Variable Coefficients Research Articles

Generalization Limits of Data-Driven Turbulence Models

AbstractMany industrial applications require turbulent closure models that yield accurate predictions across a wide spectrum of flow regimes. In this study, we investigate how data-driven augmentations of popular eddy viscosity models affect their generalization properties. We perform a systematic generalization study with a particular closure model that was trained for a single flow regime. We systematically increase the complexity of the test cases up to an industrial application governed by a multitude of flow patterns and thereby demonstrate that tailoring a model to a specific flow phenomenon decreases its generalization capability. In fact, the accuracy gain in regions that the model was explicitly calibrated for is smaller than the loss elsewhere. We furthermore show that extrapolation or, generally, a lack of training samples with a similar feature vector is not the main reason for generalization errors. There is actually only a weak correlation. Accordingly, generalization errors are probably due to a data-mismatch, i.e., a systematic difference in the mappings from the model inputs to the required responses. More diverse training sets unlikely provide a remedy due to the strict stability requirements emerging from the ill-conditioned RANS equations. The universality of data-driven eddy viscosity models with variable coefficients is, therefore, inherently limited.

Boundary Hölder continuity of stable solutions to semilinear elliptic problems in 𝐶1,1 domains

Abstract This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions n ≤ 9 n\leq 9 , for C 1 , 1 C^{1\smash{,}1} domains. We consider equations − L ⁢ u = f ⁢ ( u ) -Lu=f(u) in a bounded C 1 , 1 C^{1\smash{,}1} domain Ω ⊂ R n \Omega\subset\mathbb{R}^{n} , with u = 0 u=0 on ∂ Ω \partial\Omega , where 𝐿 is a linear elliptic operator with variable coefficients and f ∈ C 1 f\in C^{1} is nonnegative, nondecreasing, and convex. The stability of 𝑢 amounts to the nonnegativity of the principal eigenvalue of the linearized equation − L − f ′ ⁢ ( u ) -L-f^{\prime}(u) . Our result is new even for the Laplacian, for which [X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math. 224 (2020), 2, 187–252] proved the Hölder continuity in C 3 C^{3} domains.

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.

Correction: Periodic solitons of generalized coupled nonlinear Schrödinger equations with variable coefficients

Correction: Periodic solitons of generalized coupled nonlinear Schrödinger equations with variable coefficients

Numerous Analytical Solutions to a (2 + 1)-Dimensional Schrödinger Equation with Variable Coefficients in Inhomogeneous Media

Numerous Analytical Solutions to a (2 + 1)-Dimensional Schrödinger Equation with Variable Coefficients in Inhomogeneous Media

The Impact of Unnatural Death Stigmatization on Neighboring Property Prices: Propensity Score Matching and Difference-In-Differences Analysis.

This study primarily investigated the impacts of unnatural in-house deaths on nearby property prices. Propensity score matching was used to assess the homogeneity of experimental and control groups, and the difference-in-differences (DID) method was used for their comparison. Data for real estate transactions in Taipei City from 2012 to 2022 were retrieved from the Ministry of the Interior's real estate actual transaction price registration query system. The DID interaction variable coefficient of 0.066 was significant at the 10% level. A comparison of the experimental and control groups showed a 6.8% property price difference following an unnatural in-house death. This suggests that an unnatural in-house death has a positive impact on property prices nearby. There were four unnatural deaths at the Xining Public Housing from 2003 to 2012 and four at the Jinxing Building from 1984 to 2012. The high incidence of events and the disclosure of information may have depreciated nearby property prices. The study results show that the stigmatization of unnatural in-house deaths reduces nearby property prices. The depreciation of implicit prices reverted and followed a U-shape pattern after several years.

Lie Group Classification of a Class of Variable Coefficient Boiti–Leon–Manna–Pempinelli Equations

The Boiti–Leon–Manna–Pempinelli (BLMP) equation with coefficients being functions of time is considered. Since the coefficient functions are arbitrary, we have a class of BLMP equations. Symmetry analysis is carried out for this class. We derive the equivalence group admitted by the class and we present the enhanced Lie group classification. Lie symmetries are used to construct similarity reductions. Reduction operators that are not equivalent to Lie ones are also constructed.

A comprehensive investigation into the effect of vertical component of ground motion on response of sliding unanchored blocks

This study aims to quantify the effect of the vertical component of ground motion on the seismic response of sliding rigid blocks, in terms of maximum and residual displacements. A multi-stripe analysis approach is employed to examine the response with and without the vertical component under various levels of ground motion intensity. First, a parametric study is conducted using a Coulomb friction model with a constant coefficient of friction that does not account for any dependence of the friction coefficient on velocity or pressure. It is shown that the effect of the vertical component on maximum and residual displacement is more pronounced for ground motions that can barely initiate sliding. Moreover, the influence of the vertical component on the residual sliding displacement is higher than that on the maximum displacement. Subsequently, to account for the friction coefficient dependence on pressure and sliding velocity, two interface cases (steel-steel, and concrete-concrete) are analyzed. It is found that even when a variable coefficient of friction is used, the overall trends observed for a constant coefficient of friction do not change.

Efficient second-order accurate exponential time differencing for time-fractional advection-diffusion-reaction equations with variable coefficients

Efficient second-order accurate exponential time differencing for time-fractional advection-diffusion-reaction equations with variable coefficients

Transient Thermal Analysis of a Kiln Using Different Numerical Approaches by Means of CFD

A transient study for the prediction of the temperature of hot gases inside a kiln loaded with bricks, using Computational Fluid Dynamics (CFD), is presented in this work. The prediction of transient thermal performance is obtained through four different approaches: the Properties Fitted to Temperature Polynomials (PFTP) model, the density changes according to the Boussinesq Approximation (BA) model with a Constant Thermal Expansion Coefficient (CTEC), the BA model with a Variable Thermal Expansion Coefficient (VTEC) and the density changes according to the Ideal Gas Law (IGL) model. Furthermore, the effects of a layer of ceramic fiber as insulation are investigated. The numerical results of the transient temperature of the hot gases inside the kiln were validated with experimental data. The mean relative error of the transient temperature was 5.0%, 32.6%, 26.4% and 4.4% for the PFTP model, BA model with a CTEC, BA model with a VTEC and IGL model, respectively. It was concluded that IGL model is the best approach to predict the transient temperature of the kiln. Finally, an increase in the temperature of the hot gases of 12.7% was obtained by applying the IGL model and using a layer of 0.07 m of ceramic fiber as insulation. The results of this work could be helpful to improve the design and performance of kilns in further works.

A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients

In this research, we develop an innovative and efficient numerical method for solving a nonlinear one-dimensional time-fractional convection-diffusion-reaction equation (TFCDRE) with a variable coefficient. This method integrates the Crank-Nicolson finite difference scheme with extended cubic B-spline (ExCBS) basis functions. The Caputo fractional derivative is applied for temporal discretization, while the ExCBS functions are utilized for spatial discretization. The stability of the method was discussed by the von Neumann method, which shows unconditional stability; and the convergence analysis secure an order of $O(h^2+\triangle t^{2-\alpha})$. We demonstrated the efficiency and simplicity of our method through three numerical experiments and verified its accuracy using the absolute error $(L_2)$ and maximum error $(L_\infty)$ norms in temporal and spatial dimensions. The results are also graphically represented and show substantial agreement with the approximated and exact solution. We confirmed the effectiveness and accuracy of our approach over the local discontinuous Galerkin finite element and the compact finite difference methods, respectively, from the literature.

Exact solutions of a third order nonlinear Schrodinger equation with time variable coefficients

Exact solutions of a third order nonlinear Schrodinger equation with time variable coefficients

Theoretical and numerical analysis of problems with a boundary turning point and a variable diffusion coefficient

Theoretical and numerical analysis of problems with a boundary turning point and a variable diffusion coefficient

The impact of pilot policies for low carbon city construction on the development of urban circular economy multiple time-point difference-in-differences model

IntroductionAs a means of environmental regulation, the low-carbon city pilot policy aims to address climate change and alleviate environmental resource pressures in economic development. The development of low-carbon cities often requires changes to the existing industrial and energy structures, which will have a significant impact on the regional economy.MethodsThe study is based on panel data from 273 prefecture level cities in China from 2012 to 2022, and uses the Analytic Hierarchy Process and Entropy Comprehensive Evaluation Method to construct an evaluation index system for urban circular benefits. Finally, the multiple time point differences model and mediation effect model were used to evaluate the impact and influencing factors of policies on urban economy.ResultsThe results indicate that at the 1% level, the explanatory variable coefficient significantly improves the level of urban circular economy development after adding control variables. The construction of low-carbon pilot cities has improved the performance of circular economy development by about 6.42%. The mediating effect of industrial structure rationalization is significantly positive at the 1% significance level, indicating that the construction of low-carbon pilot cities has greatly promoted the coordinated development of industrial relations within China’s low-carbon development regions.DiscussionIt can be seen that the low-carbon city pilot policy has a significant promoting effect on China’s economic development and the development of urban circular economy. Therefore, it is recommended to further expand the pilot scope of low-carbon cities and fully leverage their promoting role.

The dynamic behaviors between double-hump solitons in birefringent fibers

In this paper, we research the fractional coupled Hirota equations with variable coefficients describing the collisions of two waves in deep oceans and the propagation of ultrashort light pulses in birefringent fibers and successfully acquire the double-hump one-soliton, two-solitons and N-solitons solutions via the Hirota bilinear method. At the same time, the Bäcklund transformation and the corresponding soliton solutions are also obtained. Based on the precise forms of the solitons solutions, we gain double-hump solitons images with different shapes including U-shape, V-shape and wave-type by assigning proper functions to the group velocity dispersion and the third-order dispersion and analyze the interaction dynamics of double-hump solitons. It is worth noting that the Hirota bilinear operators involved here are fractional rather than integer, which has never appeared in previous literatures.

An efficient temporal approximation for weakly singular time-fractional semilinear diffusion-wave equation with variable coefficients

An efficient temporal approximation for weakly singular time-fractional semilinear diffusion-wave equation with variable coefficients

Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth‐Order Problems in Complex Regions

ABSTRACTIn this article, an efficient spectral Galerkin method, which is based on a mixed scheme, is proposed and studied for solving fourth‐order problems in complex regions. The fundamental idea behind this approach is to transform the initial problem into an equivalent form in cylindrical coordinates and to reshape the computational domain into a product‐type rectangular one, which facilitates the utilization of spectral methods. However, when considering the equivalent fourth‐order form directly in cylindrical coordinates, it introduces intricate pole conditions and variable coefficients, posing challenges to both theoretical analysis and algorithm implementation. To address this, we employ the orthogonality of Fourier series to further decompose it into a sequence of decoupled two‐dimensional fourth‐order eigenvalue problems. For each such problem, we introduce an auxiliary function to transform it into an equivalent second‐order coupled system. Building on this, we formulate a mixed variational formulation and discrete scheme, and prove the error estimates for eigenvalue and eigenfunction approximations. Furthermore, we extend this algorithm to the two‐dimensional complex domains. Finally, a series of numerical examples are presented, and the numerical results validate the effectiveness of the algorithm and the correctness of the theoretical results.

Investigation of oceanic wave solutions to a modified (2+1)-dimensional coupled nonlinear Schrödinger system

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized [Formula: see text]-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering.

Measurement and Validation of Market Power in China’s Log Import Trade—Empirical Analysis Based on PTM Model and AIDS Model

China is the world’s largest importer of logs, possessing the scale to exert significant influence in the international market. This paper uses a fixed-effect variable coefficient Pricing-to-Market panel model to measure China’s market power in log import trade. It also utilizes the Almost Ideal Demand System model from an elasticity perspective to explore the market behavior characteristics of various source countries in China’s log import trade, thereby validating the mechanism of market power. The results indicate that: ① China’s main trading partners can be categorized into four groups according to their market power in the log import trade. Specifically, China holds superlative market power in log imports from Indonesia, Malaysia, and Myanmar; holds strong market power in log imports from Russia, the Democratic Republic of the Congo, and Mozambique; holds weak market power in log imports from Papua New Guinea, Equatorial Guinea, France, Germany, Australia, and New Zealand; and holds no market power in Japan, Cameroon, and the United States. ② As China’s expenditure on log imports increases, there is a tendency to purchase high-quality precious wood and a greater concern for the legality of market transactions. Consequently, China is anticipated to augment its imports from source countries with no or weak market power. ③ The simple price elasticity of log imports from each source country is negative. Source countries with stronger market power tend to increase prices to achieve higher total revenue, while those with weaker market power are more inclined to lower prices to achieve the same. ④ Log products from various source countries are complementary in the Chinese market, indicating that China’s substantial demand for logs relies on the simultaneous supply from multiple countries and diverse wood types. Based on the existence or absence of market power in China’s log import trade, this paper provides targeted insights into enhancing international market power and reducing trade losses.

Note on the diffusive prey-predator model with variable coefficients and degenerate diffusion

It is of interest to understand effects of variable coefficients and degenerate diffusion on the longtime behaviors of solutions of reaction diffusion equations. Recently, Yang and Yao (2024) studied a classical prey-predator model and proved that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties. In this note we shall simplify the proof of Yang and Yao (2024) and delete the condition (4).