Skip to main content

differential equations - NDSolve Dankwerts boundary conditions with changing discrete variable


I'm modeling a advection-diffusion-reaction plug flow reactor (2nd order ODE) at steady state: $$0=-v \frac{\delta p[z]}{\delta z}+D\frac{\delta ^2 p[z]}{\delta z^2}-k p[z]$$ With Dankwerts boundary conditions:


$$ v p\mid _{0-} = vp\mid _{0+} - D \frac{\delta p}{\delta z}\mid _{0+} ~~~~~~ z=0$$ $$\frac{\delta p}{\delta z} = 0 ~~~~~~~~~~~~~~~ z=l$$


And here is my code:


params = {Dif -> 0.8, k -> 0.5, v -> 0.25, l -> 5, p1in -> 300};
NDSolve[{0 == -k p[z] - v p'[z] + Dif p''[z], p1in v == v p[0] - Dif Derivative[1][p][0], Derivative[1][p][l] == 0} /. params, p, {z, 0, 5}]


I am trying to have $v$ change at certain points along the length... so I tried changing v to v[z] and using WhenEvent and DiscreteVariable like this:


params = {Dif -> 0.8, k -> 0.5(*,v\[Rule]0.25*), l -> 5, p1in -> 300};
NDSolve[ {0 == -k p[z] - v[z] p'[z] + Dif p''[z], p1in v[z] == v[z] p[0] - Dif Derivative[1][p][0], Derivative[1][p][l] == 0, v[0] == 0.25, WhenEvent[z == 1, v[z] -> 0.50]} /. params, {p, v}, {z, 0, 5}, DiscreteVariables -> {v}]

But it keeps reading v[z] as a dependent variable (not surprisingly) and giving me an error.... I am pretty sure it has to do with the boundary conditions p1in v[z] == v[z] p[0] - Dif Derivative[1][p][0] because this approach works for a regular 1st order ODE in NDSolve


Question is, how can I make v a changing discrete variable for the 2nd order ODE that I have above? i.e. make it work for my boundary conditions.




Thanks @bbgodfrey for the if statement approach (below). As per his comment on using a Piecewise statement I tried doing it and worked well like this:


params = {Dif -> 0.8, k -> 0.5, l -> 5, p1in -> 300, v -> 0.1, vv -> 5};

s2 = NDSolve[{0 == -k p[z] - Piecewise[{{v, z < 1}, {vv, z >= 1}}] p'[z] + Dif p''[z], p1in v == v p[0] - Dif Derivative[1][p][0], Derivative[1][p][l] == 0} /. params, p, {z, 0, 5}];

Answer



As I noted in my comment, replacing v by an If or Piecewise function works. For instance,


params = {Dif -> 0.8, k -> 0.5, v -> 0.25, l -> 5, p1in -> 300};
s2 = ParametricNDSolve[{0 == -k p[z] - If[z < 1, v, vv] p'[z] +
Dif p''[z], p1in v == v p[0] - Dif Derivative[1][p][0],
Derivative[1][p][l] == 0} /. params, p, {z, 0, 5}, {vv}];
Plot[Evaluate[Table[p[vv][z] /. s2, {vv, {.25, .5, 1, 2}}]], {z, 0, 5}]

enter image description here



This approach is generalizable to multiple changes in the "constants". Generally, WhenEvent is need only for changes to occur when a dependent function reaches some value.


Comments

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...