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Fitting curve by using normalized function


Because of lack of data, I only have normalized data to fit. (namely the data value are divided by the max value of data and the peak value becomes unit)


The data is shown as follow:


r76 = {{542.6701856282341`, 
    0.07764989004174758`}, {626.6173842143535`, 
    0.29349832955371236`}, {656.0895281008903`, 
    0.5913314613326053`}, {728.212293943863`, 
    1.`}, {782.1313241848961`, 
    0.9743890490336464`}, {851.0031790196008`, 
    0.913194695825656`}, {902.2419705010758`, 

    0.9251181107734547`}, {925.9565995696244`, 
    0.8783386058472558`}, {1054.3921434193162`, 
    0.8614126945081949`}, {1089.1125500560115`, 
    0.7714787065217009`}, {1177.2955761199148`, 
    0.6820289616346121`}, {1256.2853554074297`, 
    0.6688860100912675`}, {1366.9471353702515`, 
    0.6517552032847401`}, {1430.506693892375`, 
    0.6350631327313485`}, {1449.10501292244`, 
    0.5724617566068132`}, {1487.1390312262042`, 
    0.5160313138756719`}, {1693.2356816458084`, 

    0.4359499374813523`}, {1726.442918377672`, 
    0.3981071705534973`}, {1760.2208884864226`, 
    0.34071744173828206`}, {1829.7137765757996`, 
    0.23848704296299775`}, {2029.69014110562`, 
    0.17927557609349742`}, {2082.6290742109622`, 
    0.12878318301060854`}, {2165.1476130003107`, 
    0.10950567327768596`}, {2325.1744775818124`, 
    0.0895597094679642`}, {2448.8431013825957`, 
    0.07324681288106921`}, {2716.647998214203`, 
    0.05990523673095968`}};


Then, I am trying to fit the data via the function given as:


Vi[n_, i_] := 
 Vi[n, i] = (-1)^(i + n/2) Sum[ 
     k^(n/2) (2 k)! /( (n/2 - k)! k! (k - 1)! (i - k)! (2 k - 
            i)! ), { k, Floor[ (i + 1)/2 ], Min[ i, n/2] } ] // N; 
Stehfest[F_, s_, t_, n_: 16] :=
 If[n > 16, Message[Stehfest::optimalterms, n];
        If[ OddQ[n], Message[Stehfest::odd, n];
                "Enter an even number of terms",

                If[n > 32, Message[Stehfest::terms, n];
                    " Try a smaller value for n. Maximum allowable n is 32 ",
                    Log[2]/t Sum[ Vi[n, i]*F /. s -> i Log[2]/t , {i, 1, n} ] ]],
        If[ OddQ[n], Message[Stehfest::odd, n];
                "Enter an even number of terms", 
    If[n > 32, Message[Stehfest::terms, n];
                    " Try a smaller value for n. Maximum allowable n is 32.",
                    Log[2]/t Sum[ 
       Vi[n, i]*F /. s -> i Log[2]/t , {i, 1, n} ] ]]]  // N; 


C1[r_, t_, al_, td_, tc_] := Re[Stehfest[E^(r/(2*al*(1 + p*tc)) + (p*r*td)/(2*al*(1 + p*tc)))*
 AiryAi[(2*2^(1/3)*((Q + 2*p*Q*td + p^2*Q*td^2)/(4*al^2*Q*(1 + p*tc)^2) + 
     (r*(8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 8*al*b*p^3*Pi*R*tc*td*\[Phi]))/
      (4*al^2*Q*(1 + p*tc)^2)))/((8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 
      8*al*b*p^3*Pi*R*tc*td*\[Phi])/(al^2*Q*(1 + p*tc)^2))^(2/3)]*
 (M/(E^(rl/(2*al*(1 + p*tc)) + (p*rl*td)/(2*al*(1 + p*tc)))*Q*
    AiryAi[(2*2^(1/3)*((Q + 2*p*Q*td + p^2*Q*td^2)/(4*al^2*Q*(1 + p*tc)^2) + 
        (rl*(8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 8*al*b*p^3*Pi*R*tc*td*\[Phi]))/
         (4*al^2*Q*(1 + p*tc)^2)))/((8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 
         8*al*b*p^3*Pi*R*tc*td*\[Phi])/(al^2*Q*(1 + p*tc)^2))^(2/3)] - 

   al*E^(rl/(2*al*(1 + p*tc)) + (p*rl*td)/(2*al*(1 + p*tc)))*Q*(1/(2*al*(1 + p*tc)) + (p*td)/(2*al*(1 + p*tc)))*
    AiryAi[(2*2^(1/3)*((Q + 2*p*Q*td + p^2*Q*td^2)/(4*al^2*Q*(1 + p*tc)^2) + 
        (rl*(8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 8*al*b*p^3*Pi*R*tc*td*\[Phi]))/
         (4*al^2*Q*(1 + p*tc)^2)))/((8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 
         8*al*b*p^3*Pi*R*tc*td*\[Phi])/(al^2*Q*(1 + p*tc)^2))^(2/3)] - 
   (E^(rl/(2*al*(1 + p*tc)) + (p*rl*td)/(2*al*(1 + p*tc)))*(8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 
      8*al*b*p^2*Pi*R*td*\[Phi] + 8*al*b*p^3*Pi*R*tc*td*\[Phi])*AiryAiPrime[
      (2*2^(1/3)*((Q + 2*p*Q*td + p^2*Q*td^2)/(4*al^2*Q*(1 + p*tc)^2) + (rl*(8*al*b*p*Pi*R*\[Phi] + 
            8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 8*al*b*p^3*Pi*R*tc*td*\[Phi]))/(4*al^2*Q*(1 + p*tc)^2)))/
       ((8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 8*al*b*p^3*Pi*R*tc*td*\[Phi])/

         (al^2*Q*(1 + p*tc)^2))^(2/3)])/(2^(2/3)*al*(1 + p*tc)^2*
     ((8*al*b*p*Pi*R*\[Phi] + 8*al*b*p^2*Pi*R*tc*\[Phi] + 8*al*b*p^2*Pi*R*td*\[Phi] + 8*al*b*p^3*Pi*R*tc*td*\[Phi])/
       (al^2*Q*(1 + p*tc)^2))^(2/3)))), p, t, 16]]

with


b = 1; Q = 2.5/60; \[Phi] = 0.2; rl = 0.05; M = 10; R = 1;

I got some troubles here. The function value is not normalized and the peak value would change when the parameters changes; thus, I cannot use the peak value of the function to normalize during the data fitting.


Therefore, the fitting process do not work.


nlm1 = NonlinearModelFit[r76, 

   C1[7.6, t, al, td, tc], {{al, 0.1}, {td, 101}, {tc, 1}}, t];

 LogLogPlot[nlm1[t], {t, r76[[1, 1]], r76[[Length[r76], 1]]}], 
 Frame -> True]

In short, how can I use the non-normalized function to fit the normalized data?


Does anyone know how to deal with this? Thanks.




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