Talk:Burgers' equation

Factor before the integral
It is $$\ln(a \cdot b) = \ln(a) + \ln(b)$$ and thus the factor $$(4 \pi \nu t)^{-1/2}$$ in the solution of the initial value problem
 * $$u(x,t)=-2\nu\frac{\partial}{\partial x}\ln\Bigl\{(4\pi\nu t)^{-1/2}\int_{-\infty}^\infty\exp\Bigl[-\frac{(x-x')^2}{4\nu t} -\frac{1}{2\nu}\int_0^{x'}u(x,0)dx\Bigr]dx'\Bigr\}.$$

could be splitted from the integral. The differentiation $$\frac{\partial}{\partial x}$$ then removes that summand. Therefore the solution is given by
 * $$u(x,t)=-2\nu\frac{\partial}{\partial x}\ln\Bigl\{\int_{-\infty}^\infty\exp\Bigl[-\frac{(x-x')^2}{4\nu t} -\frac{1}{2\nu}\int_0^{x'}u(x,0)dx\Bigr]dx'\Bigr\}$$

which is simpler (but hides the origin of the solution, which is the heat equation). --Hero Wanders (talk) 22:37, 14 March 2008 (UTC)

Conservation form
Tt might be nice to write the Burgers' equation in conservation form at some point in the article, since that is such a fundamental idea. Also, a good reference for the article is "Numerical Methods for Conservation Laws" by LeVeque. Lavaka (talk) 18:39, 17 June 2008 (UTC)

Animation
The animation is of solutions of a two dimensional Burgers-type equation, which is not discussed on this page. Wouldn't it be more clear if there were simply an animation of a solution to Burgers' equation? 129.215.104.198 (talk) 11:59, 27 January 2014 (UTC)

Assessment comment
Substituted at 10:32, 29 April 2016 (UTC)

Redundancy?
The section Heat equation reiterates almost exactly Viscous Burgers' equation. The only difference is that in the latter the change of variables is given explicitly as $$ -2d \phi^{-1} \partial_x \phi$$ whereas in the former it is written as $$ -2\alpha \partial_x \log y$$ which is confusing because (a) the same notation is used for the original and the replaced variable and (b) $$\ \alpha $$ is not a consistent notation with the rest of the article.

Of course, the "diffusion equation" derived in Viscous Burgers' equation is identical to the "heat equation" derived in Heat equation so this adds nothing but confusion. I suggest to delete the section Heat equation. — Preceding unsigned comment added by 140.247.52.35 (talk) 20:50, 22 May 2017 (UTC)

Simulations to be Removed?
The displayed simulations seem odd. They allegedly show a 3D plot with a "moving" solution to Burger's eq up to "shock". What seems strange is: the "simulation" and its movie-frames seem to correspond to time, the amplitude (z-axis) seems to correspond to values of the solution u. So, why does one need 2 dimensions (bottom plane) for spacial domain for a 1D variable x (as discussed in the article)?

Request A: removal of the 2nd simulation: identical to the first one.

Request B: removal of both simulations. Axis not labelled properly (e.g., z-axis with u, bottom). Dimension-wise contradictory. If movie=time, then one dimension too many at the bottom.

LMSchmitt 23:29, 8 October 2020 (UTC), Updated LMSchmitt 00:06, 9 October 2020 (UTC)


 * I think they are correct, but for the 2D equation (i.e., 2 spatial dimensions), whereas the article only discusses the 1 spatial dimension case. So simulations are fine, but don't correspond to what is discussed in the article. I agree, they are weird, and should match what is in the article. Lavaka (talk) 13:40, 25 November 2022 (UTC)

Change Title for section 2.1
The title for sec 2.1 should be Linear initial condition. Reason: the section does NOT discuss the "Complete integral" of Burgers eq in general. The topic is the solution in case of a linear initial condition. It then is concluded that in this very special case, the explicit solution is also a complete integral.

LMSchmitt 00:29, 9 October 2020 (UTC)

False estimate for the breaking time
The formula for the breaking time is wrong in that sense that -1/min(f') is not well-defined. In sec 1, it is not excluded that min(f')=0. In the latter case, that would give an undefined or +/-oo estimate for the breaking time. ???

LMSchmitt 11:45, 9 October 2020 (UTC)