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Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations. (English) Zbl 1306.65294

Summary: A Legendre collocation method, an easy-to-use variant of the spectral methods for the numerical solution of a class of fractional integro-differential equations (FIDE’s), is researched. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Legendre Gauss quadrature formula. An efficient convergence analysis of the proposed method is given and rate of convergence is established in the \(L^2\)-norm. Due to the fact that the solutions of FIDE’s usually have a weak singularity at origin, we use a variable transformation to change the original equation into a new equation with a smooth solution. We prove that after this regularization technique,the numerical solution of the new equation by adopting the Legendre collocation method has exponentially rate of convergence. Numerical results are presented which clarify the high accuracy of the proposed method.

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
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[1] Bagley, R. L.; Torvik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27, 201-210 (1983) · Zbl 0515.76012
[2] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent II, Geophys. J. R. Astron. Soc., 13, 529-539 (1967)
[3] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[4] Olmstead, W. E.; Handelsman, R. A., Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Rev., 18, 275-291 (1976) · Zbl 0323.45008
[5] Diethelm, K., The Analysis of Fractional Differential Equations (2010), Springer-Verlag: Springer-Verlag Berlin
[6] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amesterdam · Zbl 1092.45003
[7] Podlubny, I., Fractional Differential Equations (1999), Academic Press · Zbl 0918.34010
[8] Brunner, H., Collocation Methods for Volterra and Related Functional Equations (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1059.65122
[9] Lubich, Ch., Runge-Kutta theory for the volterra and Abel integral equations of the second kind, Math. Comp., 41, 87-102 (1983) · Zbl 0538.65091
[10] Jiang, Y.; Ma, J., Spectral collocation methods for Volterra-integro differential equations with noncompact kernels, J. Comput. Appl. Math., 244, 115-124 (2013) · Zbl 1263.65134
[11] Nazari, D.; Shahmorad, S., Application of the fractionl differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, J. Comput. Appl. Math., 234, 883-891 (2010) · Zbl 1188.65174
[12] Rawashdeh, E. A., Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput., 176, 1-6 (2006) · Zbl 1100.65126
[13] Awawdeh, F.; Rawashdeh, E. A.; Jaradat, H. M., Analytic solution of fractional integro-differential equations, Ann. Univ. Craiova, Math. Comput. Sci. Ser., 38, 1-10 (2011) · Zbl 1240.45015
[14] Mittal, R. C.; Nigam, R., Solution of fractional calculus and fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. Mech., 4, 2, 87-94 (2008)
[15] Zhu, Li; Fan, Qibin, Numerical solution of nonlinear fractional order volterra integro differential equations by SCW, Commun. Nonlinear Sci. Numer. Simul., 18, 15, 1203-1213 (2013) · Zbl 1261.35152
[16] Huang, L.; Li, X. F.; Zhao, Y. L.; Duan, X. Y., Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl., 62, 1127-1134 (2011) · Zbl 1228.65133
[17] Ma, Xiaohua; Huang, C., Numerical solution of fractional integro-differential equations by Hybrid collocation method, Appl. Math. Comput., 219, 6750-6760 (2013) · Zbl 1290.65130
[18] Khader, M. M.; Sweilam, N. H., On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method, Appl. Math. Model., 27, 24, 819-828 (2013) · Zbl 1427.65419
[19] Ma, Xiaohua; Huang, C., Spectral collocation method for linear fractional integro-differential equations, Appl. Math. Model., 38, 4, 1434-1448 (2014) · Zbl 1427.65421
[20] Eslahchi, M. R.; Dehghan, M.; Parvizi, M., Application of the collocation method for solving nonlinear fractional integro-differential equations, J. Comput. Appl. Math., 257, 105-128 (2014) · Zbl 1296.65106
[21] Mokhtary, P.; Ghoreishi, F., The \(L^2\)-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations, Numer. Algorithms, 58, 475-496 (2011) · Zbl 1270.65078
[22] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., A new Jacobi operational matrix: an application for solving fractional differential equations, Appl. Math. Model., 36, 4931-4943 (2011) · Zbl 1252.34019
[23] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods Fundamentals in Single Domains (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1093.76002
[24] Hesthaven, J.; Gottlieb, S.; Gottlieb, D., (Spectral Methods for Time-Dependent Problems. Spectral Methods for Time-Dependent Problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21 (2007), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1111.65093
[25] Shen, J.; Tang, T.; Wang, L., Spectral Methods, Algorithms, Analysis and Applications (2011), Springer-Verlag: Springer-Verlag Berlin
[26] Chen, Y.; Tang, T., Convergence analysis of the Jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel, Math. Comp., 79, 147-167 (2010) · Zbl 1207.65157
[27] Baratella, P.; Orsi, A. P., A new appoach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math., 163, 401-418 (2004) · Zbl 1038.65144
[28] Liu, Ya-Ping; Lu, T., High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equations, Appl. Math. Comput., 178, 441-451 (2006) · Zbl 1104.65127
[29] Liu, Ya-Ping; Lu, T., Machanical quadrature method and their extrapolation for solving first kind Abel integral equations, J. Comput. Appl. Math., 201, 300-313 (2007) · Zbl 1113.65123
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