Partial Differential Equations

Entering now the vast field of partial differential equations, we immediately announce that our discussion shall be restricted to those types of equations that are of major importance in physics. These are the quasilinear PDEs of second order, which may be written in the general form $${a_{11}}\frac{{{\partial ^2}u}}{{\partial {x^2}}} + 2{a_{12}}\frac{{{\partial ^2}u}}{{\partial x\partial y}} + {a_{22}}\frac{{{\partial ^{2u}}}}{{a{y^2}}} + f\left( {x,y,u,\frac{{\partial u}}{{\partial x}},\frac{{\partial u}}{{\partial y}}} \right) = 0$$ (5.1) (“Quasilinear” means that the second derivatives of u appear in linear order only).