| Dr. Jim Sochacki, 26 February 2007 |
| Newton, Maclaurin, Picard
and Padé: Four Horsemen of Differential Equations |
ABSTRACT: Polynomial functions and rational functions (quotients of polynomial functions) are immensely popular in mathematics and science. In this talk I will discuss a strong relationship between these two type of functions using differential equations. Newton gave us differential equations. Unfortunately, many of these are difficult to solve. Macluarin later showed how we can approximate the solutions to these differential equations using polynomials. Picard later presented the theory for understanding differential equations. Pade then showed another technique for solving these differential equations. In this talk I will show how the ideas of these four mathematicians are similar and how they are related to polynomial and rational functions. I will also show how these ideas lead to a quite simple and general method for solving many differential equations efficiently. I will show the method on several famous differential equations, including Newton's planetary motion problem and the double pendulum with animations for visualization. This method is quite easy and can be understood by undergraduates who have had two semesters of calculus. |
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