Numerically integrate a time dependent differential equation

Hi - I need to numerically integrate this equation: dCDF/dt=1/tr where tr = t0 * exp(bP/Tc). t0, b, and P are constants and Tc is defined as: Tc= {Te^4+K * [307.59-190.96 (ln⁡〖t/24〗 )^0.24 ]}^(1/4) where Te, K are constants and t is changing between 1 to 300 days. The initial condition is CDF = 0 at t=ti. I want to integrate the dCDF/dt=1/tr in the forward time direction starting from t=ti until the CDF approach a finite value of 0.05. I appreciate your help.

[Torsten]

What is the unit for t in your expression for Tc ? hours ? days ?

[Alireza Mofidi]

In this empricial fit 307.59-190.96 〖(lnt)〗^0.24 which used in Tc equation the t is in days but in the final form equation for CDF the t is expressed in hours rather than days.

[Torsten]

Then my answer below should be correct.

Answer 1

[Torsten]

bP = 1;
Te = 1;
K = 1;
t0 = 1;
ti = 24;
CDF = @(x)integral(@(t) 1/t0*exp(-bP./(Te^4+K*(307.59-190.96*(log(t/24)).^0.24)).^0.25),ti,x)

CDF = function_handle with value:
    @(x)integral(@(t)1/t0*exp(-bP./(Te^4+K*(307.59-190.96*(log(t/24)).^0.24)).^0.25),ti,x)

x = ti:ti:300*ti;
F = arrayfun(@(x)CDF(x),x)

F = 1×300

0 10.9214 21.4913 31.9095 42.2291 52.4761 62.6657 72.8082 82.9106 92.9783 103.0154 113.0253 123.0105 132.9732 142.9153 152.8384 162.7438 172.6327 182.5061 192.3650 202.2102 212.0423 221.8622 231.6703 241.4672 251.2534 261.0294 270.7955 280.5523 290.2999

plot(x,F)

[Alireza Mofidi]

Thanks for your responce. My main question is how can I solve the equation iteratively in a for loop until the CDF value approach the value of 0.05 with initial condition of CDF=0 @ t=ti.

[Torsten]

You must supply the parameters used in the equation. Then CDF can be computed automatically.

I don't know what you mean by "how can I solve the equation iteratively in a for loop until the CDF value approach the value of 0.05". You solve a differential equation for CDF - there is no need for a for loop or iteration. The solution can be computed as done above.

[Torsten]

@Alireza Mofidi comment moved here:

Yes, I understand. But what I'm looking for is to find out at which "t" the CDF will approach 0.05. I think my question was confusing.

[Torsten]

As you can see, with the parameters I used for bP ,Te, K and t0, CDF tends to infinity as time grows. Insert the "correct" empirical parameters, run the code again and check the CDF curve shown. If it does not behave as you expect, recheck your model.