Suppose $a,b,x:\mathbb{R}\mapsto\mathbb{R}$ are three functions of which $a$ and $b$ are known and $x$ is unknown. Suppose they are related by the integral equation.
$$\int_{-\infty}^\infty a(t-s)\,x(s)\,ds = b(t)~,\forall\ t\in\mathbb{R}~.$$
Assume $a(t)$ is a symmetric function ($a(t) = a(-t)$) so that it has a Fourier transform $A(\omega)$ that is real. Assume also that $A(\omega)$ is positive everywhere. If the Fourier transform of $b(t)$ is $B(\omega),$ then we can solve for the Fourier transform of $x(t)$, as
$$A(\omega)X(\omega) = B(\omega)~,$$
from which we can write out $x(t)$ explicitly by using the inverse Fourier transform.
Is there such an explicit solution for $x$ when $a:[-1,1]\mapsto\mathbb{R}$ is a symmetric function, $b,x:[0,1]\mapsto\mathbb{R}$ and are related by
$$\int_{0}^1 a(t-s)\,x(s)\,ds = b(t)~,\forall\ t\in [0,1]~.$$
You can impose any regularity condition on $a(t)$ if that helps in obtaining a solution.