Quasilinear Problems

Abstract

Quasilinear differential equations are usually understood to be equations in which the highest derivative appears linearly. In the case of first order ODE systems, they are of the form

$$ C\left( y \right) \cdot y' = f\left( y \right), $$

((6.1))

where C(y) is a n × n -matrix. In the regions where C(y) is invertible, Eq. (6.1) can be written as

$$ y = C{\left( y \right)^{ - 1}} \cdot f\left( y \right) $$

((6.1))

and every ODE-code can be applied by solving at every function call a linear system. But this would destroy, for example, a banded structure of the Jacobian and it is therefore often preferable to treat Eq. (6.1) directly. If the matrix C is everywhere of rank m (m < n), Eq. (6.1) represents a quasilinear differential-algebraic system.