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Quasilinear PDE definition?

Here it's written that:

Definition 3: A partial differential equation is said to be quasilinear if it is linear with respect to all the highest order derivatives of the unknown function.

What does mean to be linear with respect to all the highest order derivatives?

Does it mean that the terms containing the highest order derivatives are linear, but there might be other terms that are nonlinear?

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2 Answers 2

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Yes, that's it. The term $$\frac{\partial u}{\partial y}\frac{\partial ^2u}{\partial y^2}$$ is linear with respect to $\frac{\partial ^2u}{\partial y^2}$, but the term $$\left(\frac{\partial ^2u}{\partial y^2}\right)^2$$ is not linear.

If you have a second order derivative, it does not matter if the equation has a term like $u^2$ to be quasilinear. You just need to check the highest order derivatives (second order ones in the examples of the link).

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To understand the meaning of quasilinear, it is helpful to start with the first-order case. On reference.wolfram.com, quasilinear is defined as follows:

The PDE is said to be quasilinear if it can be expressed in the form $$a(x, y, u(x, y)) \frac{\partial u(x,y)}{\partial x}+b(x, y, u(x, y)) \frac{\partial u(x, y)}{\partial y}=c(x, y, u(x, y))$$

where $a, b$ and $c$ can be nonlinear functions.

For higher order equations, we can have nonlinear functions of some of the derivates, so long as they are not the highest order. For example, the following PDE is second order and quasilinear because it is linear it terms of the highest order derivative, $u_{xx}$: $$ u_{xx} + u_y^2 = 0.$$ It is not linear, however, because the $u_y^2$ term.

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