Linear Equations Research Articles

ON WELL-POSED BOUNDARY CONDITIONS AND ENERGY STABLE FINITE-VOLUME METHOD FOR THE LINEAR SHALLOW WATER WAVE EQUATION

Abstract We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite-volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

APOS Framework Didactize Equivalence Linear Simultaneous Equations in Senior High School Mathematics

Many students followed rote-learned procedural rules without thinking about the meaning of quadratic equations, and the Chief examiners’ reports attributed the trend to poor teaching methods. In certain instances, candidates cannot use even the conventional methods to factorize quadratics equations due to a lack of understanding of the zero-product property in graphing, factorizing, and completing the squares or quadratic formula. In the worst circumstances, the candidates failed to scaffold higher-order quadratic equations using the Conjugale and Equivalence Linear Simultaneous Equation methods. However, the action, process, object, and schema theory which came out of the constructivist learning theory and created by Dubinsky can be applied to teach mathematics. With the aid of the APOS framework, this study sought to digitize the Equivalence Linear Simultaneous Equations in senior high schools. In this mixed-method embedded design, there were 286 first-year students selected from one school. All the students received four phases of the APOS framework. The four phases were collected based on the Actions for Factorization, Processes for Quadratic Formula, Object for Conjugale, and Schema for ELSE method. The data, which was both qualitatively and quantitatively, was analyzed using IBM SPSS Statistics (Version 26) deterministic analytics software. The data collection covered four contact periods of a total duration of four hours. The results showed that students in the Actions and Processes were not statistically significant. However, the results were statistically significant at the Object and Schema phases. It was concluded that students’ learning through the APOS framework improved their academic performance. The positive effect was the triumphant mental constructions involving encapsulation and interiorization in the conjugale and ELSE methods. It was, therefore, recommended that the framework be promoted to find solutions to many other complex mathematics problems.

Explicit Group Iterative Method with Semi-Approximate Implicit Approach for Solving One-Dimensional Burgers’ Equation

This paper is considering a semi-approximate approach in investigating the Burgers’ problem. The process of the discretization of Burgers’ problem has taken place which it starts with the second-order finite difference in the discretize process together with the linearization part by using the semi-approximate implicit scheme in a way to achieve the approximation equation, thus generating the corresponding linear system equations. Besides, the Gauss-Seidel (GS), Successive Over-Relaxation (SOR) and explicit group (EG) iterative method has combined together with the SOR iterative method, namely as 4-point EGSOR (4EGSOR) has been introduced in this study for resolving the linear system. To assess the proficiency of the suggested methods on the approximation equation, the numerical test has been conducted by considering three parameters, which are computational time, iteration number and maximum absolute error. The findings indicate that the 4EGSOR method outperforms both SOR and GS iterative methods.

Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems

In this paper we present several new contributions to the oscillation theory of linear differential equations, in particular of linear Hamiltonian systems, when the traditional Legendre condition is absent. Following our recent work (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 2023), we introduce the multiplicity of a generalized right focal point and derive the corresponding local Sturmian separation theorem. We also examine the relation between the existence of finitely many generalized right focal points, or in the special case the nonexistence of generalized right focal points, with the Legendre condition. As the main tools we use new notions of the minimal multiplicities at a given point and the dual comparative index — an object from matrix analysis or differential geometry (Maslov index theory). Furthermore, we study local limit properties of the dual comparative index and the comparative index and apply them for deriving new oscillation results phrased in terms of the generalized right and left focal points. The investigation of the interplay between generalized right and left focal points leads to conditions characterizing the situation, when in the local Sturmian separation theorem the corresponding multiplicities attain the minimal possible value. This also provides a generalization of the concepts of the right and left proper focal point defined by Kratz (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 2007) to the setting, which does not impose the Legendre condition. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm–Liouville differential equations of arbitrary even order.

Multiscale Damage Identification Method of Beam-Type Structures Based on Node Curvature

This paper proposes a multiscale damage identification method for beam-type structures based on node curvature. Firstly, based on the assumption that micro-damage has little effect on stress redistribution and the basic relationship between structural bending moment and curvature, combined with the denoising function of wavelet analysis, the linear matrix equation before and after node curvature damage is solved using the singular value decomposition (SVD) method. Then, the theoretical feasibility of this method is verified with laboratory tests of a simply supported beam. Finally, the damage sensitivity and noise resistance of this method are verified using field measurements of a beam bridge. The results show that the nodal curvature serves as an indicator parameter for damage identification in beam-type structures, enabling the precise localization of damage within these structures. When utilizing a multiscale finite element model for analysis, the nodal curvature enhances the ability to identify both the location and severity of damage within small-scale elements. Furthermore, this method can provide a reference for the damage identification and health monitoring of other types of bridges.

Minimal rank weighted weak Drazin inverses

The concept of a minimal rank weak Drazin inverse for square matrices is extended to rectangular matrices. Precisely, a minimal rank weighted weak Drazin inverse is introduced and its properties are investigated. Some known generalized inverses such as the weighted Drazin inverse, the weighted core-EP inverse, and the weighted $p$-WGI are particular cases of a minimal rank weighted weak Drazin inverse. Thus, a wider class of generalized inverses is proposed. General representation forms of a minimal rank weighted weak Drazin inverse are presented as well as its canonical form. Applying the minimal rank weighted weak Drazin inverse, corresponding systems of linear matrix equations are solved and their solutions are expressed. As consequences of our results, new properties of minimal rank weak Drazin inverse are obtained.

Disentangling dynamic and stochastic modes in multivariate time series

A signal decomposition is presented that disentangles the deterministic and stochastic components of a multivariate time series. The dynamical component analysis (DyCA) algorithm is based on the assumption that an unknown set of ordinary differential equations (ODEs) describes the dynamics of the deterministic part of the signal. The algorithm is thoroughly derived and accompanied by a link to the GitHub repository containing the algorithm. The method was applied to both simulated and real-world data sets and compared to the results of principal component analysis (PCA), independent component analysis (ICA), and dynamic mode decomposition (DMD). The results demonstrate that DyCA is capable of separating the deterministic and stochastic components of the signal. Furthermore, the algorithm is able to estimate the number of linear and non-linear differential equations and to extract the corresponding amplitudes. The results demonstrate that DyCA is an effective tool for signal decomposition and dimension reduction of multivariate time series. In this regard, DyCA outperforms PCA and ICA and is on par or slightly superior to the DMD algorithm in terms of performance.

Mathematical connection skills of junior high school students in solving system of linear equations in two variables problems

Students mathematical connection skills are valuable dimension that teachers must put into consideration so that students can develop and link their mathematical skills with knowledge beyond Mathematics. This research aimed to analyze the mathematical connection skills of Junior High School students when solving the System of Linear Equations in Two Variables (Sistem Persamaan Linear Dua Variabel/SPLDV) problem. Students mathematical connection skills are valuable dimension that teachers must put into consideration so that students can develop and link their mathematical skills with knowledge beyond Mathematics. This research aims to analyze the mathematical connection skills of Junior High School students when solving the SPLDV problem. This research made use of Qualitative approach by which 18 students of grade VIII from SMP Argopuro 1 Panti Jember, East Java, participated. Mathematical connection skill test and interview guideline were conducted for this research. Data triangulation technique was used to process and compile data for the research report. The findings of the research discovered that students with high level of mathematical connection skills were able to relate and implement their knowledge to different kinds of situations, and students with medium level of mathematical connection skills were only able to apply their knowledge to simple situations, meanwhile students with low level of mathematical connection skills were unable to relate and implement their knowledge to a variety of situations.

An electro-elastic contact problem with an internal state variable

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive foundation. General models for elastic materials with piezoelectirec effect, called electro-elastic materials. This paper is devoted to the study of the model involving a frictionless contact between an electro-elastic body characterized by long memory and a conductive adhesive foundation. The process is mechanically quasistatic and electrically static. We model the material with a general nonlinear electro-elastic constitutive law with internal state variable and the contact with normal compliance and unilateral constraint, where the adhesion is taken into account and a regularized electrical conductivity condition. The main feature of these models consists of the fact that their variational formulation is given by a history-dependent quasivariational inequality for the displacement field and a linear variational equation for the electric potential. A weak formulation for the model is given in the form of a coupled system for the displacement, the stress, the electric displacement, the electric potential, the bonding and the variable state internal fields. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove its unique weak solution. The proof is based on nonlinear evolution equations with monotone operators, differential equations and Banach fixed point arguments.

Optimum Design of Cooling Structure for In-Wheel Motor Drive System

The high-efficient cooling structure is one of the key basic problems in the design of the highly integrated in-wheel motor (IWM) drive system. A 15 kW IWM drive system with the spiral cooling structure is taken as the research object in this article. First, the thermal characteristics under different conditions are analyzed based on the modeling and the heat source determination of the IWM drive system. Second, the highest temperature of insulation and permanent magnet and temperature difference of insulation are taken as optimization objectives. The inlet flow rate, the inlet and outlet diameters, the channel numbers, the channel height, and the channel ribs thickness of cooling structure are taken as optimization parameters to optimize based on the designed experiments, the multivariate linear regression equation and PSO method. Finally, the optimization results are verified by comparing the temperature field before and after optimization. The results show that the optimized structure can achieve better effect. The highest temperature of insulation, the highest temperature of permanent magnet and the temperature difference of insulation of IWM decreased by 9.27%, 9.52%, and 20.93%, respectively. This research work can provide some ideas and methods for the optimization design of cooling structure of IWM drive system.

Corrigendum to “Is the inverse method more effective than the balance method on learning linear equations? A cross-cultural experimental study between Australia and Malaysia”

Corrigendum to “Is the inverse method more effective than the balance method on learning linear equations? A cross-cultural experimental study between Australia and Malaysia”

Symbolic equation solving via reinforcement learning

Machine-learning methods are rapidly being adopted in a wide variety of social, economic, and scientific contexts, yet they are notorious for struggling with exact mathematics. A typical example is computer algebra, which includes tasks like simplifying mathematical terms, calculating formal derivatives, or finding exact solutions of algebraic equations. Traditional software packages for these purposes are commonly based on a huge database of rules for how a specific operation (e.g., differentiation) transforms a certain term (e.g., sine function) into another one (e.g., cosine function). These rules have usually needed to be discovered and subsequently programmed by humans. Efforts to automate this process by machine-learning approaches are faced with challenges like the singular nature of solutions to mathematical problems, when approximations are unacceptable, as well as hallucination effects leading to flawed reasoning. We propose a novel deep-learning interface involving a reinforcement-learning agent that operates a symbolic stack calculator to explore mathematical relations. By construction, this system is capable of exact transformations and immune to hallucination. Using the paradigmatic example of solving linear equations in symbolic form, we demonstrate how our reinforcement-learning agent autonomously discovers elementary transformation rules and step-by-step solutions.

Implementation of Problem Based Learning Model to Improve the Ability to Solve SPLDV in Grade 9 Students

This research focuses on the challenges in learning mathematics, especially in understanding and solving the system of linear equations of two variables, which is often an obstacle for students. This study aims to improve students' ability to solve SPLDV as well as increase students' motivation and engagement in learning. This study analyzed the effectiveness of Problem Based Learning (PBL) model in improving the ability to solve the System of Linear Equations of Two Variables (SPLDV) in class VIII students of SMPN 54 Surabaya. This study involved 34 students consisting of 18 male students and 16 female students using a Classroom Action Research (PTK) approach carried out in two cycles, each of which consisted of planning, action implementation, observation, and reflection stages. The instruments used were learning outcome tests, observation sheets, and student questionnaires. The results showed a significant increase in students' SPLDV problem solving ability, with the average score of students increasing from 65 in the pretest to 75 in the first cycle post-test, and reached 85 in the second cycle post-test. The proportion of students who met the Minimum Completion Criteria (KKM) increased from 50% in the first cycle to 85% in the second cycle. Further analysis showed that female students outperformed male students, with an average score of 87 and 82. Student engagement also increased, reaching 90% in the second cycle. This study concludes that the PBL model effectively improved SPLDV problem solving skills in grade 9 students and recommends its wider application in mathematics education.

An Assessment of Water Quality and Pollution Sources in a Source Region of Northwest China

China prioritizes ensuring drinking water safety, particularly in the water-scarce northwest region. This study, utilizing water quality data from 52 village and town water sources since August 2022, assesses water quality, with a specific focus on key indicators related to organic pollution sources. This study provides a scientific foundation for enhancing water quality in these sources. Employing category factor analysis for classification and grading, principal component analysis for qualitative analysis of key evaluation indicators, and the absolute principal component linear regression equation for quantitative calculation of pollution sources, this study reveals that all 52 water sources meet quality standards. Principal component analysis categorizes pollution sources as diverse types of organic compounds in surface water. Source analysis calculations highlight decay-type organic substances as major contributors to increased water color and permanganate index, with pollution contribution rates of 54.78% and 31.31%, respectively. Fecal-type organic substances dominate the increase in dissolved total solids and total coliforms, with pollution contribution rates of 56.65% and 40.16%, respectively. Additionally, high-molecular-weight organic substances exhibit lower concentrations in the water. This article presents a systematic water quality assessment methodology, which is used for the first time to qualitatively assess the types of water sources and to quantitatively trace specific sources of organic pollution in source water in northwest China. This systematic study’s results, involving initial assessment followed by traceability, recommend the adoption of a simple contact filtration and disinfection process to enhance water quality in the region.

Overview of Artificial Neural Network-Based Solution for Ordinary and Partial Differential Equations by Feed Forward Method Using Python

In the modern era, models such as Neural Networks are increasingly used to deal with ordinary and partial differential equations (ODEs-PDEs) because of their related complexity. The main aim of the paper is to focus on Artificial Neural Networks as a method to solve these equations since it is possible to approximate complex nonlinear relationships with great accuracy and adaptively reduce computational costs, thus enhancing the accuracy of solutions within highly dimensional and nonlinear problems. The traditional methods are not applicable to partial differential equations due to the problem of nonlinearity and high dimensionality. However, the Feed Forward method implemented in Python is quite a powerful alternative that can easily approximate complicated functions and very well have a hold on nonlinear equations. This research paper contains differential equations as nonlinear and linear equations and Feed Forward neural network models for both solutions of ODEs and PDEs. Python enables the integration of AI-based algorithms for the enhancement of more accurate and efficient results. This work compares the existing solutions based on ANNs and further highlights the possible advantages of using ANNs in ordinary as well as complex differential equations problems. The results indicate that ANNs have immense potential for further developing computational algorithms applicable to solving ODEs and PDEs, particularly in real-time applications that demand high precision and adaptability.

Numerical simulation of shale gas reservoir based on ILU-GMRES method

ABSTRACT The numerical simulation of shale gas reservoirs requires transforming mathematical models into large linear equation systems, and solving large linear equation systems requires a lot of time. A novel ILU-GMRES method for solving large linear equations is presented, which treats the sub matrix discretized at nodes as the basic elements of the coefficient matrix. By combining the incomplete LU decomposition method (ILU) with the generalized minimum residue method (GMRES), the discretized large linear equations are iteratively solved. A comparison was made between the new method and other methods for calculating the actual shale gas reservoir model. The results reveal that the new method has a faster calculation speed in solving linear equation systems. The calculation time of the novel ILU-GMRES method is 1029s, the calculation time of the generalized minimum residual method (GMRES) method is 2709s, the calculation time of the Gauss–Seidel method (GS) is 3786s, and the calculation time of the successive over relaxation method (SOR) method is 3386s, the calculation time of the preprocessing conjugate gradient method (PCG) is 3115s. The calculation speed of the novel method is almost 2.6 times faster than the GMRES method, the novel method is more effective in the numerical simulation of shale gas.

A Method for Analyzing Transverse Stress in Link Slabs of Simply Supported Steel–Concrete Composite Bridges

The cracking of link slabs in jointless bridges presents significant challenges due to the complexity of their stress conditions. This study focused on analyzing the transverse stresses in link slabs of jointless steel–concrete composite bridges. Utilizing linear elasticity theory and partial differential equations of plates, the deflection and stress distribution functions for the link slabs were determined. The validity of these analytical solutions was confirmed through comparisons with finite element models and load tests. Results from both the load tests and the finite element model indicate that the upper face of the girder end link slabs experiences maximum tensile stresses in both transverse and longitudinal directions. The stress values obtained from the analytical method align well with these results, showing that the total stress, when considering transverse stresses, reaches 107% of the longitudinal stresses alone. Furthermore, a 40% reduction in longitudinal girder spacing or a 50% increase in girder end length can lead to link slab stresses of 128% and 145% of the longitudinal stresses, respectively. This finding suggests that even loads lower than those designed based solely on longitudinal stresses can result in cracking. Therefore, it is recommended that transverse stresses be considered in the design of link slabs for jointless bridges. Relying solely on conventional longitudinal stress analyses may underestimate actual stress conditions and contribute to the formation of cracks.

Stabilisation of linear waves with inhomogeneous Neumann boundary conditions

ABSTRACT We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilisation of solutions with decay rates characterised by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.

Angular variation of SAR polarimetric parameters over multiyear ice

ABSTRACT Backscatter and derived parameters from Synthetic Aperture Radar (SAR) may vary across the swath as the incident angle increases from the near to the far range, especially in the ScanSAR mode. Previous studies characterize this angular variation in the form of linear regression for commonly used ice types, namely first-year (FYI) and multi-year (MYI). The present study took advantage of the appearance of three grounded multi-year ice floes, viewed at different radar incidence angles, in a series of 22 RADARSAT-2 fully-polarimetric images, acquired over the Resolute Passage in the Canadian Arctic, during the period from 21 October to 28 December 2017. The ice properties remained invariant during the observation period; hence, variation of radar parameters of MYI can be attributed to change of incidence angle only. Quantification of angular variation of the selected SAR parameters was performed. They include conventional orthogonal backscatter coefficients, derived parameters from the incoherent decomposition of Cloude-Pottier, Yamaguchi, and Touzi, as well as a set of parameters from the compact polarimetry mode. Results specify the parameters that reveal variation with incidence angle and present the linear regressions equations. The explanation for the behaviour of each parameter, in terms of showing or not showing an angular trend, is offered. Results will be useful when using the polarimetric parameters in the sea ice classification scheme.

Use of Environmental DNA to Evaluate the Spatial Distribution of False Kelpfish (Sebastiscus marmoratus) in Nearshore Areas of Gouqi Island

This study aims to explore the spatial distribution of false kelpfish (Sebastiscus marmoratus) in the mussel farming area, artificial reef areas of Gouqi Island (Shengsi, China), and natural areas using eDNA detection methods. Surface and bottom water samples were collected at 12 stations in November 2022 and April 2023, totaling 52 samples. We used species-specific primers and probes for quantitative PCR (qPCR) detection of Sebastiscus marmoratus eDNA. The eDNA concentrations differed seasonally (p < 0.05) and did not differ (p > 0.05) among the three sampling areas and two water layers. The greatest eDNA concentrations occurred in the surface layer during the spring. Higher concentrations of Sebastiscus marmoratus eDNA were also found in the mussel aquaculture area. Temperature exhibited a significant positive correlation with Sebastiscus marmoratus eDNA concentration (p < 0.05). Additionally, we developed linear equations predicting the relationship between environmental factors and environmental factors, providing a reference for future fishery resource surveys.