The solvability and explicit solutions of singular integral–differential equations of non-normal type via Riemann–Hilbert problem

Abstract

In this article, we study non-normal type singular integral–differential equations involving convolution kernel and Cauchy kernel in class {0}, and we prove the existence of solutions and the Noether conditions. By using theory of Fourier analysis, the equations under consideration are transformed into boundary value problems for analytic functions (i.e., Riemann–Hilbert problems) with discontinuous coefficients. For such problems, we propose one method different from classical one, and we obtain the analytic solutions and the conditions of Noether solvability. At nodal points, all cases as regards the regularity and the asymptotic behaviors of the solution are discussed in detail. In particular, we generalize the case of normal type discussed in the literature (Li, 2017 [7,8], 2020, 2019) to the case of non-normal type.