
Exact Solutions of Mathematical Equations
Introduction
Ordinary differential equations,
partial differential equations,
integral equations,
functional equations, and other
equations are encountered in various fields of mathematics, physics,
mechanics, chemistry, biology, economics, and numerous applications.
Exact (closedform) solutions to
mathematical equations play an important role in the proper understanding of
qualitative features of many phenomena and processes in various areas of
natural science. Exact solutions of
nonlinear differential equations graphically demonstrate and allow unraveling
the mechanisms of many complex nonlinear phenomena such as spatial localization
of transfer processes, multiplicity or absence of steady states under various
conditions, existence of peaking regimes, and many others. Furthermore,
simple solutions are often used in teaching many courses as specific examples
illustrating basic tenets of a theory that admit mathematical formulation (in
heat and mass transfer theory, hydrodynamics, gas dynamics, wave theory,
nonlinear optics, and other fields).
Even those special exact solutions that
do not have a clear physical meaning can be used as test problems to verify
the consistency and estimate errors of various numerical, asymptotic, and
approximate analytical methods. Importantly, exact solutions can also serve
as a basis for perfecting and testing computer algebra software packages for
solving differential equations (Mathematica, Maple, MATLAB, CONVODE, and
others).
It is
significant that many equations of physics, chemistry, and biology contain
empirical parameters or empirical functions. Exact solutions allow
researchers to design and run experiments, by creating appropriate natural
(initial and boundary) conditions, to determine these parameters or
functions.


