Numerical Differential Equations
Numerical Differential Equations, also known as numerical integration, is a part
of numerical analysis that deals with solving differential equations that cannot be solved
by analytical means. Many times algorithms are used in order to make approximations for
these differential equations.
Our first founding father would be Leonhard Euler. Euler lived in the 1700’s and
is responsible for numerous contributions including findings in analysis, number theory,
applied mathematics, physics and logic among other things. Euler published his method
for solving numerical differential equations in 1768
Karl Heun was a German mathematician who was born in the 1850s and passed in
1929. During his life he was a professor and lectured on the theory behind different
aspects of differential equations. His major contribution to differential equations is used
in mathematical physics in the context of integrable systems.
Brook Taylor was an English Mathematician who seemed to have all the right
tools to add on to already established mathematical principles. For example, the Taylor
series was probably first discovered by Archimedes and was built upon into the 17th
century when James Gregory worked on the topic, but the general method for
constructing the series of functions was provided by Brook Taylor. The Taylor Series is
an approximation of a function as a sum of terms calculated from the values of the
derivatives at specific points.
Our last two founding fathers we will mention in the same breath because their
works together in the field of numerical analysis deserve mentioning. Runge was a
German born mathematician who spent time all over the world and studied under Karl
Weierstrass. Kutta was another German mathematician who studied in Munich and also
spent a year at Cambridge. The Runge-Kutta method is used to solve differential
equations numerically. In 1895 Runge published the first Runge-Kutta method and ten
years later Kutta published his popular fourth-order Runge-Kutta method.
Where is modeling with differential equations needed? It pops up pretty much
anywhere-engineering, physics, other physical sciences, even economics. Wherever
these models can be used, that is where numerical methods for solving them can be used.
The common methods to apply the methods are Euler’s Method, Runge-Kutta Method
(1st, 2nd, 4th orders), and Finite Difference Methods.
Differential equations are also used in modeling pendulums. Pendulums have
many significant applications in the world of science. All these applications are
surrounding the mathematics that is applied to understanding of pendulum motion.
Pendulums have many applications in time keeping. All of applications have a basis in
differential equations. The formulas for these significant applications are derived from
differential equations. Another significant application of the pendulum in the world is the
measurement of the acceleration due to gravity. This application is also based around the
equations to find the acceleration. This math is once again based around differential
equations. The other applications for seismology, internal guidance systems and chaotic
motion all can be applied as a result of differential equations.
The goal of this project is to investigate numerical methods for approximating
solutions of differential equations.
Numerical Computing with Matlab - George Forsythe
A First Course in Differential Equations with Modeling
Applications, 7th Edition - Dennis G. Zill
Classical Dynamics of Particles and Systems, Thornton and
Marion, 5th Edition