Abstract
In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smoothness of the coefficients and the right-hand side of the equation.
Highlights
In [1,2,3,4], the quadrature-differences methods for the various classes of the periodic singular integrodifferential equations with Hilbert kernels were justified
If for the equations with Hilbert kernels, the same uniform grid is used both for the approximation of the derivatives and integrals and as collocation nodes, for the equations with the Cauchy kernel, we must use two different grids: the roots of the special
The problem is stated in the spaces of weighted quadratically integrable functions; the “second kind” [8] of theory of the approximation methods is used, and the rate of convergence is restricted by the order of smoothness of the desired function, coefficients and the right-hand side of the equation
Summary
In [1,2,3,4], the quadrature-differences methods for the various classes of the periodic singular integrodifferential equations with Hilbert kernels were justified. Note that for the first order equations, this method was justified in [5] It is known (see, e.g., [6,7]) that the theories of the singular integral equations in periodic (with the Hilbert kernel) and non-periodic (with the Cauchy kernel) cases differ greatly, due to the discontinuity of the contour in the latter case. The problem is stated in the spaces of weighted quadratically integrable functions; the “second kind” [8] of theory of the approximation methods is used, and the rate of convergence is restricted by the order of smoothness of the desired function, coefficients and the right-hand side of the equation. The convergence of the method is proven, and the rate of convergence is obtained
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