On the Efficient Discretization of Integral Equations of the Third Kind

Economical discretization schemes for the finite‐section method

“…Let us compare the obtained estimates of Card for schemes I and II with estimates for standard piecewise-constant scheme [3]. As the result of comparison we obtain that proposed schemes is more economical for θ 1 + θ 2 < 2 and in the other case the informational expenses coincide up to multiplier.…”

“…Here, J 1 is the Bessel function of order 1 and x is the unknown distribution of the particle size. As is known [3], conditions (2) hold for κ = 3 and β = 4. The kernel a(t, τ ) = [ …”

“…Following [3], [4], we set M = 2 m , N = 2 n , where m, n are some integers, and consider the system of nodes…”

“…Earlier, this problem was studied in [3] for the standard piecewise-constant discretization scheme. The purpose of this paper is to construct new, more economical, piecewise-constant discretization schemes.…”

“…As the result of comparison we obtain that proposed schemes is more economical for θ 1 + θ 2 < 2 and in the other case the informational expenses coincide up to multiplier. In [3], one more piecewise-constant scheme for case μ > λ was studied. In comparison with mentioned scheme from [3], scheme I is more economical for θ 1 < θ 2 < 1 and θ 1 = θ 2 < 1 ; in the other case, the values for Card are of the same order.…”