Solve set of differential equations

Jacob86

I am looking for the solution of [tex] C_{3.p}(t) [/tex]

Edit: sorry I forgot to add the explantions for the symbols as HallsofIvy pointed rightfully out.
- There are n+m+p equations with n+m+p variables of C
- A, B, C, D, E, n, m, p are constants


using the following set of equations:

Part I
[tex]
\frac {dC_{1.1}} {dt} \frac {V_1} {n} = A . {C_{2.m}} - B . C_{1.1}
[/tex]
[tex]
\frac {dC_{1.2}} {dt} \frac {V_1} {n} = B . C_{1.1} - B . C_{1.2}
[/tex]
[tex]
\frac {dC_{1.3}} {dt} \frac {V_1} {n} = B . C_{1.2} - B . C_{1.3}
[/tex]
... up to n equations for C1
[tex]
\frac {dC_{1.n}} {dt} \frac {V_1} {n} = B . C_{1.(n-1)} - B . C_{1.n}
[/tex]

Part II
[tex]
\frac {dC_{2.1}} {dt} \frac {V_2} {n} = B . C_{1.n} + C . C_{3.m} - D . C_{2.1}
[/tex]
[tex]
\frac {dC_{2.2}} {dt} \frac {V_2} {n} = D . C_{2.1} - D . C_{2.2}
[/tex]
[tex]
\frac {dC_{2.3}} {dt} \frac {V_2} {n} = D . C_{3.2} - D . C_{2.3}
[/tex]
... up to m equations for C2
[tex]
\frac {dC_{2.m}} {dt} \frac {V_2} {n} = D . C_{2.(m-1)} - D . C_{2.m}
[/tex]

Part III
[tex]
\frac {dC_{3.1}} {dt} \frac {V_3} {n} = E . C_{2.m} - E . C_{3.1}
[/tex]
[tex]
\frac {dC_{3.2}} {dt} \frac {V_3} {n} = E . C_{3.1} - E . C_{3.2}
[/tex]
[tex]
\frac {dC_{3.3}} {dt} \frac {V_3} {n} = E . C_{3.2} - E . C_{3.3}
[/tex]
... up to n equations for C3
[tex]
\frac {dC_{3.p}} {dt} \frac {V_3} {n} = E . C_{3.p-1} - E . C_{3.p}
[/tex]

HallsofIvy

Are V1, V2, V3, n, p, A, and B constants? I'm afraid no one will be able to help you if you don't explain your notation. Exactly how many "C"s are there?

Unstable

If you have still interest I can maybe help you. Your proposed systems are so-called transit compartment models (zero order inflow and first order outflow). These equations have actually a very nice solution.

Jacob86

I am very keen on finding the solution to this problem! Any help is welcome. Thanks.

Unstable

Okay! I have to admit that I never saw a coupled version of transit compartments but this is hopefully not a problem. Out of curiosity, where does this system come from? What is the background? From the notation I would guess that this has something to do with delayed concentration terms.

I don’t know how experienced you are but you have to get an idea of the structure:

Therefore, the first step is to solve one part of the systems. Consider e.g. part I. Forget about the constants and write the part I as following (don't forget the initial values!)

[itex]
x’_1 (t) = k_{in}(t) – k \cdot x_1(t) \quad x_1(0)=x_1^0
\\
x'_2(t) = k \cdot x_1(t) – k \cdot x_2(t) \quad x_2(0)=x_2^0
\\
…
[/itex]

Then solve the first equation. In a first step maybe set the inflow [itex] k_{in} \equiv 0 [/itex]. Substitute the calculated solution [itex] x_1(t) [/itex] into the second equation and analytically solve again. Doing this you will discover that these solutions are actually the probability density function of the gamma distribution (up to a factor).

If you could perform and understand this, a large step is made!

Jacob86

Unstable, thanks for helping me out here!

The background of this problem is indeed as you mentioned some sort fo delayed concentration. It is actually a three tank system with recirculation streams from the second to the first and from the third to the second tank. The V constancts represent the volume of the respective tanks, whilst the A,B,.. constants are values for the flow rates. The number of equations n,m and p represent the theoretical tanks for each actual tank.
This diagram should give a better representation of the problem.


Unfortunately I am not as experienced with math as I would like. As the sequence of equtions I have always contain at least 2 variables I dont see a straight forward substitution, but than again i am far from a mattamatician. I work as a consultant process engineer...

Some additional hints/help would be useful.
Thanks,
Jacob

Unstable

Okay! Then one question: Is it really necessary to solve this system analytically (with pen and paper). Do you really want know how C3p(t) in formula(!) looks like? Or would be a numerical solution (estimated with a computer) also okay?

What is the aim of the project? Do you have values for the constants? Do you have to estimate the constants from data? Often in these transit systems the task is to estimate the length of the cascades n,m,p based on data?

All this tasks could be performed without solving the system by hand / analytically.
But if you really want to solve this with pen and paper and you never did this before with other ODEs it is very difficult. Sorry for this stupid question, but could you solve the x’_1(t) equation with kin=0 and x_1^0 = 0 from my previous post. I have to know this for further discussion.

Chestermiller

This tanks-in-series problem with recycles would typically have to be solved numerically. I assume that the C's are concentrations. I also assume, as implied by your problem formulation that there are no chemical reactions occurring. Is that correct? The way to solve a problem like this numerically is to define a solution vector y = C₁₁,...,C¹ⁿ,C₂₁,...,C_2m,C₃₁,...,C_3p. Then express the differential equation in the form

dy/dt = M y

where M is a matrix of constant coefficients. You then solve this set of m+n+p first order linear ODEs in m+n+p unknowns, subject to your specified initial conditions using an automatic ODE solver, preferably one with a stiff equation option, such as the GEAR package.