  Partial differential equations are fundamental tools in mathematics,sciences
and engineering. This book is mainly an exposition of the various algebraic
techniques of solving partial differential equations for exact solutions
developed by the author in recent years, with emphasis on physical equations
such as: the Calogero-Sutherland model of quantum many-body system in
one-dimension, the Maxwell equations, the free Dirac equations, the generalized
acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and
Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave
equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the
equation of geopotential forecast, the nonlinear Schrodinger equation and
coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson
equations of three-dimensional packets of surface waves, the equation of the
dynamic convection in a sea, the Boussinesq equations in geophysics, the
incompressible Navier-Stokes equations and the classical boundary layer
equations.
  In linear partial differential equations, we focus on finding all the
polynomial solutions and solving the initial-value problems. Intuitive
derivations of Lie symmetry of nonlinear partial differential equations are
given. These symmetry transformations generate sophisticated solutions with
more parameters from relatively simple ones. They are also used to simplify our
process of finding exact solutions. We have extensively used moving frames,
asymmetric conditions, stable ranges of nonlinear terms, special functions and
linearizations in our approaches to nonlinear partial differential equations.
The exact solutions we obtained usually contain multiple parameter functions
and most of them are not of traveling-wave type.
