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Review Article

A numerical approach to solve hyperbolic telegraph equations via Pell–Lucas polynomials

ORCID Icon & ORCID Icon
Article: 2255404 | Received 20 Mar 2023, Accepted 27 Aug 2023, Published online: 20 Sep 2023

References

  • Sharifi S, Rashidinia J. Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method. Appl Math Comput. 2016;281:28–38.
  • Nazir T, Abbas M, Yaseen M. Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric Bsplines approach. Cogent Math. 2017;4:Article ID 1382061. doi:10.1080/23311835.2017.1382061
  • Jebreen HB, Cano YC, Dassios I. An efficient algorithm based on the multi-wavelet Galerkin method for telegraph equation. AIMS Math. 2020;6:1296–1308. doi:10.3934/math.2021080
  • Bicer KE, Yalcinbas S. Numerical solution of telegraph equation using Bernoulli collocation method. Proc Natl Acad Sci India Sect A Phys Sci. 2019;89:769–775. doi:10.1007/s40010-018-0535-1
  • Zarebnia M, Parvaz R. A new approach for solution of telegraph equation. Int J Nonlinear Anal Appl. 2021;12:385–396.
  • Zarebnia M, Parvaz R. An approximation to the solution of one-dimensional hyperbolic telegraph equation based on the collocation of quadratic B-spline functions. Comput Methods Differ Equ. 2021;9:1198–1213.
  • Adebayo OA, Saheed OA, Adebayo AA, et al. Laplace transform collocation method for solving hyperbolic telegraph equation. Int J Eng Math. 2017;2017:1–9.
  • Dugassa M, File G, Aga T. Numerical solution of second order one dimensional linear hyperbolic telegraph equation. Ethiop J Educ Sci. 2018;14:39–52.
  • Nagaveni K. Haar wavelet collocation method for solving the telegraph equation with variable coefficients. Int J Appl Eng Res. 2020;15:235–243.
  • Bahsi AK, Yalcinbas S. A new algorithm for the numerical solution of telegraph equations by using Fibonacci polynomials. Math Comput Appl. 2016;21:15.
  • Wang H, He Q, Luo Z. A reduced order extrapolating technique of solution coefficient vectors to collocation spectral method for telegraph equation. Adv Differ Equ. 2020;2020:1–16. doi:10.1186/s13662-019-2438-0
  • Dehghan M, Shokri A. A numerical method for solving the hyperbolic telegraph equation. Numer Methods Partial Differ Equ. 2008;24:1080–1093. doi:10.1002/(ISSN)1098-2426
  • Yousefi SA. Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation. Numer Methods Partial Differ Equ. 2010;26:535–543.
  • Saadatmandi A, Dehghan M. Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method. Numer Numer Methods Partial Differ Equ. 2010;26:239–252. doi:10.1002/num.v26:1
  • 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 Bound Elements. 2010;34:51–59. doi:10.1016/j.enganabound.2009.07.002
  • Mohyud-Din ST, Yildirim A, Kaplan Y. Homotopy perturbation method for one-dimensional hyperbolic equation with integral conditions. J Phys Sci. 2010;65:1077–1080.
  • Salkuyeh DK, Ghehsareh HR. Convergence of the variation literation method for the telegraph equation with integral conditions. Numer Methods Partial Differ Equ. 2011;27:1442–1455. doi:10.1002/num.20590
  • Raftari B, Yildirim A. Analytical solution of second-order hyperbolic telegraph equation by variational iteration and homotopy perturbation methods. Results Math. 2012;61:13–28. doi:10.1007/s00025-010-0072-y
  • Pekmen B, Tezer-Sezgin M. Differential quadrature solution of hyperbolic telegraph equation. J Appl Math. 2012;2012:18.
  • Mohebbi A. A fourth-order finite difference scheme for the numerical solution of 1D linear hyperbolic equation. Commun Numer Anal. 2013;2013:1–11.
  • Javidi M. Chebyshev spectral collocation method for computing numerical solution of telegraph equation. Comput Methods Differ Equ. 2013;1:16–29.
  • Berwal N, Panchal D, Parihar CL. Haar wavelet method for numerical solution of telegraph equations. Ital J Pure Appl Math. 2013;30:317–328.
  • Sari M, Gunay A, Gurarslan G. A solution to the telegraph equation by using DGJ method. Int J Nonlinear Sci. 2014;17:57–66.
  • Yüzbaşı Ş. A collocation approach for solving two-dimensional second-order linear hyperbolic equations. Appl Math Comput. 2018;338:101–114.
  • Yüzbaşı Ş. A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions. Int J Comput Methods. 2017;14:Article ID 1750015. doi:10.1142/S0219876217500153
  • Yüzbaşı Ş, Karaçayır M. A Galerkin-Type method to solve one-dimensional telegraph equation using collocation points in initial and boundary conditions. Int J Comput Methods. 2018;15:Article ID 1850031. doi:10.1142/S0219876218500317
  • Yüzbaşı Ş. Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction. Appl Math Comput. 2016;287:83–93.
  • Yüzbaşı Ş, Şahin N. Numerical solutions of singularly perturbed one-dimensional parabolic convection-diffusion problems by the Bessel collocation method. Appl Math Comput. 2013;220:305–315.
  • Yüzbaşı Ş, Yıldırım G. A collocation method to solve the parabolic-type partial integro-differential equations via Pell–Lucas polynomials. Appl Math Comput. 2022;421:Article ID 126956.
  • Bulbul B, Sezer M. Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients. Int J Comput Math. 2011;88:533–544. doi:10.1080/00207161003611242
  • Isik OR, Sezer M, Guney Z. Bernstein series solution of linear second-order partial differential equations with mixed conditions. Math Meth Appl Sci. 2014;37:609–619. doi:10.1002/mma.2817
  • Yüksel G, Isik OR, Sezer M. Error analysis of the Chebyshev collocation method for linear second-order partial differential equations. Int J Comput Math. 2015;92:2121–2138. doi:10.1080/00207160.2014.966099
  • Yüksel G, Sezer M. A Chebyshev series approximation for linear second-order partial differential equations with complicated conditions. Gazi Univ J Sci. 2013;26:515–525.
  • Ahmed HF, Hashem WA. A novel spectral technique for 2D fractional telegraph equation models with spatial variable coefficients. J Taibah Univ Sci. 2022;16(1):885–894. doi:10.1080/16583655.2022.2123089
  • Akram T, Abbas M, Iqbal A, et al. Novel numerical approach based on modified extended cubic B-spline functions for solving non-linear time-fractional telegraph equation. Symmetry. 2020;12(7):1154. doi:10.3390/sym12071154
  • Anac H, Merdan M, Kesemen T. Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method. SN Appl Sci. 2020;2:1112. doi:10.1007/s42452-020-2625-3
  • Bahia G, Ouannas A, Batiha IM, et al. The optimal homotopy analysis method applied on nonlinear time-fractional hyperbolic partial differential equations. Numer Methods Partial Differ Equ. 2021;37:2008–2022. doi:10.1002/num.v37.3
  • Bhowmik SK, Karakoc SB. Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method. Numer Methods Partial Differ Equ. 2019;35:2236–2257. doi:10.1002/num.v35.6
  • Chun C, Abbasbandy S. New application of variational iteration method for analytic treatment of nonlinear partial differential equations. World Appl Sci J. 2012;16(12):1677–1681.
  • Dehghan M, Ghesmati A. Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation. Eng Anal Bound Elem. 2010;34(4):324–336. doi:10.1016/j.enganabound.2009.10.010
  • Dehghan M, Lakestani M. The use of Chebyshev cardinal functions for solution of the second-order one-dimensional telegraph equation. Numer Methods Partial Differ Equ. 2009;25(4):931–938. doi:10.1002/num.v25:4
  • Dehghan M, Mohebbi A. High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation. Numer Methods Partial Differ Equ Int J. 2009;25(1):232–243. doi:10.1002/num.v25:1
  • Dehghan M, Salehi R. A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraph equation. Math Methods Appl Sci. 2012;35(10):1220–1233. doi:10.1002/mma.v35.10
  • Dehghan M, Yousefi SA, Lotfi A. The use of He's variational iteration method for solving the telegraph and fractional telegraph equations. Int J Numer Methods Biomed Eng. 2011;27(2):219–231. doi:10.1002/cnm.v27.2
  • Dwivedi KD, Rajeev A, Das S, et al. Finite difference/collocation method to solve multi term variable-order fractional reaction-advection-diffusion equation in heterogeneous medium. Numer Methods Partial Differ Equ. 2021;37:2031–2045. doi:10.1002/num.v37.3
  • Entezari M, Abbasbandy S, Babolian E. Numerical solution of fractional partial differential equations with normalized Bernstein wavelet method. Appl Appl Math. 2019;14:890–909.
  • Eslahchi MR, Esmaili S, Namaki N, et al. Application of finite difference method in solving a second-and fourth-order PDE blending denoising model. Math Sci. 2023;17:93–106.
  • Gazi Karakoc SB, Neilan M. A C0 finite element method for the biharmonic problem without extrinsic penalization. Numer Methods Partial Differ Equ. 2014;30:1254–1278. doi:10.1002/num.v30.4
  • Gul T, Khan MA, Khan A, et al. Fractional-order three-dimensional thin-film nanofluid flow on an inclined rotating disk. Eur Phys J Plus. 2018;133(12):500. doi:10.1140/epjp/i2018-12315-4
  • Gumgum S, Kurul E, Savasaneril NB. Chebyshev collocation method for the two-dimensional heat equation. CMMA. 2018;3:1–8. doi:10.14426/mm.v3i2
  • Hajiketabi M, Abbasbandy S. The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: application to the heat equation. Eng Anal Bound Elem. 2018;87:36–46. doi:10.1016/j.enganabound.2017.11.008
  • Hajishafieiha J, Abbasbandy S. A new method based on polynomials equipped with a parameter to solve two parabolic inverse problems with a nonlocal boundary condition. Inverse Probl Sci Eng. 2020;28(5):739–753. doi:10.1080/17415977.2019.1696326
  • Hamid M, Usman M, Haq RU, et al. Linearized stable spectral method to analyze two-dimensional nonlinear evolutionary and reaction-diffusion models. Numer Methods Partial Differ Equ. 2022;38:243–261. doi:10.1002/num.v38.2
  • 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. 2015;193:55–65. doi:10.1016/j.cpc.2015.03.021
  • Jiwari R. Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine-Gordon equation. Numer Methods Partial Differ Equ. 2021;37(3):1965–1992. doi:10.1002/num.v37.3
  • Jiwari R, Pandit S, Mittal R. A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. Appl Math Comput. 2012;218(13):7279–7294.
  • Koç DA, Öztörk Y, Gülsu M. A numerical algorithm for solving one-dimensional parabolic convection–diffusion equation. J Taibah Univ Sci. 2023;17(1):Article ID 2204808. doi:10.1080/16583655.2023.2204808
  • Kumar D, Singh J, Kumar S. Analytic and approximate solutions of space–time fractional telegraph equations via Laplace transform. Walailak J Sci Technol. 2014;11(8):711–728.
  • Kumar D, Singh J, Mehmet HB, et al. An effective computational approach to local fractional telegraph equations. Nonlinear Sci Lett A. 2017;8(2):200–206.
  • Kumbinarasaiah S, Mulimani M. A novel scheme for the hyperbolic partial differential equation through Fibonacci wavelets. J Taibah Univ Sci. 2022;16(1):1112–1132. doi:10.1080/16583655.2022.2143636
  • Kurulay M, Bayram M. A novel power series method for solving second order partial differential equations. Eur J Pure Appl Math. 2009;2:268–277.
  • Li YX, Muhammad T, Bilal M, et al. Fractional simulation for Darcy–Forchheimer hybrid nanoliquid flow with partial slip over a spinning disk. Alex Eng J. 2021;60(5):4787–4796. doi:10.1016/j.aej.2021.03.062
  • Merdan M, Anac H, Bekir Yazici Z, et al. Solving of some random partial differential equations by using differential transformation method and Laplace–Padé method. GÜFBED/GUSTIJ. 2019;9:108–118.
  • Mohyud-Din ST, Aslam Noor M. Homotopy perturbation method for solving partial differential equations. Z Naturforsch. 2009;64:157–170. doi: 10.1515/zna-2009-3-402
  • Mohyud-Din ST, Yildirim A, Hosseini MM, et al. A study on systems of variable-coefficient singular parabolic partial differential equations. World Appl Sci J. 2010;10:1321–1327.
  • Ozdemir N, Secer A, Bayram M. An algorithm for numerical solution of some nonlinear multi-dimensional parabolic partial differential equations. J Comput Sci. 2021;56:Article ID 101487. doi:10.1016/j.jocs.2021.101487
  • Pandit S, Kumar M, Tiwari S. Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients. Comput Phys Commun. 2015;187:83–90. doi:10.1016/j.cpc.2014.10.013
  • Rohila R, Mittal RC. A numerical study of two-dimensional coupled systems and higher order partial differential equations. Asian-Eur J Math. 2018;12:1950071 (12 pages).
  • Saeed U, Umair M. A modified method for solving non-linear time and space fractional partial differential equations. Eng Comput. 2019;36:2162–2178. doi:10.1108/EC-01-2019-0011
  • Soradi-Zeid S, Mesrizadeh M, Cattani C. Radial basis solutions of second-order quasi-linear hyperbolic boundary value problem. Numer Methods Partial Differ Equ. 2021;37:1916–1927. doi:10.1002/num.v37.3
  • Taherkhani S, Khalilsaraye IN, Ghayebi B. Numerical solution of the diffusion problem of distributed order based on the Sinc-collocation method. Math Sci. 2023;17:133–144.
  • Usman M, Hamid M, Khalid MSU, et al. A robust scheme based on novel–operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics. Numer Methods Partial Differ Equ. 2020;36:1566–1600. doi: 10.1002/num.v36.6
  • Usman M, Hamid M, Liu M. Higher-order algorithms for stable solutions of fractional time-dependent nonlinear telegraph equations in space. Numer Methods Partial Differ Equ. 2022;38:1293–1318. doi:10.1002/num.v38.5
  • Yadav OP, Jiwari R. A finite element approach for analysis and computational modelling of coupled reaction diffusion models. Numer Methods Partial Differ Equ. 2019;35(2):830–850. doi:10.1002/num.v35.2
  • Zamani-Gharaghoshi H, Dehghan M, Abbaszadeh M. A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction-diffusion systems on surfaces. Eng Comput. 2023;1–10.
  • El-Azab MS, El-Gamel M. A numerical algorithm for the solution of telegraph equations. Appl Math Comput. 2007;190:757–764.
  • Horadam AF, Mahon Bro JM. Pell and Pell–Lucas polynomials. Fibonacci Q. 1985;23:7–20.
  • Horadam AF, Swita B, Filipponi P. Integration and derivative sequences for Pell and Pell–Lucas polynomials. Fibonacci Q. 1994;32:130–35.
  • Şahin M, Sezer M. Pell–Lucas collocation method for solving high-order functional differential equations with hybrid delays. Celal Bayar Univ J Sci. 2018;14:141–149.
  • Yüzbaşı Ş, Yıldırım G. Pell–Lucas collocation method for numerical solutions of two population models and residual correction. J Taibah Univ Sci. 2020;14:1262–1278. doi:10.1080/16583655.2020.1816027