Mechanics, Materials Science & Engineering, March 2016
ISSN 2412-5954
A Fast Algorithm to Solve Nonlinear Hypersingular Integral Equations Arising in a Crack Problem M.R. Capobianco1, G. Criscuolo2,a 1 Istituto per le Applicazioni del Calcolo "Mauro Picone" - CNR, Sede di Napoli, Via Pietro Castellino 111, 80131 Napoli, Italy e-mail: mariarosaria.capobianco@cnr.it 2 Sant'Angelo, Edificio T, Via Cintia, 80126 Napoli, Italy a
giuliana.criscuolo@unina.it DOI 10.13140/RG.2.1.5051.5605
Keywords: nonlinear hypersingular integral equation, collocation method, Newton method, GMRES algorithm.
ABSTRACT. A fast algorithm related to the generalized minimal residual algorithm (GMRES) is proposed to approximate solution of a nonlinear hypersingular integral equation arising in a crack problem. At first, a collocation method is proposed and developed in weighted Sobolev space. Then, the Newton-Kantorovjch method is used for solving the obtained system of nonlinear equations.
Introduction. In several applications in mechanics, electrodynamics, aerodynamics, we have to deal with hypersingular integral equations (HSIEs). Therefore, an active development of numerical methods for solving such equations is very interesting in modern numerical analysis. Among the numerous works devoted to developing efficient numerical schemes for approximate solution of singular and hypersingular integral equations, we mention [2]-[11], [15], and the references cited therein. In particular, many problems in fracture mechanics give rise to the nonlinear hypersingular integral equations (see for example [17], [19], [21]). Nevertheless, approximate solution methods for these latter equations are less elaborated although their increasing interest. Among the papers devoted to these approximate methods we recall [3], [4], [7], [9], [16], [19]. The problem to solve for all x equation using the normal derivative of the standard Green representation. This nonlinear integral equation is solved using Newton's method and a spline-based Galerkin method. A collocation method for approximate solution of nonlinear hypersingular integro-differential equation
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