The Existence of a Generalized Solution of an m -Point Nonlocal Boundary Value Problem

David Devadze

Communications in Mathematics (2017)

  • Volume: 25, Issue: 2, page 159-169
  • ISSN: 1804-1388

Abstract

top
An m -point nonlocal boundary value problem is posed for quasilinear differential equations of first order on the plane. Nonlocal boundary value problems are investigated using the algorithm of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems for a generalized analytic function. The conditions for the existence and uniqueness of a generalized solution in the space are considered.

How to cite

top

Devadze, David. "The Existence of a Generalized Solution of an $m$-Point Nonlocal Boundary Value Problem." Communications in Mathematics 25.2 (2017): 159-169. <http://eudml.org/doc/294797>.

@article{Devadze2017,
abstract = {An $m$-point nonlocal boundary value problem is posed for quasilinear differential equations of first order on the plane. Nonlocal boundary value problems are investigated using the algorithm of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems for a generalized analytic function. The conditions for the existence and uniqueness of a generalized solution in the space are considered.},
author = {Devadze, David},
journal = {Communications in Mathematics},
keywords = {Nonlocal boundary value problem; generalized solution},
language = {eng},
number = {2},
pages = {159-169},
publisher = {University of Ostrava},
title = {The Existence of a Generalized Solution of an $m$-Point Nonlocal Boundary Value Problem},
url = {http://eudml.org/doc/294797},
volume = {25},
year = {2017},
}

TY - JOUR
AU - Devadze, David
TI - The Existence of a Generalized Solution of an $m$-Point Nonlocal Boundary Value Problem
JO - Communications in Mathematics
PY - 2017
PB - University of Ostrava
VL - 25
IS - 2
SP - 159
EP - 169
AB - An $m$-point nonlocal boundary value problem is posed for quasilinear differential equations of first order on the plane. Nonlocal boundary value problems are investigated using the algorithm of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems for a generalized analytic function. The conditions for the existence and uniqueness of a generalized solution in the space are considered.
LA - eng
KW - Nonlocal boundary value problem; generalized solution
UR - http://eudml.org/doc/294797
ER -

References

top
  1. Ashyralyev, A., Gercek, O., Nonlocal boundary value problem for elliptic-parabolic differential and difference equations, Discrete Dyn. Nat. Soc., 2008, 16p, (2008) MR2457148
  2. Beals, R., 10.1090/S0002-9904-1964-11166-6, Bull. Amer. Math. Soc., 70, 5, 1964, 693-696, (1964) Zbl0125.33301MR0167700DOI10.1090/S0002-9904-1964-11166-6
  3. Beridze, V., Devadze, D., Meladze, H., On one nonlocal boundary value problem for quasilinear differential equations, Proceedings of A. Razmadze Mathematical Institute, 165, 2014, 31-39, (2014) Zbl1315.35101MR3300597
  4. Berikelashvili, G., Construction and analysisi of difference schemes of the rate convergence, Memoirs of Diff. Euqat. and Math. Physics, 38, 2006, 1-36, (2006) MR2258555
  5. Berikelashvili, G., On the solvability of a nonlocal boundary value problem in the weighted Sobolev spaces, Proc. A. Razmadze Math. Inst., 119, 1999, 3-11, (1999) Zbl0963.35045MR1710953
  6. Berikelashvili, G., To a nonlocal generalization of the Dirichlet problem, Journal of Inequalities and Applications, 2006, 1, 2006, 6p, Art.ID 93858. (2006) Zbl1108.35040MR2215469
  7. Bitsadze, A.V., Samarskií, A.A., On some simple generalizations of linear elliptic boundary problems, Dokl. Akad. Nauk SSSR, 185, 1969, 739-740, Translation in Sov. Math. Dokl., 10 (1969), 398--400. (1969) Zbl0187.35501MR0247271
  8. Browder, F.E., 10.2307/2373156, Amer. J. Math, 86, 1964, 735-750, (1964) Zbl0131.09702MR0171989DOI10.2307/2373156
  9. Canon, J.R., 10.1090/qam/160437, Quart. Appl. Math., 21, 2, 1963, 155-160, (1963) MR0160437DOI10.1090/qam/160437
  10. Carleman, T., Sur la théorie des équations intégrales et ses applications, 1932, F{ü}ssli, Zurich, (1932) Zbl0006.40001
  11. Courant, R., Hilbert, D., Methods of mathematical physics, 1953, Interscience Publishers, New York, (1953) Zbl0053.02805MR0065391
  12. Devadze, D., Beridze, V., 10.1007/s10958-016-3057-x, Journal of Mathematical Sciences, 218, 6, 2016, 731-736, (2016) Zbl1354.49051MR3563651DOI10.1007/s10958-016-3057-x
  13. Beridze, D. Sh. Devadze and V. Sh., Optimality conditions for quasi-linear differential equations with nonlocal boundary conditions, Uspekhi Mat. Nauk, 68, 4, 2013, 179-180, Translation in Russian Math. Surveys, 68 (2013), 773--775. (2013) MR3154819
  14. Devadze, D., Dumbadze, M., An Optimal Control Problem for a Nonlocal Boundary Value Problem, Bulletin of The Georgian National Academy Of Sciences, 7, 2, 2013, 71-74, (2013) Zbl1311.49054MR3136884
  15. Diaz, J.I., Rakotoson, J-M., 10.1007/BF00376255, Arch.Rational. Mech. Anal., 134, 1, 1996, 53-95, (1996) MR1392309DOI10.1007/BF00376255
  16. Gordeziani, D.G., On methods for solving a class of non-local boundary-value problems, Ed. TSU, Tbilisi, 1981, 32p, (1981) MR0626523
  17. Gordeziani, D.G., Avalishvili, G.A., On the constructing of solutions of the nonlocal initial boundary value problems for one dimensional medium oscillation equations, Mathem. Mod., 12, 1, 2000, 93-103, (2000) MR1773208
  18. Gordeziani, D.G., Djioev, T.Z., On solvability of one boundary value problem for a nonlinear elliptic equations, Bull. Georgian Acad. Sci, 68, 2, 1972, 189-292, (1972) MR0372414
  19. Gordeziani, G., Gordeziani, N., Avalishvili, G., Non-local boundary value problem for some partial differential equations, Bulletin of the Georgian Academy of Sciences, 157, 1, 1998, 365-369, (1998) MR1652066
  20. Gordezian, D., Gordeziani, E., Davitashvili, T., Meladze, G., On the solution of some non-local boundary and initial-boundary value problems, GESJ: Computer Science and Telecommunications, 2010, 3, 161-169, 
  21. Gulin, A.V., Marozova, V.A., A family of selfjoint difference schemes, Diff. Urav., 44, 9, 2008, 1249-1254, (2008) MR2484534
  22. Gurevich, P.L., Asymptotics of Solution for nonlocal elliptic problems in plane bounded domains, Functional Differential Equations, 10, 1-2, 2003, 175-214, (2003) MR2017408
  23. Gushchin, A.K., Mikhailov, V.P., On the stability of nonlocal problems for a second order elliptic equation, Math. Sb., 185, 1, 1994, 121-160, (1994) MR1264079
  24. Il'in, V.A., Moiseev, E.I., A two-dimensional nonlocal boundary value problem for the Poisson operator in the differential and the difference interpretation, Mat. Model., 2, 8, 1990, 139-156, (Russian). (1990) MR1086120
  25. Ionkin, N.I., Solution of boundary-value problem in heat-conduction theory with non-classical boundary conditions, Diff, Urav., 13, 1977, 1177-1182, (1977) MR0603292
  26. Kapanadze, D.V., On a nonlocal Bitsadze-Samarskiĭ boundary value problem, Differentsial'nye Uravneniya, 23, 3, 1987, 543-545, (Russian). (1987) MR0886591
  27. Mandzhavidze, G.F., Tuchke, V., Some boundary value problems for first-order nonlinear differential systems on the plane, Boundary value problems of the theory of generalized analytic functions and their applications Tbilis. Gos. Univ., Tbilisi, 1983, 79-124, (Russian). (1983) MR0745292
  28. Paneyakh, B.P, Some nonlocal boundary value problems for linear differential operators, Mat. Zametki, 35, 3, 1984, 425-434, (Russian). (1984) MR0741809
  29. Pao, C.V., 10.1006/jmaa.1995.1384, J. Math. Anal. Appl., 195, 3, 1995, 702-718, (1995) Zbl0851.35063MR1356638DOI10.1006/jmaa.1995.1384
  30. Sapagovas, M.P., 10.1134/S0012266108070148, Differential Equations, 44, 7, 2008, 1018-1028, (2008) MR2479950DOI10.1134/S0012266108070148
  31. Sapagovas, M.P., Chegis, R.Yu., On some Boundary value problems with a non-local condition, Differentsial'nye Uravneniya, 23, 7, 1987, 1268-1274, (Russian). (1987) MR0903982
  32. Shakeris, F., Dehghan, M., 10.1016/j.camwa.2008.03.055, Computers & Mathematics with Applications, 56, 9, 2008, 2175-2188, (2008) MR2466739DOI10.1016/j.camwa.2008.03.055
  33. Shelukin, V.V., A non-local in time model for radionuclide propagation in Stokes fluid, Dinamika Splosh. Sredy, 107, 1993, 180-193, (1993) MR1305005
  34. Skubachevskin, A.L., On a spectrum of some nonlocal boundary value problems, Mat. Sb., 117, 7, 1982, 548-562, (Russian). (1982) MR0651145
  35. Vekua, I.N., Generalized analytic functions. Second edition, 1988, Nauka, Moscow, (Russian). (1988) MR0977975
  36. Vasylyk, V.B., Exponentially convergent method for the m -point nonlocal problem for an elliptic differential equation in banach space, J. Numer. Appl. Math., 105, 2, 2011, 124-135, (2011) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.