Numerical solution of the hyperbolic telegraph equation using cubic B-spline-based differential quadrature of high accuracy

Document Type : Research Paper

Authors

1 Department of Mathematics, Cochin University of Science and Technology, Kerala, India.

2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada.

3 1Department of Mathematics, Cochin University of Science and Technology, Kerala, India.

Abstract

By constructing a newly modified cubic B-splines having the optimal accuracy order four, we propose a numerical scheme for solving the hyperbolic telegraph equation using a differential quadrature method. The spatial derivatives are approximated by the differential quadrature whose weight coefficients are computed using the newly modified cubic B-splines. Our modified cubic B-splines retain the tridiagonal structure and achieve the fourth order convergence rate. The solution of the associated ODEs is advanced in the time domain by the SSPRK scheme. The stability of the method is analyzed using the discretization matrix. Our numerical experiments demonstrate the better performance of our proposed scheme over several known numerical schemes reported in the literature.

Keywords

References

  • [1]          S. Abbasbandy, H. R. Ghehsareh, I. Hashim, and A. Alsaedi, A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation, Engineering Analysis with Boundary Elements, 47 (2014), 10-20.
  • [2]          I. Ahmad, H. Ahmad, A. E. Abouelregal, P. Thounthong, and M. Abdel-Aty, Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences, The European Physical Journal Plus, 135(9) (2020), 1-14.
  • [3]          A. S. Alshomrani, S. Pandit, A. K. Alzahrani, M. S. Alghamdi, and R. Jiwari, A numerical algorithm based on modified cubic trigonometric b-spline functions for computational modelling of hyperbolic-type wave equations, Engineering Computations, 2017.
  • [4]          G. Arora and V. Joshi, Comparison of numerical solution of 1d hyperbolic telegraph equation using b-spline and trigonometric b-spline by differential quadrature method, Indian Journal of Science and Technology, 9(45) (2016), 1-8.
  • [5]          M. Aslefallah and D. Rostamy, Application of the singular boundary method to the two-dimensional telegraph equation on arbitrary domains, Journal of Engineering Mathematics, 118(1) (2019), 1-14.
  • [6]          A. Babu, B. Han, and N. Asharaf, Numerical solution of the viscous burgers equation using localized differential quadrature method, Partial Differential Equations in Applied Mathematics, 2021, 100044.
  • [7]          J. Banasiak and J. R. Mika, Singularly perturbed telegraph equations with applications in the random walk theory, Journal of Applied Mathematics and Stochastic Analysis, 11(1) (1998), 9-28.
  • [8]          K.  E.  Bi¸cer  and  S.  Yal¸cinba¸s,  Numerical  solution  of  telegraph  equation  using  bernoulli  collocation  method,  Pro- ceedings of the National Academy of Sciences, India Section A: Physical Sciences, 89(4) (2019), 769-775.
  • [9]          G. B¨ohme, Non-Newtonian fluid mechanics, Elsevier, 2012.
  • [10]        M. Dehghan and A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numerical Methods  for Partial Differential Equations: An International Journal, 24(4) (2008), 1080-1093.
  • [11]        M. Dehghan and A. Ghesmati, Combination of meshless local weak and strong (mlws) forms to solve the two dimensional hyperbolic telegraph equation, Engineering Analysis with Boundary Elements, 34(4) (2010), 324-336.
  • [12]        R. M. Hafez, Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coeffi- cients using shifted jacobi collocation method, Computational and Applied Mathematics, 37(4) (2018), 5253-5273.
  • [13]        E. Hesameddini and E. Asadolahifard, A new spectral galerkin method for solving the two dimensional hyperbolic telegraph equation, Computers & Mathematics with Applications, 72(7) (2016), 1926-1942.
  • [14]        Z. Hong, Y. Wang, and H. Hao, Adaptive monte carlo  methods  for  solving  hyperbolic  telegraph  equation, Journal  of Computational and Applied Mathematics, 345 (2019), 405-415.
  • [15]        R. Jiwari, S. Pandit, and R. Mittal, A differential quadrature algorithm for the  numerical  solution  of  the  second- order one dimensional hyperbolic telegraph equation, International Journal of Nonlinear Science, 13(3) (2012), 259-266.
  • [16]        R. Jiwari, S. Pandit, and R. Mittal, A differential quadrature  algorithm to solve the two dimensional linear hyper- bolic telegraph equation with dirichlet and neumann boundary conditions, Applied Mathematics and Computation, 218(13) (2012), 7279-7294.
  • [17]        R. Jiwari, Lagrange interpolation and modified cubic b-spline differential quadrature methods for solving hyperbolic partial differential equations with dirichlet and neumann boundary conditions, Computer Physics Communications, 193 (2015), 55-65.
  • [18]        R. Jiwari, Barycentric rational interpolation and local radial basis functions based numerical algorithms for multi- dimensional sine-gordon equation, Numerical Methods for Partial Differential Equations, 37(3) (2021), 1965-1992.
  • [19]        P. Jordan and A. Puri, Digital signal propagation in dispersive media, Journal of Applied Physics, 85(3) (1999), 1273-1282.
  • [20]        M. Lakestani and B. N. Saray, Numerical solution of telegraph equation using interpolating scaling functions, Computers & Mathematics with Applications, 60(7) (2010), 1964-1972.
  • [21]        J. Lin, F. Chen, Y. Zhang, and J. Lu, An accurate meshless collocation technique for solving two-dimensional  hyperbolic telegraph equations in arbitrary domains, Engineering Analysis with Boundary Elements, 108 (2019), 372-384.
  • [22]        R. Mittal and R. Jain, Numerical solutions of nonlinear burgers equation with modified cubic b-splines collocation method, Applied Mathematics and Computation, 218(15) (2012), 7839-7855.
  • [23]        R. C. Mittal and R. K. Jain, Numerical solutions of nonlinear fisher’s reaction–diffusion equation with modified cubic b-spline collocation method, Mathematical Sciences, 7(1) (2013), 1-10.
  • [24]        R. Mittal, R. Jiwari, and K. K. Sharma, A numerical scheme based on differential quadrature method to solve  time dependent burgers’ equation, Engineering Computations, 2013.
  • [25]        R. Mittal and R. Bhatia, Numerical solution  of  second  order  one  dimensional  hyperbolic  telegraph  equation  by  cubic b-spline collocation method, Applied Mathematics and Computation, 220 (2013), 496-506.
  • [26]        R. Mittal and R. Bhatia, A numerical study of two dimensional hyperbolic telegraph equation by modified b-spline differential quadrature method, Applied Mathematics and Computation, 244 (2014), 976-997.
  • [27]        R. Mittal and S. Dahiya, Numerical simulation of three-dimensional telegraphic equation using cubic b-spline differential quadrature method, Applied Mathematics and Computation, 313 (2017), 442-452.
  • [28]        R. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equa- tions, International Journal of Computer Mathematics, 86(12) (2009), 2061-2071.
  • [29]        H. J. Nieuwenhuis and L. Schoonbeek, Stability of matrices with negative diagonal submatrices,  Linear algebra  and its applications, 353(1-3) (2002), 183-196.
  • [30]        T. Nazir, M. Abbas, and M. Yaseen, Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric b-splines approach, Cogent Mathematics & Statistics, 4(1) (2017), 1382061.
  • [31]        S. Niknam and H. Adibi, A numerical solution of two-dimensional hyperbolic telegraph equation based on moving least square meshless method and radial basis functions, Computational Methods for Differential Equations, 2021.
  • [32]        S. Pandit, M. Kumar, and S. Tiwari, Numerical simulation of second-order one dimensional hyperbolic telegraph equation, Computer Physics Communications, 187 (2015), 83-90.
  • [33]        S. Pandit, R. Jiwari, K. Bedi, and M. E. Koksal, Haar wavelets operational matrix based  algorithm for computa-  tional modelling of hyperbolic type wave equations, Engineering Computations, 2017.
  • [34]        B. Pekmen and M. Tezer-Sezgin, Differential quadrature solution of hyperbolic telegraph equation, Journal of Applied Mathematics, 2012.
  • [35]        D. Rostamy, M. Emamjome, and S. Abbasbandy, A meshless technique based on the pseudospectral radial basis functions method for solving the two-dimensional hyperbolic telegraph equation, The European Physical Journal Plus, 132(6) (2017), 1-11.
  • [36]        M. Shamsi and M. Razzaghi, Numerical solution of the controlled duffing oscillator by the interpolating scaling functions, Journal of electromagnetic waves and applications, 18(5) (2004), 691-705.
  • [37]        S. Sharifi and J. Rashidinia, Numerical solution of hyperbolic telegraph equation  by  cubic  b-spline  collocation  method, Applied Mathematics and Computation, 281 (2016), 28-38.
  • [38]        C. Shu. Differential Quadrature and Its Application in Engineering, Springer Science & Business Media, 2000.
  • [39]        B. K. Singh and P. Kumar, An algorithm based on a new dqm with modified extended cubic b-splines for numerical study of two dimensional hyperbolic telegraph equation, Alexandria engineering journal, 57(1) (2018), 175-191.
  • [40]        B. K. Singh, J. P. Shukla, and M. Gupta, Study of one dimensional hyperbolic telegraph equation via a hybrid cubic b-spline differential quadrature method, International Journal of Applied and Computational Mathematics, 7(1) (2021), 1-17.
  • [41]        R. J. Spiteri and S. J. Ruuth, A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM Journal on Numerical Analysis, 40(2) (2002), 469-491.
  • [42]        N. Sweilam, A. Nagy, and A. El-Sayed, Sinc-chebyshev collocation method for time-fractional  order  telegraph  equation, Appl. Comput. Math, 19(2) (2020), 162-174.
  • [43]        P. R. Wallace, Mathematical analysis of physical problems. Courier Corporation, 1984.
  • [44]        F. Wang and Y. Wang, A coupled pseudospectral-differential quadrature method for a class of hyperbolic telegraph equations, Mathematical Problems in Engineering, 2017.
  • [45]        V. Weston and S. He, Wave splitting of the telegraph equation in r3 and its application to inverse scattering, Inverse Problems, 9(6) (1993), 789.
  • [46]        M. Zarebnia and R. Parvaz, An approximation to the solution of one-dimensional hyperbolic telegraph equation based on the collocation of quadratic b-spline functions, Computational Methods for Differential Equations, 9(4) (2021), 1198-1213.
  • [47]        Y. Zhou, W. Qu, Y. Gu, and H. Gao, A hybrid meshless method  for the solution of the second  order hyperbolic  telegraph equation in two space dimensions, Engineering Analysis with Boundary Elements, 115 (2020), 21-27.
Computational Methods for Differential Equations
Volume 10, Issue 4
October 2022
Pages 837-859

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History

  • Receive Date: 02 September 2021
  • Revise Date: 08 January 2022
  • Accept Date: 14 January 2022

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