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The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces.

Singular integral equations (SIE) and singular integro-differential equations with Cauchy kernels (SIDE) and systems of such equations model many problems in elasticity theory, aerodynamics, mechanics, thermoelasticity and queuing analysis (see [

In the past, there was a lot of research in literature devoted to an approximate solution of SIE and SIDE by collocation and mechanical quadrature methods. The equations are defined on the unit circle centered at the origin or on the real axis, see for example [

It should be noted that conformal mapping from the arbitrary smooth closed contour to the unit circle does not solve the problem. Moreover, it makes it more difficult. In the present paper we consider the collocation and mechanical quadrature methods for the approximate solution of weakly SIDE. We use the Fejér points as collocation knots. In Section

We note that the convergence of the collocation method, reduction method and mechanical quadrature method for SIDE and systems of such equations in generalized Hölder spaces has been obtained in [

Let

Let

Let

In the complex space

We search for a solution of (

Thus, we obtain the following SLAE from (

We formulate one result from [

The SIE (

In formulas (

Let

The proof can be found in [

Let the assumptions of Lemma

The proof of this lemma can be found in [

Define

The differential operator

From Lemma

The operator

The proofs of Lemmas

The convergence of collocation method and mechanical quadrature method are given in the following theorems.

Let the following conditions be satisfied:

the functions

the index of the function

the operator

the points

The

Using the conditions of Theorem

Using formulas (

The collocation method for SIE was considered in [

Let all conditions of Theorem

It is easy to verify that SLAE (

In this paper, we have proposed the numerical schemes of the collocation method and mechanical quadrature method for solving of weakly SIDE. The equations are defined on an arbitrary smooth closed contour. The convergence of these methods was proved in Lebesgue spaces.

The author would like to thank the Deanship of Scientific Research at King Faisal Universit, Saudi Arabia, for supporting this work by Grant no. 120037.