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numerical integration - NDSolve diffusion equation over/underdetermined


I have a feeling the solution to my problem is very simple… but my knowledge of differential equations is pretty weak.


I am trying to solve a scalar diffusion equation (used in NMR spectroscopy, but from what I understand, modeled after heat and Fickian diffusion). I've tried following model equations, but whenever I ask Mathematica to evaluate, it returns the original equation. What am I missing here?


Here is my code:


With[{Dif = 2300, pde = D[u[t, r], t] == Dif/r*D[u[t, r]*r, r, r] - 

 1/(0.0945 E^(0.000212+0.4077r))*(u[t, r]+1.2004)}, 
 soln = NDSolve[{pde, u[0, r] == 0, Derivative[0, 1][u][t, 0] == 0, 
 Limit[u[t, r], r -> \[Infinity]] == 0}, u, {t, 0, 10}, {r, 0, 500}]]

Upon evaluation, it returns an error stating that the equation is "overdetermined". Removing a single one of the 3 boundary conditions results in an "underdetermined" error.


Is this possible to solve using NDSolve, or do I need to break down the equation?


Thanks for the help!



Answer



As Andrew stated above, changing the lower limit to something non-zero takes care of the problem.


The following code gives a speedy answer:



With[{Dif = 2300}, pde = D[u[t, r], t] == Dif/r*D[u[t, r]*r, r, r] - 
 1/(0.0945 E^(0.000212 + 0.4077 r))*(u[t, r] + 1.2004); 
 soln = NDSolve[{pde, u[0, r] == 0, Derivative[0, 1][u][t, 10^-8] == 0, 
 u[t, 500] == 0}, u, {t, 0, 10}, {r, 10^-8, 500}]]

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