Numerical Analysis 10. Approximation of ordinary differential equations (ODEs) скачать в хорошем качестве

Numerical Analysis 10. Approximation of ordinary differential equations (ODEs)

This video provides a detailed lecture on the numerical approximation of ordinary differential equations (ODEs). Key Topics Covered: Initial Value Problems (IVPs): Introduction to first-order explicit differential equations [01:27]. Euler's Method: Definition: A first-order numerical procedure for solving ODEs using the recursion Derivation: Explained through Taylor polynomial approximation, numerical differentiation, and numerical integration [12:12]. Error Analysis: Demonstrates that the method's error is linear in terms of the step size (h) [29:06]. Rounding Errors: Analysis of how hardware rounding can cause errors to grow significantly if the step size is too small [47:02]. Taylor's Method: A generalization of Euler's method using higher-order Taylor expansions to achieve faster convergence [53:48]. Runge-Kutta Methods: Midpoint & Modified Euler Methods: Second-order methods that use average slopes to improve accuracy without complex derivative calculations [01:20:02]. Classical Runge-Kutta (RK4): A widely used fourth-order method that requires four function evaluations per step and provides high precision even with larger step sizes [01:26:36]. Gemini is AI and can make mistakes, including about people. Your privacy & GeminiOpens in a new window