Numerical solution of time-fractional telegraph equation by using a new class of orthogonal polynomials
DOI:
https://doi.org/10.5269/bspm.44010Abstract
‎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.
References
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29. 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
30. 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
31. 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
32. 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
33. 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
34. 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
35. 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
36. 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
37. V. S. Chelyshkov, Alternative orthogonal polynomials and quadratures, Electron. Trans. Numer. Anal., 25 (7) (2006), 17-26.
38. 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
39. E. Suli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University press, 2003. https://doi.org/10.1017/CBO9780511801181
40. 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.
2. 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
3. A. Jeffrey, Advanced engineering mathematics, Harcourt Academic Press, 2002.
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. M. Javidi, Chebyshev spectral collocation method for computing numerical solution of telegraph equation, Comput. Methods Differ. Equ. 1 (2013) 16-29.
11. B. Pekmen, M. Tezer-Sezgin , Differential quadrature solution of hyperbolic telegraph equation, J. Appl. Math. (2012) 18. https://doi.org/10.1155/2012/924765
12. 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
13. 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.
14. 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
15. 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
16. 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.
17. S. G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Langhorne, 1993.
18. C. Cattani, H. M. Srivastava, X. J. Yang, Fractional Dynamics. de Gruyter, Berlin, 2016. https://doi.org/10.1515/9783110472097
19. 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
20. 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
21. A. Saadatmandi, M. Mohabbati, Numerical solution of fractional telegraph equation via the tau method. Math. Rep., 17 (2015), 155-166.
22. 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
23. 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
24. 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
25. 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.
26. 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
27. 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
28. 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
29. 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
30. 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
31. 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
32. 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
33. 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
34. 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
35. 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
36. 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
37. V. S. Chelyshkov, Alternative orthogonal polynomials and quadratures, Electron. Trans. Numer. Anal., 25 (7) (2006), 17-26.
38. 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
39. E. Suli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University press, 2003. https://doi.org/10.1017/CBO9780511801181
40. 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.
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