Applied Numerical Methods Using MATLAB, 2nd Edition

6Ordinary Differential Equations

Chapter Outline

• 6.1 Euler's Method
• 6.2 Heun's Method – Trapezoidal Method
• 6.3 Runge‐Kutta Method
• 6.4 Predictor‐Corrector Method
  • 6.4.1 Adams‐Bashforth‐Moulton Method
  • 6.4.2 Hamming Method
  • 6.4.3 Comparison of Methods
• 6.5 Vector Differential Equations
  • 6.5.1 State Equation
  • 6.5.2 Discretization of LTI State Equation
  • 6.5.3 High‐order Differential Equation to State Equation
  • 6.5.4 Stiff Equation
• 6.6 Boundary Value Problem (BVP)
  • 6.6.1 Shooting Method
  • 6.6.2 Finite Difference Method
  • Problems

Differential equations (DEs) are mathematical descriptions of how the variables and their derivatives (rates of change) with respect to one or more independent variable affect each other in a dynamical way. Their solutions show us how the dependent variable(s) will change with the independent variable(s). Many problems in natural sciences and engineering fields are formulated into a scalar differential equation or a vector differential equation, i.e. a system of differential equations.

In this chapter, we look into several methods of obtaining the numerical solutions to ordinary differential equations (ODEs) in which all dependent variables (x) depend on a single independent variable (t). First, the initial value problems (IVPs) will be handled with several methods including Runge‐Kutta method and predictor‐corrector methods in Sections 6.1 to 6.5. The last section (Section 6.6) will introduce the shooting method and the finite difference method for solving ...