Mech300 Numerical Methods, Hong Kong University of Science and Technology. 1 Part Seven Ordinary Differential Equations

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 2 Basics Example: the falling parachutist v: dependent variable (function) t: independent variable Differential equation: an equation composed of an unknown function and its derivatives Ordinary differential equation: if there is only one independent variable Partial differential equation: if there are two or more independent variables Order of ODE: the order of the highest derivative in the equation Example: second order ODE Reduction of order: higher-order ODE can be reduced to a system of 1 st -order ODE

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 3 Why Study Differential Equations? Many physical phenomena are best formulated mathematically in terms of their rate of change (which is derivative)! Example: motion of a swinging pendulum

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 4 ODE and Engineering Practice Fundamental laws Empirical observations ODE Solutions Analytical/numerical methods Sequence of the application of ODEs for engineering problems Independent variable: spatial and temporal

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 5 Noncomputer Methods for Solving ODEs One particular useful analytical integration technique: linearization a n (x)y (n) + a n-1 (x)y (n-1) + … +a 1 (x)y’+ a 0 (x)y = f(x) Differential equation Integration Solution conversion Analytical integration techniques This can be solved analytically! Sin  ≈  if  is small (non-linear) (linear)

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 6 Solution by Integration differentiation For an n th -order ODE, n conditions are required to obtain a unique solution Multiple solutions integration Initial-value problem All n conditions are specified at a same value of x n conditions occur at different x Boundary-value problem

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 7 Overall Structure Initial-value problem