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calculus and analysis - How to collect terms with z-derivative?


my equations are very long (several pages). Here I will provide simple example:


eq = (g[x, y, z, t])^2*D[D[f[x, y, z, t], {x, 1}] {z, 1}] + \[Alpha]*

f[x, y, z, t]*D[D[g[x, y, z, t], {y, 1}], {z, 2}] + 
g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}]+D[f[x,y,z,t],{z,2}]

So, it has several functions, constant parameters, and consists of the sum of some terms. Each term is the product of some number of these functions and it's derivatives.


I want to collect, or sort, these terms according to z-derivative. So, first, I want to sort these terms according to the highest z-derivative of the function f[x,y,z,t]. So, in the example above, first term should be g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}]+D[f[x,y,z,t],{z,2}], as long as it has second derivatives of the function f[x,y,z,t] over z. After that it should be (g[x, y, z, t])^2*D[D[f[x, y, z, t], {x, 1}] {z, 1}], and then f[x, y, z, t]*D[D[g[x, y, z, t], {y, 1}], {z, 2}].


Note, that the derivative could be taken over several arguments, like g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}].


I looked through the examples of Collect, but didn't find a way to specify z-derivative, as a second argument. Also it would be good, if you point out, how to show only the terms with z derivatives.


Thanks in advance, Mikhail



Answer



I'm sure there is an easier and shorter way of doing this so consider this a starting point.



A strange thing I noticed is that when I copied and pasted your equation into a notebook it turned into a list.


eq = g[x, y, z, t]^2*D[D[f[x, y, z, t], {x, 1}] {z, 1}] + 
   α*f[x, y, z, t]*D[D[g[x, y, z, t], {y, 1}], {z, 2}] + 
  g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}] + 
  D[f[x, y, z, t], {z, 2}]

Mathematica graphics


I continue to be mystified by this however we need to take it into account.


Step 1 - Strip curly brackets


eq = eq[[1]]



Mathematica graphics


Step 2 - Break it into parts


eqList = eq[[#]] & /@ Range[Length[eq]]


Mathematica graphics


This effectively breaks it at the plus sign.


Step 3 - Sort the parts


This is the significant portion. We will use the value of the zth derivative to do the sorting.


sortedEqList = 
 Sort[eqList, 
  Total@Cases[#1, 

      Derivative[i_Integer, j_Integer, k_Integer, l_Integer][f | g][x,
         y, z, t] -> k, {0, Infinity}] > 
    Total@Cases[#2, 
      Derivative[i_Integer, j_Integer, k_Integer, l_Integer][f | g][x,
         y, z, t] -> k, {0, Infinity}] &]

Mathematica graphics


Step 4 - Join the sorted parts


If you merely rejoin the parts using Plus they will be sorted back to the original order so use Inactive.


Inactive[Plus] @@ sortedEqList


Mathematica graphics


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