ChE 250 Numeric Methods

1
ChE 250 Numeric Methods

• Lecture 22
• Part 7 Ordinary Differential Equations
• Chapter 25 Runge-Kutta Method
• 20070402

2
Todays Lecture

• Part 7 Introduction
• Ordinary Differential Equations
• Initial value problems
• Boundary Problems
• Chapter 25 R-K Methods
• Eulers (review)
• Error and Step Size

3
Ordinary Differential Equations

• Ordinary in this term means only one independent
  variable in the equation(s)
• Partial Differential Equations, have multiple
  independent variables
• ODEs can be divided into linear and non-linear
  categories
• e.g. (linear)

4
Ordinary Differential Equations

• For any ODE there are an infinite number of
  solutions
• To find a particular solution we need to
  determine the constants of integration
• We use boundary conditions to solve for the
  particular equation
• For an n-th order ODE, we need n boundary
  conditions

5
Ordinary Differential Equations

• If the boundary conditions are all given at one
  value of the independent variable, the problem is
  called an initial Value Problem
• Usually parameters are known and we are solving
  for the state at a different value of the
  independent variable
• If the boundary conditions are at two values of
  the independent variable, it is known as a
  Boundary Value Problem
• Can be used to solve for the value of parameters
  (very tricky)

6
ODEs

• Pendulum Example
• Questions?

7
R-K Methods

• To review Eulers method
• We used this method to determine the velocity
  versus position of a projectile with g(x) a
  function of altitude
• We also had the Parachutist example

8
ODEs

• Eulers can be improved by
• Decreasing the step size, h
• Using more sophisticated methods to derive the
  slope, higher order Taylor expansion
• More commonly, the R-K method is used
• Recurrence relations are used to calculate the
  R-K constants

9
ODEs
10
ODEs

• Example 27.9
• Adaptive R-K Fehlberg
• ODE45 function
• ODE45 uses variable step size, so only the
  endpoints are specified

11
Preparation for 4Apr

• Chapter 27 Boundary Value Problems
• Homework
• Chapter 21
• 9, 13, 20
• Chapter 22
• 3, 7, 14
• Chapter 23
• 1, 9, 12, 13, 14, 27
• Chapter 24
• 4, 5, 12