Partial Differential Equations Question: State if the following PDEs are linear homogeneous, linear nonhomogeneous, or nonlinear:

Question

State if the following PDEs are linear homogeneous, linear nonhomogeneous, or nonlinear:

$u_{t}+u_{x}=sin(x)u$

$u_{tt}-u_{xx}=e^{t}u_{t}$

$u_{tt}-u_{xx}=x^{2}$

$u_{xx}+u_{yy}=u_{x}u_{y}$

As for the first equation, I think it's linear homogeneous, and the second one is linear non-homogeneous...but I'm not sure about the last two! Would appreciate any help! Thank you in advance!

Answer 1

Keep it simple.

First group all the terms involving $u$ to get an expression of the form $$ \mathcal{D}(u)=f(t,x,y) $$ if $f(t,x,y)=0$ your equation is said homogeneous, otherwise it is said non-homogeneous

Then consider the homogeneous equation $\mathcal{D}(u)=0$ (ignoring the eventual term $f$). By definition the equation is said linear if given two solutions $u, v$ and two scalars $\alpha, \beta$ then $\alpha u+\beta v$ is also a solution. In other term you must check that $\mathcal{D}(\alpha u+\beta v)=\alpha \mathcal{D}(u) + \beta \mathcal{D}(v)$. If the equation is not linear it is said non-linear.

1. $\mathcal{D}(u)=u_t+u_x-sin(x)u$ and $f=0$, thus linear homogeneous.

2. $\mathcal{D}(u)=u_{tt}-u_{xx}-e^tu_t$ and $f=0$, thus linear homogeneous.

3. $\mathcal{D}(u)=u_{tt}-u_{xx}$ and $f=x^2$, thus linear non-homogeneous.

4. $\mathcal{D}(u)=u_{xx}-u_{yy}-u_xu_y$ and $f=0$ , thus non-linear homogeneous.

Answer 2

A linear PDE is one that is of first degree in all of its field variables and partial derivatives. Hence, all of them are linear.

1. $u_{t}+u_{x}=sin(x)u$

   LINEAR HOMOGENEOUS.

2. $u_{tt}-u_{xx}=e^{t}u_{t}$

   LINEAR HOMOGENEOUS.

3. $u_{tt}-u_{xx}=x^{2}$

   LINEAR NON-HOMOGENEOUS.

4. $u_{xx}+u_{yy}=u_{x}u_{y}$

   LINEAR HOMOGENEOUS.

Answer 3

PDEs are linear if there is a linear operator $\mathscr{D}$ that operates only on $u$, and they are homogeneous if all terms contained within the PDE are accounted for in $\mathscr{D}u.$ $$ \mathscr{D}u = 0 \text{ is homogeneous} \\ \mathscr{D}u = f \text{ is nonhomogeneous}. $$