Explicit solutions of singular integro-differential equations with Carleman shift via Riemann–Hilbert methods

The purpose of this paper is to discuss the solutions of two classes of singular integro-differential equations with Carleman shift. Most of the current solutions for such equations are approximate ones obtained via numerical approximation, and there are not many results for explicit solutions. Unlike classical solution methods, we employ the Riemann–Hilbert method to tackle the equations by transforming them into a boundary value problem. This transformation is achieved through the Fourier transform and a special type of Möbius transformations, enabling us to derive both the explicit solution and the Noether solvability condition.