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A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation

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Abstract

This study is devoted to the numerical investigation of linear and nonlinear hyperbolic telegraph equation. We have proposed a wavelet collocation method based on Legendre polynomials for approximating the solution. Both the spatial and temporal variables, along with their derivatives, are approximated using the Legendre wavelet and its integration. The present approach is simple, consistent and straightforward. To assure the theoretical consistency of the method, an estimate for the upper bound of the error norm is provided. We have proved an exponential order of convergence which is better than the methods available in the literature. Some numerical experiments are carried out to justify the theoretical results and the outcomes confirm the computational efficiency of the proposed method.

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References

  1. Jiwari, R.: Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions. Comput. Phys. Commun. 193, 55–65 (2015)

    MathSciNet  Google Scholar 

  2. Mittal, R., Bhatia, R.: A numerical study of two dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method. Appl. Math. Comput. 244, 976–997 (2014)

    MathSciNet  Google Scholar 

  3. Jordan, P., Puri, A.: Digital signal propagation in dispersive media. J. Appl. Phys. 85(3), 1273–1282 (1999)

    Google Scholar 

  4. Koksal, M.E., Senol, M., Unver, A.K.: Numerical simulation of power transmission lines. Chin. J. Phys. 59, 507–524 (2019)

    MathSciNet  Google Scholar 

  5. Banasiak, J., Mika, J.R.: Singularly perturbed telegraph equations with applications in the random walk theory. Int. J. Stoch. Anal. 11, 9–28 (1998)

    MathSciNet  Google Scholar 

  6. Evans, D.J., Bulut, H.: The numerical solution of the telegraph equation by the alternating group explicit (AGE) method. Int. J. Comput. Math. 80(10), 1289–1297 (2003)

    MathSciNet  Google Scholar 

  7. Jordan, P., Meyer, M.R., Puri, A.: Causal implications of viscous damping in compressible fluid flows. Phys. Rev. E 62(6), 7918 (2000)

    Google Scholar 

  8. Weston, V., He, S.: Wave splitting of the telegraph equation in R\(^3\) and its application to inverse scattering. Inverse Prob. 9(6), 789 (1993)

    MathSciNet  Google Scholar 

  9. Alharbi, W., Petrovskii, S.: Critical domain problem for the reaction-telegraph equation model of population dynamics. Mathematics 6(4), 59 (2018)

    Google Scholar 

  10. Mohanty, R.: New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations. Int. J. Comput. Math. 86(12), 2061–2071 (2009)

    MathSciNet  Google Scholar 

  11. Bülbül, B., Sezer, M.: Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients. Int. J. Comput. Math. 88(3), 533–544 (2011)

    MathSciNet  Google Scholar 

  12. Mittal, R., Bhatia, R.: A collocation method for numerical solution of hyperbolic telegraph equation with Neumann boundary conditions. Int. J. Comput. Math. 2014, 526814 (2014)

    Google Scholar 

  13. Dehghan, M., Ghesmati, A.: Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method. Eng. Anal. Boundary Elem. 34(1), 51–59 (2010)

    MathSciNet  Google Scholar 

  14. Hosseini, M., Mohyud-Din, S.T., Nakhaeei, A.: New Rothe-wavelet method for solving telegraph equations. Int. J. Syst. Sci. 43(6), 1171–1176 (2012)

    MathSciNet  Google Scholar 

  15. Dosti, M., Nazemi, A.: Quartic B-spline collocation method for solving one-dimensional hyperbolic telegraph equation. J. Inf. Comput. Sci. 7(2), 83–90 (2012)

    Google Scholar 

  16. Heydari, M.H., Hooshmandasl, M., Ghaini, F.M.: A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type. Appl. Math. Model. 38(5–6), 1597–1606 (2014)

    MathSciNet  Google Scholar 

  17. Abbasbandy, S., Ghehsareh, H.R., Hashim, I., Alsaedi, A.: A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation. Eng. Anal. Boundary Elem. 47, 10–20 (2014)

    MathSciNet  Google Scholar 

  18. Sari, M., Gunay, A., Gurarslan, G.: A solution to the telegraph equation by using DGJ method. Int. J. Nonlinear Sci. 17(1), 57–66 (2014)

    MathSciNet  Google Scholar 

  19. Arslan, D.: The numerical study of a hybrid method for solving telegraph equation. Appl. Math. Nonlinear Sci. 5(1), 293–302 (2020)

    MathSciNet  Google Scholar 

  20. Elgindy, K.T.: High-order numerical solution of second-order one-dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method. Numer. Methods Partial Differ. Equ. 32(1), 307–349 (2016)

    MathSciNet  Google Scholar 

  21. Bülbül, B., Sezer, M.: A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation. Appl. Math. Lett. 24(10), 1716–1720 (2011)

    MathSciNet  Google Scholar 

  22. Acebrón, J.A., Ribeiro, M.A.: A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions. J. Comput. Phys. 305, 29–43 (2016)

    MathSciNet  Google Scholar 

  23. Inc, M., Akgül, A., Kılıçman, A.: Numerical solutions of the second-order one-dimensional telegraph equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal. 2013, 768963 (2013)

    MathSciNet  Google Scholar 

  24. Raftari, B., Khosravi, H., Yildirim, A.: Homotopy analysis method for the one-dimensional hyperbolic telegraph equation with initial conditions. Int. J. Numer. Methods Heat Fluid Flow 23(2), 355–372 (2013)

    MathSciNet  Google Scholar 

  25. Abd-Elhameed, W., Doha, E., Youssri, Y., Bassuony, M.: New Tchebyshev-Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations. Numer. Methods Partial Differ. Equ. 32(6), 1553–1571 (2016)

    MathSciNet  Google Scholar 

  26. Mardani, A., Hooshmandasl, M., Hosseini, M., Heydari, M.: Moving least squares (MLS) method for the nonlinear hyperbolic telegraph equation with variable coefficients. Int. J. Comput. Methods 14(03), 1750026 (2017)

    MathSciNet  Google Scholar 

  27. Lakestani, M., Saray, B.N.: Numerical solution of telegraph equation using interpolating scaling functions. Comput. Math. Appl. 60(7), 1964–1972 (2010)

    MathSciNet  Google Scholar 

  28. Mohanty, R., Jain, M.K., Arora, U.: An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions. Int. J. Comput. Math. 79(1), 133–142 (2002)

    MathSciNet  Google Scholar 

  29. Dehghan, M., Ghesmati, A.: Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation. Eng. Anal. Boundary Elem. 34(4), 324–336 (2010)

    MathSciNet  Google Scholar 

  30. Dehghan, M., Shokri, A.: A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions. Numer. Methods Partial Differ. Equ. Int. J. 25(2), 494–506 (2009)

    MathSciNet  Google Scholar 

  31. Ding, H., Zhang, Y.: A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation. J. Comput. Appl. Math. 230(2), 626–632 (2009)

    MathSciNet  Google Scholar 

  32. Alshomrani, A.S., Pandit, S., Alzahrani, A.K., Alghamdi, M.S., Jiwari, R.: A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. Eng. Comput. 34(4), 1257–1276 (2017)

    Google Scholar 

  33. Rashidinia, J., Jokar, M.: Application of polynomial scaling functions for numerical solution of telegraph equation. Appl. Anal. 95(1), 105–123 (2016)

    MathSciNet  Google Scholar 

  34. Yüzbaşı, Ş: Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction. Appl. Math. Comput. 287, 83–93 (2016)

    MathSciNet  Google Scholar 

  35. Kirli, E., Irk, D., Zorsahin Gorgulu, M.: High order accurate method for the numerical solution of the second order linear hyperbolic telegraph equation. Numer. Methods Partial Differ. Equ. 39(3), 2060–2072 (2023)

    MathSciNet  Google Scholar 

  36. Ashyralyev, A., Koksal, M.E.: On the numerical solution of hyperbolic PDEs with variable space operator. Numer. Methods Partial Differ. Equ. Int. J. 25(5), 1086–1099 (2009)

    MathSciNet  Google Scholar 

  37. Mohanty, R.: An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation. Appl. Math. Lett. 17(1), 101–105 (2004)

    MathSciNet  Google Scholar 

  38. Babu, A., Han, B., Asharaf, N.: Numerical solution of the hyperbolic telegraph equation using cubic B-spline-based differential quadrature of high accuracy. Comput. Methods Differ. Equ. 10(4), 837–859 (2022)

    MathSciNet  Google Scholar 

  39. Asif, M., Haider, N., Al-Mdallal, Q., Khan, I.: A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations. Numer. Methods Partial Differ. Equ. 36(6), 1962–1981 (2020)

    MathSciNet  Google Scholar 

  40. Razzaghi, M., Yousefi, S.: The Legendre wavelets operational matrix of integration. Int. J. Syst. Sci. 32(4), 495–502 (2001)

    MathSciNet  Google Scholar 

  41. Shamsi, M., Razzaghi, M.: Solution of Hallen’s integral equation using multiwavelets. Comput. Phys. Commun. 168(3), 187–197 (2005)

    Google Scholar 

  42. Lakestani, M., Razzaghi, M., Dehghan, M.: Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations. Math. Probl. Eng. 2006, 096184 (2006)

    MathSciNet  Google Scholar 

  43. Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44(2), 141–183 (1991)

    MathSciNet  Google Scholar 

  44. Faheem, M., Khan, A., El-Zahar, E.: On some wavelet solutions of singular differential equations arising in the modeling of chemical and biochemical phenomena. Adv. Differ. Equ. 2020, 526 (2020)

    MathSciNet  Google Scholar 

  45. Khan, A., Faheem, M., Raza, A.: Solution of third-order Emden-Fowler-type equations using wavelet methods. Eng. Comput. 38(6), 2850–2881 (2021)

    Google Scholar 

  46. Faheem, M., Khan, A., Wong, P.J.: A Legendre wavelet collocation method for 1D and 2D coupled time-fractional nonlinear diffusion system. Comput. Math. Appl. 128, 214–238 (2022)

    MathSciNet  Google Scholar 

  47. Sahu, P.K., Ray, S.S.: Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Appl. Math. Comput. 256, 715–723 (2015)

    MathSciNet  Google Scholar 

  48. Zhou, F., Xu, X.: Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets. Appl. Math. Comput. 247, 353–367 (2014)

    MathSciNet  Google Scholar 

  49. Faheem, M., Raza, A., Khan, A.: Wavelet collocation methods for solving neutral delay differential equations. Int. J. Nonlinear Sci. Numer. Simul. 23(7–8), 1129–1156 (2022)

    MathSciNet  Google Scholar 

  50. Faheem, M., Raza, A., Khan, A.: Collocation methods based on Gegenbauer and Bernoulli wavelets for solving neutral delay differential equations. Math. Comput. Simul. 180, 72–92 (2021)

    MathSciNet  Google Scholar 

  51. Kumar, S., Ahmadian, A., Kumar, R., Kumar, D., Singh, J., Baleanu, D., Salimi, M.: An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. Mathematics 8(4), 558 (2020)

    Google Scholar 

  52. Verma, A.K., Rawani, M.K., Cattani, C.: A numerical scheme for a class of generalized Burgers’ equation based on Haar wavelet nonstandard finite difference method. Appl. Numer. Math. 168, 41–54 (2021)

    MathSciNet  Google Scholar 

  53. Walnut, D.F.: Multiresolution Analysis, pp. 163–214. Birkhäuser Boston, Boston (2004). https://doi.org/10.1007/978-1-4612-0001-7_7

    Book  Google Scholar 

  54. Sharifi, S., Rashidinia, J.: Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method. Appl. Math. Comput. 281, 28–38 (2016)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors are thankful to the Editor and the anonymous reviewers for their constructive and fruitful comments which improved the presentation and quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Maulana Azad National Urdu University, Hyderabad, Telangana, 500032, India

    Basharat Hussain

  2. Department of Mathematics, National Institute of Technology Andhra Pradesh, Tadepalligudem, Andhra Pradesh, 534101, India

    Mo Faheem

  3. Department of Mathematics, Jamia Millia Islamia, New Delhi, Delhi, 110025, India

    Arshad Khan

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  1. Basharat Hussain
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Hussain, B., Faheem, M. & Khan, A. A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation. J. Appl. Math. Comput. 70, 3661–3684 (2024). https://doi.org/10.1007/s12190-024-02098-0

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