MHB Classifying Equation 1: Linear, Semilinear or Quasilinear?

[Julio1]

Classify the equation $u_{x_1}+u_{x_2}=u^{3/2}$ depending on if is linear, semilinear or quasilinear.

Hello MathHelpBoards :). The equation isn't linear? How prove this?

 

[Ackbach]

Well, for a first-order pde in one dependent variable and two independent variables to be linear, you have to be able to write it as follows:
$$a(x_1,x_2) \, \frac{\partial u(x_1,x_2)}{\partial x_1}+b(x_1,x_2) \, \frac{\partial u(x_1,x_2)}{\partial x_2}+c(x_1,x_2) \, u(x_1,x_2)=d(x_1,x_2).$$
The RHS, being what it is, precludes being able to write it like this. Now the quasi-linear criteria is that you can write it like this:
$$a(x_1,x_2,u(x_1,x_2)) \, \frac{\partial u(x_1,x_2)}{\partial x_1}+b(x_1,x_2,u(x_1,x_2)) \, \frac{\partial u(x_1,x_2)}{\partial x_2}=c(x_1,x_2,u(x_1,x_2)).$$
Can you write your pde in this form?

 

[Julio1]

Hello Ackbach :). I have clear that isn't linear, because the term $u^{3/2}$ isn't of first grade. Is this the explained?

But I don't how show that the equation is or not semilinear or quasi-linear. Can any help me please?

 

[Ackbach]

How about $a=b=1$, and then $c$ would have to equal...?

 

[Julio1]

OK, then the equation is linear, because the operators $u_x$ and $u_y$ are linear. Therefore the equation isn't semilinear, because the equation is linear. Is quasilinear, because have the form for the definition.

This is right? :)

 

[Ackbach]

I would agree that the equation is quasi-linear. Your first sentence contradicts your latter sentences! Just because the operators on the LHS are linear (which is true) doesn't mean the equation is linear!

 

[Julio1]

I would agree that the equation is quasi-linear. Your first sentence contradicts your latter sentences! Just because the operators on the LHS are linear (which is true) doesn't mean the equation is linear!

Thanks! Sorry, is a mistake.

The equation is quasi-linear because have the form for the definition, have that $c=u^{3/2}=(u(x_1,x_2))^{3/2}.$

The equation is linear because have the form:

$\dfrac{\partial u}{\partial x_1}(x_1,x_2)+\dfrac{\partial u}{\partial x_2}(x_1,x_2)+(-1)u^{3/2}=0$ where $a=b=1, c=-1.$

The equation isn't semilinear, because haven't principal part linear :).

I correct now?

 

[Ackbach]

Yep, looks good to me!

 

[Julio1]

Thanks Ackbach :)!