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Can anyone explain the Quasi Linear Equation

A first order PDE is called quasilinear if its coefficients depend on the variable u.

A example would help.

doraemonpaul
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Khanak
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  • http://www.dm.unibo.it/home/citti/html/AnalisiMM/quasilinearfirstorder.pdf – Khanak Feb 17 '13 at 21:47
  • I can't seem to find any quasi linear equations that i can solve. I have to know the Burgers Equation as well. Are they related? – Khanak Feb 17 '13 at 21:48
  • Ok, thanks, I had not seen it was first order. – Julien Feb 17 '13 at 21:50
  • From what I see in your link, they're related since the Burgers equation is an example of quasilinear first oder pde. – Julien Feb 17 '13 at 21:51
  • For example I have a burgers equation but i am not sure on how to solve it. I read wikipedia on how to solve burgers equation with a side condition but i had not achieved much. – Khanak Feb 17 '13 at 21:52
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    What does everyone want to solve Burgers' equation these days? Also here and here. You can read about quasilinear equations of 1st and 2nd order in the notes by Erich Miersemann. –  Feb 17 '13 at 22:13

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The definition you asked about works only in some specific cases, I think.

A quasi-linear equation, informally, is a PDE whose highest order terms appear only as individual terms, multiplied by lower order terms (I found this definition in this question). So the Burgers' equation is a quasi-linear first-order PDE because you can write it as $$(1, u) \cdot (u_t, u_x)=0 .$$

A more specific definition of quasi-linear PDE is the following. Let's first write a PDE in its general form: $$F(D^k(u), D^{k-1}(u), \ldots, Du, u, x) =0$$ where $x$ is in some open set $\mathcal{U} \subseteq \mathbb{R}^n$, $u: \mathcal{U} \to \mathbb{R}$ is the unknown function and $F: \mathbb{R}^{n^k} \times \mathbb{R}^{n^{k-1}} \times \ldots \times \mathbb{R}^{n} \times \mathbb{R} \times \mathcal{U} \to \mathbb{R}$ is given.

[$D^k(u)$ stands for the $k$-th multi-index derivative of $u$. That's a generalized way to write a derivative of a scalar multi-variable function.]

A PDE is quasi-linear if you can write it as: $$\sum_{|\alpha|=k} a_{\alpha} (D^{k-1} (u), \ldots, Du, u, x) D^{\alpha} (u) + a_0 (D^{k-1} (u), \ldots, Du, u, x) = 0.$$

Considering the Burgers' equation, we have $k=1$, $a_1 (Du, u, x) = (1, u)$ and $Du = (u_t, u_x)$. With a little bit of patience you can see that this extended definition actually works.

Source: L.C. Evans, Partial Differential Equations (2010), AMS

Francesco Mattiacci
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