Download presentation

Presentation is loading. Please wait.

Published byClaribel McKenzie Modified over 4 years ago

1
Dr. Jie Zou PHY 33201 Chapter 9 Ordinary Differential Equations: Initial-Value Problems Lecture (I) 1 1 Besides the main textbook, also see Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 20, McGraw Hill, 2008.

2
Dr. Jie Zou PHY 33202 Outline Introduction: Some definitions Engineering and Scientific Applications One-step Runge-Kutta (RK) Methods (1) Euler’s Method The method (algorithm) Error analysis (next lecture) Stability (next lecture)

3
Dr. Jie Zou PHY 33203 Introduction: Some definitions Differential equation: An equation involving the derivatives or differentials of the dependent variable. Ordinary differential equation: A differential equation involving only one independent variable. Example: For the bungee jumper, Partial differential equation: A differential equation involving two or more independent variables (with partial derivatives). Order of a differential equation: The order of the highest derivative in the equation. Example: For an unforced mass-spring system with damping-a second-order equation:

4
4 Introduction: Some definitions (cont.) For an nth-order differential equation, n conditions are required to obtain a unique solution. Initial-value problem: All conditions are specified at the same value of the independent variable (e.g., at x or t = 0). Example: For the bungee jumper, Boundary-value problem: Conditions are specified at different values of the independent variable. Example: Particle in an infinite square well Initial Condition Fig. PT6.3 (Ref. by Chapra): Solutions for dy/dx = -2x 3 + 12x 2 – 20x + 8.5 with different constants of integration, C. Boundary Conditions

5
Dr. Jie Zou PHY 33205 Engineering and scientific applications Fig. PT6.1 (Ref. by Chapra): The sequence of events in the development and solution of ODEs for engineering and science.

6
6 Euler’s method Let’s look at the Bungee-Jumper’s example: Solve an ODE-initial-value problem (1) Step 1: Finite-difference approximation for dv/dt (2) Step 2: Substitute Eq. (2) in Eq. (1) (3) Step 3: Notice that dv/dt at t i = g- c d v(t i ) 2 /m, (3) becomes Euler’s method (a one-step method) Fig. 1.4 (Ref. by Chapra): Numerical solution by Euler’s method.

7
Dr. Jie Zou PHY 33207 Another look at Euler’s method Solving ODE: dy/dt = f(t,y) All one-step methods (Runge-Kutta methods) have the general form: : an increment function for extrapolating from an old value y i to a new value y i+1. One-step methods: use information from one pervious point i to extrapolate to a new value. h: Step size = t i+1 – t i. Euler’s method: = f(t i,y i ), the 1 st derivate of y at t i y i+1 = y i + f(t i,y i )h Fig. 20.1 (Ref. by Chapra): Euler’s method

8
Dr. Jie Zou PHY 33208 Example: Euler’s method Example 20.1 (Ref.): Use Euler’s method to integrate y’ = 4e 0.8t – 0.5y from t = 0 to t = 4 with a step size of 1. The initial condition at t = 0 is y = 2. Note that the exact solution can be determined analytically as y = (4/1.3)(e 0.8t – e -0.5t ) + 2e -0.5t

9
Dr. Jie Zou PHY 33209 Results ty true y Euler | t | (%) 02.00000 ---- 16.194635.0000019.28 214.8439211.4021623.19 333.6771725.5132124.24 475.3389656.8493124.54 Fig. 20.2 (Ref. by Chapra) Table 20.1 (Ref. by Chapra)

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google