Partial Differential Research Articles

High order Morley elements for biharmonic equations on polytopal partitions

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H2 norm and the usual L2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension.

Quantifying the contributions of climatic and human factors to vegetation net primary productivity dynamics in East Africa

As an important part of the terrestrial ecosystem, vegetation dynamics are subject to impacts from both climate change and human activities. Clarifying the driving mechanisms of vegetation variation is of great significance for regional ecological protection and achieving sustainable development goals. Here, net primary productivity (NPP) was used to investigate the spatiotemporal variability of vegetation dynamics from 2000 to 2020 in East Africa, and its correlations with climate factors. Furthermore, we utilized partial derivatives analysis and set up different scenarios to distinguish the relative contributions of climatic and human factors to NPP changes. The results revealed that NPP exhibited a significant increase with 4.16 g C/m2/a from 2000 to 2020 in East Africa, and an upward trend was detected across 71.06% of the study area. The average contributions of precipitation, temperature, and solar radiation to the NPP inter-annual variations in East Africa were 2.02, −1.09, and 0.31 gC⋅m–2⋅a–1, respectively. Precipitation made the greatest positive contribution among all of the climatic factors, while temperature made strong negative contributions. The contributions of climate change and human activities to NPP changes were 1.24 and 2.34 gC⋅m–2⋅a–1, respectively. Moreover, the contribution rate of human activities to NPP increase was larger than that of climate change, while the role of climate change in NPP decrease was larger than that of human activities. The findings of the study can provide new evidence for a deeper understanding of ecosystem stability and carbon cycling in East Africa, as well as a reference for decision-making and scientific support for ecological environmental protection.

Mathematical models for analysis of temperature regimes in vehicle braking systems

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in the braking systems of vehicles, which are geometrically depicted as isotropic spatial heat-active media that are subject to internal local thermal heating, have been developed. With the use of classical methods, it is not possible to obtain analytical solutions of linear and nonlinear boundary value problems of mathematical physics in a closed form. This is especially the case when the right-hand sides of differential equations with partial derivatives and boundary conditions are discontinuous functions. The given approach is based on the application of the apparatus of generalized functions to describe the local concentration of thermal influence. This made it possible to apply the integral transformation and, on this basis, to obtain analytical solutions of both linear and nonlinear boundary value problems. In the case of a nonlinear boundary value problem, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a linearized second-order differential equation with partial derivatives and boundary conditions with a discontinuous right-hand side were obtained. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. On the basis of the developed mathematical models, a computational algorithm was created and on this basis, software tools were created, using which the heat exchange processes in the middle of the brake structures for the selected materials of the brake pads were analyzed in terms of their effectiveness, as well as the determination of the optimal temperature values for the effective operation of the braking system of vehicles. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active media with internal heating make it possible to analyze their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and individual elements, but also the entire structure.

Limit theorems for local polynomial estimation of regression for functional dependent data

<p>Local polynomial fitting exhibits numerous compelling statistical properties, particularly within the intricate realm of multivariate analysis. However, as functional data analysis gains prominence as a dynamic and pertinent field in data science, the exigency arises for the formulation of a specialized theory tailored to local polynomial fitting. We explored the intricate task of estimating the regression function operator and its partial derivatives for stationary mixing random processes, denoted as $ (Y_i, X_i) $, using local higher-order polynomial fitting. Our key contributions include establishing the joint asymptotic normality of the estimates for both the regression function and its partial derivatives, specifically in the context of strongly mixing processes. Additionally, we provide explicit expressions for the bias and the variance-covariance matrix of the asymptotic distribution. Demonstrating uniform strong consistency over compact subsets, along with delineating the rates of convergence, we substantiated these results for both the regression function and its partial derivatives. Importantly, these findings rooted in reasonably broad conditions that underpinned the underlying models. To demonstrate practical applicability, we leveraged our results to compute pointwise confidence regions. Finally, we extended our ideas to the nonparametric conditional distribution, and obtained its limiting distribution.</p>

Универсальный метод Монте-Карло для процессов Леви и его экстремумов

The article proposes a universal approach to constructing Monte Carlo methods for pricing options with payoffs depending on the joint distribution of the final position of the Lévy process X_T and its infimum J_T (or supremum S_T). We derive approximate formulas for the conditional cumulative distribution functions of the Lévy process "P"(X_T<x|S_T=y) (P(X_T<x|J_T=y)), which are expressed through the partial derivative with respect to y of the joint cumulative distribution function "P"(X_T<x,S_T<y) (P(X_T<x,J_T<y)) and the density of the infimum (or supremum) at the final moment of time. By applying the Laplace transform to the joint cumulative distribution function of the Lévy process and its extremum, we use the approximate Wiener–Hopf factorization to represent the image of its partial derivative. By inverting the Laplace transform using the Gaver–Stehfest algorithm, we find the desired conditional cumulative distribution function. The developed algorithm for simulating the joint position of the Lévy process and its extremum at a given point in time consists of two key stages. At the first stage, we simulate the extremum value of the L´evy process based on the approximation of its cumulative distribution function "P"(S_T<x) (or P(J_T<x)). In the second step, we simulate the final value of the Lévy process based on the approximation of the conditional cumulative distribution function of the final position of the Lévy process relative to its extremum. The universality of the Monte Carlo method we developed lies in the implementation of a uniform approach for a wide class of Lévy processes, in contrast to classical approaches, when simulations are essentially based on the features of the probability distribution associated with the simulated random process or its extrema. In our approach, it is enough to know the characteristic exponent of the Lévy process. The most time-consuming computational unit for simulating a random variable based on a known cumulative distribution function can be effectively implemented using neural networks and accelerated through parallel computing. Thus, on the one hand, the approach we propose is suitable for a wide class of Lévy models, on the other hand, it can be combined with machine learning methods.

A MULTIPOINT BOUNDARY VALUE PROBLEM IN TIME FOR A $2B$-PARABOLIC EQUATION WITH DEGENERACY

One of the most important issues in the general theory of differential equations with partial derivatives is establishing the solvability of boundary value problems. Among the boundary value problems for equations with partial derivatives, problems with nonlocal boundary conditions occupy an important place. Such interest in such problems is caused both by their rich practical application (the process of diffusion, moisture distortion in soils, plasma physics, etc.), and by the needs of the general theory of boundary value problems. A general multipoint boundary value problem for nonuniformly $2b$-parabolic equations with degeneracy is studied. The coefficients of parabolic equations and boundary conditions allow power degeneracy of arbitrary order in terms of time variable and spatial variables at some set of points. To solve the given multipoint boundary value problem, solutions of problems with smooth coefficients in Hölder spaces with the appropriate norm are studied. With the help of interpolation inequalities and a priori estimates, estimates of the solution of auxiliary problems and their derivatives in special Gelder spaces are established. Using the theorems of Ross and Archel, a convergent sequence is distinguished from the compact sequence of solutions of the auxiliary problems, the limiting value of which is the solution of the multipoint boundary value problem in time for the $2b$-parabolic equation with degeneracy. Estimates of the solution of the given problem are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of features of the coefficients of the equations and the boundary conditions. With certain restrictions on the right-hand side of the equation and boundary conditions, an integral image of the given problem is obtained.

Analyzing the structure of solutions for weakly singular integro-differential equations with partial derivatives

<p>In this work, we analyze the approximate solution of a specific partial integro-differential equation (PIDE) with a weakly singular kernel using the spectral Tau method. It present a numerical solution procedure for this PIDE, which is transferred into a Volterra–Fredholm integral equation (VFIE), and the spectral method is performed on VFIE. In some illustrated examples, we show that the VFIE problem has high numerical stability with respect to the original form of the PIDE problem. For this aim, we apply the spectral Tau method in two cases, first for the problem in the form of VFIE and then also for the problem in the form of PIDE. The remarkable numerical results obtained from the VFIE problem form compared to those gained from the PIDE problem form show the efficiency of the proposal method. Also, we prove the convergence theorem of the numerical solution of the Tau method for the VFIE problem, and then it is generalized to the PIDE problem.</p>

On analytical study of heat transfer phenomenon in special-shape soldering iron

The work presents the results of the analytical solution of the phenomenon of heat transfer in soldering iron of a special shape. For analysis, differential equations in partial derivatives from many variables were used, where initial and boundary conditions were given, according to experimental data. Partial equations were obtained in the process of solving by the method of separating variables. The resulting finite function of coordinate variables in the cylindrical coordinate system and time is derived from the original equation.

Gyroscopically coupled in-plane and transverse vibrations of an annular free-clamped microplate

In this work, we construct and study a model of coupled plane-transverse oscillations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertia forces caused by the rotation of the system along an axis located in the plane of the plate. Equations of vibrations in partial derivatives are obtained using the variational principle of Hamilton - Ostrogradsky. Assuming the smallness of the angular velocity of rotation with respect to the frequency of the working skew-symmetric flexural form of the plate oscillations, an approximate analytical solution is found for both the radial and circumferential, and transverse components of the displacement field in the free oscillation mode. Using the Galerkin projection method, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric modes of the plate vibrations. It is found that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into the regime of amplitude-modulated beats. Analytical expressions are found both for the frequency of the slow beat envelope and for the relative depth of their amplitude modulation. The fundamental possibility of determining the modulus of the projection of the angular velocity vector onto the plane of the plate from the measured value of the envelope frequency is shown. The problem of choosing the optimal geometric shape of the resonator from the point of view of maximizing the sensitivity of the system to changes in the value of the angular velocity of rotation is studied. The question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of the amplitude modulation of the beat regime is considered.

Discussion on the development of sports industry in China based on the perspective of “low carbon economy”

Abstract Low-carbon economy is an important part of national strategic objectives, which not only refers to an ecological and environmental economic model but also is the main direction for the development of the sports industry. Based on the characteristics and components of the low carbon economy, this paper utilizes the environmental Kuznets curve and decoupling theory to set the goal of low carbon economy development in the sports industry and explore the influence mechanism of low carbon economy on the structural development of the sports industry. Starting from the perspective of the development of the sports industry structure, a multi-period double-difference model is designed to calculate the net effect of a low-carbon economy on the sports industry structure. The relationship between the low carbon economy and the structural change of the sports industry is visualized using non-linear fitting and partial derivatives. The evaluation model of sports industry development capacity is constructed to evaluate the development of the sports industry from two perspectives: economic performance and economic value added. The experimental results show that the impact of a low carbon economy on the structure of the sports industry is divided into three stages, namely, coefficient value <0.15, coefficient value in [0.15, 0.25), and coefficient value ≤0.25. The low-carbon economy has a marginal positive effect on the sports industry in the second stage. The calculation of EVA value shows that the EVA value of the subject sports enterprises is greater than 0 from 2017 to 2023, which indicates that the sports industry realizes value creation along with the development of the low-carbon economy.

Метод приближенного решения уравнений в частных производных

The article considers a partial differential equation of the form ∂u/∂t=f(t,x,y,u,∂u/∂x,∂u/∂y,(∂^2 u)/(∂x^2 ),(∂^2 u)/(∂y^2 ),(∂^2 u)/∂x∂y),(x,y)∈D⊂R^2,t≥0, with respect to an unknown function u, defined in a domain D of spatial variables x,y and for t≥0. A method for finding an approximate solution is proposed. The equation under consideration is replaced by an approximate one by introducing the shift operator S:D→D, which allows replacing at each step of the calculations the unknown values of the function u(x,y,t) on the right side with the values u(S(x,y),t), obtained at the previous step. The idea of the proposed method goes back to the idea of the Tonelli method, known for differential equations with respect to functions of one variable (with ordinary, not partial derivatives). The advantages of the proposed method are the simplicity of the obtained iteration relation and the possibility of application to a wide class of equations and boundary conditions. In the article, iteration formulas are obtained for solving a boundary value problem with the Dirichlet condition for spatial variables and with an initial or boundary condition for the variable t. Based on the proposed method, an approximate solution is obtained for a specific initial-boundary value problem for the heat conductivity equation in a square domain.

Rethinking Attentive Object Detection via Neural Attention Learning.

Visual attention advances object detection by attending neural networks to object representations. While existing methods incorporate empirical modules to empower network attention, we rethink attentive object detection from the network learning perspective in this work. We propose a NEural Attention Learning approach (NEAL) which consists of two parts. During the back-propagation of each training iteration, we first calculate the partial derivatives (a.k.a. the accumulated gradients) of the classification output with respect to the input features. We refine these partial derivatives to obtain attention response maps whose elements reflect the contributions to the final network predictions. Then, we formulate the attention response maps as extra objective functions, which are combined together with the original detection loss to train detectors in an end-to-end manner. In this way, we succeed in learning an attentive CNN model without introducing additional network structures. We apply NEAL to the two-stage object detection frameworks, which are usually composed of a CNN feature backbone, a region proposal network (RPN), and a classifier. We show that the proposed NEAL not only helps the RPN attend to objects but also enables the classifier to pay more attention to the premier positive samples. To this end, the localization (proposal generation) and classification mutually benefit from each other in our proposed method. Extensive experiments on large-scale benchmark datasets, including MS COCO 2017 and Pascal VOC 2012, demonstrate that the proposed NEAL algorithm advances the two-stage object detector over state-of-the-art approaches.

Zhang-Zhang Polynomials of Phenylenes and Benzenoid Graphs

The aim of this paper is to study some variations of the Zhang-Zhang polynomial for phenylenes, which can be obtained as special cases of the multivariable Zhang-Zhang polynomial. Firstly, we prove the equality between the first Zhang-Zhang polynomial of a phenylene and the generalized Zhang-Zhang polynomial of some benzenoid graph, which enables us to prove also the equality between the first Zhang-Zhang polynomial and the generalized cube polynomial of the resonance graph. Next, some results on the roots of the second Zhang-Zhang polynomial of phenylenes are provided and another expression for this polynomial is established. Finally, we give structural interpretation for (partial) derivatives of different Zhang-Zhang polynomials.

A Bus Privacy Preserving Decentralized Power Flow Algorithm Considering Neighbor Partial Derivative Information

A Bus Privacy Preserving Decentralized Power Flow Algorithm Considering Neighbor Partial Derivative Information

Reinforcement Learning-Based Adaptive Optimal Control for Nonlinear Systems With Asymmetric Hysteresis.

This article investigates the adaptive optimal tracking problem for a class of nonlinear affine systems with asymmetric Prandtl-Ishlinskii (PI) hysteresis nonlinearities based on actor-critic (A-C) learning mechanisms. Considering the huge obstacles arising from the uncertainty of hysteresis nonlinearity in actuators, we develop a scheme for the conflict between the construction of Hamilton functions and hysteresis nonlinearity. The actuator hysteresis forces the input into a hysteresis delay, thus preventing the Hamilton function from getting the current moment's input instantly and thus making optimization impossible. In the first step, an inverse model is constructed to compensate for the hysteresis model with a shift factor. In the second step, we compensate for the control input by designing a feedback controller and incorporating the estimation and approximation errors into the Hamilton error. Optimal control, the other part of the actual control input, is obtained by taking partial derivatives of the Hamiltonian function after the nonlinearities have been circumvented. At the end, a simulation is given to validate the developed solution.

Solving first order fuzzy partial differential equations using fuzzy Aboodh transform

In this article, we extend the concept of the fuzzy Aboodh transform to the solution of fuzzy partial differential equations using Hukuhara differentiability. In order to create the solution to first-order fuzzy partial differential equations, we first provide the fuzzy Aboodh transform and then present the basic definitions and theorems for the fuzzy Aboodh transform of fuzzy partial derivatives. Finally, the method is illustrated with an example to show the ability of the proposed method.

The Nelder–Mead Method-Based Improved Parameter Estimation of Single-Phase Induction Motors

Abstract This work presents a comprehensive method for precisely determining the equivalent circuit characteristics of single-phase induction motors (SPIM), including both direct and indirect steps. First, a DC test, a no-load test and a locked-rotor test are performed to ascertain the primary electrical characteristics of both the main and auxiliary windings. Next, the indirect phase consists of iteratively modifying the mechanical characteristics, such as the inertial moment and friction factor, in a motor simulation model in Simulink until they match the previously determined electrical parameters. In addition, the motor parameter estimate process can be improved by applying the Nelder–Mead optimisation approach, which eliminates the need to calculate partial derivatives of a cost function. The study also applies the scalar control to the SPIM. Ultimately, the efficacy of the suggested methodology is confirmed through a comparison of simulated and actual outcomes.

REDUCTION OF A LINEAR MATRIX OPERATOR OF DIFFERENTIATION IN MANY VARIABLES TO AN OPERATOR WITH AN ORDINARY DIFFERENTIATION OPERATOR

Abstract. In the theory of equations with a linear partial differential operator of the first order, the question of differentiation and integration of vector functions is associated with considering them alternately along the characteristics of the operator, and the representation of their integrals by characteristics in the form of functions of many variables depended on the characteristic first integrals of the operator. The difficulty in this matter was to manage the characteristics and the first integrals for each of their variables so that they were in their places. Such control was implemented by projectors having pairwise ductile properties. The main results of this study are: 1) the use of the reduction method to reduce the matrix operator in partial derivatives to an operator with ordinary differentiations for one variable and 2) the use of projectors in matters of the composition of functions from many variables and the characteristics of the operator for each of the variables. These novelties of the research have a serious perspective.

Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions

<p>In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.</p>

Enhancing Angular Stability in Magnitude and Phase of Frequency Selective Slabs Using Partial Derivatives Expansion Based on Equivalent Medium Theory

Enhancing Angular Stability in Magnitude and Phase of Frequency Selective Slabs Using Partial Derivatives Expansion Based on Equivalent Medium Theory