I know this question mayn't be new, but here is my problem: I have to solve this integral equation (numerically, of course, except from very few special cases):
$$\phi_{\nu}(t) = 1 - q\int_0^t \frac{d\psi_{\nu}}{d t'} \phi_{\nu}(t - t')\ dt'$$
Where of course I know what $\psi$ is. I can choose $q = 1$.
I tried to take a look at some past answers, but I did not find what I was looking for. The problem may be that this is a convolution integral.
This problem can be solved using the Laplace transform of a function $f$: $$ \hat{f}(s):=\int_0^\infty e^{-s t} f(t)\, dt. $$ Your equation can be written as $$ f(t)=1-q\int_0^t f(t-t') g(t')\, dt'. $$ Taking the Laplace transform of both sides and using the convolution property of the Laplace transform one gets $$ \hat{f}(s) = 1/s - q\, \hat{f}(s)\,\hat{g}(s). $$ Solving this for $\hat{f}$ gives $$ \hat{f}(s)=(1/s) \left(1+q\,\hat{g}(s)\right)^{-1}.$$ Now use the inverse transform to find $f(t)$.
g[s]. Given that expression, it may be possible to useInverseLaplaceTransformto obtain the desired answer. Am alternative approach is to discretize your integral equation to form a matrix equation, which can be inverted to obtain a numerical answer. $\endgroup$