Numerical solution of time-fractional telegraph equation by using a new class of orthogonal polynomials
Resumo
In this article, an efficient numerical method based on a new class of orthogonal polynomials, namely Chelyshkov polynomials, has been presented to approximate solution of time-fractional telegraph (TFT) equations. The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations. The error analysis of the proposed collocation method is also investigated. A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations. Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method.
Downloads
Referências
N. Mollahasani, M. M. Moghadam, K. Afrooz, A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling, 40(4) 2016, 2804-2814. https://doi.org/10.1016/j.apm.2015.08.020
E. Orsingher, L. Beghin, Time-fractional telegraph equations and telegraph processes with Brownian time. Probability Theory and Related Fields, 128 (1) (2004) 141-160. https://doi.org/10.1007/s00440-003-0309-8
A. Jeffrey, Advanced engineering mathematics, Harcourt Academic Press, 2002.
M. Lakestani, B. N. Saray, Numerical solution of telegraph equation using interpolating scaling functions, Comput. Math. Appl., 60 (2010), pp. 1964-1972. https://doi.org/10.1016/j.camwa.2010.07.030
R. C. Cascaval, E. C. Eckstein, C. L. Frota, J. A. Goldstein, Fractional telegraph equations, Math. Anal. Appl., 276 (2002), pp. 145-159. https://doi.org/10.1016/S0022-247X(02)00394-3
M. Dehghan, A Shokri , A numerical method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ. 24 (2008) 1080-1093. https://doi.org/10.1002/num.20306
A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differ. Equ. 26 (2010) 239-252. https://doi.org/10.1002/num.20442
S. A. Yousefi, Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ. 26 (2010) 535-543. https://doi.org/10.1002/num.20445
S. T. Mohyud-Din, A. Yıldırım , Y. Kaplan, Homotopy perturbation method for one-dimensional hyperbolic equation with integral conditions, J. Phys. Sci. 65 (2010) 1077-1080. https://doi.org/10.1515/zna-2010-1210
M. Javidi, Chebyshev spectral collocation method for computing numerical solution of telegraph equation, Comput. Methods Differ. Equ. 1 (2013) 16-29.
B. Pekmen, M. Tezer-Sezgin , Differential quadrature solution of hyperbolic telegraph equation, J. Appl. Math. (2012) 18. https://doi.org/10.1155/2012/924765
S. Sharifi, J. Rashidinia, Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method. Applied Mathematics and Computation, 281 (2016), 28-38. https://doi.org/10.1016/j.amc.2016.01.049
N. Berwal , D. Panchal , C. L. Parihar, Haar waveleet method for numerical solution of telegraph equations, Ital. J. Pure Appl. Math. 30 (2013) 317-328.
S. Yuzbasi, Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction, Applied Mathematics and Computation, 287(2016), 83-93. https://doi.org/10.1016/j.amc.2016.04.036
M. Dehghan, A. Ghesmati, Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Eng. Anal. Bound. Elements 34 (2010) 51-59. https://doi.org/10.1016/j.enganabound.2009.07.002
I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198. Academic press, 1998.
S. G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Langhorne, 1993.
C. Cattani, H. M. Srivastava, X. J. Yang, Fractional Dynamics. de Gruyter, Berlin, 2016. https://doi.org/10.1515/9783110472097
E. Shivanian, Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation. Math. Methods Appl. Sci. (2015). https://doi.org/10.1002/mma.3604
A. H. Bhrawy, M. A. Zaky, J. A. T. Machado, Numerical solution of the two-sided space and time fractional telegraph equation via Chebyshev tau approximation. J. Optim. Theory Appl. (2016). https://doi.org/10.1007/s10957-016-0863-8
A. Saadatmandi, M. Mohabbati, Numerical solution of fractional telegraph equation via the tau method. Math. Rep., 17 (2015), 155-166.
M. Suleman, T. M. Elzaki, J. U. Rahman, Q. Wu, A Novel Technique to Solve Space and Time Fractional Telegraph Equation. Journal of Computational and Theoretical Nanoscience, 13(3) 2016, 1536-1545. https://doi.org/10.1166/jctn.2016.5078
J. Chen, F. Liu, and V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, Journal of Mathematical Analysis and Applications, 338 (2) 1364-1377, 2008. https://doi.org/10.1016/j.jmaa.2007.06.023
N. H. Sweilam, A. M. Nagy, A. A. El-Sayed, Solving Time-Fractional Order Telegraph Equation Via Sinc-Legendre Collocation Method. Mediterranean Journal of Mathematics, 13(6) (2016), 5119-5133. https://doi.org/10.1007/s00009-016-0796-3
D. Kumar, J. Singh, S. Kumar, Analytic and Approximate Solutions of Space-Time Fractional Telegraph Equations via Laplace Transform. Walailak Journal of Science and Technology (WJST), 11(8) 2013, 711-728.
S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Applied Mathematical Modelling, 38(13) 2014 3154-3163. https://doi.org/10.1016/j.apm.2013.11.035
M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi, Two-dimensional Legendre wavelets for solving time-fractional telegraph equation. Advances in Applied Mathematics and Mechanics, 6(02) (2014), 247-260. https://doi.org/10.4208/aamm.12-m12132
S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Applied Mathematics and Computation, 170 (2) 1126-1134, 2005. https://doi.org/10.1016/j.amc.2005.01.009
A. Yildirim, He's homotopy perturbation method for solving the space- and time-fractional telegraph equations, International Journal of Computer Mathematics, 87(13) 2010, 2998-3006. https://doi.org/10.1080/00207160902874653
A. Prakash, Analytical method for space-fractional telegraph equation by homotopy perturbation transform method, Nonlinear Engineering, 5(2) 2016, 123-128. https://doi.org/10.1515/nleng-2016-0008
A. Sevimlican, An approximation to solution of space and time fractional telegraph equations by He's variational iteration method, Mathematical Problems in Engineering 2010 (2010). https://doi.org/10.1155/2010/290631
A. H. Bhrawy, M. A. Zaky and J. A. T. Machado, Numerical solution of the two-sided space-time fractional telegraph equation via Chebyshev tau approximation, Journal of Optimization Theory and Applications 174(1) (2017), 321-341. https://doi.org/10.1007/s10957-016-0863-8
S. Yuzbasi, A shifted Legendre method for solving a population model and delay linear Volterra integro-differential equations. International Journal of Biomathematics, 10(7) (2017), p.1750091. https://doi.org/10.1142/S1793524517500917
N. Razmjooy and M. Ramezani, Analytical solution for optimal control by the second kind Chebyshev polynomials expansion, Iranian Journal of Science and Technology, Transactions A: Science 41(4) (2017), 1017-1026. https://doi.org/10.1007/s40995-017-0336-4
H. Singh, K. Rajesh Pandey D. Baleanu, Stable numerical approach for fractional delay differential equations, Few-Body Systems 58(6) (2017), 156. https://doi.org/10.1007/s00601-017-1319-x
P. Muthukumar and B.Ganesh Priya, Numerical solution of fractional delay differential equation by shifted Jacobi polynomials. International Journal of Computer Mathematics, 94(3) (2017), 471-492. https://doi.org/10.1080/00207160.2015.1114610
V. S. Chelyshkov, Alternative orthogonal polynomials and quadratures, Electron. Trans. Numer. Anal., 25 (7) (2006), 17-26.
E. Gokmen, G. Yuksel, M. Sezer, A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays. Journal of Computational and Applied Mathematics, 311 (2017), 354-363. https://doi.org/10.1016/j.cam.2016.08.004
E. Suli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University press, 2003. https://doi.org/10.1017/CBO9780511801181
A. Sadeghian, M. H. Heydari, M. R. Hooshmandasl, S. M. Karbassi, Numerical Solution of Fractional Telegraph Equation Using the Second Kind Chebyshev Wavelets Method, J. Appl. Environ. Biol. Sci., 5(9S) 2015, 64-74.
Copyright (c) 2021 Boletim da Sociedade Paranaense de Matemática

This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).