Asymptotic solution to convolution integral equations on large and small intervals

We consider convolution integral equations on a finite interval with a real-valued kernel of even parity, a problem equivalent to finding a Wiener-Hopf factorisation of a notoriously difficult class of $2\times 2$ matrices. The kernel function is assumed to be sufficiently smooth and decaying for large values of the argument. Without loss of generality, we focus on a homogeneous equation and we propose methods to construct explicit asymptotic solutions when the interval size is large and small. The large interval method is based on a reduction of the original equation to an integro-differential equation on a half-line that can be asymptotically solved in a closed form. This provides an alternative to other asymptotic techniques that rely on fast (typically exponential) decay of the kernel function at infinity which is not assumed here. We also consider the problem on a small interval and show that finding its asymptotic solution can be reduced to solving an ODE. In particular, approximate solutions could be constructed in terms of readily available special functions (prolate spheroidal harmonics). Numerical illustrations of the obtained results are provided and further extensions of both methods are discussed.