Solve integral (convolution) equation

Question

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$

and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$

How to find a closed expression for $\phi(t)$?

Answer 1

Taking the Laplace transform on both sides of the equation $$ s\,(\mathcal{L}u)(s)-u(0)=(\mathcal{L}\phi)(s)\,(\mathcal{L}u)(s), $$ and $$ (\mathcal{L}\phi)(s)=s-\frac{1}{(\mathcal{L}u)(s)}, $$ $$ \phi=\mathcal{L}^{-1}(s)-\mathcal{L}^{-1}\Bigl(\frac{1}{(\mathcal{L}u)(s)}\Bigr). $$ Unfortunately it seems that there is no closed form for $\mathcal{L}(u)(s)$ in terms of elementary functions. A naïve asymptotic analysis suggests that $$ (\mathcal{L}\phi)(s)\sim -A\quad\text{as }s\to\infty. $$

Answer 2

The first terms of the series expansion of the function $\phi(t)$ can be computed as shown in attachment :