Solvability of some classes of singular integral equations of convolution type via Riemann–Hilbert problem

Abstract

In this paper, we study methods of solution for some kinds of convolution type singular integral equations with Cauchy kernel. By means of the classical boundary value problems for analytic functions and of the theory of complex analysis, we deal with the necessary and sufficient conditions of solvability and obtain the general solutions and the conditions of solvability for such equations. All cases as regards the index of the coefficients in the equations are considered in detail. Especially, we discuss some properties of the solutions at the nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equation and boundary value problems for analytic functions (that is, Riemann-Hilbert problems). Therefore, the classical theory of integral equations is extended.

Generalized boundary value problems for analytic functions with convolutions and its applications

“…The proof of the necessity is similar to that of the necessity as in Theorem 2.4 of Li. 12 The proof is complete.…”

“…Similar to the solving method of (5.16), we take Fourier transforms to (6.11) and obtain (6.14) is the classical BVPAF with discontinuous coefficients and its solution is similar to the method used in Li and Colliander et al 12,15 Once H ± (x) are determined, we substitute them to (6.12) and then determine (x) and Ψ ± (x). Finally, we can obtain a solution of Equation 6.7, and further discussions will be omitted also.…”

The solvability of a kind of generalized Riemann–Hilbert problems on function spaces $H_{\ast }$

“…This paper mainly deals with the solvability and explicit solutions of the R-HP with two unknown functions. Indeed, it is possible to solving the problem mentioned above in Clifford analysis, which is similar to that in [20][21][22][23][24][25][26][27]. We omit further discussion.…”