Applied Numerical Methods Using MATLAB, 2nd Edition

9Partial Differential Equations

Chapter Outline

• 9.1 Elliptic PDE
• 9.2 Parabolic PDE
  • 9.2.1 The Explicit Forward Euler Method
  • 9.2.2 The Implicit Backward Euler Method
  • 9.2.3 The Crank‐Nicholson Method
  • 9.2.4 Using the MATLAB function ‘pdepe()’
  • 9.2.5 Two‐Dimensional Parabolic PDEs
• 9.3 Hyperbolic PDES
  • 9.3.1 The Explicit Central Difference Method
  • 9.3.2 Two‐Dimensional Hyperbolic PDEs
• 9.4 Finite Element Method (FEM) for Solving PDE
• 9.5 GUI of MATLAB for Solving PDES – PDEtool
  • 9.5.1 Basic PDEs Solvable by PDEtool
  • 9.5.2 The Usage of PDEtool
  • 9.5.3 Examples of Using PDEtool to Solve PDEs
  • Problems

What is a partial differential equation (PDE)? It is a class of differential equations involving more than one independent variable. In this chapter, we consider a general second‐order PDE in two independent variables x and y, which is written as

(9.0.1)

subject to the boundary conditions (BCs) given by

(9.0.2)

These PDEs are classified into three groups:

• elliptic PDE: if B² − 4AC < 0
• parabolic PDE: if B² − 4AC = 0
• hyperbolic PDE: if B² − 4AC > 0

These three types of PDE are associated with equilibrium states, diffusion states, and oscillating system, respectively. We will study some numerical methods for solving these PDEs, since their analytical solutions are usually difficult ...