Education Technology

Numerical Methods and Series

Activity Overview

In this activity, students take a closer look at numerical methods used to solve initial-value differential equations, including the methods used internally by the calculator. They also learn to solve numerical computations for the equations where the calculator can not be used.

Before the Activity

  • See the attached PDF file for detailed instructions for this activity
  • Print pages 77 - 94 from the attached PDF file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:
    Graphical Euler's Method:

  • Enter the differential equation
  • Change the values of the number of Euler steps, number of steps between plotted points, change in t for plotted points, and graph

  • Numerical Euler's method:
  • Enter the differential equation and select Euler format
  • Use six different values of h the stepsize
  • Confirm the expected relationship between h and size of the final t-value
  • Estimate the number of Euler steps needed to have a final accuracy of 1E-7

  • Solution Series:
  • Use differentiation to get higher derivatives
  • Solve the equation using Taylor series expansion, knowing the derivative at t = 0
  • Note the method of using Taylor series involves too much symbolic work while the Runge-Kutta method is simpler
  • Use the Runge-Kutta method to solve the differential equation

  • Numerical Runge-Kutta Method
  • Write a small program to implement the classical third-order Runge-Kutta method to solve the differential equation, and also include lists of the local error (for the first step) and global error for variety of stepsizes
  • Vary the values of stepsize and total number of Euler steps to get a more accurate outcome

  • Numerical internal RK Method
  • Re-enter the equation and change the parameters and initial conditions
  • Repeatedly change the difTol and explore the effect on the global error, and time

  • Graphical internal RK Methods
  • Enter the system of equations and graph them with separate styles
  • Use the difTol function to set the parameters
  • Use eval command to compare the final computed values for the different tolerances
  • After the Activity

    Students complete the problems on the exercise page.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary