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Power Series Approximation to Solutions of Nonlinear Systems of Differential Equations

Robert C. Pickett
Organization: Physics Department, Bemidji State University
Robert K. Anderson
Organization: Physics Department, Bemidji State University
Gordon E. Lindgren
Organization: Physics Department, Bemidji State University
Journal / Anthology

American Journal of Physics
Year: 1993
Volume: 61
Issue: 1
Page range: 81-85

An excellent article in the American Journal of Physics, by Fairen, Lopez, and Conde develops power series approximations for various systems of nonlinear differential equations. The methods discussed can be applied to solve a wide range of problems. The most important limitations to the method, which is especially severe for some differential equations, results from using floating point arithmetic to determine the (approximate) finite powers series. Increasing the order of the series at first increases the accuracy of the result but after some order is reached, floating point errors result in a dramatic decrease in accuracy. This limitation can be fairly easily overcome by employing modern computer hardware and software. This note develops the necessary Mathematica code for two examples, the Morse oscillator, and the gravitational two-body problem; it was implemented on a NeXT computer.

One feature of Mathematica, the ability to use symbolic programing to achieve exact power series expressions (exact to arbitrary order), makes this language ideal for such problems. Also, calculations involving only rational numbers yield exact results in Mathematica; so, if coefficients are limited to rational numbers, floating point errors are completely eliminated.

*Applied Mathematics > Numerical Methods > Approximation Theory
*Mathematics > Calculus and Analysis > Differential Equations