1. Euler’s Method CS 170: Computing for the Sciences and Mathematics

2. Administrivia Last time (in P265)  Basics of Algorithms and computation Today  More Maple  Euler’s method  HW #6 Due!  HW #7 assigned

3. Euler’s method Simplest simulation technique for solving differential equation Intuitive Some other methods faster and more accurate Error on order of ∆t  Cut ∆t in half  cut error by half

4. Euler’s Method Euler’s Method is a simulation technique. Example: unconstrained growth dP/dt = 0.1P with P 0 = 100 P(t) = P(t - ∆t ) + growth(t) ∆t (new = old + change) growth(t) is dP/dt = 0.1P(t - ∆t ) (change = r*P old )

5. Example dP/dt = 0.1P with P 0 = 100 and ∆t = 8

6. Approximating Unconstrained Growth initialize simulationLength, population, growthRate, ∆t numIterations  simulationLength / ∆t for i going from 1 to numIterations do the following: growth  growthRate * population population  population + growth * ∆t t  t + ∆t

7. Approximating Unconstrained Growth (Better) initialize simulationLength, population, growthRate, ∆t numIterations  simulationLength / ∆t for i going from 1 to numIterations do the following: growth  growthRate * population population  population + growth * ∆t t  i*∆t

8. Error Analytical solution: P = 100 e 0.10t After 100 time units: P = 2,202,647 Smaller ∆t  better estimation ∆t = 1estimate of P = 1,378,061 ∆t = 0.25estimate of P = 1,947,808 ∆t = 0.01estimate of P = 2,191,668 ∆t = 0.005estimate of P = 2,197,149

9. HOMEWORK! Homework 7  Maple Tutorial 2  Complete the worksheet’s questions and turn it in  Also complete Module 5.2’s Project 1 on page 167.  Due 11/1/2010 Thursday’s Class in P115