Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 12 (2012), pp. 44-52
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The method of parametric families of continuous solutions construction for the Volterra integral equations of the first kind arising in the theory of developing systems is proposed. The kernels of these equations admit a first-order discontinuities on the monotone increasing curves. The explicit characteristic algebraic equation is constructed. In the regular case characteristic equation has no positive roots and solution of the integral equation is unique. In irregular case the characteristic equation has natural roots and the solution contains arbitrary constants. The solution can be unbounded if characteristic equation has zero root. It is shown that the number of arbitrary constants in the solution depends on the multiplicity of positive roots of the characteristic equation. We prove existence theorem for parametric families of solutions and built their asymptotics with logarithmic power polynomials. Asymptotics can be specified numerically or using the successive approximations.
Keywords:
Volterra integral equation of the first kind, asymptotics, discontinuous kernel, logarithmic power polynomials, succesive approximations.
@article{VYURU_2012_12_a4,
author = {D. N. Sidorov},
title = {Solution to the {Volterra} {Integral} {Equations} of the {First} {Kind} with {Discontinuous} {Kernels}},
journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
pages = {44--52},
publisher = {mathdoc},
number = {12},
year = {2012},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VYURU_2012_12_a4/}
}
TY - JOUR AU - D. N. Sidorov TI - Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels JO - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie PY - 2012 SP - 44 EP - 52 IS - 12 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/VYURU_2012_12_a4/ LA - ru ID - VYURU_2012_12_a4 ER -
%0 Journal Article %A D. N. Sidorov %T Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels %J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie %D 2012 %P 44-52 %N 12 %I mathdoc %U http://geodesic.mathdoc.fr/item/VYURU_2012_12_a4/ %G ru %F VYURU_2012_12_a4
D. N. Sidorov. Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 12 (2012), pp. 44-52. http://geodesic.mathdoc.fr/item/VYURU_2012_12_a4/