A Collocation Method for a Class of Nonlinear Singular Integral Equations with a Carleman Shift

The paper is concerned with the applicability of the collocation method to a class of nonlinear singular integral equations with a Carleman shift preserving orientation on simple closed smooth Jordan curve in the generalized Holder space   L H  .


INTRODUCTION
Nonlinear singular integral equations are widely used and connected with applications in several field of engineering mechanics like structural analysis, fluid mechanics and aerodynamics.This leads to the necessity to derive solutions for the nonlinear singular integral equations arising in applications, by using some approximate and constructive methods, (see [16] ).
The successful development of the theory of singular integral equations (SIE) naturally stimulated the study of singular integral equations with shift (SIES).The Noether theory of singular integral operators with shift (SIOS) is developed for a closed and open contour (see [12][13][14], 17] and others).
The classical and more recent results on the solvability of NSIE should be generalized to corresponding equations with shift, (see [23]).The theory of SIES is an important part of integral equations because of its recent applications in many field of physics and engineering,(see [7,13,14]) We consider a simple closed smooth Jordan curve L in the complex plane with equation   where s-arc coordinate accounts from some fixed point, llength of the curve L .Denote by  D and Consider the conformal mappings   Now, consider the following NSIES: Under the following conditions , respectively.and the homeomorphism L L  : and the derivative   0 without shift has been studied in [4] by modified Newton-Kantorovich method in the generalized Holder space In this paper the polynomial collocation method has been applied to NSIES (1.
where [12,18].We denote by the space of all continuous functions   t u defined on L with the norm: Definition 2.3 [5,9].We denote by , with the norm: Definition 2.4 [5,14].Let denotes to the operator of singular integration to which we associate the Cauchy projection operators where I is the identity operator on Lemma 2.1 [5].The singular operator S is abounded operator on the space ) (L H  and satisfies the inequality where 0  is a constant defined as follows : Lemma 2.2 [5].The shift operator W is a linear bounded continuously invertible operator on the space ) (L H  and satisfies the inequality where is Frechet differentiable at every fixed point In the sphere 2) The equation (2.8) reduces to the following SIES, for the for initial value o u and the arbitrary function Using Definition 2.4 the dominant equation of equation (2.10) reduces to the following singular integral operator with shift : (2.11) Theorem 2.1 [1,5,14].The singular integral functional operator M is Noetherian on where The index of a Noetherian operator M is given by (2.12) Theorem 2.2 [5,21].Let the conditions of Lemma 2.3 and Theorem 2.1 be satisfied and 2) has a unique solution * u in the sphere


, to which the successive approximations: of modified Newton method converges and the rate of convergence is given by the inequality: where the coefficients k  are defined from the system of nonlinear algebraic equation with shift (SNAES) Introduce the operator   We can rewrite SNAES (3.2) in the operator form: Consider, the coordinates of the vector Analogous to Lemma 2.3 the following lemma is valid.

Lemma 3.1.[6]
Let the conditions of Lemma 2.3 be satisfied.Then the operator n  is Frechet differentiable at every fixed point in the sphere   For this aim, we consider the SALES: corresponding to the SIES: According to the collocation method, we seek an approximate solution of equation (2.10) as the form : where the coefficients k  are defined from SLAES: The SLAES (3.8) can be rewritten as following form: on L , the index 0   and the operator '   has a linear inverse in where 1 d and 2 d are constants do not depend on n.

Proof.
From [8], we can write equation (2.10) in the following form: Denote by n X to be the (2n+1)-dimensional subspace of the space , and let n Q be the projection operator into the set of interpolation polynomial of degree n with respect to the collocation points n j t j 2 , 0 ,  .Then the system (3.9) can be written in n X as a linear operator ) where Now, we determine the difference , from (3.10),(3.12)we have (3.13) where n  is polynomial of the best uniform approximation of the function  with degree not exceeding n.From [6,8] and inequality (2.7), we have Hence, we get Let   t J n be the polynomial of best uniform approximation to the function Then from [6], we have   and the rate of convergence is given by the inequality:  

D 0 L and  0 L
the interior and exterior domain of L respectively and let the origin be   D 0 .Denote by 0 L the unite circle with the center at the origin and let  the interior and exterior domain of 0 L respectively.

1 
is a positive constant.Now, we show that the system of linear algebraic equations with shift (SLAES):


, the system (3.9) has the unique solution  

From [ 6 .Theorem 3 . 2
the linear operator n j,  has bounded inverse, that is the SLAES (3.4) under condition (3.5) has the unique solution Thus the following theorem is proved.Let the coordinate of the vector 

SOME AUXILIARY RESULTS Definition 2.1. We denote by
r,

which have partial derivatives up to second order with respect to u and satisfy the following condition
Method for a Class of Nonlinear Singular Integral Equations 125