A spline method for the solution of integral equations of the third kind

On one polynomial method of solving integral equations of the third kind

“…Spline methods [7,9], as well as the variants of the collocation method [8], based on special polynomials, have a saturation property, i.e., the refinement of structural properties of the source data affects the precision of approximate solutions only up to a certain limit. On the other hand, the proposed method is applicable for a narrower class of ETKs compared to the methods, developed in [7,8], since it requires the fulfillment of stronger conditions on the kernel and the right-hand side.…”

“…A number of results in this area was obtained in [3][4][5][6][7][8][9]. In [3], the author constructs a complete theory of solvability of the considered equations, as well as suggests and theoretically substantiates the methods of their approximate solving in the space of type D of distributions, based on Dirac delta function, and in partial cases of zeros of the coefficient u(t) in the space of type V , constructed by meas of the functional of a "finite part of Hadamard integral".…”

“…In [3], the author constructs a complete theory of solvability of the considered equations, as well as suggests and theoretically substantiates the methods of their approximate solving in the space of type D of distributions, based on Dirac delta function, and in partial cases of zeros of the coefficient u(t) in the space of type V , constructed by meas of the functional of a "finite part of Hadamard integral". In [4][5][6][7][8][9] the authors develop and substantiate a number of methods of exact and approximate solving the equations of the form (1) in the space of type V in the most general case of zeros of the coefficient u(t).…”

“…In this paper, based on the ideas and the arguments of the works [3][4][5][6][7][8][9], we construct and substantiate in the sense of [10] (Chap. 1) a special variant of subdomain-collocation method of solving the equations of the form (1) in the space of distributions of type V in the general case of zeros of the coefficient u(t).…”

Special version of the subdomain method for a class of integral equations of the third kind in the space of distributions

“…Therefore, development of theoretically substantiated effective methods of their approximate solution in classes of distributions is an actual and actively developing area of mathematical analysis and numerical mathematics. Some results in the field are obtained in [3][4][5][6][7][8][9]. besides, in [3] a comprehensive theory of solvability is given, and some special approximate methods that are based on using polynomial or splines are developed in the space D{p 1 , p 2 ; m, τ } and for some partial cases of locations, outside the interval, of zeroes of the coefficient for the desired function (further, for short, we name it simply "the coefficient") in the space of type V .…”