A shifted Chebyshev operational matrix method for pantograph-type nonlinear fractional differential equations. (English) Zbl 1539.34008
Summary: In this study, we investigate and analyze an approximation of the Chebyshev polynomials for pantograph-type fractional-order differential equations. First, we construct the operational matrices of pantograph and Caputo fractional derivatives using Chebyshev interpolation. Then, the obtained matrices are utilized to approximate the fractional derivative. We also provide a detailed convergence analysis in terms of the weighted square norm. Finally, we describe and discuss the results of three numerical experiments conducted to confirm the applicability and accuracy of the computational scheme.
© 2023 John Wiley & Sons Ltd.
© 2023 John Wiley & Sons Ltd.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
Caputo derivative; Chebyshev polynomial; collocation method; fractional pantograph differential equation; operational matrixReferences:
| [1] | H.Sun, Y.Zhang, D.Baleanu, W.Chen, and Y.Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul.64 (2018), 213-231. · Zbl 1509.26005 |
| [2] | A.Ahmadi, B. M.Vinagre, Y.Chen, and S. H.HosseinNia, Linear fractional order controllers: a survey in the frequency domain, Ann. Rev. Control47 (2019), 51-70. |
| [3] | A. N.Chatterjee and B.Ahmad, A fractional‐order differential equation model of COVID‐19 infection of epithelial cells, Chaos, Solitons Fract.2021 (2021), 110952. · Zbl 1486.92211 |
| [4] | G.Ling, Z.Fanhai, T.Ian, B.Kevin, and G. E.Karniadakis, Efficient multistep methods for tempered fractional calculus: algorithms and simulations, SIAM J. Sci. Comput.41 (2019), no. 4, 2510-2535. · Zbl 07099350 |
| [5] | Y.Zhou, C.Zhang, and H.Wang, Boundary value methods for Caputo fractional differential equations, J. Comput. Math.39 (2020), no. 1, 108-129. · Zbl 1474.65237 |
| [6] | R.Douaifia, S.Bendoukha, and S.Abdelmalek, A newton interpolation based predictor-corrector numerical method for fractional differential equations with an activator-inhibitor case study, Math. Comput. Simul.187 (2021), 391-413. · Zbl 1540.65208 |
| [7] | M. S.Hashemi, A.Atangana, and S.Hajikhah, Solving fractional pantograph delay equations by an effective computational method, Math. Comput. Simul.177 (2020), 295-305. · Zbl 1510.65130 |
| [8] | C.Wang, Z.Wang, and L.Wang, A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative, J. Sci. Comput.76 (2018), no. 1, 166-188. · Zbl 1402.65172 |
| [9] | S. A.Rakhshan and S.Effati, A generalized Legendre-Gauss collocation method for solving nonlinear fractional differential equations with time varying delays, Appl. Numer. Math.146 (2019), 342-360. · Zbl 1448.34147 |
| [10] | P. T.Toan, T. N.Vo, and M.Razzaghi, Taylor wavelet method for fractional delay differential equations, Eng. Comput.37 (2021), 231-240. |
| [11] | A.Rayal and S.Ram Verma, Numerical analysis of pantograph differential equation of the stretched type associated with fractal‐fractional derivatives via fractional order Legendre wavelets, Chaos, Solitons Fract.139 (2020), 110076. · Zbl 1490.65129 |
| [12] | R.Amin, K.Shah, M.Asif, and I.Khan, A computational algorithm for the numerical solution of fractional order delay differential equations, Appl. Math. Comput.402 (2021), 125863. · Zbl 1510.65126 |
| [13] | K.Rabiei and Y.Ordokhani, Solving fractional pantograph delay differential equations via fractional‐order Boubaker polynomials, Eng. Comput.35 (2019), 1431-1441. |
| [14] | S.Sabermahani, Y.Ordokhani, and S.‐A.Yousefi, Fractional‐order Fibonacci‐hybrid functions approach for solving fractional delay differential equations, Eng. Comput.36 (2020), 795-806. |
| [15] | S.Nemati, P.Lima, and S.Sedaghat, An effective numerical method for solving fractional pantograph differential equations using modification of hat functions, Appl. Numer. Math.131 (2018), 174-189. · Zbl 1446.65042 |
| [16] | M. M.Alsuyuti, E. H.Doha, S. S.Ezz‐Eldien, and I. K.Youssef, Spectral Galerkin schemes for a class of multi‐order fractional pantograph equations, J. Comput. Appl. Math.384 (2021), 113157. · Zbl 1456.65129 |
| [17] | C.Yang and X.Lv, Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation, Math. Methods Appl. Sci.44 (2021), no. 1, 153-165. · Zbl 1512.65168 |
| [18] | S. S.Ezz‐Eldien, Y.Wang, M. A.Abdelkawy, M. A.Zaky, A. A.Aldraiweesh, and J. T.Machado, Chebyshev spectral methods for multi‐order fractional neutral pantograph equations, Nonlinear Dyn.100 (2020), no. 4, 3785-3797. · Zbl 1516.34016 |
| [19] | Z.Avazzadeh, M. H.Heydari, and M. R.Mahmoudi, An approximate approach for the generalized variable‐order fractional pantograph equation, Alex. Eng. J.59 (2020), no. 4, 2347-2354. |
| [20] | C.Yang and J.Hou, Jacobi spectral approximation for boundary value problems of nonlinear fractional pantograph differential equations, Numer. Algo.86 (2021), no. 3, 1089-1108. · Zbl 1458.65072 |
| [21] | C.Yang, J.Hou, and X.Lv, Jacobi spectral collocation method for solving fractional pantograph delay differential equations, Eng. Comput.38 (2022), no. 3, 1985-1994. |
| [22] | M. A.Zaky and I. G.Ameen, A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra‐Fredholm integral equations with smooth solutions, Numer. Algo.84 (2020), no. 1, 63-89. · Zbl 1453.65198 |
| [23] | M. A.Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non‐smooth solutions, J. Comput. Appl. Math.357 (2019), 103-122. https://www.sciencedirect.com/science/article/pii/S0377042719300834 · Zbl 1415.41001 |
| [24] | I. G.Ameen, M. A.Zaky, and E. H.Doha, Singularity preserving spectral collocation method for nonlinear systems of fractional differential equations with the right‐sided Caputo fractional derivative, J. Comput. Appl. Math.392 (2021), 113468. · Zbl 1467.65075 |
| [25] | M. A.Zaky and I. G.Ameen, A novel Jacobi spectral method for multi‐dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions, Eng. Comput.37 (2021), no. 4, 2623-2631. |
| [26] | M. A.Zaky, An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions, Appl. Numer. Math.154 (2020), 205-222. · Zbl 1442.65464 |
| [27] | M. A.Zaky, Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems, Appl. Numer. Math.145 (2019), 429-457. https://www.sciencedirect.com/science/article/pii/S0168927419301151 · Zbl 1427.34021 |
| [28] | W. G.Aiello, H. I.Freedman, and J.Wu, Analysis of a model representing stage‐structured population growth with state‐dependent time delay, SIAM J. Appl. Math.52 (1992), no. 3, 855-869. · Zbl 0760.92018 |
| [29] | H. T.Tuan and H.Trinh, A linearized stability theorem for nonlinear delay fractional differential equations, IEEE Trans. Autom. Control63 (2018), no. 9, 3180-3186. · Zbl 1423.34092 |
| [30] | O. P.Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn.38 (2004), no. 1, 323-337. · Zbl 1121.70019 |
| [31] | B.‐Y.Guo and Z.‐Q.Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math.30 (2009), no. 3, 249-280. · Zbl 1162.65375 |
| [32] | A.Dabiri and E. A.Butcher, Numerical solution of multi‐order fractional differential equations with multiple delays via spectral collocation methods, Appl. Math. Model.56 (2018), 424-448. · Zbl 1480.65158 |
| [33] | A. A.Kilbas, H. M.Srivastava, and J. J.Trujillo, Theory and applications of fractional differential equations, North‐Holland Mathematics Studies, Vol. 204, Elsevier Science Ltd, 2006. · Zbl 1092.45003 |
| [34] | T. J.Rivlin, An introduction to the approximation of functions, 1981 ed., Dover Publications Inc., New York, 2003. · Zbl 0489.41001 |
| [35] | M. I.Syam, M.Sharadga, and I.Hashim, A numerical method for solving fractional delay differential equations based on the operational matrix method, Chaos, Solitons Fract.147 (2021), 110977. · Zbl 1486.76005 |
| [36] | M. S.Hafshejani, S. K.Vanani, and J. S.Hafshejani, Numerical solution of delay differential equations using Legendre wavelet method, World Appl. Sci. J.13 (2011), 27-33. |
| [37] | U.Saeed, M. U.Rehman, and M. A.Iqbal, Modified Chebyshev wavelet methods for fractional delay‐type equations, Appl. Math. Comput.264 (2015), no. 1, 431-442. · Zbl 1410.65286 |
| [38] | C.Canuto, M. Y.Hussaini, A.Quarteroni, and T. A.Zang, Spectral methods: fundamentals in single domains, Springer, 2006. · Zbl 1093.76002 |
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