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Integration of Interpolation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg105896] Integration of Interpolation
  • From: blamm64 <blamm64 at charter.net>
  • Date: Thu, 24 Dec 2009 00:15:01 -0500 (EST)
Hi All,

Well, my problem with NDSolve diagnosing delay differential algebraic
equations where there are no explicit delay arguments is because I
defined a "function of an InterpolatingFunction"; it never gets
evaluated apparently because Integrate cannot deal with it unless the
InterpolatingFunction is not "changed" in any way.

Now, I searched the forums and found some posts regarding essentially
what my post is titled, but I could only find instances where the user
was basically trying to get rid of integrations of
InterpolatingFunctions being tied to a particular argument, and what I
want is just the precise reverse of that.

I have an dataset I interpolate.  The independent argument is pressure
difference.  I have a quantity tied to the integral of that function
derived from the dataset.  The quantity is displacement and it is tied
to time-dependent pressure difference "sent through" that function
(volumetric flow rate as a function of pressure difference across it.

I "played around" and came up with the following.  Please consider:

Model

fmodel[delp_]=30 (-(1/(delp+1)^3)+1);

Data

qdata = Table[{delp,fmodel[delp]},{delp,0,1000,1}]; (* note this is
not actual data I have *)

Interpolation

qFConIN = Interpolation[qdata,InterpolationOrder->2];

Interpolation with time dependent variables in the needed form p3[t] -
p4[t] "pre-loaded"

dp3p4 = p3[t]-p4[t];

qFC[t_]=qFConIN[dp3p4]
Out[5]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]]

Interpolation with time dependent variables "pre-loaded", but no
independent argument (t) on l.h.s.

qFCnoT=qFConIN[dp3p4]
Out[6]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]]

Derivatives (zero and higher) work fine (see below after setting
explicit, but dummy, p3[t] and p4[t])

dfmodeldt=D[fmodel[dp3p4],t];

dqFCdt=D[qFC[t],t]
Out[8]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]] (p3'[t]-
(p4'[t])

dqFCnoTdt=D[qFCnoT,t]
Out[9]= InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]] (p3'[t]-
(p4'[t])

Integration of Model

xPfmodel[t_]=Integrate[fmodel[dp3p4],{s,0,t}];

Intergration of "pre-loaded" interpolation (will FAIL)
Next is what I wanted, but this is what is causing NDSolve to diagnose
the diffeqs as delay (I have several functions of xP[t] as terms in
the diffeqs, e.g. fDoor[xP[t]]*qFC[t]/areaPh ... where fDoor is a
force as a function of displacement function ... qFC[t] gets evaluated
just fine ... and qFC[t]/areaPh is a fluid velocity, and qFC[t] is the
pre-defined regulation of that flow which in turn constrains the
displacement xP[t] )


xP[t_]=Integrate[qFC[s],{s,0,t}]
The output here is Integralsign, limits 0 to t, InterpolatingFunction
[{{0,1000}},<>][p3[s]-p4[s]] ds

Now here's where it gets interestingly strange
Even with strange leading t, this integration with "pre-loaded" time-
dependent arguments (but no independent argument t on the
interpolation function), apparently works (see below)

xP2[t_]=Integrate[qFCnoT,{s,0,t}]
Out[12]= t InterpolatingFunction[{{0,1000}},<>][p3[t]-p4[t]]

Now define "dummy" pressures, as functions of time.  In actuality, p3
[t] and p4[t] can vary according to the physics modeled by the diffeqs
(they are two of five time dependent variables in my diffeqs).  All we
need ensure here is their initial value difference is zero

p3[t_]=30+4 t;
p4[t_]=30;

Derivatives work fine (a bit higher InterpolationOrder would make
disparity less)

N[{dqFCdt/.t->2,dqFCnoTdt/.t->2,D[fmodel[dp3p4],t]/.t->2},6]
Out[15]= {0.0571875,0.0571875,0.0548697}

Integral of model at p3[6] - p4[6]

xPfmodel[6]
Out[16]= 562464/3125

APPARENTLY works

In[17]:= xP2[6]
Out[17]= 562464/3125

FAILS

xP[2]
The output is Integralsign, limits 0 to 2, InterpolatingFunction
[{{0,1000}},<>][4 s] ds

Now, I tried several different models, and several different p3 and p4
functions, and I always got agreement between xPfmodel and xP2.

But that really does not *prove* anything.  My problem (one of them
anyway) is I don't know enough Mathematica semantics to decide if xP2
will always do what I need.  I never need the time derivative of xP
since I already have it explicitly (qFC[t]).

Can anyone enlighten me?  What I need to know is the *proper* way to
get what I need done which is integrate and InterpolatingFunction,
rather, a function of an InterpolatingFunction.  What is the *correct*
way of getting what is represented by xP[t_] = Integrate[qFC[s], {s,
0,t}] (but obviously does not work)?

Thanks,
-Brian L.
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