Finite difference methods are one of the primary tools used for numerically analyzing differential equations, both ODE and PDE. In introductory differential equations classes, students are posed with problems like this:

Find the solution of the following DE.

Students are lead to believe that analytical methods, performed on paper or by a computer, are the right tools for solving DE. This could not be further from the truth. In the modern era, numerical methods have completely replaced analytical methods. The problem that applied mathematicians, scientists, and engineers actually care about is this one:

Numerically approximate the solution of the following DE.

Unlike analytical methods, which are severely limited and can only address the simplest DE problems, numerical methods can be used to study an incredible variety of genuinely complex applications.

• Fluid dynamics and wave mechanics.
• Heat conduction and the diffusion of particles.
• Options pricing in a stock market.
• Electromagnetic fields.
• Disease outbreaks and population dynamics.
• Biochemistry and physiology.

Besides being useful in a practical sense, from a purely mathematical perspective, Finite Difference methods lead to fascinating questions about convergence, and the stability and accuracy of computing algorithms. Numerical analysis is a deep field, filled with many interesting theoretical questions.