Paper Key : IRJ************745
Author: Kirti,Pardeep Beniwal
Date Published: 04 Jul 2024
Abstract
Partial Differential Equations (PDEs) are fundamental tools in the mathematical modeling of phenomena across various scientific and engineering disciplines, including fluid dynamics, electromagnetic theory, and financial mathematics. Traditionally, solutions to these equations are sought through analytical methods or numerical techniques such as finite differences, finite elements, and spectral methods. However, these conventional approaches often struggle with complex geometries, nonlinearities, and the need for high precision across broad domains. Bernstein polynomials have emerged as a powerful alternative for numerically solving PDEs. Their propertiesnon-negativity, partition of unity, and ease of differentiation and integrationmake them particularly suitable for approximation tasks. Bernstein polynomials are foundational in computer-aided geometric design due to their strong stability and best-approximation characteristics under certain norms. Hybrid methods that integrate Bernstein polynomials with numerical techniques like collocation, Galerkin, or tau methods have demonstrated enhanced convergence rates and accuracy. These methods have been successfully applied to a wide range of problems, from simple heat transfer to complex fluid flows, achieving reduced computational costs and improved precision. Despite some challenges, such as determining the optimal polynomial degree and managing computational efficiency, ongoing research continues to address these issues, broadening the applicability and effectiveness of Bernstein polynomials in solving PDEs.
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