Explicit solutions of Cauchy singular integral equations with weighted Carleman shift

Abstract

Existence and uniqueness of solutions, as well as their explicit representations, are obtained for singular integral equations with weighted Carleman shift which cannot be reduced to binomial boundary value problems.

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

“…We will omit the proofs for the cases of the other convolutions, because they are similar to this one, using, of course, the corresponding different identities. The following identities help us to obtain the factorizations for the other convolutions, when we consider = 1 and k = 2, through the decomposition presented for the factorization (17). Namely, we have…”

“…Such class of integral equations has brought about a great number of results in mathematical analysis, assembled in a huge number of works along the last decades. 2,9,[15][16][17][18][19][20][21] We are speaking about the class of Wiener-Hopf plus Hankel integral equations. Namely, in the present work, we will obtain the L 2 -solution of a Wiener-Hopf plus Hankel equation, in terms of a Fourier-type series.…”

Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case

“…In recent years, many papers devoted to particular investigations and containing solutions in explicit form of SIES have been published (see [3,4,5,8,17,18,19,20,21]). The most general and important class among the SIES reducible to Riemann boundary value problems that can provide explicit solutions in a certain sense, is that of singular integral equations with a Carleman shift.…”

“…In the general case of Carleman shift W , the equation of the form (1.3) attracted the attention of many authors. Namely, under the assumption that the coefficient a(t) does not vanish on Γ, the papers [3,4,7,8,28,31] studied the solvability and obtained explicit solutions of their corresponding equations (1.3) by means of Riemann boundary value problems. Our main goal is to analyse the solvability and obtain eventual solutions of (1.3) when the function-coefficient a(t) vanishes on Γ in the sense that it has isolated zero-points, i.e.…”

On the Solvability of Singular Integral Equations with Reflection on the Unit Circle

“…In much of the cases, the so-called Gohberg-Krupnik-Litvinchuk identity (see e.g., [16,17,20]) and other explicit operator equivalence relations (c.f., e.g., [5,17]) are main ingredients for such analysis. In this way, the solvability of a (scalar) SIES associated with the SIOS is equivalently reformulated as a matrix factorization problem for corresponding matrices (which are built based on the new matrix coefficients); for these and other methods see, for instance, [6,10,19,20,23].…”