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 Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, Issue 12, Pages 44–52 (Mi vyuru56)

Mathematical Modelling

Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels

D. N. Sidorov

Energy Systems Institute SB RAS, Irkutsk State University (Irkutsk, Russian Federation)

Abstract: 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.

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UDC: 517.983
MSC: 93A30, 45D05, 45M05

Citation: D. N. Sidorov, “Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 12, 44–52

Citation in format AMSBIB
\Bibitem{Sid12} \by D.~N.~Sidorov \paper Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels \jour Vestnik YuUrGU. Ser. Mat. Model. Progr. \yr 2012 \issue 12 \pages 44--52 \mathnet{http://mi.mathnet.ru/vyuru56} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. D. N. Sidorov, A. N. Tynda, I. R. Muftakhov, “Chislennoe reshenie integralnykh uravnenii Volterra I roda s kusochno-nepreryvnymi yadrami”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 7:3 (2014), 107–115
2. I. R. Muftahov, D. N. Sidorov, “Solvability and numerical solutions of systems of nonlinear Volterra integral equations of the first kind with piecewise continuous kernels”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 9:1 (2016), 130–136
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