Product integration for finite-part singular integral equations: Numerical asymptotics and convergence acceleration

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We consider equations of the form f{hook}(x)=g(x)+f{hook}bak(x,t) f{hook}(t) (t-x)αdt, a > x > b, for integral α> 1 and where the integral is taken in the Hadamard or finite-part sense (Davis and Rabinowitz (1984, pp. 11-13), Hadamard (1952, pp. 133-141), Paget (1981, p.447)). Such equations occur in fracture mechanics, gas radiation and fluid flow (Kaya and Erdogan (1987)). In particular, we study a variety of examples with α=2, k(x,t)=1 and [a, b]=[0,1]. In order to investigate the dependence of error upon the differentiability of the solution we construct equations whose solutions are x;v for many rational and integral values of v in the interval [-2, 10]. In addition, we compute cases where the solution function is discontinuous, or continuous but not differentiable, at an interior point of the interval. Based solely upon the empirical error analysis, a single extrapolation technique was developed which universally enhanced convergence. © 1992.

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Journal of Computational and Applied Mathematics