Home Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative
Article
Licensed
Unlicensed Requires Authentication

Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative

  • Abdul Majeed EMAIL logo , Mohsin Kamran and Noreen Asghar
Published/Copyright: June 4, 2021

Abstract

This article focusses on the implementation of cubic B-spline approach to investigate numerical solutions of inhomogeneous time fractional nonlinear telegraph equation using Caputo derivative. L1 formula is used to discretize the Caputo derivative, while B-spline basis functions are used to interpolate the spatial derivative. For nonlinear part, the existing linearization formula is applied after generalizing it for all positive integers. The algorithm for the simulation is also presented. The efficiency of the proposed scheme is examined on three test problems with different initial boundary conditions. The influence of parameter α on the solution profile for different values is demonstrated graphically and numerically. Moreover, the convergence of the proposed scheme is analyzed and the scheme is proved to be unconditionally stable by von Neumann Fourier formula. To quantify the accuracy of the proposed scheme, error norms are computed and was found to be effective and efficient for nonlinear fractional partial differential equations (FPDEs).


Corresponding author: Abdul Majeed, Department of Mathematics, Division of Science and Technology University of Education, Lahore, Pakistan, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors have no conflict of interest.

References

[1] P. M. Jordan and A. Puri, “Digital signal propagation in dispersive media,” J. Appl. Phys., vol. 85, pp. 1273–1282, 1999. https://doi.org/10.1063/1.369258.Search in Google Scholar

[2] R. Hilfer, Applications of Fractional Calculus in Physics, River Edge, NJ, USA, World Sci. Publishing, 2000.10.1142/3779Search in Google Scholar

[3] I. Podlubny, Fractional Differential Equations, San Diego, Academic Press, 1999.Search in Google Scholar

[4] A. Majeed, M. Abbas, K. T. Miura, M. Kamran, and T. Nazir, “Surface modeling from 2D contours with an application to craniofacial fracture construction,” Mathematics, vol. 8, no. 8, p. 1246, 2020. https://doi.org/10.3390/math8081246.Search in Google Scholar

[5] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Hoboken, NJ, USA, Wiley, 1993, p. 384.Search in Google Scholar

[6] A. Majeed, M. Kamran, M. K. Iqbal, and D. Baleanu, “Solving time fractional Burgers and Fishers equations using cubic B spline approximation method,” Adv. Differ. Equ., vol. 1, pp. 1–15, 2020.10.1186/s13662-020-02619-8Search in Google Scholar

[7] A. Majeed, M. Kamran, N. Asghar, and D. Baleanu, “Numerical approximation of inhomogeneous time fractional Burgers–Huxley equation with B-spline functions and Caputo derivative,” Eng. Comput., vols 1–16, 2021.10.1007/s00366-020-01261-ySearch in Google Scholar

[8] A. Majeed, M. Kamran, and M. Rafique, “An approximation to the solution of time fractional modified Burgers’ equation using extended cubic B-spline method,” Comput. Appl. Math., vol. 39, no. 4, pp. 1–21, 2020. https://doi.org/10.1007/s40314-020-01307-3.Search in Google Scholar

[9] H. Jafari, N. A. Tuan, and R. M. Ganji, “A new numerical scheme for solving pantograph type nonlinear fractional integro-differential equations,” J. King Saud Univ. Sci., vol. 33, no. 1, p. 101185, 2021. https://doi.org/10.1016/j.jksus.2020.08.029.Search in Google Scholar

[10] R. M. Ganji, H. Jafari, and B. Dumitru, “A new approach for solving multi variable orders differential equations with Mittag–Leffler kernel,” Chaos, Solit. Fractals, vol. 130, p. 109405, 2020. https://doi.org/10.1016/j.chaos.2019.109405.Search in Google Scholar

[11] J. F. Gmez-Aguilar, R. F. Escobar-Jimnez, M. G. Lpez-Lpez, and V. M. Alvarado-MartDnez, “Atangana–Baleanu fractional derivative applied to electromagnetic waves in dielectric media,” J. Electromagn. Waves Appl., vol. 30, no. 15, pp. 1937–1952, 2016.10.1080/09205071.2016.1225521Search in Google Scholar

[12] A. Coronel-Escamilla, J. F. Gmez-Aguilar, E. Alvarado-Mndez, G. V. Guerrero-Ramrez, and R. F. Escobar-Jimnez, “Fractional dynamics of charged particles in magnetic fields,” Int. J. Mod. Phys. C, vol. 27, no. 08, p. 1650084, 2016. https://doi.org/10.1142/s0129183116500844.Search in Google Scholar

[13] V. F. Morales-Delgado, M. A. Taneco-Hernndez, and J. F. Gmez-Aguilar, “On the solutions of fractional order of evolution equations,” Eur. Phys. J. Plus, vol. 132, no. 1, p. 47, 2017. https://doi.org/10.1140/epjp/i2017-11341-0.Search in Google Scholar

[14] A. Majeed, M. Kamran, M. Abbas, and J. Singh, “An efficient numerical technique for solving time fractional generalized Fisher’s equation,” Front. Phys., vol. 8, p. 293, 2020. https://doi.org/10.3389/fphy.2020.00293.Search in Google Scholar

[15] V. R. Hosseini, E. Shivanian, and W. Chen, “Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation,” Eur. Phys. J. Plus, vol. 130, no. 2, p. 33, 2015. https://doi.org/10.1140/epjp/i2015-15033-5.Search in Google Scholar

[16] O. Nikan, Z. Avazzadeh, and J. A. Tenreiro Machado, “An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model,” J. King Saud Univ. Sci., vol. 33, no. 1, p. 101243, 2021. https://doi.org/10.1016/j.jksus.2020.101243.Search in Google Scholar

[17] O. Nikan and Z. Avazzadeh, “An improved localized radial basis-pseudospectral method for solving fractional reaction–subdiffusion problem,” Results Phys., vol. 23, p. 104048, 2021. https://doi.org/10.1016/j.rinp.2021.104048.Search in Google Scholar

[18] A. Okubo, “Application of the telegraph equation to oceanic diffusion: another mathematic model,” Tech. Rep., 1971.Search in Google Scholar

[19] J. Banasiak and J. R. Mika, “Singularly perturbed telegraph equations with applications in the random walk theory,” J. Appl. Math. Stoch. Anal., vol. 11, no. 1, p. 928, 1998. https://doi.org/10.1155/s1048953398000021.Search in Google Scholar

[20] V. H. Weston and S. He, “Wave splitting of the telegraph equation in R3 and its application to inverse scattering,” Inverse Probl., vol. 9, no. 6, pp. 789–812, 1993. https://doi.org/10.1088/0266-5611/9/6/013.Search in Google Scholar

[21] N. H. Can, O. Nikan, M. N. Rasoulizadeh, H. Jafari, and Y. Gasimov, “Numerical computation of the time non-linear fractional generalized equal width model arising in shallow water channel,” Therm. Sci., vol. 24, no. Suppl. 1, pp. 49–58, 2020. https://doi.org/10.2298/tsci20049c.Search in Google Scholar

[22] O. Nikan, Z. Avazzadeh, and J. A. Tenreiro Machado, “A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer,” J. Adv. Res., 2021. https://doi.org/10.1016/j.jare.2021.03.002.Search in Google Scholar PubMed PubMed Central

[23] R. Mohanty, “New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations,” Int. J. Comput. Math., vol. 86, pp. 2061–2071, 2009. https://doi.org/10.1080/00207160801965271.Search in Google Scholar

[24] J. Chen, F. Liu, and V. Anh, “Analytical solution for the time fractional telegraph equation by the method of separating variables,” J. Math. Anal. Appl., vol. 338, no. 2, pp. 1364–1377, 2008. https://doi.org/10.1016/j.jmaa.2007.06.023.Search in Google Scholar

[25] O. Nikan, Z. Avazzadeh, and J. A. Tenreiro Machado, “Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport,” Commun. Nonlinear Sci. Numer. Simulat., vol. 99, p. 105755, 2021. https://doi.org/10.1016/j.cnsns.2021.105755.Search in Google Scholar

[26] F. Huang, “Analytical solution for the time-fractional telegraph equation,” J. Appl. Math., vol. 9, 2009, Art no. 890158.10.1155/2009/890158Search in Google Scholar

[27] M. Dehghan and A. Shokri, “A numerical method for solving the hyperbolic telegraph equation,” Numer. Methods Part. Differ. Equ., vol. 24, no. 4, pp. 1080–1093, 2008. https://doi.org/10.1002/num.20306.Search in Google Scholar

[28] A. Saadatmandi and M. Dehghan, “Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method,” Numer. Methods Part. Differ. Equ., vol. 26, no. 1, pp. 239–252, 2010. https://doi.org/10.1002/num.20442.Search in Google Scholar

[29] S. A. Yousefi, “Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation,” Numer. Methods Part. Differ. Equ., vol. 26, no. 3, pp. 535–543, 2010.10.1002/num.20445Search in Google Scholar

[30] S. Das and P. K. Gupta, “Homotopy analysis method for solving fractional hyperbolic partial differential equations,” Int. J. Comput. Math., vol. 88, no. 3, pp. 578–588, 2011. https://doi.org/10.1080/00207161003631901.Search in Google Scholar

[31] N. Mollahasani, M. Mohseni Moghadam, and K. Afrooz, “A new treatment based on hybrid functions to the solution of telegraph equations of fractional order,” Appl. Math. Model., vol. 40, no. 4, pp. 2804–2814, 2015.10.1016/j.apm.2015.08.020Search in Google Scholar

[32] W. Jiang and Y. Lin, “Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space,” Commun. Nonlinear Sci. Numer. Simulat., vol. 16, pp. 3639–3645, 2011. https://doi.org/10.1016/j.cnsns.2010.12.019.Search in Google Scholar

[33] P. Veeresha and D. G. Prakasha, “Numerical solution for fractional model of telegraph equation by using q-HATM,” arXiv 2018, arXiv: 1805.03968.Search in Google Scholar

[34] H. Al-badrani, S. Saleh, H. O. Bakodah, and M. Al-Mazmumy, “Numerical solution for nonlinear telegraph equation by modified adomian decomposition method,” Nonlinear Anal. Differ. Equ., vol. 4, pp. 243–257, 2016. https://doi.org/10.12988/nade.2016.6418.Search in Google Scholar

[35] M. Inc, A. Akgl, and A. Kiliman, “Explicit solution of telegraph equation based on reproducing kernel method,” J. Funct. Spaces Appl., vol. 2012, 2012, https://doi.org/10.1155/2012/984682.Search in Google Scholar

[36] J. Biazar, H. Ebrahimi, and Z. Ayati, “An approximation to the solution of telegraph equation by variational iteration method,” Numer. Methods Part. Differ. Equ., vol. 25, pp. 797–801, 2009. https://doi.org/10.1002/num.20373.Search in Google Scholar

[37] H. Khan, R. Shah, P. Kumam, D. Baleanu, and M. Arif, “An efficient analytical technique, for the solution of fractional-order telegraph equations,” Mathematics, vol. 7, no. 5, p. 426, 2019. https://doi.org/10.3390/math7050426.Search in Google Scholar

[38] M. Asgari, R. Ezzati, and T. Allahviranloo, “Numerical solution of time-fractional order telegraph equation by Bernstein polynomials operational matrices,” Math. Probl. Eng., vol. 115, 2016.10.1155/2016/1683849Search in Google Scholar

[39] O. Tasbozan and A. Esen, “Quadratic B-spline Galerkin method for numerical solutions of fractional telegraph equations,” Bull. Math. Sci. Appl., vol. 18, pp. 23–39, 2017. https://doi.org/10.18052/www.scipress.com/bmsa.18.23.Search in Google Scholar

[40] M. Uddin and A. Ali, “On the approximation of time-fractional telegraph equations using localized kernel-based method,” Adv. Differ. Equ., vol. 2018, no. 1, p. 305, 2018.10.1186/s13662-018-1775-8Search in Google Scholar

[41] T. Akram, M. Abbas, A. I. Ismail, N. H. M. Ali, and D. Baleanu, “Extended cubic B-splines in the numerical solution of time fractional telegraph equation,” Adv. Differ. Equ., vol. 2019, no. 1, p. 365, 2019. https://doi.org/10.1186/s13662-019-2296-9.Search in Google Scholar

[42] B. Sepehrian and Z. Shamohammadi, “Numerical solution of nonlinear time-fractional telegraph equation by radial basis function collocation method,” Iran. J. Sci. Technol. Trans. A-Science, vol. 42, no. 4, pp. 2091–2104, 2018. https://doi.org/10.1007/s40995-017-0446-z.Search in Google Scholar

[43] O. Tasbozan, A. Esen, Y. Ucar, and N. M. Yagmurlu, “A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations,” Tbilisi Math. J., vol. 8, pp. 181–193, 2015.10.1515/tmj-2015-0020Search in Google Scholar

[44] S. G. Rubin and R. A. Graves, “Cubic spline approximation for problems in fluid mechanics,” Nasa TR R-436, Washington, DC, 1975.Search in Google Scholar

[45] I. Dag, D. Irk, and B. Saka, “A numerical solution of Burgers equation using cubic B-splines,” Appl. Math. Comput., vol. 163, pp. 199–211, 2005.10.1016/j.amc.2004.01.028Search in Google Scholar

[46] T. S. El-Danaf and A. R. Hadhoud, “Parametric spline functions for the solution of the one time fractional burger equation,” Appl. Math. Model., vol. 36, pp. 4557–4564, 2012. https://doi.org/10.1016/j.apm.2011.11.035.Search in Google Scholar

Received: 2020-01-16
Revised: 2021-04-24
Accepted: 2021-05-12
Published Online: 2021-06-04
Published in Print: 2022-08-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 16.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2020-0013/html
Scroll to top button