HISTORY OF EULER METHOD
Leonhard Euler
Leonhard Euler was one of the giants of 18th Century mathematics. Like the Bernoulli’s, he was born in Basel, Switzerland, and he studied for a while under Johann Bernoulli at Basel University. But, partly due to the overwhelming dominance of the Bernoulli family in Swiss mathematics, and the difficulty of finding a good position and recognition in his hometown, he spent most of his academic life in Russia and Germany, especially in the burgeoning St. Petersburg of Peter the Great and Catherine the Great. (1707 - 1783)
Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music.
There are many different methods that can be used to approximate solutions to a differential equation and in fact whole classes can be taught just dealing with the various methods. We are going to look at one of the oldest and easiest to use here. This method was originally devised by Euler and is called, oddly enough, Euler’s Method.
General first order IVP; Where f(t,y) is a known function and the values in the initial condition are also known numbers. From the second theorem in the Intervals of Validity (IVP) section we know that if f and fy are continuous functions then there is a unique solution to the IVP in some interval surrounding.
Modifications and extensions
A simple modification of the Euler method which eliminates the stability problems noted in the previous section is the backward Euler method: yn+1= yn+ h ftn+1, yn+1.
This differs from the (standard, or forward) Euler method in that the function f is evaluated at the end point of the step, instead of the starting point. The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has yn+1 on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly. Other modifications of the Euler method that help with stability yield the exponential Euler method or the semi-implicit Euler method.More complicated methods can achieve a higher order (and more accuracy). One possibility is to use more function evaluations. This is illustrated by the midpoint method which is already mentioned in this article: yn+1= yn+ h f \Big tn+ \tfrac12 h, yn+ \tfrac12 h ftn, yn\Big
This leads to the family of Runge–Kutta methods.
The other possibility is to use more past values, as illustrated by the two-step Adams-Bashforth method: yn+1= yn+ \tfrac32 h ftn, yn- \tfrac12 h ftn-1, yn-1.
This leads to the family of linear multistep methods.