Timeline for Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?
Current License: CC BY-SA 3.0
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| Nov 24, 2014 at 14:40 | comment | added | Carlo Beenakker | I'm not sure what you mean by "the first term", probably you mean the boundary term of a partial integration? Then you have to send $y$ to $\pm\infty$, and the boundary term vanishes. The answer to the second question is "yes, it's obvious you need a delta function" because you are trying to write a local relation in a nonlocal form, and the delta function restores locality. | |
| Nov 24, 2014 at 14:34 | comment | added | Michael Andrew Bentley | please could I just clarify a few things? Firstly, when I do integration by parts I get the first term as $v(y)u(y,t)\delta (x-y)$. For every $x$ not equal to $y$ this is zero and the term disappears, but at $x=y$ it is $\inf$, so why can we ignore it? Secondly, how do you know to use the delta function in defining $h(y,x)$, was it an 'ansatz' or is it something obvious that I am missing? Sorry for my ignorance, it's just I'm a biologist by training and this is all new to me. | |
| Nov 24, 2014 at 11:15 | comment | added | Michael Andrew Bentley | I just had to learn about the Dirac delta function! This is amazing, thank you so much for your help. | |
| Nov 24, 2014 at 11:09 | vote | accept | Michael Andrew Bentley | ||
| Nov 24, 2014 at 6:28 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
added 11 characters in body
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| Nov 24, 2014 at 5:50 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |