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THE HYPERBOLIC SYSTEM OF EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS

Year 2012, Volume 2, Issue 2, 154 - 178, 01.12.2012

Abstract

The initial value problem for the hyperbolic system of equations with nonlocal boundary conditions is studied. The positivity of the space operator A generated by this problem in interpolation spaces is established. The structure interpolation spaces of this state operator is studied. The positivity of this space operator in H¨older spaces is established. In applications, the stability estimates for the hyperbolic system of equations with nonlocal boundary conditions are obtained.

References

  • Fattorini, H. O., (1985), Second Order Linear Differential Equations in Banach Spaces, Elsevier Science Publishing Company, North-Holland.
  • Goldstein, J. A., (1985), Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York.
  • Krein, S. G., (1966), Linear Differential Equations in Banach space, Nauka, Moscow, (Russian). English transl.: (1968), Linear Differential Equations in Banach space, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI..
  • Pogorelenko, V. and Sobolevskii, P. E., (1998), The ”counter-example” to W. Littman counter-example of Lp -energetic inequality for wave equation, Functional Differential Equations 4, no. 1-2, 165-133.
  • Sobolevskii, P. E., (1964), On the equations of the second order with a small parameter at the highest derivatives, Uspekhi. Mat. Nauk, 19, no. 6, 217-219. (Russian).
  • Sobolevskii, P. E. and Semenov, S., (1983), On some approach to investigation of singular hyperbolic equations, Dokl. Akad. Nauk SSSR, 270, no. 1, 555-558. (Russian).
  • Vasilev, V. V. and Krein, S. G. and Piskarev, S., (1990), Operator Semigroups, Cosine Operator Functions, and Linear Differential Equations, Mathematical Analysis, Vol. 28. (Russian). (1990), Itogi Nauki i Tekhniki, 204, 87-202, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. Translated in J. Soviet Math. 54, (1991), no. 4, 1042–1129.
  • Ashyralyev, A. and Fattorini, H. O., (1992), On uniform difference-schemes for 2 d-order singular pertubation problems in Banach spaces, SIAM J. Math. Anal., 23, no. 1, 29-54.
  • Ashyraliyev, M., (2008), A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space, Numerical Functional Analysis and Optimization, 29, no. 7-8, 750-769.
  • Yurtsever, A. and Prenov, R., (2000), On stability estimates for method of line for Şrst order system of differential equations of hyperbolic type, in: Some Problems of Applied Mathematics (Editors by A. Ashyralyev and A. Yurtsever), Fatih University Publications, Istanbul, pp. 198-205.
  • Ashyralyev, A. and Sobolevskii, P. E., (2004), New Difference Schemes for Partial Differential Equa- tions, Operator Theory Advances and Applications, Birkh¨auser Verlag, Basel, Boston, Berlin.
  • Dehghan, M., (2003), On the numerical solution of the diffusion equation with a nonlocal boundary condition, Mathematical Problems in Engineering, no.2, 81-92.
  • Cannon, J. R., Perez Esteva, S. and van der Hoek, J., (1987), A Galerkin procedure for the diffiusion equation subject to the speciŞcation of mass, SIAM J. Numerical Analysis, 24, no.3, 499-515.
  • Gordeziani, N., Natani, P. and Ricci, P. E., (2005), Finite-difference methods for solution of nonlocal boundary value problems, Computers and Mathematics with Applications, 50, 1333-1344.
  • Dautray, R. and Lions, J. L., (1988), Analyse Mathematique et Calcul Numerique Pour les Sciences et les Technique, Volume 1-11, Masson, Paris.
  • Ashyralyev, A. and Ozdemir, Y., (2005), Stability of difference schemes for hyperbolic-parabolic equa- tions, Computers and Mathematics with Applications, 50, no. 8-9, 1443-1476.
  • Hersch, R. and Kato, T., (1982), High accuracy stabe difference schemes for well-posed initial value problems, SIAM J. Numer Anal. 19, no. 3, 599-603.
  • Brenner, P., Crouzeix, M. and Thomee, V., (1982), Single step methods for inhomogeneous linear differential equations in Banahch space. R.A.I.R.O. Analyse numerique, Numer.Anal.16 no. 1, 5-26. [19] Godunov, S. K., (1962), Numerical Methods of Solution of the Equation of Gasdynamics, NSU, Novosibirsk (Russian).
  • Tikhonov, A. N. and Samarskii, A. A., (1977), Equations of Mathematical Physics, Nauka, Moscow (Russian).
  • Ashyralyev, A. and Sobolevskii, P. E., (1994), Well-Posedness of Parabolic Difference Equations, Operator Theory Advances and Applications, Birkh¨auser Verlag, Basel, Boston, Berlin.
  • Agarwal, R., Bohner, M. and Shakhmurov, V. B., (2005), Maximal regular boundary value problems in Banach-valued weighted spaces, Boundary Value Problems, 1, 9-42.
  • Shakhmurov, V. B., (2004), Coercive boundary value problems for regular degenerate differential- operator equations, Journal of Mathematical Analysis and Applications, 292, no. 2, 605-620.
  • A. Ashyralyev, for a photograph and biography, see TWMS Journal of Applied and Engineering Mathematics, Volume 2, No.1, 2012.

Year 2012, Volume 2, Issue 2, 154 - 178, 01.12.2012

Abstract

References

  • Fattorini, H. O., (1985), Second Order Linear Differential Equations in Banach Spaces, Elsevier Science Publishing Company, North-Holland.
  • Goldstein, J. A., (1985), Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York.
  • Krein, S. G., (1966), Linear Differential Equations in Banach space, Nauka, Moscow, (Russian). English transl.: (1968), Linear Differential Equations in Banach space, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI..
  • Pogorelenko, V. and Sobolevskii, P. E., (1998), The ”counter-example” to W. Littman counter-example of Lp -energetic inequality for wave equation, Functional Differential Equations 4, no. 1-2, 165-133.
  • Sobolevskii, P. E., (1964), On the equations of the second order with a small parameter at the highest derivatives, Uspekhi. Mat. Nauk, 19, no. 6, 217-219. (Russian).
  • Sobolevskii, P. E. and Semenov, S., (1983), On some approach to investigation of singular hyperbolic equations, Dokl. Akad. Nauk SSSR, 270, no. 1, 555-558. (Russian).
  • Vasilev, V. V. and Krein, S. G. and Piskarev, S., (1990), Operator Semigroups, Cosine Operator Functions, and Linear Differential Equations, Mathematical Analysis, Vol. 28. (Russian). (1990), Itogi Nauki i Tekhniki, 204, 87-202, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. Translated in J. Soviet Math. 54, (1991), no. 4, 1042–1129.
  • Ashyralyev, A. and Fattorini, H. O., (1992), On uniform difference-schemes for 2 d-order singular pertubation problems in Banach spaces, SIAM J. Math. Anal., 23, no. 1, 29-54.
  • Ashyraliyev, M., (2008), A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space, Numerical Functional Analysis and Optimization, 29, no. 7-8, 750-769.
  • Yurtsever, A. and Prenov, R., (2000), On stability estimates for method of line for Şrst order system of differential equations of hyperbolic type, in: Some Problems of Applied Mathematics (Editors by A. Ashyralyev and A. Yurtsever), Fatih University Publications, Istanbul, pp. 198-205.
  • Ashyralyev, A. and Sobolevskii, P. E., (2004), New Difference Schemes for Partial Differential Equa- tions, Operator Theory Advances and Applications, Birkh¨auser Verlag, Basel, Boston, Berlin.
  • Dehghan, M., (2003), On the numerical solution of the diffusion equation with a nonlocal boundary condition, Mathematical Problems in Engineering, no.2, 81-92.
  • Cannon, J. R., Perez Esteva, S. and van der Hoek, J., (1987), A Galerkin procedure for the diffiusion equation subject to the speciŞcation of mass, SIAM J. Numerical Analysis, 24, no.3, 499-515.
  • Gordeziani, N., Natani, P. and Ricci, P. E., (2005), Finite-difference methods for solution of nonlocal boundary value problems, Computers and Mathematics with Applications, 50, 1333-1344.
  • Dautray, R. and Lions, J. L., (1988), Analyse Mathematique et Calcul Numerique Pour les Sciences et les Technique, Volume 1-11, Masson, Paris.
  • Ashyralyev, A. and Ozdemir, Y., (2005), Stability of difference schemes for hyperbolic-parabolic equa- tions, Computers and Mathematics with Applications, 50, no. 8-9, 1443-1476.
  • Hersch, R. and Kato, T., (1982), High accuracy stabe difference schemes for well-posed initial value problems, SIAM J. Numer Anal. 19, no. 3, 599-603.
  • Brenner, P., Crouzeix, M. and Thomee, V., (1982), Single step methods for inhomogeneous linear differential equations in Banahch space. R.A.I.R.O. Analyse numerique, Numer.Anal.16 no. 1, 5-26. [19] Godunov, S. K., (1962), Numerical Methods of Solution of the Equation of Gasdynamics, NSU, Novosibirsk (Russian).
  • Tikhonov, A. N. and Samarskii, A. A., (1977), Equations of Mathematical Physics, Nauka, Moscow (Russian).
  • Ashyralyev, A. and Sobolevskii, P. E., (1994), Well-Posedness of Parabolic Difference Equations, Operator Theory Advances and Applications, Birkh¨auser Verlag, Basel, Boston, Berlin.
  • Agarwal, R., Bohner, M. and Shakhmurov, V. B., (2005), Maximal regular boundary value problems in Banach-valued weighted spaces, Boundary Value Problems, 1, 9-42.
  • Shakhmurov, V. B., (2004), Coercive boundary value problems for regular degenerate differential- operator equations, Journal of Mathematical Analysis and Applications, 292, no. 2, 605-620.
  • A. Ashyralyev, for a photograph and biography, see TWMS Journal of Applied and Engineering Mathematics, Volume 2, No.1, 2012.

Details

Primary Language English
Journal Section Research Article
Authors

Allaberen ASHYRALYEV This is me
Department of Mathematics, Fatih University 34500, B¨uy¨uk¸cekmece, ˙Istanbul, Turkey and Department of Mathematics, ITT University 74400, Ashgabat, Turkmenistan


Rahat PRENOV This is me
Department of Mathematics, ITT University 74400, Ashgabat, Turkmenistan

Publication Date December 1, 2012
Published in Issue Year 2012, Volume 2, Issue 2

Cite

Bibtex @ { twmsjaem761708, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2012}, volume = {2}, number = {2}, pages = {154 - 178}, title = {THE HYPERBOLIC SYSTEM OF EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS}, key = {cite}, author = {Ashyralyev, Allaberen and Prenov, Rahat} }