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A UNIFORMLY STABLE SOLVABILITY OF NLBVP FOR PARAMETERIZED ODE

Year 2021, , 50 - 69, 30.12.2021
https://doi.org/10.47086/pims.975424

Abstract

Nonlocal boundary value problem of the first kind for an odinary linear second order differential equation with positive parameter at the highest derivative is considered. The existence and uniqueness, as well as, a uniformly stable estimate of classical solution is established under accurate condition on coefficients and location of nonlocal data carriers of multipoint boundary value condition. An essentiality of the revealed condition is confirmed by ill-posed problem examples.

References

  • Reference1 A. N. Tikhonov, On the dependence of the solutions of differential equations on a small parameter, Mat. Sb., 22(64) (1948), 193-204.
  • Reference2 A. V. Bitsadze and A. A. Samarskii, On some simple generalizations of linear elliptic boundary problems, USSR Academy of Science Reports, 185(4) (1969), 739-740.
  • Reference3 V. A. Il'in and E. I. Moiseev, A nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations, Differ. Uravn., 23(7) (1987), 1198-1207.
  • Reference4 V. A. Il’in and E. I. Moiseev, An a priori estimate for the solution of a problem associated with a nonlocal boundary value problem of the first kind, Differ. Equ., 24(5) (1988), 519-526.
  • Reference5 R. Chegis, Numerical solution of problems with small paprameter at higher derivatives and nonlocal conditions, Liet matem. rink., 28(1) 1988, 144-152.
  • Reference6 D. M. Dovletov, On the nonlocal boundary value problem of the first kind in differential and difference interpretation, Differ. Equ., 25(8) (1989), 917-924.
  • Reference7 D. M. Dovletov, On some nonlocal boundary value problem in differential and difference interpretation, The dissertation of the candidate for physical and mathematical sciences, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow (1989), 1-128 (In Russian language: The Russian State Library, The General Digital Catalogue. Website links www.rsl.ru/en, www.rsl.ru : search request for the paper is "Dovletov Dovlet Meydanovich").
  • Reference8 D. M. Dovletov, Uniformly difference schemes for nonlocal boundary value problem with a small parameter, "The Differential Equations and Applications". Proceedings of the scientific and practical conference. (Perfomed by The Academy of Sciences of Turkmenistan, Institute of Mechanics and Mathematics, Turkmenistan State University) Ashgabat, Turkmenistan, 1993(2) (1993), 68-72.
  • Reference9 N. Adzic, Spectral approximation and nonlocal boundary value problems, Novi Sad J. Math.,30(3) (2000), 1-10.
  • Reference10 D. Arslan and M. Cakir, A numerical solution of singularly perturbed convection-diffusion nonlocal boundary problem, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2) (2019), 1482-1491.
  • Reference11 V.A. Il’in and E.I. Moiseev, Second kind nonlocal boundary value problem for Sturm-Liouville operator, Differ. Equ., 23(8) (1987), 1422-1431.
  • Reference12 E.P. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Dublin, Boole Press (1980).
  • Reference13 D.M. Dovletov, Nonlocal boundary value problem in terms of flow for Sturm-Liouville operator in differential and difference statements, e-Journal of Analysis and Applied Mathematics, 2018(1) (2018), 37-55.
  • Reference14 M. Kumar, P. Singh and H.K. Mishra, A recent survey on computational technique for solving singulalrly perturbed boundary value problems, International Journal of Computer Mathematics, 84(10) (2007), 1439-1463.
Year 2021, , 50 - 69, 30.12.2021
https://doi.org/10.47086/pims.975424

Abstract

References

  • Reference1 A. N. Tikhonov, On the dependence of the solutions of differential equations on a small parameter, Mat. Sb., 22(64) (1948), 193-204.
  • Reference2 A. V. Bitsadze and A. A. Samarskii, On some simple generalizations of linear elliptic boundary problems, USSR Academy of Science Reports, 185(4) (1969), 739-740.
  • Reference3 V. A. Il'in and E. I. Moiseev, A nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations, Differ. Uravn., 23(7) (1987), 1198-1207.
  • Reference4 V. A. Il’in and E. I. Moiseev, An a priori estimate for the solution of a problem associated with a nonlocal boundary value problem of the first kind, Differ. Equ., 24(5) (1988), 519-526.
  • Reference5 R. Chegis, Numerical solution of problems with small paprameter at higher derivatives and nonlocal conditions, Liet matem. rink., 28(1) 1988, 144-152.
  • Reference6 D. M. Dovletov, On the nonlocal boundary value problem of the first kind in differential and difference interpretation, Differ. Equ., 25(8) (1989), 917-924.
  • Reference7 D. M. Dovletov, On some nonlocal boundary value problem in differential and difference interpretation, The dissertation of the candidate for physical and mathematical sciences, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow (1989), 1-128 (In Russian language: The Russian State Library, The General Digital Catalogue. Website links www.rsl.ru/en, www.rsl.ru : search request for the paper is "Dovletov Dovlet Meydanovich").
  • Reference8 D. M. Dovletov, Uniformly difference schemes for nonlocal boundary value problem with a small parameter, "The Differential Equations and Applications". Proceedings of the scientific and practical conference. (Perfomed by The Academy of Sciences of Turkmenistan, Institute of Mechanics and Mathematics, Turkmenistan State University) Ashgabat, Turkmenistan, 1993(2) (1993), 68-72.
  • Reference9 N. Adzic, Spectral approximation and nonlocal boundary value problems, Novi Sad J. Math.,30(3) (2000), 1-10.
  • Reference10 D. Arslan and M. Cakir, A numerical solution of singularly perturbed convection-diffusion nonlocal boundary problem, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2) (2019), 1482-1491.
  • Reference11 V.A. Il’in and E.I. Moiseev, Second kind nonlocal boundary value problem for Sturm-Liouville operator, Differ. Equ., 23(8) (1987), 1422-1431.
  • Reference12 E.P. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Dublin, Boole Press (1980).
  • Reference13 D.M. Dovletov, Nonlocal boundary value problem in terms of flow for Sturm-Liouville operator in differential and difference statements, e-Journal of Analysis and Applied Mathematics, 2018(1) (2018), 37-55.
  • Reference14 M. Kumar, P. Singh and H.K. Mishra, A recent survey on computational technique for solving singulalrly perturbed boundary value problems, International Journal of Computer Mathematics, 84(10) (2007), 1439-1463.
There are 14 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Articles
Authors

Dovlet Dovletov 0000-0003-0834-2163

Publication Date December 30, 2021
Acceptance Date September 23, 2021
Published in Issue Year 2021

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