Nonlinearity

One can classify the degree of nonlinearity in a PDE according to the following categories (from most linear to most nonlinear):

• Linear equations
• Semilinear equations
• semilinear-with-derivatives equations
• Quasilinear equations
• Fully nonlinear equations

It should not be surprising that as an equation becomes more nonlinear, it becomes more difficult to analyze. In particular, iterative methods tend to work well only for semilinear equations, as well as some (but not all) semilinear-with-derivative equations.