Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     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},
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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/

[1] N. A. Sidorov, B. V. Loginov, A. Sinitsyn, M. Falaleev, Lyapunov–Schmidt methods in nonlinear analysis and applications, Kluwer Academic Publishers, Dordrecht, 2002, 548 pp. | MR | Zbl

[2] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev type equations and degenerate semigroups of operators, VSP, Utrecht–Boston–Köln–Tokyo, 2003, 216 pp. | MR | Zbl

[3] Aparcin A. S., Nonclassic Volterra Integral Equations of the First Kind: Theory and Numerical Methods, Nauka, Novosibirsk, 1999, 193 pp.

[4] Markova E. V., Sidler I. V., Trufanov V. V., “On Modeling of Evolving Glushkov Systems and Applications in Power Industry”, Avtomatika i telemehanika, 2011, no. 7, 20–28 | MR

[5] A. M. Denisov, A. Lorenzi, “On a special Volterra integral equation of the first kind”, Boll. Un. Mat. Ital. B, 7:9 (1995), 443–457 | MR

[6] Jacenko Ju. P., Integral Models of the Systems with Memory Under Control, Naukova dumka, Kiev, 1991, 218 pp. | MR | Zbl

[7] G. C. Evans, “Volterra's Integral Equation of the Second Kind with Discontinuous Kernel”, Transactions of the American Mathematical Society, 11:4 (1910), 393–413 | MR | Zbl

[8] Kromov A. P., “Integral Operators with Kernels that are Discontinuous on Broken Lines”, Sbornik: Mathematics, 197:11 (2006), 1669–1696 | DOI | DOI | MR | Zbl

[9] Magnickij N. A., “Asymptotic Solution of the Volterra Integral Equation of the First Kind”, DAN SSSR, 269:1 (1983), 29–32 | MR | Zbl

[10] Sidorov N. A., Sidorov D. N., “Existence and Construction of Generalized Solutions of Nonlinear Volterra Integral Equations of the First Kind”, Differential Equations, 42:9 (2006), 1312–1316 | DOI | MR | Zbl

[11] Sidorov N. A., Trufanov A. V., “Nonlinear Operator Equations with a Functional Perturbation of the Argument of Neutral Type”, Differential Equations, 45:12 (2009), 1840–1844 | DOI | MR | Zbl

[12] Sidorov N. A., Trufanov A. V., Sidorov D. N., “Existence and Structure of Solution of Integral-functional Volterra Equation of the First Kind”, Izv. IGU, ser. Matematika, 1 (2007), 267–274 | Zbl

[13] N. A. Sidorov, M. V. Falaleev, D. N. Sidorov, “Generalized solutions of Volterra integral equations of the first kind”, Bull. Malays. Math. Soc., 29:2 (2006), 1–5 | MR

[14] Sidorov N. A., Falaleev M. V., Sidorov D. N., “Generalized Solutions of Volterra Integral Equations of the First Kind”, Bull. Malays. Math. Soc., 29:2 (2006), 1–5

[15] Sidorov N. A., Sidorov D. N., Krasnik A. V., “Solution of Volterra Operator-integral Equations in the Nonregular Case by the Successive Approximation Method”, Differential Equations, 46:6 (2010), 882–891

[16] Trenogin V. A., Functional Analysis, Fizmatlit, M., 2007, 488 pp.

[17] Gel'fond A. O., Finite Differences Calculus, Fizmatlit, M., 1959, 400 pp. | MR

[18] Sidorov D. N., Sidorov N. A., “Monotone Majorants Method in the Theory of Nonlinear Volterra Equations”, Izv. IGU, ser. Matematika, 4:1 (2011), 97–108 | MR