A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains
The author develops a numerical method of collocation type for a class of integral equations \[ y^{\delta}f(y)-y^{\lambda}\log^{l}y\int_{a}^b {k_0(x,y)f(x)w(x)}dx = g(y), \] where \((a,b)=(0,\infty)\), \(f\) is unknown, \(k\) and \(g\) are given functions, \(w(x)=|x|^{\alpha}\exp\{-|x|^\beta\}\). The case of \((a,b)=(-\infty, \infty)\) is reduced to a system of two integral equations on the semi-axes \((0,\infty)\). The singular equation of the third kind is transformed in an equation of the second kind (on the semi-axes) with smooth functions via the change of variables \(x=t^{q/\lambda}\) and \(y=s^{q/\lambda}\). The author suggests to approximate the solution of the regularized equation by a Nyström method which is based on the application of the truncated Gaussian rule to the equation's integral. The obtained functional system should be changed by a collocation system with special knots. The proposed method is theoretically proved and three numerical examples are presented. It is also productive to mention a method of normal splines for the linear integral-differential equations with arbitrary degenerate main parts, including the cases of unbounded intervals: \textit{V. K. Gorbunov} [U.S.S.R. Comput. Math. Math. Phys. 29, No.~1, 145--154 (1989; Zbl 0702.65106)], \textit{V. K. Gorbunov, V. V. Petrischev} and \textit{V. Y. Sviridov} [Lecture Notes in Computer Science 2658, 492--499 (2003; Zbl 1143.45002)], \textit{V. K. Gorbunov} and \textit{V. Yu. Sviridov} [Appl. Numer. Math. 59, No.~3--4, 656--670 (2009; Zbl 1160.65045)].
Fredholm integral equations of the third kind
scientific article; zbMATH DE number 5611790
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