Problems For Linear Equations Research Articles

A Transfer-Learning-Like Neural Dynamics Algorithm for Arctic Sea Ice Extraction

Sea ice plays a pivotal role in ocean-related research, necessitating the development of highly accurate and robust techniques for its extraction from diverse satellite remote sensing imagery. However, conventional learning methods face limitations due to the soaring cost and time associated with manually collecting sufficient sea ice data for model training. This paper introduces an innovative approach where Neural Dynamics (ND) algorithms are seamlessly integrated with a recurrent neural network, resulting in a Transfer-Learning-Like Neural Dynamics (TLLND) algorithm specifically tailored for sea ice extraction. Firstly, given the susceptibility of the image extraction process to noise in practical scenarios, an ND algorithm with noise tolerance and high extraction accuracy is proposed to address this challenge. Secondly, The internal coefficients of the ND algorithm are determined using a parametric method. Subsequently, the ND algorithm is formulated as a decoupled dynamical system. This enables the coefficients trained on a linear equation problem dataset to be directly generalized to solve the sea ice extraction challenges. Theoretical analysis ensures that the effectiveness of the proposed TLLND algorithm remains unaffected by the specific characteristics of various dataset. To validate its efficacy, robustness, and generalization performance, several comparative experiments are conducted using diverse Arctic sea ice satellite imagery with varying levels of noise. The outcomes of these experiments affirm the competence of the proposed TLLND algorithm in addressing the complexities associated with sea ice extraction.

Two-Point Boundary-Value Problem for Linear Stochastic Differential Equations

Two-Point Boundary-Value Problem for Linear Stochastic Differential Equations

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Two-point boundary-value problems for differential equations with generalized piecewise-constant argument

UDC 517.9 We consider a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. The new general solution, boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of Cauchy problems for linear ordinary differential equations on the subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem.

Unique Solvability of the Zaremba Problem for Linear Second Order Elliptic Equations with Drift

Unique Solvability of the Zaremba Problem for Linear Second Order Elliptic Equations with Drift

Uniqueness results for inverse source problems for semilinear elliptic equations

We study inverse source problems associated to semilinear elliptic equations of the form Δux+ax,u=Fx on a bounded domain Ω⊂Rn , n⩾2 . We show that it is possible to use nonlinearity to recover both the source F and the nonlinearity a(x,u) simultaneously and uniquely for a class of nonlinearities. This is in contrast to inverse source problems for linear equations, which always have a natural (gauge) symmetry that obstructs the unique recovery of the source. The class of nonlinearities for which we can uniquely recover the source and nonlinearity, includes a class of polynomials, which we characterize, and exponential nonlinearities.For general nonlinearities a(x,u) , we recover the source F(x) and the Taylor coefficients ∂uka(x,u) up to a gauge symmetry. We recover general polynomial nonlinearities up to the gauge symmetry. Our results also generalize results of Feizmohammadi and Oksanen (2020 J. Differ. Equ. 269 4683–719), Lassas et al (2020 Rev. Mat. Iberoam. 37 1553–80) by removing the assumption that u≡0 is a solution. To prove our results, we consider linearizations around possibly large solutions.Our results can lead to new practical applications, because we show that many practical models do not have the obstruction for unique recovery that has restricted the applicability of inverse source problems for linear models.

A new approach to the fractional Abel k−integral equations and linear fractional differential equations in a fuzzy environment

This paper introduces some recent approaches to the fuzzy fractional Abel integral equations with respect to another function, which is called the fuzzy fractional Abel k−integral equations. The problem proposed here allows for the interpolation of different types of fuzzy classical fractional Abel integral equations, and the solvability of each type of integral equation is also discussed. Additionally, a new approach to the linear fuzzy fractional differential equations in the space of generalized fuzzy functions is introduced. Unlike previous approaches with gH-differentiability, this approach avoids making any prior assumptions about the solution's monotonic behavior during the solving process. It also overcomes the problem of multiple solutions for fuzzy fractional differential equations. To demonstrate the benefits of this approach, two initial value problems of linear fuzzy differential equations are compared using two different concepts of fractional derivatives: the Caputo fractional gH-differentiability and the Caputo fractional derivative in the generalized fuzzy space. Furthermore, some applications are provided in a viscoelastic material model and a population growth with harvesting model.

An inverse source problem for a pseudoparabolic equation with memory

<abstract><p>This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine <italic>lsqnonlin</italic>. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.</p></abstract>

Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems

In this article, we present a new algorithmic implementation of exact three-point difference schemes for a certain class of singular Sturm–Liouville problems. We demonstrate that computing the coefficients of the exact scheme at any grid node xj requires solving two auxiliary Cauchy problems for the second-order linear ordinary differential equations: one problem on the interval [xj−1,xj] (forward) and one problem on the interval [xj,xj+1] (backward). We have also proven the coefficient stability theorem for the exact three-point difference scheme.

A study of the multipoint boundary value problem for nonlinear differential equations of arbitrary fractional order based on the fixed point theorem for composite nonlinear operators

Abstract The existence and distinctiveness of solutions, as well as the iterative approximations of the distinctive answer to the initial boundary value issue of nonlinear differential equations, may be solved with the help of the theory of nonlinear operators, which offers a strong theoretical guarantee and a fundamental tool. In this work, we examine the fixed point theory for composites nonlinear operators, which may offer a fresh approach to solving linear differential equations having arbitrary order fractional multipoint boundary value problems. The existing result of the boundary value issue’s solution is confirmed by the fixed point theorem for composites nonlinear operators, and the four-point boundary value problem for fractional-order linear equations of the Riemann-Liouville type is examined. A group of fractional-order nonlinear differential equations with deviating quantities are examined to examine the presence of an unusual positive solution for this boundary value problem. The existence of this unique positive solution is determined by the use of both the mixed monotone operator and the combined nonlinear operator fixed point theorem. Ultimately, the nonlinear Bagley-Torvik equation with a fractional order variable coefficient four-point boundary value problem is converted into a Fredholm-Hammerstein integral the formula of the second type with weakly singular or continuous kernels, and the fixed point principle is used to demonstrate the uniqueness of the solution in the space of continuous functions. Approximate solutions are provided for the second class of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels. The outcomes demonstrate the applicability and validity of the fixed point theorem to composite nonlinear operators, offering a fresh approach to solving the multipoint boundary value issue for nonlinear differential equations of any fractional order.

Model order reduction for optimality systems through empirical gramians

In the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE) discretization. This leads to a specific linear but high-dimensional time variant (LTV) dynamical system. To reduce the size of the LTV system, we apply a tailored reduced order modeling technique based on empirical gramians and derived directly from the first-order optimality system. For testing purpose, we focus on two specific examples: a multiobjective optimization and a closed-loop optimal control problem. Our proposed methodology results to be better performing than a standard proper orthogonal decomposition (POD) approach for the above mentioned examples.

Impulsive Stability for Positive Linear Differential-Algebraic Equations With Time-Varying Delay

This paper deals with the impulsive stability problem for positive linear differential-algebraic equations (DAEs) with a time-varying delay. First, a sufficient condition for the positivity of the considered system is addressed. Then, a new exponential stability criterion for positive linear DAEs with a time-varying delay is obtained based on a suitable change of variables. According to our results, a stable impulse-free system can keep its original stability property under certain destabilizing impulsive perturbations. It should be noted that it is the first time the exponential stability results for impulsive positive linear DAEs are given. For this purpose, delay-range dependent sufficient conditions for such systems are stated in the paper. An additional numerical example is provided to demonstrate the results' efficacy.

Results on Ulam‐type stability of linear differential equation with integral transform

The main theme of this study is to implement the Sumudu integral transform technique to solve the stability problem of linear differential equations. Another important aspect of this paper is to investigate the Ulam–Hyers and Ulam–Hyers–JRassias stability of linear differential equations by using Sumudu transform method. Further, the results are extended to the Mittag‐Leffler–Ulam–Hyers and Mittag‐Leffler–Ulam–Hyers–JRassias stability of these differential equations. As an application point of view, the Sumudu transform is used to find Ulam stabilities of differential equations arising in field‐controlled DC servo motor with position control.

The Utilization of Artificial Intelligence Based Chatbot in Interactive Learning Media

Abstract: After the spread of the COVID-19 virus, the world of learning is no longer the same as it was before the pandemic, especially at the university level, which requires students to be more active in finding lots of learning references via the internet. Active learning has a close relationship with the use of artificial intelligence technology, in this study researchers designed an interactive learning media that applied Artificial Intelligence (AI) Assisted Chatbot. The multimedia development method used in this research is the Multimedia Development Lifecycle (MDLC) method. Interactive learning media is built using Android Studio, and the retrievalbased chatbot is built with Python, Tensorflow, Hard (Extension for Tensorflow), NumPy, and Matplotlib. The researcher applies the White box and Black box testing methods to test whether the learning media that have been made have worked according to the user's needs. The chatbot model that has been applied to this learning media when tested with the BLEU metricgets a result of 0.1117, this shows the chatbot produces good answers and provides credible learning references. There is a difference in student learning outcomes after using chatbot-based learning media in basic electronics subjects, the utilization of chatbotbased learning media is able to improve student learning outcomes, especially in the student weaknesses section, the chatbot can provide detailed explanations and guide students in solving linear equation problems. Keywords: Artificial Intelligence; Chatbot; Interactive Learning Media; Retreival-based Chatbot;

Algorithmic realization of exact three‐point difference scheme for singular Sturm–Liouville problem

AbstractA new algorithmic realization of exact three‐point difference schemes for a class of singular Sturm–Liouville problems is presented. It is shown that to compute the coefficients of the exact scheme in an arbitrary grid node, it is necessary to solve two auxiliary initial value problems for the second order linear ordinary differential equations: one problem on the interval (forward) and one problem on the interval (backward). Finally, the coefficient stability of the exact three‐point difference scheme is proved.

Comparison of Speed Completion System Linear Equations Using the Elimination Method and Using Arithmetic Sequences

There are many ways to solve linear system equation problems, such as using Gaussian Elimination or the elimination method. To solve linear system equation problems, we can transform them into an arithmetic sequence. This method can make it easier to answer the questions quickly. The objective of this research is to provide a comparison of problem-solving using the elimination method and using arithmetic sequences. The method used in this research is quantitative. Data collection involves solving linear system equation problems and recording the time it takes to solve them using both the elimination method and the arithmetic sequence method. The data used consists of UNBK (National Computer-Based Exam) questions and university selection questions, comparing the time it takes to solve problems using the substitution method and the arithmetic sequence method. The conclusion from this research is that the use of arithmetic sequences in solving linear equations is easier and faster, making it applicable when answering exams with limited time duration.

Two-point boundary value problem for linear stochastic differential equations

We present the extension of the two-point boundary problem in the case where the problem doesn’t always have a unique solution or the condition on boundary coefficients described in [Linear stochastic differential equations with boundary conditions, Probab. Th. Rel. Fields, 82, 489–526 (1989)] is not satisfied.

Utilization of Collaborative Problem Solving of Mathematical Representation in Solving One Variable Linear Equation Problems

Penelitian ini bertujuan untuk melihat dan mendeskripsikan kemampuan representasi matematis siswa pada materi Persamaan Linier Satu Variabel (PLSV) setelah diterapkan model Collaborative Problem Solving (CPS). Metode penelitian yang digunakan adalah deskriptif dengan analisis data secara kuantitatif dan kualitatif sesuai dengan indikator representasi matematis. Penelitian ini dilaksanakan di SMP Negeri 6 Indralaya Utara di kelas VII.1 dengan melibatkan sebanyak 16 siswa sebagai subjeknya. Pengumpulan data dilakukan dengan memberikan tes tertulis berupa soal uraian dan hasil wawancara dengan siswa. Teknik analisis data yang digunakan meliputi memeriksa tes uraian merujuk pada indikator representasi matematis yaitu visual, simbolik dan verbal. Data yang telah terkumpul kemudian dianalisis berdasarkan pedoman penskoran yang merujuk pada indikator representasi matematis. Hasil analisis data selanjutnya dikelompokkan menjadi tiga kategori yaitu tinggi, sedang, dan rendah. Berdasarkan hasil analisis data didapatkan bahwa kemampuan representasi matematis siswa pada materi persamaan linier satu variabel setelah diterapkannya model CPS berada dikategori sedang dengan nilai rata-rata sebesar 59,687 dan skor rata-rata untuk indikator representasi visual yaitu sebesar 71,093, representasi simbolik 60,156, dan verbal 35,937.This study aims to look at and describe students' mathematical representation abilities in the One Variable Linear Equation (PLSV) material after applying the Collaborative Problem Solving (CPS) model. The research method used is descriptive with quantitative and qualitative data analysis according to the indicators of mathematical representation. This research was conducted at SMP Negeri 6 Indralaya Utara in class VII.1 involving 16 students as subjects. Data collection was carried out by giving written tests in the form of description questions and the results of interviews with students. The data analysis technique used includes examining the description test referring to indicators of mathematical representation, namely visual, symbolic and verbal. The data that has been collected is then analyzed based on scoring guidelines that refer to indicators of mathematical representation. The results of the data analysis are then grouped into three categories, namely high, medium, and low. Based on the results of data analysis, it was found that students' mathematical representation abilities in the material of one-variable linear equations after the implementation of the CPS model were in the moderate category with an average score of 59.687 and an average score for visual representation indicators that was 71.093, symbolic representation 60.156, and verbal 35.937.

The Homotopy Perturbation Method for Solving Nonlocal Initial-Boundary Value Problems for Parabolic and Hyperbolic Partial Differential Equations

To obtain approximate-exact solutions to nonlocal initial-boundary value problems (IBVPs) of linear and nonlinear parabolic and hyperbolic partial differential equations (PDEs) subject to initial and nonlocal boundary conditions of integral type, the homotopy perturbation method (HPM) is utilized in this study. The HPM is used to solve the specified nonlocal IBVPs, which are then transformed into local Dirichlet IBVPs. Some examples demonstrate how accurate and efficient the HPM.