ABSTRACT
This chapter presents the main ideas used in numerical approximations to solutions of first order differential equations. It demonstrates the most powerful technique in applied mathematics—iteration and the reader on programming an algorithm. An in-depth treatment of numerical analysis requires careful attention to error bounds and estimates, the stability and convergence of methods, and machine error introduced during computation. Since both differentiation and integration are infinite processes involving limits that cannot be carried out on computers, they must be discretized instead. Most of them include a sequence of relatively simple steps related to each other—called recurrence. Numerical algorithms define a sequence of discrete approximate values to the actual solution recursively or iteratively. Many mathematical models that attempt to interpret physical phenomena often can be formulated in terms of the rate of change of one or more variables and as such naturally lead to differential equations. Equations for sequences of values arise frequently in many applications.
