Classifying a PDE's order and linearity

[docnet]

Homework Statement

State the system of PDEs order and whether they are linear, semi-linear, quasilinear, or fully nonlinear.

Relevant Equations

The equations of conservation laws.
$\partial_t u_1 = \partial_x (f_1(u_1, u_2))$
$\partial_ t u_2 = \partial_x (f_2(u_1, u_2))$

please see below.

 

[docnet]

We have the following equations of conversation laws.

$\partial_t u_1 = \partial_x (f_1(u_1, u_2))$
$\partial_ t u_2 = \partial_x (f_2(u_1, u_2))$This is a system of first order PDEs, because the highest derivatives are of order 1. The nature of the functions $f_1$ and $f_2$ are not given, so we assume the linearity of the PDEs depend on these functions. If the functions $f_1$ and $f_2$ are linear in both $u_1$ and $u_2$, the PDEs are linear. if the functions $f_1$ and $f_2$are not linear in $u_1$ and $u_2$, we have quasilinear PDEs because the coefficients of the highest derivatives depend on lower derivatives.

 

[Delta2]

What if say $f_1(u_1,u_2)=\sin u_1+\frac{1}{\cos u_2}$ isn't then the PDEs full non linear?

 

[docnet]

@Delta2
Then we have

$\partial_t u_1 = \partial_x (\sin u_1+\frac{1}{\cos u_2})$
$\partial_t u_1 = cos(u_1)\partial_x u_1 +sec(u_2)tan(u_2)\partial_x u_2$

and our highest derivatives $\partial_x u_1$ and $\partial_x u_2$ depend on trigonometric functions of $u_1$ and $u_2$, which means our PDE is quasilinear.

We learned a PDE is quasilinear if the coefficients of highest derivatives only depend on lower derivatives. a PDE is fully nonlinear if the dependence on highest derivatives is nonlinear, for example, when we have $(\partial_x u_1)^2$ in a term. Am I misunderstanding the definitions?

 

[Delta2]

We learned a PDE is quasilinear if the coefficients of highest derivatives only depend on lower derivatives. a PDE is fully nonlinear if the dependence on highest derivatives is nonlinear, for example, when we have (∂xu1)2 in a term. Am I misunderstanding the definitions?

I see, hmm, then if those are your definitions of quasilinear and non linear then I think you are correct.

 

[docnet]

@Delta thank you

So we have a system of PDEs whose properties depend on our functions $f_1$ and $f_2$.

(1) If $f_1$ and $f_2$ are both linear in both $u_1$ and $u_2$, our PDEs are linear with constant coefficients.

(2) If $f_1$ and $f_2$ result in derivatives whose coefficients are functions of $u_1$ and $u_2$, our PDEs are quasilinear, because the coefficients of the highest derivatives are functions of $u_1$ and $u_2$.

For example, for $f_1 = (\sin u_1+\frac{1}{\cos u_2})$ we have $\partial_x (\sin u_1+\frac{1}{\cos u_2}) = cos(u_1)\partial_x u_1 +sec(u_2)tan(u_2)\partial_x u_2$.

(3) If $f_1$ and $f_2$ result in derivatives whose coefficients are functions $f(x, t)$, our PDEs are semi-linear, because the coefficients of the highest derivatives only depend on $x$ and $t$.

(4) If $f_1$ and $f_2$ result in PDEs whose dependence on highest derivatives is nonlinear, then we have fully nonlinear PDE.

I am not sure if the case is possible, for I cannot think of an example.

 

[Delta2]

No case (4) is not possible for these PDEs because from the chain rule we get $$\partial_x f_1(u_1,u_2)=\partial_{u_1}f_1\partial_x u_1+\partial_{u_2}f_1\partial_x u_2$$ so the derivatives $\partial_x u_1,\partial_x u_2$ appear as they are, without being raised to powers or anything else.

 

[docnet]

I am sorry, for my last post to make sense, I should have explained that
$u_1, u_2 : t × x → R$ and $f_1, f_2 : R^2 → R$

No case (4) is not possible for these PDEs because from the chain rule we get $$\partial_x f_1(u_1,u_2)=\partial_{u_1}f_1\partial_x u_1+\partial_{u_2}f_1\partial_x u_2$$ so the derivatives $\partial_x u_1,\partial_x u_2$ appear as they are, without being raised to powers or anything else.

Thank you! I understand now.