Toronto Math Forum
APM3462015F => APM346Home Assignments => HA1 => Topic started by: Emily Deibert on September 29, 2015, 08:39:18 PM

1.
http://www.math.toronto.edu/courses/apm346h1/20159/PDEtextbook/Chapter1/S1.P.html#problem1.P.1 (http://www.math.toronto.edu/courses/apm346h1/20159/PDEtextbook/Chapter1/S1.P.html#problem1.P.1)
a) Linear homogeneous
b) Nonlinear (quasilinear) homogeneous
c) Linear homogeneous
d) Nonlinear (quasilinear) inhomogeneous
e) Nonlinear homogeneous (though I am not sure, as this one seems like it could also be called inhomogeneous semilinear?) I think I have decided that this one is semilinear. Indeed
f) Nonlinear inhomogeneous
g) Nonlinear inhomogeneous
h) Nonlinear homogeneous Linear homogeneous
i) Nonlinear inhomogeneous
j) Nonlinear homogeneous

Similar answers except for:
e) semi linear is always inhomogeneous?
shouldn't h) be linear homogeneous?

If an equation has a variable coefficient, will it also be considered as semilinear? (From chapter 1.3. Classification of equations, it looks like that coefficient as variable also results into a semilinear form.)
\begin{equation} Lu := a_{11}(x,y) u_{xx} + 2a_{12}(x,y) u_{xy} + a_{22}(x,y) u_{yy} = f(x,y,u,u_x,u_y)\end{equation}
Then questions like a) are semilinear homogeneous, which is a subgroup of nonlinear equation?

Andrew, thank you for pointing out my typo in (h)! It is of course homogeneous.

We apply homogeneous/inhomogeneous classification only for linear equations. I corrected in text crossing out homogeneous/inhomogeneous when this is not applicable and typing in red the correct answers when needed