Solution Of Problem Research Articles

Bounded solutions in anisotropic degenerate parabolic problems with a singular term

Bounded solutions in anisotropic degenerate parabolic problems with a singular term

Taylor theory in quantum calculus: a general approach

Let the function β be strictly increasing and continuous on an interval I ⊂ ℝ. The β-difference operator is defined by Dβ f (t) = (f(β(t)) − f(t)/(β(t) – t), where t ≠ β(t), and Dβ f (t) = f′ t(t) when t = s 0 is a fixed point of the function β. This quantum operator is a generalization of q-Jackson, Hahn, power and other quantum operators. As a convenience of the β-function: β(t) turns into the probability distribution function with the probability measure 1, and the sample space ℝ, in the case of its conditions are relaxed to be increasing and continuous from the right, that is, lim t →∞ β(t) = 1 and lim t →∞ β(t) = 0, and also by using the Lebesgue-Stieltjes measure of the interval [a, b] to be β(b) − β(a). In this paper, we investigate a β-Taylor’s formula associated with the operator Dβ when the function β has a unique fixed point s 0 ∈ I, which may allow for more flexible and accurate approximations of functions. An estimation of its remainder is given. Additionally, the β-power series is defined. Furthermore, as application, the β-expansion form of some fundamental functions is introduced. Finally, we find the unique solution of the β-shifting problem.

The overdetermined Cauchy problem for the hyperbolic Gellerstedt equation

Abstract Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.

A Direct Contour Control Method for Free-Form Surface Machining Trajectories Based on Coordinate Transformation

The requirement for high-speed and high-precision contouring with free-form surfaces in the CNC machining process is increasing. The contour error is an essential criterion for the quality of CNC machining. For the problem of contour error control, the present research is mainly based on position tracking, and the control method directly aiming at contour control has not been well studied. This paper proposed a new type of coordinate transformation-based direct contour control method (CT-DCC) for free-form surface machining trajectories. By real-time monitoring of the contour error and the foot point of the free-form parametric trajectory, the contouring task is decomposed into the contour error control task in the normal direction and the velocity control problem in the feed direction. Using coordinate transformation, the output commands of the two tasks are converted to the axial motion command. CT-DCC completely eliminates position tracking in the traditional control to achieve direct control of the contour error, which provides a new solution for the contour control problem of free-form surface machining. Both the axial velocity control and the axial torque control-based CT-DCC scheme were studied, and the two-axis and three-axis direct contour controllers were tested. The simulation and experiment results show the feasibility of the proposed method.

Lipschitz stability of an inverse conductivity problem with two Cauchy data pairs

Abstract In 1996 Seo proved that two appropriate pairs of current and voltage data measured on the surface of a planar homogeneous object are sufficient to determine a conductive polygonal inclusion with known deviating conductivity. Here we show that the corresponding linearized forward map is injective, and from this we deduce Lipschitz stability of the solution of the original nonlinear inverse problem. We also treat the case of an insulating polygonal inclusion, in which case a single pair of Cauchy data is already sufficient for the same purpose.

APPROACH TO CREATION OF GEOLOGICAL AND RESERVOIR SIMULATION MODEL AND METHODS FOR HISTORY MATCHING OF RESERVOIR ENERGY STATE

Under conditions of complex geological structure and substantial share of hard-to-recover oil reserves, reservoir simulation is a key tool in reservoir management. Direct problems solved by reservoir simulation include forecasting of the production performance, control and management of reservoir development, and localization of residual oil reserves. Inverse problem solution is useful for updating the knowledge of geological structure, reservoir quality, and historical production data by means of history matching. The quality of history matching is fundamental to ensure reliable future forecasts and localization of residual oil zones. The main approaches to appropriate history matching rely on analysis and adjustment of well logging and geological parameters, as well as checking and correction of errors, inadequate historical data, application of physics-based and reasonable approaches and tools. The prerequisite for adequate history matching and predictive ability of the resultant reservoir simulation model is the compliance of simulated and actual energy state of the reservoir. In this paper, the initial stage of reservoir simulation is performed, the approach to history matching of reservoir energy state aimed at improving the quality of further predictions is described as exemplified by one of the production areas of Russian oil field. The main causes of poor history match of formation and bottomhole pressures are revealed. These causes are grouped as follows: — geological causes/reservoir quality, which include permeability field distribution, reservoir connectivity, and determination of rock and fluid compressibility;— technical causes (during reservoir simulation), that is model limitations in case of larger areas, the need to use sector models, consideration of buffer region;— technological causes, including inadequate historical data or lack of data, detection of behind-the-casing flows and production casing leaks in wellbore, etc. For each group of factors that have a substantial impact on reservoir pressure, recommendations and methods of energy state history matching on geological and reservoir simulation models are provided, examples are given.

Nonexistence of solutions to quasilinear Schrödinger equations with a parameter

This paper is concerned with the class of quasilinear Schrödinger equations: which models the self‐channeling of a high‐power ultra short laser in matter provided , where is a parameter, and . Under some appropriate assumptions on , we establish the nonexistence of solutions for the above problem.

Influence of Rotation and Viscosity on Parallel Rolls of Electrically Conducting Fluid

Rayleigh–Bénard convection is a fundamental fluid dynamics phenomenon that significantly influences heat transfer in various natural and industrial processes, such as geophysical dynamics in the Earth’s liquid core and the performance of heat exchangers. Understanding the behavior of conductive fluids under the influence of heating, rotation, and magnetic fields is critical for improving thermal management systems. Utilizing the Boussinesq approximation, this study theoretically examines the nonlinear convection of a planar layer of conductive liquid that is heated from below and subjected to rotation about a vertical axis in the presence of a magnetic field. We focus on the onset of stationary convection as the temperature difference applied across the planar layer increases. Our theoretical approach investigates the formation of parallel rolls aligned with the magnetic field under free–free boundary conditions. To analyze the system of nonlinear equations, we expand the dependent variables in a series of orthogonal functions and express the coefficients of these functions as power series in a parameter ϵ. A solution for this nonlinear problem is derived through Fourier analysis of perturbations, extending to O(ϵ8), which allows for a detailed visualization of the parallel rolls. Graphical results are presented to explore the dependence of the Nusselt number on the Rayleigh number (R) and Ekman number (E). We observe that both the local Nusselt number and average Nusselt number increase as the Ekman number decreases. Furthermore, the flow appears to become more deformed as E decreases, suggesting an increased influence of external factors such as rotation. This deformation may enhance mixing within the fluid, thereby improving heat transfer between different regions.

Performance Comparison of AI Platforms in Solving Computer Science Problems

In the rapidly evolving landscape of artificial intelligence (AI), its significance and accelerated development are undeniable. AI has emerged as a cornerstone technology with profound implications across various domains, driving innovation and reshaping the way we approach complex problems. Particularly, the utilization of AI in coding tasks has garnered substantial attention, given its potential to streamline development processes and enhance the efficiency of software engineering practices. Against this backdrop, this paper presents a detailed comparative analysis of four different AI platforms, namely ChatGPT, Gemini, Blackbox, and Microsoft Copilot, in addressing key challenges within the realm of computer science, spanning natural language processing, image processing, and cybersecurity. The study focuses on leveraging the C++ programming language to develop solutions for these multifaceted problems across the aforementioned platforms. Each platform's outputs are meticulously evaluated on various parameters including accuracy, execution time, code size, and time complexity to provide a comprehensive understanding of their performance. Furthermore, an iterative optimization methodology is employed, entailing three rounds of refinement for the code produced by each platform, with the resultant outputs subjected to comparative analysis in each iteration. Through this rigorous approach, the paper not only elucidates the efficacy of different AI platforms in addressing diverse computational challenges but also underscores the iterative enhancement process on AI platforms for refining code quality and performance across multiple domains within computer science.

Distributed power system coordination via parametric optimization and ADMM

In this paper we present an efficient model predictive control algorithm for distributed coordination of an electrical power system comprising many spatially distributed controllable units. The coordination problem is formulated using parametric solutions of local optimization problems corresponding to individual subsystems leading to favorable problem structure which can be split across all subsystems and nodes in the network. The structure of the obtained coordination problem is exploited to develop a very efficient solution algorithm based on an ADMM technique. The key features of the overall approach are: (i) private data of individual subsystems are protected, (ii) simple and efficient on-line computations, (iii) parallelization of computation across all subsystems and all nodes in the network. Efficiency of the proposed control strategy is demonstrated on a number of numerical case studies of varying size.

Match-based solution of general parametric eigenvalue problems

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

Determination of the coefficient of thermal effusivity ε for a semi-confined rod

The article provides methods and formulas for calculating the coefficient of thermal effusivity ε (coefficient of heat accumulation) of solid bodies using known solutions of direct heat conduction problem. The work describes a method by which the final formulas are simplified to algebraic equations using the relative temperature θM of heating semi-confined rod from the end. The temperature Tc at the beginning of the rod is measured — x0 and time — τ0 from the start of heating. For calculation, the amount of heat Q accumulated during heating by a heater with a constant temperature at the end of the rod is used for the entire heating time. The technique is physically implemented during the experiment by bringing the ends of two semi-confined rods into contact. A flat, low-inertia electric heater is placed between the rods. The amount of heat ∆Q used in the calculations is half of the total heat of the heater for the entire heating time.

Entropy solutions to the fully nonlocal diffusion equations

AbstractWe consider the fully nonlocal diffusion equations with nonnegative ‐data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time‐space fractional Laplacian equations.

Optimization of calculations in modeling the breaking of plasma oscillations

The proposed method of optimization of calculations is of a combined nature. First, using a piecewise uniform grid, a difference scheme of the second order of accuracy of the McCormack type is constructed, and then — to weaken the stability condition — an implicit version of the scheme is used, which even in the nonlinear case allows an iteration-free implementation. When constructing a grid, a preliminary study is required, which is based on the analytical properties of the solution of the differential problem. Based on the calculations carried out, it can be concluded that the potential reduction in the amount of calculations is dozens of times without a noticeable loss of accuracy.

A NEW DUAL HEURISTIC ALGORITHM FOR FINDING THE INITIAL BASIC FEASIBLE SOLUTION FOR A TRANSPORTATION PROBLEM

Because determining the best initial basic feasible solution (IBFS) for a transportation problem is so crucial, numerous authors have expended a great deal of energy developing effective algorithms that will result in the lowest possible cost of moving products from a given source to a destination. The goal of this work was to develop an efficient dual algorithm for finding an initial basic feasible solution to a transportation problem (TP). Two distinct algorithms that produce the same IBFS make up our suggested approach. Compared to some popular methods in the literature, Using four numerical examples, the Row Minimum Method (RMM), Column Minimum Method (CMM), Least Cost Method (LCM), Extremum Difference Method (EDM), Northwest Corner Method (NWCM), Vogel's Approximation Method (VAM), etc. In comparison to the other heuristic techniques compared with the optimal dictate solution modified distribution (MODI), the proposed heuristic approach (PS-DESPAN) approximation was shown to provide a better starting solution (a solution that is extremely close to the optimal solution)

Sharp estimates of solution of an elliptic problem on a family of open non‐convex planar sectors

Sharp estimates of solution of an elliptic problem on a family of open non‐convex planar sectors

A Toolbox of Strategies to Improve Test-Taking Skills: A Comparison of Student Perceptions.

Changes in the national examination structures have renewed interest in the development of test-taking strategies for professional students in the health sciences. It is often assumed that these high-achieving students have developed proficient test-taking skills and abilities prior to admittance. However, the assessments in these programs and for national licensure require an elevated level of reasoning and integration with clinical concepts. It was hypothesized that the implementation of an immersive test-taking strategies program would improve dental student perceptions of their abilities. A "toolbox" of four methods was implemented which included: 1. An introductory video that provided students with ten tips for approaching high-level exam questions 2. Problem solution videos interspersed with class practice problems to walk students independently through rationales 3. Collaborative group assessments in which students implemented the strategies in teams to prepare for exams and 4. Unit exam debriefings to review the question rationales. While all methods were positively reviewed on surveys, students indicated that the problem solution videos and the collaborative group assessments were more helpful and improved their test-taking skills significantly more than the other strategies (P < 0.01, one-way ANOVA and Bonferroni post hoc test). Students felt they had developed strong test-taking strategies (average of 4.21 on a 5-point scale, SD 0.76) and felt more prepared for the Integrated National Board Dental Examination (4.48, SD 0.66). These results suggest that a multi-pronged approach with frequent opportunities to practice test-taking strategies can improve student perceptions of their ability to master high-level and integrated assessment questions.

Achromatic solutions of the color constancy problem: the Helmholtz-Kohlrausch effect explained.

Achromatic solutions of the color constancy problem: the Helmholtz-Kohlrausch effect explained.

Weak long‐range diffusion

AbstractWe study the spreading solutions of the nonlinear diffusion equation when the far‐field diffusivity is small. The method of strained coordinates is used to construct a uniform asymptotic correction to the similarity solution of the unperturbed problem. The equation provides a possible analogue to similar models of fluid jets and plumes.

Interval-Valued Multiobjective Programming Problems Based on Convex Cones

The new solution concepts of interval-valued multiobjective optimization problems using ordering cone are proposed in this paper. An equivalence relation is introduced to divide the collection of all bounded closed intervals into the equivalence classes. The family of all equivalence classes is also called a quotient set. In this case, this quotient set can turn into a vector space under some suitable settings for vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. It means that an ordering in the quotient set can be defined to study the Pareto optimal solution in multiobjective optimization problems. In this paper, we consider the multiobjective optimization problem such that its coefficients are taken to be the bounded closed intervals. With the help of the convex cone, we can study the Pareto optimal solutions of the multiobjective optimization problem with interval-valued coefficients.