Approximate Solution as a Polynomial-Minimum of Residue | Finite Element Analysis | SNS Institutions

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In this video, we dive deep into the method of Approximate Solution as a Polynomial Minimisation of Residue, a key numerical approach used when exact analytical solutions to differential equations are difficult or impossible to obtain. The core idea is to assume a polynomial trial function that satisfies the given boundary or initial conditions, substitute it into the governing differential equation, and then minimize the resulting residual (error) over the domain. By carefully choosing the polynomial basis and applying minimization techniques, we construct an approximate solution that closely represents the true physical behavior of the system. This method is closely related to weighted residual techniques such as the Galerkin method, least squares method, and collocation method, which are widely used in engineering analysis, applied mathematics, and computational physics. Throughout the video, the theory is explained step by step, starting from the formulation of the problem, selection of trial functions, derivation of governing equations for the unknown coefficients, and final construction of the approximate solution. Practical examples are included to show how the method works in real applications, helping you build intuition and confidence to apply polynomial residual minimization to solve boundary value problems and engineering models in your own studies or projects.