ABSTRACT
This chapter looks at different methods to find an approximation of the solution of a boundary value problem. It seeks to provide the flavour and the basics of numerical algorithms for first order differential equations, and certainly not to develop the most efficient algorithm, or even to give an overview of the different types of algorithms. The chapter considers first order differential equations of the type. It must be understood that the algorithms used in computer programs are very sophisticated and complicated. The granddaddy of a whole family of algorithms was named after the greatest mathematician in the history of Switzerland, Leonard Euler. There are many improvements of the Euler algorithm. One type is the so-called Runge-Kutta algorithms which came about originally in the period 1895 – 1901 due to the research of C. Runge, W. Kutta, and K. Heun, with further improvements since then.
