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An adaptive pseudospectral method for fractional order boundary value problems. (English) Zbl 1261.34009
Summary: An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP) which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE). By dividing the interval of the problem into subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG) collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples, including the Bagley-Torvik equation, the proposed method is shown to be efficient and accurate.

34A08Fractional differential equations
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Hackensack, NJ, USA, 2012. · Zbl 1244.49028 · doi:10.1016/j.amc.2012.02.080
[2] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[3] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, New York, NY, USA, 2008. · Zbl 1243.68062 · doi:10.1016/j.jpdc.2007.05.007
[4] K. Moaddy, I. Hashim, and S. Momani, “Non-standard finite difference schemes for solving fractional-order Rössler chaotic and hyperchaotic systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1068-1074, 2011. · Zbl 1228.65119 · doi:10.1016/j.camwa.2011.03.059
[5] A. Pedas and E. Tamme, “Piecewise polynomial collocation for linear boundary value problems of fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 13, pp. 3349-3359, 2012. · Zbl 1245.65104 · doi:10.1016/j.cam.2012.03.002
[6] Q. M. Al-Mdallal, M. I. Syam, and M. N. Anwar, “A collocation-shooting method for solving fractional boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3814-3822, 2010. · Zbl 1222.65078 · doi:10.1016/j.cnsns.2010.01.020
[7] N. J. Ford and J. A. Connolly, “Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 382-391, 2009. · Zbl 1166.65066 · doi:10.1016/j.cam.2008.04.003
[8] S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488-494, 2006. · Zbl 1096.65131 · doi:10.1016/j.amc.2005.11.025
[9] G. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A, vol. 374, no. 25, pp. 2506-2509, 2010. · Zbl 1237.34007 · doi:10.1016/j.physleta.2010.04.034
[10] S. Abbasbandy, “An approximation solution of a nonlinear equation with Riemann-Liouville’s fractional derivatives by He’s variational iteration method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 53-58, 2007. · Zbl 1120.65133 · doi:10.1016/j.cam.2006.07.011
[11] J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters A, vol. 375, no. 38, pp. 3362-3364, 2011. · Zbl 1252.49027 · doi:10.1016/j.physleta.2011.07.033
[12] J. H. He, G. C. Wu, and F. Austin, “The variational iteration method which should be followed,” Nonlinear Science Letters A, vol. 1, no. 1, pp. 1-30, 2010.
[13] M. Lakestani, M. Dehghan, and S. Irandoust-pakchin, “The construction of operational matrix of fractional derivatives using B-spline functions,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1149-1162, 2012. · Zbl 1276.65015 · doi:10.1016/j.cnsns.2011.07.018
[14] M. U. Rehman and R. A. Khan, “A numerical method for solving boundary value problems for fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 894-907, 2012. · Zbl 1243.65095 · doi:10.1016/j.apm.2011.07.045
[15] E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931-4943, 2012. · Zbl 1252.34019
[16] M. U. Rehman and R. Ali Khan, “The Legendre wavelet method for solving fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4163-4173, 2011. · Zbl 1222.65063 · doi:10.1016/j.cnsns.2011.01.014
[17] O. Abdulaziz, I. Hashim, and S. Momani, “Application of homotopy-perturbation method to fractional IVPs,” Journal of Computational and Applied Mathematics, vol. 216, no. 2, pp. 574-584, 2008. · Zbl 1142.65104 · doi:10.1016/j.cam.2007.06.010
[18] I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 674-684, 2009. · Zbl 1221.65277 · doi:10.1016/j.cnsns.2007.09.014
[19] A. Al-Rabtah, S. Momani, and M. A. Ramadan, “Solving linear and nonlinear fractional differential equations using spline functions,” Abstract and Applied Analysis, vol. 2012, Article ID 426514, 9 pages, 2012. · Zbl 1235.65015 · doi:10.1016/j.amc.2012.07.047
[20] D. Baleanu, H. Mohammadi, and S. Rezapour, “Positive solutions of an initial value problem for nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 837437, 7 pages, 2012. · Zbl 1242.35215 · doi:10.1155/2012/837437
[21] J.-H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters A, vol. 376, no. 4, pp. 257-259, 2012. · Zbl 1255.26002 · doi:10.1016/j.physleta.2011.11.030
[22] M. Moshrefi-Torbati and J. K. Hammond, “Physical and geometrical interpretation of fractional operators,” Journal of the Franklin Institute B, vol. 335, no. 6, pp. 1077-1086, 1998. · Zbl 0989.26004 · doi:10.1016/S0016-0032(97)00048-3
[23] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent,” Journal of the Royal Australian Society, vol. 13, Part 2, pp. 529-539, 1967.
[24] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Scientific Computation, Springer, Berlin, Germany, 2006. · Zbl 1093.76002
[25] M. Maleki, I. Hashim, and S. Abbasbandy, “Solution of time-varying delay systems using an adaptive collocation method,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1434-1448, 2012. · Zbl 1291.65246
[26] P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 294-298, 1984. · Zbl 1203.74022 · doi:10.1115/1.3167615
[27] Y. \cCenesiz, Y. Keskin, and A. Kurnaz, “The solution of the Bagley-Torvik equation with the generalized Taylor collocation method,” Journal of the Franklin Institute, vol. 347, no. 2, pp. 452-466, 2010. · Zbl 1188.65107 · doi:10.1016/j.jfranklin.2009.10.007
[28] S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248-1255, 2007. · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068