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Get a Piecewise representation of integrals of piecewise functions?

Sunil Nandihalli

Posted 8 years ago

Hi, I am unable to obtain piecewise representations of integrals of piecewise functions beyond the first integral. I am pasting the code used below. For better readability, I have also attached the pdf of the notebook. I would appreciate any help in resolving this issue. Thanks and regards, Sunil f[intervals_]:= Function[{x}, Plus@@Map[Function[{a,b,c},UnitStep[x-a]UnitStep[b-x]c]@@#&,intervals]] g:=f[{{0,1,1},{2,3,-1},{4,5,-1}}] assum:= 0<=t<=10 && t\[Element]Rationals && p\[Element]Rationals && q\[Element]Rationals && r\[Element]Rationals ig[t_] := PiecewiseExpand[FullSimplify[Integrate[g[p],{p,0,t}],Assumptions->assum]] iig[t_]:= PiecewiseExpand[FullSimplify[Integrate[ig[q],{q,0,t}],Assumptions-> assum]] iiig[t_]:= PiecewiseExpand[FullSimplify[Integrate[iig[r],{r,0,t}],Assumptions->assum]] Plot[{g[t],ig[t],iig[t],iiig[t]},{t,0,10}] Attachments:

Hi,

I am unable to obtain piecewise representations of integrals of piecewise functions beyond the first integral. I am pasting the code used below. For better readability, I have also attached the pdf of the notebook. I would appreciate any help in resolving this issue.

Thanks and regards,

Sunil

f[intervals_]:= Function[{x}, Plus@@Map[Function[{a,b,c},UnitStep[x-a]UnitStep[b-x]c]@@#&,intervals]]

g:=f[{{0,1,1},{2,3,-1},{4,5,-1}}]


assum:= 0<=t<=10 && t\[Element]Rationals && p\[Element]Rationals && q\[Element]Rationals && r\[Element]Rationals

ig[t_] := PiecewiseExpand[FullSimplify[Integrate[g[p],{p,0,t}],Assumptions->assum]]

iig[t_]:= PiecewiseExpand[FullSimplify[Integrate[ig[q],{q,0,t}],Assumptions-> assum]]

iiig[t_]:= PiecewiseExpand[FullSimplify[Integrate[iig[r],{r,0,t}],Assumptions->assum]]

Plot[{g[t],ig[t],iig[t],iiig[t]},{t,0,10}]

POSTED BY: Sunil Nandihalli

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Sunil Nandihalli

Posted 8 years ago

Hi Gianiuca, Sorry, when I tried again it worked. I must have missed something the previous time. Thanks for your help.

POSTED BY: Sunil Nandihalli

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 8 years ago

It is expanded for me (version 11.3): In[7]:= iiig[t] // InputForm Out[7]//InputForm= Piecewise[{{t^3/6, Inequality[0, Less, t, LessEqual, 1] \|\| t < 0}, {-3 + 2t, Inequality[3, Less, t, LessEqual, 4]}, {(-79 + 39t - 3t^2)/6, t > 5}, {(1 - 3t + 3t^2)/6, Inequality[1, Less, t, LessEqual, 2]}, {(9 - 15t + 9t^2 - t^3)/6, Inequality[2, Less, t, LessEqual, 3]}, {(46 - 36t + 12*t^2 - t^3)/6, Inequality[4, Less, t, LessEqual, 5]}}, 0]

It is expanded for me (version 11.3):

In[7]:= iiig[t] // InputForm

Out[7]//InputForm=
Piecewise[{{t^3/6, Inequality[0, Less, t, LessEqual, 1] || t < 0}, 
  {-3 + 2*t, Inequality[3, Less, t, LessEqual, 4]}, {(-79 + 39*t - 3*t^2)/6, t > 5}, 
  {(1 - 3*t + 3*t^2)/6, Inequality[1, Less, t, LessEqual, 2]}, 
  {(9 - 15*t + 9*t^2 - t^3)/6, Inequality[2, Less, t, LessEqual, 3]}, 
  {(46 - 36*t + 12*t^2 - t^3)/6, Inequality[4, Less, t, LessEqual, 5]}}, 0]

POSTED BY: Gianluca Gorni

Sunil Nandihalli

Posted 8 years ago

Hi Gianluca, Thanks a lot for the reply. Can you explain why the last integral is still not piecewise-expanded? f[intervals_] := Function[{x}, Evaluate[ Plus @@ Map[ Function[{a, b, c}, UnitStep[x - a] UnitStep[b - x] c] @@ # &, intervals]]]; g = f[{{0, 1, 1}, {2, 3, -1}, {4, 5, -1}}]; Clear[t]; ig[t_] = PiecewiseExpand@ Integrate[g[p], {p, 0, t}, Assumptions -> Element[t, Reals]] Clear[t]; iig[t_] = PiecewiseExpand @ Integrate[ig[q], {q, 0, t}, Assumptions -> Element[t, Reals]] Clear[t]; iiig[t_] = PiecewiseExpand @ Integrate[iig[q], {q, 0, t}, Assumptions -> Element[t, Reals]] Clear[t]; Plot[{g[t], ig[t], iig[t], iiig[t]}, {t, 0, 10}]

Hi Gianluca,

Thanks a lot for the reply. Can you explain why the last integral is still not piecewise-expanded?

f[intervals_] := 
  Function[{x}, 
   Evaluate[
    Plus @@ Map[
      Function[{a, b, c}, UnitStep[x - a] UnitStep[b - x] c] @@ # &, 
      intervals]]];
g = f[{{0, 1, 1}, {2, 3, -1}, {4, 5, -1}}];
Clear[t];
ig[t_] = PiecewiseExpand@
  Integrate[g[p], {p, 0, t}, Assumptions -> Element[t, Reals]]
Clear[t];
iig[t_] = 
 PiecewiseExpand @
  Integrate[ig[q], {q, 0, t}, Assumptions -> Element[t, Reals]]
Clear[t];
iiig[t_] = 
 PiecewiseExpand @ 
  Integrate[iig[q], {q, 0, t}, Assumptions -> Element[t, Reals]]
Clear[t];
Plot[{g[t], ig[t], iig[t], iiig[t]}, {t, 0, 10}]

POSTED BY: Sunil Nandihalli

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 8 years ago

This way it seems to work: f[intervals_] := Function[{x}, Evaluate[ Plus @@ Map[ Function[{a, b, c}, UnitStep[x - a] UnitStep[b - x] c] @@ # &, intervals]]]; g = f[{{0, 1, 1}, {2, 3, -1}, {4, 5, -1}}]; Clear[t]; ig[t_] = PiecewiseExpand@ Integrate[g[p], {p, 0, t}, Assumptions -> Element[t, Reals]] iig[t_] = Integrate[ig[q], {q, 0, t}, Assumptions -> Element[t, Reals]] iiig[t_] = Integrate[iig[q], {q, 0, t}, Assumptions -> Element[t, Reals]] Plot[{g[t], ig[t], iig[t], iiig[t]}, {t, 0, 10}]

This way it seems to work:

f[intervals_] := 
  Function[{x}, 
   Evaluate[
    Plus @@ Map[
      Function[{a, b, c}, UnitStep[x - a] UnitStep[b - x] c] @@ # &, 
      intervals]]];
g = f[{{0, 1, 1}, {2, 3, -1}, {4, 5, -1}}];
Clear[t];
ig[t_] = PiecewiseExpand@
  Integrate[g[p], {p, 0, t}, Assumptions -> Element[t, Reals]]
iig[t_] = Integrate[ig[q], {q, 0, t}, Assumptions -> Element[t, Reals]]
iiig[t_] = 
 Integrate[iig[q], {q, 0, t}, Assumptions -> Element[t, Reals]]
Plot[{g[t], ig[t], iig[t], iiig[t]}, {t, 0, 10}]

POSTED BY: Gianluca Gorni

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