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On the convergence of a quadrature-difference method for linear singular integro-differential equations with discontinuous coefficients. (Russian) Zbl 0728.65115
The author proposes some computational methods and proves some quadrature methods for linear singular integro-differential equations with Hilbert kernel and with discontinuous coefficients and discontinuous right-hand side.
Some estimates of convergence of the numerical solution to the exact solution in the norm of the space \(W^ m_ n\) are given. The approximate values of the solution in the nodes are determined from a system of linear algebraic equations. The coefficients of this system are equal to the coefficients of the initial equation, when the coefficients are continuous and are equal to an average value in the interval between two nearest nodes in the case of an integrable discontinuity.
The approach is in some analogy to the works of H. Iokk [Izv. Akad. Nauk Ehst. SSR, Fiz., Mat. 22, No.1, 31-36 (1973; Zbl 0261.65054) and ibid., No.3, 227-232 (1973; Zbl 0285.65051)] and of Yu. R. Agachev [Kazan Kazanskij gosudarst. Univ. UdSSR (1986; 1987)].

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations