ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2019. Vol. 27

Classic Solutions of Boundary Value Problems for Partial Differential Equations with Operator of Finite Index in the Main Part of Equation

N. A. Sidorov

This paper is an attempt to give the review of a part of our results in the area of singular partial differential equations. Using the results of the theory of complete generalized Jordan sets we consider the reduction of the PDE with the irreversible linear operator B of finite index in the main differential expression to the regular problems. Earlier we and other authors applied similar methods to the development of Lyapunov alternative method in singular analysis and numerous applications in mechanics and mathematical physics. In this paper, we show how the problem of the choice of boundary conditions is connected with the B-Jordan structure of coefficients of PDE. The estimation of various approaches shows that the most efficient approach for solving this problem is the functional approach combined with the alternative Lyapunov method, Jordan structure coefficients and skeleton decomposition of irreversible linear operator in the main part of the equation. On this base, the problem of the correct choice of boundary conditions for a wide class of singular PDE can be solved. The aggregated theorems of existence and uniqueness of classical solutions can be proved with continuously depending of experimental definite function. The theory is illustrated by considering the solution of some integro – differential equations with partial derivatives.

About the Authors

Nikolay Sidorov, Dr. Sci. (Phys.–Math.), Prof., Irkutsk State University, 1, K. Marx st., Irkutsk, 664003, Russian Federation, e-mail: sidorovisu@gmail.com

For citation

Sidorov N.A. Classic Solutions of Boundary Value Problems for Partial Differential Equations with Operator of Finite Index in the Main Part of Equation. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 27, pp. 55-70. https://doi.org/10.26516/1997-7670.2019.27.55

degenerate PDE, Jordan set, Banach space, Noether operator, boundary value problems
35L05, 45D05
  1. Falaleev M.V., Sidorov N.A. Continuous and generalized solutions of singular differential equations. Lobachevskii Journal of Mathematics, 2005, vol. 20, pp. 31-45
  2. Leontyev R.Yu. Nonlinear equations in Banach spaces with a vector parameter in irregular cases. Irkutsk, Irkutsk State University Publ., 2011, 101 p. (in Russian)
  3. Loginov B.V. Bibliographic Index of Works (compiled by O. Gorshanina). Ulyanovsk, UlSTU Publ., 2008, 59 p., Series ”ULTU Scientists).
  4. Lusternik L.A. Some issues of nonlinear functional analysis. Russian math. surveys, 1956, vol. 11, no. 6, pp.145-168. (in Russian)
  5. Muftahov I.R., Sidorov D.N., Sidorov N.A. On the Perturbation method. Vestnik YuUrGU. Ser. Mat. Model. Progr., 2015, vol. 8, no. 2, pp. 69–80. https://doi.org/10.14529/mmp150206
  6. Orlov S.S. Generalized solutions of high-order integro-differential equations in Banach spaces. Irkutsk, Irkutsk State University Publ., 2014, 149 p. (in Russian)
  7. Rendon L., Sinitsyn A.V., Sidorov N.A. Bifurcation points of nonlinear operators: existence theorems, asymptotics and application to the Vlasov-Maxwell system. Rev. Colombiana Math, 2016, vol. 50, no. 1, pp. 85–107. https://doi.org/10.15446/recolma.v50n1.62200
  8. Sidorov N.A., Blagodatskaya E.B. Differential equations with a Fredholm operator in the leading differential expression. Soviet math. Dokl., 1992, vol. 44, no. 1. pp. 302-303.
  9. Sidorov N.A., Romanova O.A., Blagodatskaya E.B. Partial differential equations with an operator of finite index at the principal part. Differ. Equations, 1993, vol. 30, no. 4. pp. 676-678. (in Russian)
  10. Sidorov N.A., Blagodatskaya E.B. Differential equations with a Fredholm operator in the leading differential expression. AN SSSR, Irkutsk Computer Thentr Preprint, 1991, no. 1. 36 p. (in Russian)
  11. Sidorov N.A., Loginov B.V., Sinithyn A.V., Falaleev M.V. Lyapunov-Schmidt methods in nonlinear analysis and applications. Springer, ser. Mathematica and applications, 2003, vol. 550, 558 p. https://doi.org/10.1007/978-94-017-2122-6
  12. Sidorov N.A., Sidorov D.N. Existence and construction of generalized solutions of nonlinear Volterra integral equations of the first kind. Differ. Equations, 2006, vol. 42, no. 9, pp.1312–1316. https://doi.org/10.1134/S0012266106090096
  13. Sidorov N.A., Romanova O.A. Difference-differential equations with fredholm operator in the main part. The Bulletin of Irkutsk State University. Series Mathematics, 2007, vol. 1, no.1, pp. 254-266. (in Russian)
  14. Sidorov N.A., Romanova O.A. On the construction of the trajectory of a single dynamic system with initial data on hyperplanes. The Bulletin of Irkutsk State University. Series Mathematics, 2015, no. 12, pp. 93–105. (in Russian)
  15. Sidorov D.N., Sidorov N.A. Solution of irregular systems using skeleton decomposition of linear operators. Vestnik YuUrGU. Ser. Mat. Model. Progr., 2017, vol. 10, no. 2, pp. 63–73. https://doi.org/10.14529/mmp170205
  16. Sidorov N.A., Sidorov D.N. On the Solvability of a Class of Volterra Operator Equations of the First Kind with Piecewise Continuous Kernels. Math. Notes, 2014, vol. 96, no. 5, pp. 811–826. https://doi.org/10.4213/mzm10220
  17. Sidorov N.A., Leontiev R.Yu., Dreglea A.I. On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods. Math. Notes, 2012, vol. 91, no. 1, pp. 90–104. https://doi.org/10.4213/mzm8771
  18. Sidorov N.A., Sidorov D.N. Small solutions of nonlinear differential equations near branching points. Russian Math. (Iz. VUZ), 2011, vol. 55, no. 5, pp.43–50. https://doi.org/10.3103/S1066369X11050070
  19. Sidorov N.A. Parametrization of simple branching solutions of full rank and iterations in nonlinear analysis. Russian Math. (Iz. VUZ), 2001, vol. 45, no. 9, pp.55–61.
  20. Sidorov N.A. Explicit and implicit parametrizations in the construction of branching solutions by iterative methods. Sb. Math. J., 1995, vol. 186, no. 2, pp.297–310. https://doi.org/10.1070/SM1995v186n02ABEH000017
  21. Sidorov N.A. A class of degenerate differential equations with convergence. Math. Notes, 1984, vol. 35, no. 4, pp.300–305. https://doi.org/10.1007/BF01139992
  22. Sidorov D.N., Sidorov N.A. Convex majorants method in the theory of non-linear Volterra equations. Banach J. Math. Anal, 2012, vol. 6, no. 1, pp.1–10. https://doi.org/10.15352/bjma/1337014661
  23. Sidorov N.A., Sidorov D.N. Solving the Hammerstein integral equation in the irregular case by successive approximations. Siberian Math. J., 2010, vol. 51, no. 2, pp. 325–329. https://doi.org/10.1007/s11202-010-0033-4
  24. Sidorov D.N. Integral Dynamical Models: Singularities, Signals and Control. World scientific series, nonlinear science, Series A, vol. 87, ed. by L. O. Chua, 2015, 243 p. https://doi.org/10.1142/9278
  25. Sviridyuk G.A., Fedorov V.E. Linear Sobolev type equations and degenerate semigroups of operators. Utrecht, VSP, 2003, 228 p.
  26. Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov problem of the Sobolev type equations. The Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, pp. 104-125.
  27. Vainberg M.M., Trenogin V.A. The theory of branches of solutions of nonlinear equations. Wolters-Noordhoff, Groningen, 1974, 302 p.

Full text (english)