Methods of numerical solution of singular integral equations with Cauchy-type kernels
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Ioakimidis, Nikolaos
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Abstract
The direct methods are the most efficient methods of numerical solution of singular integral equations with Cauchy-type kernels, which appear in several physics and engineering problems. Here the fundamental already available recent results concerning these methods, which were found mainly during the last fifteen years, are presented in brief. More explicitly, after an introduction to singular integral equations both of the second kind (with variable or constant coefficients) and of the first kind, the four best known direct methods of numerical solution of singular integral equations, i.e. (i) the Galerkin method, (ii) the collocation method, (iii) the quadrature–collocation method and (iv) the quadrature method, are described in some detail. The numerical integration rules for Cauchy-type principal value integrals used in the quadrature–collocation and the quadrature methods are also presented. Moreover, the natural interpolation formula used in the quadrature method and the convergence of this method are also mentioned. A direct iterative method for the numerical solution of singular integral equations is also described. Finally, fifteen mathematical problems concerning possible generalizations of the above methods are also reported. An extensive bibliography is also included in this technical report.
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Cauchy-type singular integral equations, Direct methods, Numerical solution, Galerkin method, Collocation method, Quadrature–collocation method, Quadrature method, Numerical integration rules, Quadrature rules, Cauchy-type integrals, Principal value integrals, Natural interpolation formula, Convergence, Iterative methods