Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Existence and Uniqueness of Solutions of Initial Value Problems

  • Gianne Derks
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_193

Definition of the Subject

Many problems in physics, engineering, biology, economics, etc., can be modeled as relationsbetween observables or states and their derivatives, hence as differential equations. When onlyderivatives with respect to one variable play a role, the differential equation is called anordinary differential equation. The field of differential equations has a long history, startingwith Newton and Leibniz in the seventeenth century. In the beginning of the study of differentialequations, the focus is on finding explicit solutions as the emphasis is on solving the underlyingphysical problems. But soon one starts to wonder: If a starting point for a solution ofa differential equation is given, does the solution always exist? And if such a solution exists, howlong does it exist and is there only one such solution? These are the questions of existence anduniqueness of solutions of initial value problems. The first existence result is given in the middleof the nineteenth...

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Gianne Derks
    • 1
  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK