S.O.S. Mathematics CyberBoard

Hi
I'm new here. I'm having a difficult problem with reaction kinetics. I have a first order differential equation of the form

dx/dt = k'*{(a-2x)^n}*{(b-2x)^m}

where "m", "n" and "k'" are unknown constants and I need to evaluate these. Constants "a" and "b" are known. I have experimental data for "t" and "x"

t x
8 10
13 20
19 30
26 40
33 50
43 60
54 70
69 80
87 90
110 100
140 110
204 120
310 127
inf 144.5

at t(0), x = 0

I would be grateful if someone can suggest methods to solve this

thank you

Hi !
The analytic solution of the equation involves a special function, the incomplete Beta function (see attachment).
Compution of the optimum values of parameters k, m, n can been carried out thanks to non-linear "least squares" regression process. Computer softwares exist in specialized packages for statisticians.
In case of not available such tool, more information should be usefull :
Are k, n, m integers or reals ? (method of solving would be different)
What exactly are the values of the known coefficients a and b ? (in order to compare to the range of x given in the numerical table)

Hi Jacquelin
Thanks for your reply!!

I have matlab but am not conversant with programming in it. I was going through some possible solutions and I found one where the author says that I can use nonlinear curve fit routine and call the ordinary differential equation solver (runge-kutta) from within it. But as I said I'm new to matlab and programming am trying to find a way to write this program.

k, n and m are positive real numbers. n and m can have values ranging from "0 - 4". "k" can have any positive value (its usually in the range of 10^5 to 10^7).
a = 406
b = 289

Hi Leaderey,

if you intened to built by yourself a regression program in the case of differential equation, I am afraid that will be rather arduous. It requires a background in the theory and skill in software.
May be, if you don't need much accuracy and if you accept a long computing time, then you could write a program on the principle of groping (scanning of a limited range of the parameters).
A more smart method, not simple but less complicated than the case of differential equation, consists first to get ride of the differential equation. The integration of dx/dt = k*{(a-2x)^n}*{(b-2x)^m} leads to t as a function of x. This function can be expressed in terms of Incomplete Beta function t = F(x, a, b, k, n, m) in wich the Incomplete Beta appears explicitelly among other terms. So, a non-linear regression process can be applied. It's possible, nevertheless, not easy to implement. For exemple :
http://mathworld.wolfram.com/NonlinearL ... tting.html

Hi !

There is probably a mistake in your wording : a confusion between the coefficient k and 1/k.
For example, in the case x=50
dx/dt = k*[(a-2x)^n]*[(b-2x)^m] The smallest value of dx/dt is = k ( because for any n>0 and m >0 then (406-2*50)^n is >1 and (289-2*50)^m is >1. So, dx/dt > k )
If we consider your given range of 10^5 < k < 10^7 , then dx/dt > 10^5
On the other hand, the order of magnitude of dx/dt is :
(60-40) / (t(60)-t(40)) = (50-40) / (43-26) = 1.18 which is inconsistent with dx/dt > 10^5
So, I suppose that the equation is dx/dt =(1/K) *[(a-2x)^n]*[(b-2x)^m] with k=1/K

The result of the least square fitting depends on what we are looking for. Since the relationship is far to be linear, the optimum values of K, n and m are very different if we intend to obtain the smallest mean relative deviation or the smallest mean absolute deviation.
My results are shown below :

SEARCH FOR THE SMALLEST MEAN ABSOLUTE DEVIATION :
K = 47430
n = 0.505
m = 1.538
Mean absolute deviation = 5.98
Mean relative deviation = 15.4%

Experimental values : (xe, te) . Computed values : tc

xe ; te ; tc
10 ; 8 ; 4.017
20 ; 13 ; 8.642
30 ; 19 ; 14.023
40 ; 26 ; 20.363
50 ; 33 ; 27.944
60 ; 43 ; 37.171
70 ; 54 ; 48.65
80 ; 69 ; 63.328
90 ; 87 ; 82.78
100 ; 110 ; 109.847
110 ; 140 ; 150.256
120 ; 204 ; 217.749
127 ; 310 ; 301.543

SEARCH FOR HE SMALLEST MEAN RELATIVE DEVIATION :
K = 48400
n = 1.157
m = 0.797
Mean absolute deviation = 8.7
Mean relative deviation = 9.8%

xe ; te ; tc
10 ; 8 ; 5.393
20 ; 13 ; 11.47
30 ; 19 ; 18.374
40 ; 26 ; 26.292
50 ; 33 ; 35.471
60 ; 43 ; 46.248
70 ; 54 ; 59.1
80 ; 69 ; 74.721
90 ; 87 ; 94.168
100 ; 110 ; 119.158
110 ; 140 ; 152.723
120 ; 204 ; 200.939
127 ; 310 ; 250.671