A note on solving Cauchy integral equations of the second kind by projection
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 on such thatConsider hat functions in such thatDefine the projection from into itself byWe recall that by [2]The solution φ of (4) satisfiesLetWe getso thatHenceLet us approximate the solution of (5) by (P for Kantorovich) such that Theorem 1
Sloan projection
Using (5), Theorem 3 The inverse operator exists for n large enough, it is uniformly bounded for n large enough, andwhere Proof We remark that It follows from (8) that By (8),and the result follows. Since T is compact, , and hence is finite. □
Galerkin projection
By using Galerkin projection from (5), Theorem 4 The inverse operator exists for n large enough, it is uniformly bounded for n large enough, andwhere Proof We have From (9), By (6), Hence,and we get the desired result. Since T is compact, , and hence . □
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