EqWorld

The World of Mathematical Equations

Exact Solutions Methods Software For Authors Math Forums

Exact Solutions of Mathematical Equations

• Introduction
• Algebraic Equations and Systems of Algebraic Equations
• Ordinary Differential Equations
• Systems of Ordinary Differential Equations
• First-Order Partial Differential Equations
• Linear Partial Differential Equations of Mathematical Physics
• Nonlinear Partial Differential Equations of Mathematical Physics
• Systems of Partial Differential Equations
• Nonlinear Delay Partial Differential Equations
• Integral Equations
• Functional Equations
• Equation Index
• Equation Archive
• Basic Handbooks
• Interesting Papers

Introduction

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.