 EqWorld The World of Mathematical Equations Editor: Andrei D. Polyanin Russian version

## 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 (closed-form) 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.   The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.