Solving a system of integro-differential equations

I have a system of integro-differential equations that I need to solve. I am having some trouble with it. The system is as below:

This is a complicated system, and I am struggling with how to tackle the integrals, mainly how to pass the previous values in a solver like ode23s. Please note that for the infinite sum, it is reasonable to truncate it after some terms (e.g. n=0 to 5) which would be decided by the user. Would appreciate help in solving this system.

[Walter Roberson]

You cannot deal with an infinite sum in finite time -- not unless you can find a closed form for the infinite sum.

When I look through the set of equations, I do not think you are going to be able to find a closed form for the infinite sum. relies on which is expressed only in the form of its derivative so you would have to integrate that. But it depends upon so you would have to integrate ... but what you have for is its derivative ... that is defined in terms of the infinite sum...

Maybe there is some clever approach that could work, but I am not at all confident that there is.

I think you are probably going to have to use numeric approximations, but even then I have my doubts it can work.

Now if you were to substitute an integral for the infinite sum, there miight maybe be more hope of getting a workable approximation.

[Shoubhanik]

Sorry, I should have been clearer. So it is possible to truncate the series upto finite terms. We can assume that the series is truncated after some number of terms, decided by the user.

[Walter Roberson]

Have you tried setting up the system symbolically, with some small fixed number of terms for the infinite series (for example, 5 terms to experiment with), and then following the steps shown in the first example for odeFunction and see if MATLAB is able to figure out what the resulting ode system should look like?

Answer 1

[Torsten]

Define

dI_n / dt = P1*exp(-lambda_n^2*D_a*t) , I_n(0) = 0

as additional ODEs to be solved. Then

I_n = integral_{tau = 0}^{tau = t} P1(tau)*exp(-lambda_n(tau)^2*D_a*tau) dtau

[Shoubhanik]

What I really need is to pass the vector of past solutions of P1 and the times at which they were evaluated, for any time t. If I do as you suggested, does that not mean that I_n would actually be solved only for the current timestep, and not all past timesteps? Does solving the additional ODE reflect the dependence on all the timesteps till now? I am confused about that.

Is there ay way to pass the vectors of past values of P1, l, and t to my ode function, where I can then use a numerical integration to evaluate the integral?

[Torsten]

If I do as you suggested, does that not mean that I_n would actually be solved only for the current timestep, and not all past timesteps?

I_n is computed by the additional ODE from t = 0 up to the time t in the upper limit of integration. And this value is needed to compute B_n(t) according to your formula.

Is there ay way to pass the vectors of past values of P1, l, and t to my ode function, where I can then use a numerical integration to evaluate the integral?

Why ? I_n , the value of the additional ODE function, is the value of the integral.

If

y(t) = integral_{s=0}^{s=t} g(s) ds,

then

dy/dt = g(t), y(0) = 0. 

It's nothing but the fundamental theorem of calculus.