AccScience Publishing / IJOCTA / Volume 9 / Issue 1 / DOI: 10.11121/ijocta.01.2019.00592
RESEARCH ARTICLE

On stable high order difference schemes for hyperbolic problems with the Neumann boundary conditions

Ozgur Yildirim1*
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1 Department of Mathematics, Yildiz Technical University, Turkey
Submitted: 16 March 2018 | Accepted: 1 November 2018 | Published: 31 January 2019
© 2019 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract

In this paper, third and fourth order of accuracy stable difference schemes for approximately solving multipoint nonlocal boundary value problems for hyperbolic equations with the Neumann boundary conditions are considered. Stability estimates for the solutions of these difference schemes are presented. Finite difference method is used to obtain numerical solutions. Numerical results of errors and CPU times are presented and are analyzed.

Keywords
Nonlocal and multipoint BVPs
Stability
Abstract hyperbolic equations
Finite difference methods
Conflict of interest
The authors declare they have no competing interests.
References

[1] Day, W. A. (1983). A decreasing property of solutions of parabolic equations with ap- plications to Thermoelasticity. Quart. Appl. Math., 40, 468-475.

[2] Gurevich, P., Jager, W., Skubachevskii, A. (2009). On periodicity of solutions for thermocontrol problems with hysteresis-type switches. SIAM J. Math. Anal., 41, 733-752.

[3] Grigorescu, I., Kang, M. (2001). Brown- ian motion on the figure eight. J. Theoret. Probab., 15, 817-844.

[4] Ozdemir, N., Avci, D., Iskender, B. B. (2011). Numerical Solutions of Two-Dimensional Space-Time Riesz-Caputo Fractional Difu- sion Equation. IJOCTA, 1, 17-26.

[5] Ozdemir, N., Agrawal, O. P., Karadeniz, D., Iskender, B. B. (2009). Fractional opti- mal control problem of An axis-symmetric difusion-wave propagation. Phys. Scr. T., 136, 014024, 5 pp.

[6] Gustafson, B., Kreiss, H. O., Oliger, J.(1995). Time dependent problems and difer- ence methods. Wiley, New York.

[7] Ashyralyev, A., Sobolevskii, P. E. (2004). New diference shemes for partial diferen- tial equations, operator theory: advances and applications. Birkh¨auser., vol 148, Basel, Boston, Berlin.

[8] Sobolevskii, P. C., Chebotaryeva, L. M.(1977). Approximate solution by method of lines of the Cauchy problem for an abstract hyperbolic equations. Izv. Vyssh. Uchebn. Zav. Matematika, 5, 103-116.

[9] Belakroum, Kh, Ashyralyev, A., Guezane- Lakoud, A. (2018). A Note on the Nonlocal Boundary Value Problem for a Third Order Partial Diferential Equation. Filomat, 32(3), 801081, 89-98.

[10] Ashyralyev, A., Agirseven, D. (2018). Bounded solutions of nonlinear hyperbolic equations with time delay. Electronic Journal of Diferential Equations, 21, 1-15.

[11] Ashyralyev, A., Akturk, S. (2017). A note on positivity of two-dimensional diferential op- erators. Filomat, 31(14), 4651663.

[12] Ashyralyev, A., Beigmohammadi, E.O.(2017). Well-posedness of a fourth order of accuracy diference scheme for Bitsadze- Samarskii-type problem. Numerical Func- tional Analysis and Optimization, 38(10), 1244-1259.

[13] Lax, P. D., Wendrof, B. (1964). Difer- ence schemes for hyperbolic equations with high order of accuracy. Commun. Pure Appl. Math., 17, 381-398.

[14] Fattorini, H. O. (1985). Second order lin- ear diferential equations in Banach spaces. North-Holland Mathematics Studies, vol. 108, North-Holland, Amsterdam, Netherlands.

[15] Krein, S. G. (1966). Linear diferential equa- tions in a Banach space. Nauka, Moscow.

[16] Sobolevskii, P. E. (1975). Diference meth- ods for the approximate solution of diferen- tial equations. Izdat. Gosud. Univ, Voronezh.

[17] Ashyralyev, A., Sobolevskii, P. E. (2005). Two new approaches for construction of the high order of accuracy diference schemes for hyperbolic diferential equations. Discrete Dyn. Nat. Soc., 2, 183-213.

[18] Yildirim, O., Uzun, M. (2015). On the nu- merical solutions of high order stable difer- ence schemes for the hyperbolic multipoint nonlocal boundary value problems. Applied Mathematics and Computation, 254, 210-218.

[19] Yildirim, O., Uzun, M. (2015). On third or- der stable diference scheme for hyperbolic multipoint nonlocal boundary value problem. Discrete Dyn. Nat. Soc., 2015, 16 pages.

[20] Yildirim O., Uzun M. (2017). On fourth order stable diference scheme for hyperbolic multi- point NBVP. Numerical Functional Analysis and Optimization, 38(10), 1305-1324.

[21] Ashyralyyev, C. (2014). High order ap- proximation of the inverse elliptic problem with Dirichlet-Neumann Conditions. Filomat, 28(5), 94762.

[22] Ashyralyyev, C. (2014). High order of accu- racy diference schemes for the inverse ellip- tic problem with Dirichlet condition. Bound. Value Probl., 2014:5, 13.

[23] Ashyralyev, A., Sobolevskii, P. E. (2001). A note on the diference schemes for hyperbolic equations. Abstr. Appl. Anal., 6, 63-70.

[24] Ashyralyev, A., Yildirim, O. (2010). On mul- tipoint nonlocal boundary value problems for hyperbolic diferential and diference equa- tions. Taiwanese J. Math., 14, 165-194.

[25] Direk, Z., Ashyraliyev, M. (2014). FDM for the integral-diferential equation of the hyper- bolic type. Adv. Difer. Equations, 2014, 1-8.

[26] Piskarev, S., Shaw, Y. (1997). On certain op- erator families related to cosine operator func- tion. Taiwanese J. Math., 1, 3585-3592.

[27] Yildirim, O., Uzun, M. (2016). On stability of diference schemes for hyperbolic multipoint NBVP with Neumann conditions. AIP Conf. Proc., 1759, Almaty, Kazakhstan.

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