Spectral collocation methods for fractional multipantograph delay differential equations. (English) Zbl 1532.65092
Summary: In this paper, we propose and analyze a spectral collocation method for the numerical solutions of fractional multipantograph delay differential equations. The fractional derivatives are described in the Caputo sense. We present that some suitable variable transformations can convert the equations to a Volterra integral equation defined on the standard interval \([-1, 1]\). Then the Jacobi-Gauss points are used as collocation nodes, and the Jacobi-Gauss quadrature formula is used to approximate the integral equation. Later, the convergence analysis of the proposed method is investigated in the infinity norm and weighted \(L^2\) norm. To perform the numerical simulations, some test examples are investigated, and numerical results are presented. Further, we provide the comparative study of the proposed method with some existing numerical methods.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65D32 | Numerical quadrature and cubature formulas |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65T60 | Numerical methods for wavelets |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 45D05 | Volterra integral equations |
| 35R07 | PDEs on time scales |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
fractional delay differential equations; Caputo derivative; spectral collocation method; convergence analysisReferences:
| [1] | Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations (2004), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 1059.65122 · doi:10.1017/CBO9780511543234 |
| [2] | Brunner, H.; Huang, Q.; Xie, H., Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. Numer. Anal., 48, 5, 1944-1967 (2010) · Zbl 1219.65076 · doi:10.1137/090771922 |
| [3] | Canuto, C.; Hussaini, MY; Quarteroni, A.; Zang, TA, Spectral Methods: Fundamentals in Single Domains (2006), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 1093.76002 · doi:10.1007/978-3-540-30726-6 |
| [4] | Chen, S.; Shen, J.; Wang, L-L, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput., 85, 300, 1603-1638 (2016) · Zbl 1335.65066 · doi:10.1090/mcom3035 |
| [5] | Chen, Y.; Tang, T., Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math., 233, 4, 938-950 (2009) · Zbl 1186.65161 · doi:10.1016/j.cam.2009.08.057 |
| [6] | Chen, Z.; Gou, Q., Piecewise Picard iteration method for solving nonlinear fractional differential equation with proportional delays, Appl. Math. Comput., 348, 465-478 (2019) · Zbl 1429.65134 |
| [7] | D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Appl.Math. Sci., Vol. 93, Springer, Berlin, Heidelberg, 1998. · Zbl 0893.35138 |
| [8] | Deng, G.; Yang, Y.; Tohidi, E., High accurate pseudo-spectral Galerkin scheme for pantograph type Volterra integro-differential equations with singular kernels, Appl. Math. Comput., 396 (2021) · Zbl 1508.65178 |
| [9] | Ezz-Eldien, SS; Wang, Y.; Abdelkawy, MA; Zaky, MA; Aldraiweesh, AA; Machado, JT, Chebyshev spectral methods for multi-order fractional neutral pantograph equations, Nonlinear Dyn., 100, 4, 3785-3797 (2020) · Zbl 1516.34016 · doi:10.1007/s11071-020-05728-x |
| [10] | Gokdogan, A.; Merdan, M.; Yildirim, A., Differential transformation method for solving a neutral functionaldifferential equation with proportional delays, Casp. J. Math. Sci., 1, 1, 31-37 (2012) · Zbl 1412.34219 |
| [11] | Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 0456.35001 · doi:10.1007/BFb0089647 |
| [12] | T. Tang J. Shen and L. Wang, Spectral Methods: Algorithms, Analyses and Applications, Springer, Berlin, Heidelberg, 2011. · Zbl 1227.65117 |
| [13] | Kufner, A.; Persson, L., Weighted Inequalities of Hardy Type (2003), New York: World Scientific, New York · Zbl 1065.26018 · doi:10.1142/5129 |
| [14] | Li, X.; Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47, 3, 2108-2131 (2009) · Zbl 1193.35243 · doi:10.1137/080718942 |
| [15] | Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [16] | Ma, X.; Huang, C., Spectral collocation method for linear fractional integro-differential equations, Appl. Math. Modelling, 38, 4, 1434-1448 (2014) · Zbl 1427.65421 · doi:10.1016/j.apm.2013.08.013 |
| [17] | Mastroianni, G.; Occorsio, D., Optimal systems of nodes for Lagrange interpolation on bounded intervals, J. Comput. Appl. Math., 134, 1-2, 325-341 (2001) · Zbl 0990.41003 · doi:10.1016/S0377-0427(00)00557-4 |
| [18] | Moghaddam, BP; Mostaghim, ZS, A numerical method based on finite difference for solving fractional delay differential equations, J. Taibah Univ. Sci., 7, 3, 120-127 (2013) · doi:10.1016/j.jtusci.2013.07.002 |
| [19] | Muthukumar, P.; Priya, BG, Numerical solution of fractional delay differential equation by shifted Jacobi polynomials, Int. J. Comput. Math., 94, 3, 471-492 (2017) · Zbl 1388.34070 · doi:10.1080/00207160.2015.1114610 |
| [20] | Nemati, S.; Lima, P.; Sedaghat, S., An effective numerical method for solving fractional pantograph differential equations using modification of hat functions, Appl. Numer. Math., 31, 174-189 (2018) · Zbl 1446.65042 · doi:10.1016/j.apnum.2018.05.005 |
| [21] | Neval, P., Mean convergence of Lagrange interpolation, Trans. Am. Math. Soc., 282, 2, 669-698 (1984) · Zbl 0577.41001 · doi:10.1090/S0002-9947-1984-0732113-4 |
| [22] | Ragozin, DL, Polynomial approximation on compact manifolds and homogeneous space, Trans. Am. Math. Soc., 150, 41-53 (1970) · Zbl 0208.14701 · doi:10.1090/S0002-9947-1970-0410210-0 |
| [23] | Ragozin, DL, Constructive polynomial approximation on spheres and projective spaces, Trans. Am. Math. Soc., 162, 157-170 (1971) · Zbl 0234.41011 |
| [24] | Rahimkhani, P.; Ordokhani, Y.; Babolian, E., Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math., 309, 493-510 (2017) · Zbl 1468.65089 · doi:10.1016/j.cam.2016.06.005 |
| [25] | Shen, J.; Tang, T., Pectral and High-Order Methods with Applications (2006), Beijing: Science Press, Beijing · Zbl 1234.65005 |
| [26] | Shi, X., Spectral collocation methods for fractional integro-differential equations with weakly singular kernels, J. Math., 2022, 3767559 (2022) · doi:10.1155/2022/3767559 |
| [27] | Shi, X.; Chen, Y., Spectral-collocation method for Volterra delay integro-differential equations with weakly singular kernels, Adv. Appl. Math. Mech., 8, 4, 648-669 (2016) · Zbl 1488.65758 · doi:10.4208/aamm.2015.m1088 |
| [28] | Shi, X.; Chen, Y.; Huang, Y.; Huang, F., Spectral collocation methods for second-order Volterra integro-differential equations with weakly singular kernels, Adv. Appl. Math. Mech., 12, 2, 480-502 (2020) · Zbl 1488.65759 · doi:10.4208/aamm.OA-2019-0056 |
| [29] | Shi, X.; Huang, F.; Hu, H., Convergence analysis of spectral methods for high-order nonlinear Volterra integrodifferential equations, Comput. Appl. Math., 38, 2, 56 (2019) · Zbl 1438.65343 · doi:10.1007/s40314-019-0827-3 |
| [30] | X. Shi and Y. W, Convergence analysis of the spectral collocation methods for two-dimensional nonlinear weakly singular Volterra integral equations, Numer. Methods Partial Differ. Equations, 58(1):57-94, 2018. · Zbl 1391.45005 |
| [31] | Shi, X.; Wei, Y.; Huang, F., Spectral collocationmethods for nonlinearweakly singular Volterra integro-differential equations, Numer. Methods Partial Differ. Equations, 35, 2, 576-596 (2019) · Zbl 1416.65236 · doi:10.1002/num.22314 |
| [32] | Tang, Z.; Tohidi, E.; He, F., Generalized mapped nodal Laguerre spectral collocation method for Volterra delay integro-differential equations with noncompact kernels, Comput. Appl. Math., 39, 4, 298 (2020) · Zbl 1476.34045 · doi:10.1007/s40314-020-01352-y |
| [33] | Tao, L.; Yong, H., Extrapolation method for solving weakly singular nonlinear Volterra integral equations of second kind, J. Math. Anal. Appl., 324, 1, 225-237 (2006) · Zbl 1115.65129 · doi:10.1016/j.jmaa.2005.12.013 |
| [34] | Tohidi, E.; Samadi, ORN; Shateyi, S., Convergence analysis of Legendre pseudospectral scheme for solving nonlinear systems of Volterra integral equations, J. Math. Anal. Appl., 2014 (2014) · Zbl 1303.65112 |
| [35] | Wang, C.; Wang, Z.; Wang, L., A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative, J. Sci. Comput., 76, 1, 166-188 (2018) · Zbl 1402.65172 · doi:10.1007/s10915-017-0616-3 |
| [36] | Wang, H.; Chen, Y.; Huang, Y.; Mao, W., A posteriori error estimates of the Galerkin spectral methods for spacetime fractional diffusion equations, Adv. Appl. Math. Mech., 12, 1, 87-100 (2020) · Zbl 1488.65521 · doi:10.4208/aamm.OA-2019-0137 |
| [37] | Yang, C.; Lv, X., Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation, Math. Meth. Appl. Sci., 44, 1, 153-165 (2021) · Zbl 1512.65168 · doi:10.1002/mma.6718 |
| [38] | Yang, Y., Jacobi spectral Galerkin methods for fractional integro-differential equations, Calcolo, 52, 4, 519-542 (2015) · Zbl 1331.65179 · doi:10.1007/s10092-014-0128-6 |
| [39] | Y. Yang, Y. Chen, and Y. Huang, A convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations, Acta. Math. Sci., Ser. B, Engl. Ed., 34(3):673-690, 2014. · Zbl 1313.65343 |
| [40] | Yang, Y.; Chen, YP; Huang, YQ; Wei, HY, Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis, Comput. Math. Appl., 73, 6, 1218-1232 (2017) · Zbl 1412.65168 · doi:10.1016/j.camwa.2016.08.017 |
| [41] | Y. Yang, G. Deng, and Emran Tohidi, High accurate convergent spectral Galerkin methods for nonlinear weakly singular Volterra integro-differential equations, Comput. Appl. Math., 40(4):118, 2021. · Zbl 1476.33008 |
| [42] | Yang, Y.; Huang, Y.; Zhou, Y., Numerical simulation of time fractional cable equations and convergence analysis, Numer. Methods Partial Differ. Equations, 34, 5, 1556-1576 (2018) · Zbl 1407.65233 · doi:10.1002/num.22225 |
| [43] | Y. Yang, S. Kang, and V.I. Vasil’ev, The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions, Electron. Res. Arch., 28(3):1161-1189, 2020. · Zbl 1448.65084 |
| [44] | Yang, Y.; Tohidi, E., Numerical solution of multi-pantograph delay boundary value problems via an efficient approach with the convergence analysis, Comput. Appl. Math., 38, 3, 127 (2019) · Zbl 1438.65258 · doi:10.1007/s40314-019-0896-3 |
| [45] | Yang, Y.; Tohidi, E.; Deng, G., A high accurate and convergent numerical framework for solving high-order nonlinear Volterra integro-differential equations, J. Comput. Appl. Math., 421 (2023) · Zbl 1498.65229 · doi:10.1016/j.cam.2022.114852 |
| [46] | Yang, Y.; Tohidi, E.; Ma, X.; Kang, S., Rigorous convergence analysis of Jacobi spectral Galerkin methods for Volterra integral equations with noncompact kernels, J. Comput. Appl. Math., 366 (2020) · Zbl 1436.45008 · doi:10.1016/j.cam.2019.112403 |
| [47] | Yuttanan, B.; Razzaghi, M.; Vo, TN, A fractional-order generalized Taylor waveletmethod for nonlinear fractional delay and nonlinear fractional pantograph differential equations, Math. Methods Appl. Sci., 44, 5, 4156-4175 (2021) · Zbl 1512.65117 · doi:10.1002/mma.7020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
