10.1007/s40096-022-00460-6

Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

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  • A. G. Atta1,
  • W. M. Abd-Elhameed2,
  • G. M. Moatimid1,
  • Y. H. Youssri*2,
  1. Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, EG
  2. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, EG
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Published in Issue 2022-03-06

How to Cite

Atta, A. G., Abd-Elhameed, W. M., Moatimid, G. M., & Youssri, Y. H. (2022). Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem. Mathematical Sciences, 17(4 (December 2023). https://doi.org/10.1007/s40096-022-00460-6
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Abstract

Abstract A new numerical scheme based on the tau spectral method for solving the linear hyperbolic telegraph type equation is presented and implemented. The derivation of this scheme is based on utilizing certain modified shifted Chebyshev polynomials of the sixth-kind as basis functions. For this purpose, some new formulas concerned with the modified shifted Chebyshev polynomials of the sixth-kind have been stated and proved, and after that, they serve to study the hyperbolic telegraph type equation with our proposed scheme. One advantage of using this scheme is that it reduces the problem into a system of algebraic equations that can be simplified using the Kronecker algebra analysis. The convergence and error estimate of the proposed technique are analyzed in detail. In the end, some numerical tests are presented to demonstrate the efficiency and high accuracy of the proposed scheme.

Keywords

  • Hyperbolic telegraph equation,
  • Chebyshev polynomials of the sixth-kind,
  • Spectral methods,
  • Kronecker algebra,
  • Hypergeometric functions,
  • Convergence analysis
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References

  1. Brio et al. (2010) Academic Press
  2. Jordan and Puri (1999) Digital signal propagation in dispersive media 85(3) (pp. 1273-1282) https://doi.org/10.1063/1.369258
  3. Roussy and Pearce (1995) Wiley
  4. Devi et al. (2020) Lagrange’s operational approach for the approximate solution of two-dimensional hyperbolic telegraph equation subject to Dirichlet boundary conditions
  5. Biçer and Yalçinbaş (2019) Numerical solution of telegraph equation using Bernoulli collocation method 89(4) (pp. 769-775) https://doi.org/10.1007/s40010-018-0535-1
  6. Ureña et al. (2020) Solving the telegraph equation in 2-D and 3-D using generalized finite difference method (GFDM) (pp. 13-24) https://doi.org/10.1016/j.enganabound.2019.11.010
  7. Asif et al. (2020) A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations 36(6) (pp. 1962-1981) https://doi.org/10.1002/num.22512
  8. Dehghan and Salehi (2012) A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraph equation 35(10) (pp. 1220-1233) https://doi.org/10.1002/mma.2517
  9. Ahmad, I., Seadawy, A.R., Ahmad, H., Thounthong, P., Wang, F.: Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method. Int. J. Nonlinear Sci. Numer. Simul., (2021)
  10. Wang, F., Hou, E, Ahmad, I., Ahmad, H., Gu, Y.: An efficient meshless method for hyperbolic telegraph equations in
  11. (1+1)
  12. usepackage{amsmath}
  13. usepackage{wasysym}
  14. usepackage{amsfonts}
  15. usepackage{amssymb}
  16. usepackage{amsbsy}
  17. usepackage{mathrsfs}
  18. usepackage{upgreek}
  19. setlength{oddsidemargin}{-69pt}
  20. begin{document}$$(1+1)$$end{document}]]>
  21. dimensions. Model. Eng. Sci., (2021)
  22. https://doi.org/10.32604/cmes.2021.014739
  23. Jebreen (2021) On the numerical solution of Fisher’s equation by an efficient algorithm based on multiwavelets 6(3) (pp. 2369-2384) https://doi.org/10.3934/math.2021144
  24. Ahmad et al. (2020) Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences 135(9) (pp. 1-14) https://doi.org/10.1140/epjp/s13360-020-00784-z
  25. Taghian, H.T., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H.: Shifted Gegenbauer–Galerkin algorithm for hyperbolic telegraph type equation. Int. J. Modern Phys. C, 2150118, (2021)
  26. Zhou et al. (2020) A hybrid meshless method for the solution of the second order hyperbolic telegraph equation in two space dimensions (pp. 21-27) https://doi.org/10.1016/j.enganabound.2020.02.015
  27. Ahmad et al. (2020) Solution of multi-term time-fractional pde models arising in mathematical biology and physics by local meshless method 12(7) https://doi.org/10.3390/sym12071195
  28. Ahmad, H., Khan, T.A., Stanimirović, P.S., Chu, Y., Ahmad, I.: Modified variational iteration algorithm-ii: convergence and applications to diffusion models. Complexity, 2020, (2020)
  29. Ahmad et al. (2020) A new analyzing technique for nonlinear time fractional cauchy reaction-diffusion model equations https://doi.org/10.1016/j.rinp.2020.103462
  30. Qu and He (2020) A spatial-temporal GFDM with an additional condition for transient heat conduction analysis of FGMs https://doi.org/10.1016/j.aml.2020.106579
  31. Qu and He (2022) A GFDM with supplementary nodes for thin elastic plate bending analysis under dynamic loading https://doi.org/10.1016/j.aml.2021.107664
  32. Qu, W., Gao, H., Gu, Y.: Integrating krylov deferred correction and generalized finite difference methods for dynamic simulations of wave propagation phenomena in long-time intervals. Adv. Appl. Math. Mech., (2021)
  33. Napoli and Abd-Elhameed (2017) An innovative harmonic numbers operational matrix method for solving initial value problems 54(1) (pp. 57-76) https://doi.org/10.1007/s10092-016-0176-1
  34. Atta et al. (2019) Generalized Fibonacci operational collocation approach for fractional initial value problems 5(1) https://doi.org/10.1007/s40819-018-0597-4
  35. Rahimkhani and Ordokhani (2019) A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions 35(1) (pp. 34-59) https://doi.org/10.1002/num.22279
  36. Youssri and Hafez (2019) Exponential Jacobi spectral method for hyperbolic partial differential equations 13(4) (pp. 347-354) https://doi.org/10.1007/s40096-019-00304-w
  37. Mohammadi et al. (2019) Haar wavelet collocation method for solving singular and nonlinear fractional time-dependent emden-fowler equations with initial and boundary conditions 13(3) (pp. 255-265) https://doi.org/10.1007/s40096-019-00295-8
  38. Doha and Abd-Elhameed (2005) Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method 181(1) (pp. 24-45) https://doi.org/10.1016/j.cam.2004.11.015
  39. Atta et al. (2020) Generalized Fibonacci operational tau algorithm for fractional bagley-torvik equation (pp. 215-224) https://doi.org/10.18576/pfda/060305
  40. Abd-Elhameed and Youssri (2019) Sixth-kind Chebyshev spectral approach for solving fractional differential equations 20(2) (pp. 191-203) https://doi.org/10.1515/ijnsns-2018-0118
  41. Dehghan and Shokri (2008) A numerical method for solving the hyperbolic telegraph equation 24(4) (pp. 1080-1093) https://doi.org/10.1002/num.20306
  42. Abd-Elhameed and Youssri (2021) New formulas of the high-order derivatives of fifth-kind Chebyshev polynomials: Spectral solution of the convection-diffusion equation https://doi.org/10.1002/num.22756
  43. Youssri (2017) A new operational matrix of caputo fractional derivatives of fermat polynomials: an application for solving the Bagley-Torvik equation 2017(1) (pp. 1-17) https://doi.org/10.1186/s13662-017-1123-4
  44. Abd-Elhameed, W.M., Machado, J.A.T., Youssri, Y.H.: Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations. Int. J. Nonlinear Sci. Numer. Simul.,
  45. https://doi.org/10.1515/ijnsns-2020-0124
  46. Abd-Elhameed et al. (2016) New Tchebyshev-Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations 32(6) (pp. 1553-1571) https://doi.org/10.1002/num.22074
  47. Doha et al. (2019) Fully Legendre spectral Galerkin algorithm for solving linear one-dimensional telegraph type equation 16(08) https://doi.org/10.1142/S0219876218501189
  48. Youssri, Y.H., Abd-Elhameed, W.M.: Numerical spectral Legendre-Galerkin algorithm for solving time fractional telegraph equation. Rom. J. Phys, 63(107), (2018)
  49. Atta et al. (2021) Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations (pp. 237-256) https://doi.org/10.1016/j.apnum.2021.05.010
  50. Hammad et al. (2020) Exponential Jacobi-Galerkin method and its applications to multidimensional problems in unbounded domains (pp. 88-109) https://doi.org/10.1016/j.apnum.2020.05.017
  51. Wang and Chen (2020) Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam https://doi.org/10.1016/j.chaos.2019.109585
  52. Doha et al. (2015) On the coefficients of differentiated expansions and derivatives of Chebyshev polynomials of the third and fourth kinds 35(2) (pp. 326-338) https://doi.org/10.1016/S0252-9602(15)60004-2
  53. Youssri et al. (2021) A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials 44(11) (pp. 9224-9236) https://doi.org/10.1002/mma.7347
  54. Habenom, H., Suthar, D.L.: Numerical solution for the time-fractional Fokker–Planck equation via shifted Chebyshev polynomials of the fourth kind. Adv. Difference Equ., 315, (2020)
  55. Doha et al. (2015) On using third and fourth kinds Chebyshev operational matrices for solving Lane-Emden type equations 60(3–4) (pp. 281-292)
  56. Masjed-Jamei,M.: Some New Classes of Orthogonal Polynomials and Special Functions: A Symmetric Generalization of Sturm-Liouville Problems and its Consequences. Ph.D thesis, University of Kassel, Department of Mathematics, Kassel, Germany, (2006)
  57. Abd-Elhameed and Youssri (2018) Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations (pp. 2897-2921) https://doi.org/10.1007/s40314-017-0488-z
  58. Xu (2002) An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials 29(2) (pp. 328-343) https://doi.org/10.1016/S0196-8858(02)00017-9
  59. Jafari et al. (2019) A novel approach for solving an inverse reaction-diffusion-convection problem 183(2) (pp. 688-704) https://doi.org/10.1007/s10957-019-01576-x
  60. Abd-Elhameed (2021) Novel expressions for the derivatives of sixth-kind Chebyshev polynomials: Spectral solution of the non-linear one-dimensional Burgers’ equation https://doi.org/10.3390/fractalfract5020053
  61. Koepf, W.:Hypergeometric summation. . Second Edition, Springer Universitext Series, 2014,
  62. http://www.hypergeometric-summation.org
  63. , (2014)
  64. Stewart, G.W.: Matrix Algorithms: Volume II: Eigensystems. SIAM, (2001)
  65. Stewart, J.:Single Variable Essential Calculus: Early Transcendentals. Cengage Learning, (2012)
  66. El-Gamel and El-Shenawy (2014) The solution of a time-dependent problem by the B-spline method (pp. 254-265) https://doi.org/10.1016/j.cam.2014.02.004
  67. Saadatmandi and Dehghan (2010) Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method 26(1) (pp. 239-252) https://doi.org/10.1002/num.20442