Started working on trying to find an analytical approximation to this integral and not getting very far. Any assistance or direction is greatly appreciated! Thanks Vince
$$\int_{0}^{t} \frac{\frac{a}{(1+bt_p)^2}e^{-(\frac{at_p}{1+bt_p})}}{\sqrt{(c + (1+bt_p)^2)^2 + d(t-t_p)}}dt_p$$
Found the above equation may be a possible Laplace form from An Introduction to Asymptotic Analysis by Simon J. A Malham (see Chap4. pg 31 onward):
$$I(x) = \int_{u}^{v} f(t)e^{x~\phi(t)}$$ with $t = t_p$ and the limits are $v=t$ and $u=0$ gives: $$f(t_p) = \frac{\frac{a}{(1+bt_p)^2}}{\sqrt{(c + (1+bt_p)^2)^2 + d(v-t_p)}} $$ and taking $x =-1$ and $\phi(t_p)$ as: $$\phi(t_p) = e^{\frac{at_p}{1+bt_p}}$$ Next I need to work through the boundary term at the limits of 0 to t (t is the final time of the filtration period which is a constant and known experimental value from my data sets): $$ -\frac{f(t_p)}{\phi^{'}(t_p)}\cdot~e^{-\phi(t_p)}$$
and new integral term: $$ \int_{u}^{v} \frac{d}{dt_p}\left(\frac{f(t_p)}{\phi^{'}(t_p)}\right)\cdot e^{x\phi(tp)}dt_p$$
Next steps are going to take me some time but will update as I progress through this problem but of course welcome any edits or additions.