Partial Differential Equations

So far in this chapter we have been exploring ways to solve ordinary differential equations. These are equations with one independent variable. A PDE is one that has more than one independent variable. An example of a PDE is the conservation of mass equation used in fluid dynamics. shown in Eq. (20.34).

Equation 20.34

As you probably guessed, solving the typical PDE is more complicated and more treacherous than solving an ODE. Another complicating factor is that many physical models involve coupled sets of PDEs. The most commonly used ways to solve PDEs are by using finite-difference or finite-element techniques. The computational domain is subdivided into smaller domains called cells . The cells will characterize a 1-, 2-, or 3-D space. The collection of cells that model the computational domain is called a grid. The PDEs are then discretized and solved at each cell . A cumulative record of the solution error is computed. The solution is iterated on until the error falls below a certain convergence criteria.

A detailed discussion of methods to solve partial differential equations is beyond the scope and intent of this book. There are entire books devoted to the subject of solving PDEs. The Java language is well suited to developing methods to solve PDEs. Classes would be defined to represent the PDEs, the computational grid, and each cell within the computational grid. The PDE solvers themselves could be written as public static methods and stored in a package that could be readily accessed by other programs.