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    $\begingroup$ In general, a differential equation cannot be solved numerically over an infinite domain, so you will need to truncate the problem in x and provide boundary conditions. Even then, `NDSolve cannot solve this problem without help. Please see my solution to a related problem here. $\endgroup$ Commented Sep 18, 2020 at 20:26
  • $\begingroup$ @bbgodfrey thank you, I'm trying to understand this numerical method of lines. Before I'd like to ask if it's possible to avoid boundary conditions. Actually in my problem I have the constraint $\int_0^\infty\rho(t,x)dx=1$ for all $t\geq0$. I could impose $\rho(t,\bar x)=0$ where $\bar x=100$ (or any suitably large number), but I have no condition to impose for $\rho(t,0)$. Furthermore from other results I expect $\rho(t,0)$ to change a lot during time passing from $0$ to a Dirac delta $\endgroup$ Commented Sep 20, 2020 at 14:57
  • $\begingroup$ It is not possible, even in principle, to avoid boundary conditions. However, you could try integrating the entire integral-differential equation over x to obtain an expression for ρ(t,0). Note, though, that you will encounter resolution problems as your solution collapses to a delta-function. Perhaps, representing rho as the sum of a numerical function to be calculated plus a one-parameter function peaking at x = 0, with that parameter also to be calculated. Adding the definitions of f[x] and g[x] might be helpful to readers. $\endgroup$ Commented Sep 20, 2020 at 16:32
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    $\begingroup$ So much for that idea of mine. Adding the definitions of f[x] and g[x] to the question would be helpful. $\endgroup$ Commented Sep 20, 2020 at 18:21
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    $\begingroup$ In your 1st block of code, when making tables, you scale various functions by $MachinePrecision. I've find that a strange choice. Can you tell why you chose that scale factor? $\endgroup$ Commented Sep 24, 2020 at 23:39