A partial differential equation (PDE) extends ODEs by adding one or more independent variables (usually spatial variables). For example, the wave equation for the ideal vibrating string adds one spatial dimension (along the axis of the string) and may be written as follows:
where denotes the transverse displacement of the string at position along the string and time , and denotes the partial derivative of with respect to .2.7 The physical parameters in this case are string tension and string mass-density . This PDE is the starting point for both digital waveguide models (Chapter 6) and finite difference schemes (§C.2.1).
Difference Equations (Finite Difference Schemes)