approximate the convective parts of equations. To develop and test modern algorithms from family of semi-Lagrangian methods, we use continuity equation from the system of Navier Stokes equations. Conservative versions of semi-Lagrangian approach are based on Gauss Ostrogradsky (divergence) theorem. It allows scientists to get conservation low (balance equation) for numerical solution in norm of L1 space. Aim of our investigation is to use different time steps in different parts of computation domain. It enables us to obtain at the same time three advantages: convergence of numerical solution to exact solution, reduction of computational complicacy, implementation of conservation low (balance equation) without weight coefficients. For this purpose we decompose computational domain into two parts (subdomains) and use different time steps in them. Main complication is design algorithm in the boundaries of computational subdomains. We use one dimensional (on space) problem to demonstrate ability of developing numerical method with described advantages. Generalization of considered approach for two- or even three-dimensional cases allows engineers to model gas flow more accurately and without artificial viscosity. It is essentially important in the parts of computational domain with high level of solution gradient.

Improved training of physics-informed neural networks for parabolic differential equations with sharply perturbed initial conditions

We propose a multi-component approach for improving the training of the physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. As an example of a parabolic problem, we consider the advection–dispersion equation (ADE) with a point (Gaussian) source initial condition. In the d-dimensional ADE, perturbations in the initial condition decay with time t as t−d/2. We demonstrate that for d≥2, this decay rate can cause a large approximation error in the PINN solution. Furthermore, localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation’s residual highly inefficient. Finally, the PINN solution of parabolic equations is sensitive to the choice of weights in the loss function. We propose a normalized form of ADE where the initial perturbation of the solution does not decrease in amplitude and demonstrate that this normalization significantly reduces the PINN approximation error. Next, we present an adaptive sampling scheme based on the analytical estimate of the solution decay rate that significantly reduces the PINN estimation error for the same number of sampling (residual) points. Finally, we develop criteria for selecting weights based on the order of magnitude of different terms in the loss function. We demonstrate the accuracy of the proposed PINN model for forward, inverse, and backward ADEs.

Modelling and simulation of wind flow: A gradient method of identifying windy region

This paper presents biharmonic modelling and simulations of surface wind flow, which identify windy locales through wind speed gradients. The bulk measured and meteosat wind speed data encapsulate the wind isotachs and wind flow gradients, which are very useful in identifying windy locales. Thus, this paper presents a biharmonic wind flow model, BWFM for the development of wind isotachs and gradients to identify locales suitable for installing solar photovoltaic power plants within the study areas. The techniques include the acquisition of wind speed data (1980–2020) from the National Aeronautic and Space Administration (NASA), development of multiple BWFM solutions (free and forced) depending on the presence and absence of forcing function, respectively. The forcing function represents the topographic and orographic features of the study areas. The spatial development of isopleth of the study areas, unveiled the isotachs. The wind speed gradients were obtained by scalar computation of 2-D wind speed gradients. Comparison of forced solution with the threshold or maximum free solution engendered the identification of windy locales. The results of the model were validated against NASA data. The average wind speed threshold isotach (2.83 m/s) and wind gradient ([Formula: see text]) for the study areas (All Regions) were established by scalar computation of free solution gradients. The study areas include Northern, Eastern, Central and Western Regions recorded the following maximum forced average wind speeds (2.725, 2.755, 2.875 and 1.794 m/s, respectively) and maximum wind flow gradients (insignificant, 0.03767, 0.08469 and infinitesimal [Formula: see text], respectively). These results are useful for identifying windy locales for installation of solar and wind facilities.

Analysis of mass controlled reaction–diffusion systems with nonlinearities having critical growth rates

We analyze semilinear reaction–diffusion systems that are mass controlled, and have nonlinearities that satisfy critical growth rates. The systems under consideration are only assumed to satisfy natural assumptions, namely the preservation of non-negativity and a control of the total mass. It is proved in dimension one that if nonlinearities have (slightly super-) cubic growth rates then the system has a unique global classical solutions. Moreover, in the case of mass dissipation, the solution is bounded uniformly in time in sup-norm. One key idea in the proof is the Hölder continuity of gradient of solutions to parabolic equation with possibly discontinuous diffusion coefficients and low regular forcing terms. When the system possesses additionally an entropy inequality, the global existence and boundedness of a unique classical solution is shown for nonlinearities satisfying a cubic intermediate sum condition, which is a significant generalization of cubic growth rates. The main idea in this case is to combine a modified Gagliardo-Nirenberg inequality and the newly developed L^p-energy method in Fitzgibbon et al. (SIAM J Math Anal 53(6):6771–6803, 2021) and Morgan and Tang (Commun Contemp Math, 2022). This idea also allows us to deal with the case of discontinuous diffusion coefficients in higher dimensions, which has only recently been touched in the context of mass controlled reaction–diffusion systems.

Numerical model supplemented by thin-layer analysis for diffusiophoresis of a particle incorporating finite ion size effects

The impact of finite-sized ions on the diffusiophoresis of a charged colloid subjected to a concentration gradient of electrolyte solution consisting monovalent or multivalent ionic species, is studied. In diffusiophoresis, the ion concentration is of O(1M). In this non-dilute electrolyte solutions, the ion–ion steric interaction is important. We have adopted the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) model to account for the ion steric interactions and the Batchelor–Green expression for the relative viscosity of suspension. We have solved the standard model numerically considering ions as point charge (PNP-model), the modified Nernst–Planck equations incorporating the ion steric interaction with constant viscosity (MNP-model), and modification of the MNP-model by incorporating the viscosity variation with the ionic volume fraction (MNPV-model). Semi-analytical expressions for mobility based on a linear perturbation technique under a thinner Debye length is presented for PNP- and MNP-models. In the MNP-model, counterion saturation in the Debye layer due to the ion steric interaction enhances the surface potential by attenuating the shielding effect, diminishes the surface conduction, and magnifies the induced electric field. These in combination create a larger mobility at a thinner Debye length compared with the PNP-model. This increment in mobility attenuates when the MNPV-model is considered. The MNPV-model is more appropriate to analyze the finite ion size effects, and it is found to yield the mobility values more close to the experimental data compared with the MNP- and PNP-model. The semi-analytical expressions for mobility based on the PNP- and MNP-models agree with the corresponding exact numerical solutions when the surface potential is in the order of thermal potential. However, a large discrepancy between the simplified expression and the exact numerical results is found for a concentrated electrolyte in which the induced electric field is large.

Application of Physics-Informed Neural Networks for forward and inverse analysis of pile–soil interaction

The application of the Physics-Informed Neural Networks (PINNs) to forward and inverse analysis of pile–soil interaction problems is presented. The main challenge encountered in the Artificial Neural Network (ANN) modeling of pile–soil interaction is the presence of abrupt changes in material properties, which results in large discontinuities in the gradient of the displacement solution. Therefore, a domain-decomposition multi-network model is proposed to deal with the discontinuities in the strain fields at common boundaries of pile–soil regions and soil layers. The application of the model to the analysis and parametric study of single piles embedded in both homogeneous and layered formations is demonstrated under axisymmetric and plane strain conditions. The performance of the model in parameter identification (inverse analysis) of pile–soil interaction is particularly investigated. It is shown that by using PINNs, the localized data acquired along the pile length –possibly obtained via fiber optic strain sensing–can be successfully used for the inversion of soil parameters in layered formations.

Shape- and orientation-dependent diffusiophoresis of colloidal ellipsoids.

We present the diffusiophoresis of ellipsoidal particles induced by ionic solute gradients. Contrary to the common expectation that diffusiophoresis is shape independent, here we show experimentally that this assumption breaks down when the thin Debye layer approximation is relaxed. By tracking the translation and rotation of various ellipsoids, we find that the phoretic mobility of ellipsoids is sensitive to the eccentricity and the orientation of the ellipsoid relative to the imposed solute gradient, and can further lead to nonmonotonic behavior under strong confinement. We show that such a shape- and orientation-dependent diffusiophoresis of colloidal ellipsoids can be easily captured by modifying theories for spheres.

Analysis of a method to compute mixed-mode stress intensity factors for non-planar cracks in three-dimensions

In this work, we present and prove results underlying a method which uses functionals derived from the interaction integral to approximate the stress intensity factors along a three-dimensional crack front. We first prove that the functionals possess a pair of important properties. The functionals are well-defined and continuous for square-integrable tensor fields, such as the gradient of a finite element solution. Furthermore, the stress intensity factors are representatives of such functionals in a space of functions over the crack front. Our second result is an error estimate for the numerical stress intensity factors computed via our method. The latter property of the functionals provides a recipe for numerical stress intensity factors; we apply the functionals to the gradient of a finite element approximation for a specific set of crack front variations, and we calculate the stress intensity factors by inverting the mass matrix for those variations.

Uniform integrability in periodic homogenization of fully nonlinear elliptic equations

This paper is devoted to the study of uniform W^{1,frac{np}{n-p}}- and W^{2,p}-estimates for periodic homogenization problems of fully nonlinear elliptic equations. We establish sharp, global, large-scale estimates under the Dirichlet boundary conditions. The main novelty of this paper can be found in the characterization of the size of the “effective” Hessian and gradient of viscosity solutions to homogenization problems. Moreover, the large-scale estimates work in a large class of non-convex problems. It should be stressed that our global estimates are new even for the standard problems without homogenization.

From Uncertainties to Solutions: A Scenario-Based Framework for an Agriculture Protection Zone in Magic Valley Idaho

As growth in the western U.S. continues to lead to the development of land, pressure is being exerted on agricultural production, and could lead to the loss of prime agricultural land. A wide array of perspectives concerning agricultural protection requires a variety of possible solutions. Diverse and plausible scenarios, driven by stakeholders, can be modeled by researchers to guide potential solutions to address key challenges within a region. This paper addresses one stakeholder-defined social-ecological system (SES) solution in the context of southern Idaho, one of the fastest-growing states in the U.S.: agricultural protection zoning. This project demonstrates a method for incorporating an Agriculture Protection Zone (APZ) within a suite of scenarios showing land protection opportunities across a range of future conditions and challenges. The results, by way of a Geodesign framework, entail suitability analyses through a series of weighted raster overlays to analyze scenario-based solutions. The suite of scenario solutions was compared to demonstrate effective proportions of the APZ. The analysis of the results, as a solution gradient, aim to inform policy makers, planners, and developers about the efficiencies of various APZ delineations as well as a methodology to demonstrate the impact of solutions based on assumptions of stakeholder-informed future scenarios.