Lecture 8-7 | Modified Euler Method | Advanced Mathematical Methods for Engineers

Overview In this module you will learn how to solve Ordinary Differential Equations (ODEs) both using analytical and numerical methods. Many engineering processes are described by either a single ODE or system of ODEs. Examples include vibrations in systems, spreading of pathogens, or chemical reaction mechanisms. You will learn how to code numerical methods to solve ODEs, solve systems of ODEs, and how to make sure that the numerical solution of ODEs is accurate, even in cases where no analytical solutions are known. Learning Objectives By the end of this module, you will be able to: 8.1 - Solve an ODE using explicit/ implicit/ modified/ midpoint Euler method using self-coded functions. 8.2 - Solve an ODE using Runge-Kutta methods using self-coded functions. 8.3 - Identify errors of ODE solution methods and their formal order. 8.4 - Solve higher-order ODEs and system of ODEs using self-coded functions. 8.5 - Identify numerical solution methods for stiff ODEs. Lecture Videos: Lecture 8.1: ODEs Overview Lecture 8.2: Analytical Solutions of ODEs Lecture 8.3: Numerical Solutions of ODEs Lecture 8.4: Euler's Explicit Method Lecture 8.5: Euler's Implicit Method Lecture 8.6: Stability Lecture 8.7: Modified Euler Method Lecture 8.8: Midpoint Euler Method Lecture 8.9: Errors Lecture 8.10: Runge-Kutta Methods Lecture 8.11: Accuracy of Numerical Solutions of ODEs Lecture 8.12: Numerical Solutions of Systems of ODEs Lecture 8.13: Numerical Solutions of Higher Order ODEs Lecture 8.14: Stiff ODEs