Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 18

An equation is called linear if in (1.3.1), is a linear function of the unknown function and its derivatives.

Thus, for example, the equation is a linear equation, while is a nonlinear equation. The nonlinear equations are often further classified into subclasses according to the type of their nonlinearity. Generally, the nonlinearity is more pronounced when it appears in higher‐order derivatives. For example, the following equations are both nonlinear

(1.3.4)

(1.3.5)

Here denotes the norm of the gradient of . While (1.3.5) is nonlinear, it is still linear as a function of the highest‐order derivative (here and ). Such a nonlinearity is called quasilinear. On the other hand, in (1.3.4), the nonlinearity is only in the unknown solution . Such equations are called semilinear.