- #1
Thomas Moore
- 12
- 2
Hi. I'm a bit confused on determining whether a certain PDE is linear or non-linear.
For example, for the wave equation, we have: u_{xx} + u_{yy} = 0, where a subscript denotes a partial derivative.
So, my textbook says to write:
$L = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$
And then it is easy to deduce that $L(u+v) = L(u) + L(v)$ and $L(c u) = cL(u)$.
But, I have no idea how to do this for the following PDEs:
1. $u_{t} - u_{xx} + u/x = 0$, the $u/x$ is throwing me off.
2. $u_{tt} - u_{xx} + u^3 = 0$, the $u^3$ term is throwing me off.
I don't know how to write this as $Lu = 0$, to determine linearity. Any help would be appreciated, thanks!
For example, for the wave equation, we have: u_{xx} + u_{yy} = 0, where a subscript denotes a partial derivative.
So, my textbook says to write:
$L = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$
And then it is easy to deduce that $L(u+v) = L(u) + L(v)$ and $L(c u) = cL(u)$.
But, I have no idea how to do this for the following PDEs:
1. $u_{t} - u_{xx} + u/x = 0$, the $u/x$ is throwing me off.
2. $u_{tt} - u_{xx} + u^3 = 0$, the $u^3$ term is throwing me off.
I don't know how to write this as $Lu = 0$, to determine linearity. Any help would be appreciated, thanks!