A note on solving Cauchy integral equations of the second kind by projection

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Abstract

The purpose of this paper is to approximate the solution of a Cauchy integral equation of the second kind, using projection, finite rank approximations and a regularization procedure. We us the Kantorovich projection, the Sloan projection, the Galerkin projection, respectively. We give a general framework and we prove the existence of the solution for a projection schemes.

Section snippets

Introduction and mathematical background

The theory of Cauchy integral equations has important applications in the mathematical modeling of many scientific fields, such as unsteady aerodynamics and aero elastic phenomena, visco-elasticity, fluid dynamics, electrodynamics (cf. [7], [8], [9], [10], [14]). Several authors have been studied Cauchy integral equations with different numerical procedures, (cf. [4], [5], [15], [16]). The theory of projection approximations is developed in [1], [2], [3]. In [12], we have studied projection

Finite rank approximations and regularization

In this section we introduce a grid (xn,j)j=0n on [-1,1] such that-1<xn,0<xn,1<<xn,n-1<xn,n<1.Consider hat functions e0,e1,e2,,en in C0([-1,1],C) such thatej(xn,k)=δj,k.Define the projection πn from C0([-1,1],C) into itself byπng(x)j=0ng(xn,j)ej(x).We recall that by [2]limnπng-g=0.The solution φ of (4) satisfiesφ=1λ(Tφ-f).LetψTφ=λφ+f.We getφ=1λ(ψ-f),so thatψ=1λ(Tψ-Tf).Henceπnψ=1λ(πnTψ-πnTf).Let us approximate the solution of (5) by ψnP (P for Kantorovich) such thatψnP=1λ(πnTψnP-πnTf).

Theorem 1

Sloan projection

Using (5),ψnS=1λ(TπnψnS-Tf).

Theorem 3

The inverse operator (Tπn-λI)-1 exists for n large enough, it is uniformly bounded for n large enough, andψ-ψnSM1Tπnψ-ψ,whereM1supn0Tπn-λI-1.

Proof

We remark that(Tπn-λI)(ψ-ψnS)=Tπnψ-λψ-TπnψnS+λψnS.

It follows from (8) that(Tπn-λI)(ψ-ψnS)=Tπnψ-λψ-Tf.

By (8),(Tπn-λI)(ψ-ψnS)=Tπnψ-Tψand the result follows. Since T is compact, limnπnT-T=0, and hence M1 is finite. 

Galerkin projection

By using Galerkin projection from (5),ψnG=1λ(πnTπnψnG-πnTf).

Theorem 4

The inverse operator (I-1λπnT)-1 exists for n large enough, it is uniformly bounded for n large enough, andψ-ψnGγπnψ-ψ,whereγsupn0I-1λπnT-1.

Proof

We haveI-1λπnTψ-πnψnG=ψ-1λπnTψ-πnψnG+1λπnTψnG.

From (9),I-1λπnTψ-πnψnG=ψ-1λπnTψ+1λπnTf=ψ-1λ(πnTψ-πnTf).

By (6),I-1λπnTψ-πnψnG=ψ-πnψ.

Hence,ψ-πnψnG=I-1λπnT-1(ψ-πnψ)and we get the desired result. Since T is compact, limnπnT-T=0, and hence γ<. 

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