Dalabaev Umurdin and Xasanova Dilfuza, University of World Economy and Diplomacy, Tashkent, Uzbekistan
This article discusses approximate solutions of linear parabolic equations with initial-boundary conditions. The primary focus is on methods that effectively find such solutions by employing a moving finite difference analog of the differential equation. This approach allows us to formulate an approximate analytical solution, significantly simplifying the computation process .By transitioning from the differential equation to an algebraic equation, we obtain a single equation, the solution of which represents an approximate analytical solution to the original problem. However, to achieve higher accuracy in this solution, we apply additional moving nodes, which enhances the results.By using multipoint moving nodes, we can form a system of algebraic equations, the solution of which provides us with an improved analytical solution. The article also presents numerical experiments that confirm the effectiveness of the proposed method and its advantages over traditional approaches.
Boundary conditions, differential equation, multipoint moving nodes, initial-value problem.
Cloves Rocha Sampaio Júnior, Retired Federal Civil Servant - Independent Researcher
In this work, we explore the number π from a new perspective, using advanced mathematical methods and the GeoGebra software for investigations in Cartesian, isometric and polar coordinates. We focus on proportional relationships in the first quadrant of the unit circle, extending the analyses to the following quadrants with precision and rigorous logic. Traditionally, π is recognized as an irrational number, intrinsic to nature and geometry. However, our research suggests the possibility of understanding π in a rational and repetitive way, through divisions, infinite series, and pertinent theorems. We propose a renewed view on π, questioning established assumptions and enriching mathematical understanding by challenging old concepts and broadening our understanding. The methodology analyzes the relationship between circumference perimeters and their diameters, radian angles, and square roots. The integration of these factors with inscribed and circumscribed polygons indicates a rational pattern for the number π. Advanced computing is essential for calculations that surpass ruler and compass capabilities. Recent studies suggest that, although traditional geometric methods do not determine an exact and repeatable value for π, advanced computational techniques and mathematical models can reveal new aspects of this essential constant.
Number π; GeoGebra Software; Pure trigonometry.
