THE HYPERBOLIC SYSTEM OF EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS
Year 2012,
Volume 2,
Issue 2,
154 - 178,
01.12.2012
Allaberen ASHYRALYEV
Rahat PRENOV
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
Allaberen ASHYRALYEV
Rahat PRENOV
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.