Research Article
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Year 2022, , 92 - 98, 01.12.2022
https://doi.org/10.33187/jmsm.1132139

Abstract

References

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  • [2] A. Din, Y. Li, T. Khan, G. Zaman, Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China, Chaos Solitons Fractals, 141 (2020), 110286.
  • [3] P. Veeresha, M. Yavuz, C. Baishya, A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators, Int. J. Optim. Control: Theor. Appl. , 11(3) (2021), 52-67.
  • [4] F. Özbağ, Numerical simulations of traveling waves in a counterflow filtration combustion model, Turk. J. Math., 46(4) (2022), 1424-1435.
  • [5] L. Zada, R. Nawaz, K. S. Nisar, M. Tahir, M. Yavuz, New approximate-analytical solutions to partial differential equations via auxiliary function method, Partial Differ. Eq. Appl. Math., 4 (2021), 100045.
  • [6] Z. Hammouch, M. Yavuz, N. Özdemir, Numerical solutions and synchronization of a variable-order fractional chaotic system, Math. Model. Numer. Simul. App., 1(1) (2021), 11-23.
  • [7] H. Eltayeb, S. Mesloub, Y. T. Abdalla, A. Kilicman, A note on double conformable Laplace transform method and singular one dimensional conformable pseudo-hyperbolic equations, Mathematics, 7(10) (2019), 949.
  • [8] S. V. Potapova, Boundary value problems for pseudo-hyperbolic equations with a variable time direction, J. Pure Appl. Math., 3(1) (2012), 75-91.
  • [9] Y. Zhang, Y. Niu, D. Shi, Nonconforming H1 -Galerkin mixed finite element method for pseudo-hyperbolic equations, American J. Comp. Math., 2 (2012), 269-273.
  • [10] Y. Liu, J. Wang, H. Li, W. Gao, S. He, A new splitting H1-Galerkin mixed method for pseudo-hyperbolic equations, Int. J. Math. Comp. Sci., 5(3) (2011), 1444-1449.
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  • [12] G. Chen, Z. Yang, Initial value problem for a class of nonlinear pseudo-hyperbolic equations, Acta Math. App. Sinica, 9(2) (1993), 166-173.
  • [13] P. A. Krutitskii, An initial-boundary value problem for the pseudo-hyperbolic equation of gravity-gyroscopic waves, J. Math. Kyoto Univ., 37(2) (1997), 343-365.
  • [14] Z. Zhao, H. Li, A continuous Galerkin method for pseudo-hyperbolic equations with variable coefficients, J. Math. Anal. App., 473(2) (2019), 1053-1072.
  • [15] M. Modanli, F. Özbağ, A. Akgülma, Finite difference method for the fractional order pseudo telegraph integro-differential equation, J. Appl. Math. Comput. Mech., 21(1) (2022), 41-54.
  • [16] H. W. Liu, L. B. Liu, An unconditionally stable spline difference scheme of O(k2 +h4) for solving the second-order 1D linear hyperbolic equation, Math. Comp. Model., 49 (2009), 1985-1993.
  • [17] M. M. Islam, M. S. Hasan, A study on exact solution of the telegraph equation by (G’/G)-expansion method, African J. Math. Comp. Sci. Res., 11(7) (2018), 103-108.
  • [18] A. Merad, A. Bouziani, Solvability the telegraph equation with purely integral conditions, J. App. Eng. Math., 3(2) (2013), 245-253.
  • [19] M. Sarı, A. G¨unay, G. G¨urarslan, A solution to the telegraph equation by using DGJ method, Int. J. Nonlinear Sci., 17(1) (2014), 57-66.
  • [20] B. Soltanalizadeh, Differential transformation method for solving one-space-dimensional telegraph equation, Comput. Appl. Math., 30(3) (2016), 639-653.
  • [21] M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numer. Methods for Partial Differ. Eq., 24(4) (2008), 1080-1093.
  • [22] M. Javidi, N. Nyamoradi, Numerical solution of telegraph equation by using LT inversion technique, Int. J. Adv. Math. Sci., 1(2) (2013), 64-77.
  • [23] A. Ashyralyev, M. Modanli, An operator method for telegraph partial differential and difference equations, Boundary V. Prob., 1(41) (2015).
  • [24] A. Ashyralyev, M. Modanli, Nonlocal boundary value problem for telegraph equations, AIP Conf. Proc., 1676 (2015).
  • [25] M. E. K¨oksal, An operator-difference method for telegraph equations arising in transmission lines, Discrete Dyn. Nature Soc., 2011(6) (2011), 17.
  • [26] A. Ashyralyev, P.E. Sobolevskii, New difference schemes for partial differential equations, Operator Theory: Advances and Applications, 148, (2004).
  • [27] V. Pogorelenko, P. E. Sobolevskii, The “counter-example” to W. Littman counter-example of Lp-energetical inequality for wave equation, Funct. Differ. Equ., 4(1-2) (1997), 165-172.
  • [28] V. A Kostin, Analytic semigroups and cosine functions, Dokl. Akad. Nauk SSSR, 307(4) (1989), 796-799.
  • [29] M. Modanli, B. Bajjah, Double Laplace decomposition method and finite difference method of time-fractional Schrödinger pseudoparabolic partial differential equation with Caputo derivative, J. Math., 2021 (2021), 10.
  • [30] M. Modanlı, F. Şimşek, Pseudo-hiperbolik telegraf kısmi diferansiyel denklemin modifiye çift Laplace metodu ile Çözümü, Karadeniz Fen Bilimleri Dergisi, 12(1) (2022), 43-50.
  • [31] M. Modanli, B. Bajjah, S. Kuşulay, Two numerical methods for solving the Schr¨odinger parabolic and pseudoparabolic partial differential equations, Adv. Math. Phys., 2022 (2022), 10.

Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation

Year 2022, , 92 - 98, 01.12.2022
https://doi.org/10.33187/jmsm.1132139

Abstract

Hyperbolic partial differential equations are frequently referenced in modeling real-world problems in mathematics and engineering. Therefore, in this study, an initial-boundary value issue is proposed for the pseudo-hyperbolic telegraph equation. By operator method, converting the PDE to an ODE provides an exact answer to this problem. After that, the finite difference method is applied to construct first-order finite difference schemes to calculate approximate numerical solutions. The stability estimations of finite difference schemes are shown, as well as some numerical tests to check the correctness in comparison to the precise solution. The numerical solution is subjected to error analysis. As a result of the error analysis, the maximum norm errors tend to decrease as we increase the grid points. It can be drawn that the established scheme is accurate and effective

References

  • [1] A. Din, Y. Li, F. M. Khan, Z. U. Khan, P. Liu, On analysis of fractional order mathematical model of Hepatitis B using Atangana-Baleanu Caputo (ABC) derivative, Fractals, 30(01) (2022), 2240017.
  • [2] A. Din, Y. Li, T. Khan, G. Zaman, Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China, Chaos Solitons Fractals, 141 (2020), 110286.
  • [3] P. Veeresha, M. Yavuz, C. Baishya, A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators, Int. J. Optim. Control: Theor. Appl. , 11(3) (2021), 52-67.
  • [4] F. Özbağ, Numerical simulations of traveling waves in a counterflow filtration combustion model, Turk. J. Math., 46(4) (2022), 1424-1435.
  • [5] L. Zada, R. Nawaz, K. S. Nisar, M. Tahir, M. Yavuz, New approximate-analytical solutions to partial differential equations via auxiliary function method, Partial Differ. Eq. Appl. Math., 4 (2021), 100045.
  • [6] Z. Hammouch, M. Yavuz, N. Özdemir, Numerical solutions and synchronization of a variable-order fractional chaotic system, Math. Model. Numer. Simul. App., 1(1) (2021), 11-23.
  • [7] H. Eltayeb, S. Mesloub, Y. T. Abdalla, A. Kilicman, A note on double conformable Laplace transform method and singular one dimensional conformable pseudo-hyperbolic equations, Mathematics, 7(10) (2019), 949.
  • [8] S. V. Potapova, Boundary value problems for pseudo-hyperbolic equations with a variable time direction, J. Pure Appl. Math., 3(1) (2012), 75-91.
  • [9] Y. Zhang, Y. Niu, D. Shi, Nonconforming H1 -Galerkin mixed finite element method for pseudo-hyperbolic equations, American J. Comp. Math., 2 (2012), 269-273.
  • [10] Y. Liu, J. Wang, H. Li, W. Gao, S. He, A new splitting H1-Galerkin mixed method for pseudo-hyperbolic equations, Int. J. Math. Comp. Sci., 5(3) (2011), 1444-1449.
  • [11] I. Fedotov, M.Y. Shatalov, J. Marais, Hyperbolic and pseudo-hyperbolic equations in the theory of vibration, Acta Mech., 227(11) (2016), 3315-3324.
  • [12] G. Chen, Z. Yang, Initial value problem for a class of nonlinear pseudo-hyperbolic equations, Acta Math. App. Sinica, 9(2) (1993), 166-173.
  • [13] P. A. Krutitskii, An initial-boundary value problem for the pseudo-hyperbolic equation of gravity-gyroscopic waves, J. Math. Kyoto Univ., 37(2) (1997), 343-365.
  • [14] Z. Zhao, H. Li, A continuous Galerkin method for pseudo-hyperbolic equations with variable coefficients, J. Math. Anal. App., 473(2) (2019), 1053-1072.
  • [15] M. Modanli, F. Özbağ, A. Akgülma, Finite difference method for the fractional order pseudo telegraph integro-differential equation, J. Appl. Math. Comput. Mech., 21(1) (2022), 41-54.
  • [16] H. W. Liu, L. B. Liu, An unconditionally stable spline difference scheme of O(k2 +h4) for solving the second-order 1D linear hyperbolic equation, Math. Comp. Model., 49 (2009), 1985-1993.
  • [17] M. M. Islam, M. S. Hasan, A study on exact solution of the telegraph equation by (G’/G)-expansion method, African J. Math. Comp. Sci. Res., 11(7) (2018), 103-108.
  • [18] A. Merad, A. Bouziani, Solvability the telegraph equation with purely integral conditions, J. App. Eng. Math., 3(2) (2013), 245-253.
  • [19] M. Sarı, A. G¨unay, G. G¨urarslan, A solution to the telegraph equation by using DGJ method, Int. J. Nonlinear Sci., 17(1) (2014), 57-66.
  • [20] B. Soltanalizadeh, Differential transformation method for solving one-space-dimensional telegraph equation, Comput. Appl. Math., 30(3) (2016), 639-653.
  • [21] M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numer. Methods for Partial Differ. Eq., 24(4) (2008), 1080-1093.
  • [22] M. Javidi, N. Nyamoradi, Numerical solution of telegraph equation by using LT inversion technique, Int. J. Adv. Math. Sci., 1(2) (2013), 64-77.
  • [23] A. Ashyralyev, M. Modanli, An operator method for telegraph partial differential and difference equations, Boundary V. Prob., 1(41) (2015).
  • [24] A. Ashyralyev, M. Modanli, Nonlocal boundary value problem for telegraph equations, AIP Conf. Proc., 1676 (2015).
  • [25] M. E. K¨oksal, An operator-difference method for telegraph equations arising in transmission lines, Discrete Dyn. Nature Soc., 2011(6) (2011), 17.
  • [26] A. Ashyralyev, P.E. Sobolevskii, New difference schemes for partial differential equations, Operator Theory: Advances and Applications, 148, (2004).
  • [27] V. Pogorelenko, P. E. Sobolevskii, The “counter-example” to W. Littman counter-example of Lp-energetical inequality for wave equation, Funct. Differ. Equ., 4(1-2) (1997), 165-172.
  • [28] V. A Kostin, Analytic semigroups and cosine functions, Dokl. Akad. Nauk SSSR, 307(4) (1989), 796-799.
  • [29] M. Modanli, B. Bajjah, Double Laplace decomposition method and finite difference method of time-fractional Schrödinger pseudoparabolic partial differential equation with Caputo derivative, J. Math., 2021 (2021), 10.
  • [30] M. Modanlı, F. Şimşek, Pseudo-hiperbolik telegraf kısmi diferansiyel denklemin modifiye çift Laplace metodu ile Çözümü, Karadeniz Fen Bilimleri Dergisi, 12(1) (2022), 43-50.
  • [31] M. Modanli, B. Bajjah, S. Kuşulay, Two numerical methods for solving the Schr¨odinger parabolic and pseudoparabolic partial differential equations, Adv. Math. Phys., 2022 (2022), 10.
There are 31 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mahmut Modanlı 0000-0002-7743-3512

Fatih Özbağ 0000-0002-5456-4261

Publication Date December 1, 2022
Submission Date June 17, 2022
Acceptance Date September 2, 2022
Published in Issue Year 2022

Cite

APA Modanlı, M., & Özbağ, F. (2022). Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation. Journal of Mathematical Sciences and Modelling, 5(3), 92-98. https://doi.org/10.33187/jmsm.1132139
AMA Modanlı M, Özbağ F. Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation. Journal of Mathematical Sciences and Modelling. December 2022;5(3):92-98. doi:10.33187/jmsm.1132139
Chicago Modanlı, Mahmut, and Fatih Özbağ. “Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation”. Journal of Mathematical Sciences and Modelling 5, no. 3 (December 2022): 92-98. https://doi.org/10.33187/jmsm.1132139.
EndNote Modanlı M, Özbağ F (December 1, 2022) Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation. Journal of Mathematical Sciences and Modelling 5 3 92–98.
IEEE M. Modanlı and F. Özbağ, “Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation”, Journal of Mathematical Sciences and Modelling, vol. 5, no. 3, pp. 92–98, 2022, doi: 10.33187/jmsm.1132139.
ISNAD Modanlı, Mahmut - Özbağ, Fatih. “Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation”. Journal of Mathematical Sciences and Modelling 5/3 (December 2022), 92-98. https://doi.org/10.33187/jmsm.1132139.
JAMA Modanlı M, Özbağ F. Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation. Journal of Mathematical Sciences and Modelling. 2022;5:92–98.
MLA Modanlı, Mahmut and Fatih Özbağ. “Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation”. Journal of Mathematical Sciences and Modelling, vol. 5, no. 3, 2022, pp. 92-98, doi:10.33187/jmsm.1132139.
Vancouver Modanlı M, Özbağ F. Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation. Journal of Mathematical Sciences and Modelling. 2022;5(3):92-8.

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