Smoothing transformation and spline collocation for linear fractional boundary value problems. (English) Zbl 1410.65284
Summary: We construct and justify a high order method for the numerical solution of multi-point boundary value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, we first regularize the solution by a suitable smoothing transformation. After that, we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. To illustrate the reliability of the proposed method some numerical results are given.
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
fractional boundary value problem; Caputo derivative; weakly singular integral equation; smoothing transformation; spline collocation methodReferences:
| [1] | Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus. Models and Numerical Methods (2012), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Singapore · Zbl 1248.26011 |
| [2] | Herrmann, R., Fractional Calculus. An Introduction for Physicists (2014), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack · Zbl 1293.26001 |
| [3] | Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity (2010), Imperial College Press: Imperial College Press London · Zbl 1210.26004 |
| [4] | Diethelm, K., The Analysis of Fractional Differential Equations. The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, vol. 2004 (2010), Springer: Springer Berlin · Zbl 1215.34001 |
| [5] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [6] | Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 |
| [7] | Diethelm, K.; Ford, N. J., Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154, 621-640 (2004) · Zbl 1060.65070 |
| [8] | Pedas, A.; Tamme, E., Spline collocation methods for linear multi-term fractional differential equations, J. Comput. Appl. Math., 236, 167-176 (2011) · Zbl 1229.65138 |
| [9] | Pedas, A.; Tamme, E., Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math., 255, 216-230 (2014) · Zbl 1291.65247 |
| [10] | Doha, E. H.; Bhrawy, A. H.; Baleanu, D.; Ezz-Eldien, S. S., On shifted Jacobi spectral approximations for solving fractional differential equations, Appl. Math. Comput., 219, 8042-8056 (2013) · Zbl 1291.65207 |
| [11] | 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 |
| [12] | Ma, X.; Huang, C., Spectral collocation method for linear fractional integro-differential equations, Appl. Math. Model., 38, 1434-1448 (2014) · Zbl 1427.65421 |
| [13] | Yan, Y.; Pal, K.; Ford, N. J., Higher order numerical methods for solving fractional differential equations, BIT Numer. Math., 54, 555-584 (2014) · Zbl 1304.65173 |
| [14] | Agarwal, R. P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109, 973-1033 (2010) · Zbl 1198.26004 |
| [15] | Ahmad, B.; Nieto, J. J.; Pimentel, J., Some boundary value problems of fractional differential equations and inclusions, Comput. Math. Appl., 62, 1238-1250 (2011) · Zbl 1228.34011 |
| [16] | Rehman, M.u.; Khan, R. A.; Asif, N. A., Three point boundary value problems for nonlinear fractional differential equations, Acta Math. Sci., 31B, 1337-1346 (2011) · Zbl 1249.34012 |
| [17] | Zhou, W. X.; Chu, Y. D., Existence of solutions for fractional differential equations with multi-point boundary conditions, Common. Nonlinear Sci. Numer. Simult., 17, 1142-1148 (2012) · Zbl 1245.35153 |
| [18] | Benchohra, M.; Hamidi, N.; Henderson, J., Fractional differential equations with anti-periodic boundary conditions, Numer. Funct. Anal. Optim., 34, 404-414 (2013) · Zbl 1283.34003 |
| [19] | Baleanu, D.; Agarwal, R. P.; Mohammadi, H.; Rezapour, S., Some existence results for a nonlinear fractional differential equation on partially ordered banach spaces, Boundary Value Problems, vol. 112 (2013) · Zbl 1301.34007 |
| [20] | Al-Mdallal, Q. M.; Syam, M. I.; Anwar, M. N., A collocation-shooting method for solving fractional boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 15, 3814-3822 (2010) · Zbl 1222.65078 |
| [21] | Ford, N. J.; Morgado, M. L., Fractional boundary value problems: Analysis and numerical methods, Fract. Calc. Appl. Anal, 14, 554-567 (2011) · Zbl 1273.65098 |
| [22] | Al-Refai, M.; Hajji, M. A., Monotone iterative sequences for nonlinear boundary value problems of fractional order, Nonlinear. Anal., 74, 3531-3539 (2011) · Zbl 1219.34005 |
| [23] | Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62, 2364-2373 (2011) · Zbl 1231.65126 |
| [24] | Bhrawy, A. H.; Al-Shomrani, M. M., A shifted Legendre spectral method for fractional-order multi-point boundary value problems, Adv. Difference Equ. 2012, 8 (2012) · Zbl 1280.65074 |
| [25] | Rehman, M.u.; Khan, R. A., A numerical method for solving boundary value problems for fractional differential equations, Appl. Math. Modell., 36, 894-907 (2012) · Zbl 1243.65095 |
| [26] | Maleki, M.; Hashim, I.; Kajani, M. T.; Abbasbandy, S., An adaptive pseudospectral method for fractional order boundary value problems, Abstr. Appl. Anal., 2012, 19 (2012), art. ID 381708 · Zbl 1261.34009 |
| [27] | Pedas, A.; Tamme, E., Piecewise polynomial collocation for linear boundary value problems of fractional differential equations, J. Comput. Appl. Math., 236, 3349-3359 (2012) · Zbl 1245.65104 |
| [28] | Pedas, A.; Tamme, E., Spline collocation for nonlinear fractional boundary value problems, Appl. Math. Comput., 244, 502-513 (2014) · Zbl 1337.65106 |
| [29] | Gracia, J. L.; Stynes, M., Central difference approximation of convection in caputo fraction derivative two-point boundary value problems, J. Comput. Appl. Math., 273, 103-115 (2015) · Zbl 1295.65081 |
| [30] | Kopteva, N.; Stynes, M., An efficient collocation method for a Caputo two-point boundary value problem, BIT Numer. Math., 55, 1105-1123 (2015) · Zbl 1337.65088 |
| [31] | Brunner, H.; Pedas, A.; Vainikko, G., Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39, 957-982 (2001) · Zbl 0998.65134 |
| [32] | Brunner, H.; van der Houven, P. J., The Numerical Solution of Volterra Equations (1986), North-Holland: North-Holland Amsterdam · Zbl 0611.65092 |
| [33] | Vainikko, G., Multidimensional Weakly Singular Integral Equations. Multidimensional Weakly Singular Integral Equations, Lecture Notes in Mathematics, vol. 1549 (1993), Springer: Springer Berlin · Zbl 0789.65097 |
| [34] | Brunner, H., Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal., 20, 1106-1119 (1983) · Zbl 0533.65087 |
| [35] | Cao, Y.; Herdman, T.; Xu, Y., A hybrid collocation method for Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal., 41, 364-381 (2003) · Zbl 1042.65106 |
| [36] | Huang, M.; Xu, Y., Superconvergence of the iterated hybrid collocation method for weakly singular Volterra integral equations, J. Integr. Eqn. Appl., 18, 83-116 (2006) · Zbl 1141.65091 |
| [37] | Ma, X.; Huang, C., Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl. Math Comput., 219, 6750-6760 (2013) · Zbl 1290.65130 |
| [38] | Ford, N. J.; Morgado, M. L.; Rebelo, M., High order numerical methods for fractional terminal value problems, Comput. Methods Appl. Math., 14, 55-70 (2014) · Zbl 1285.65049 |
| [39] | Ford, N. J.; Morgado, M. L.; Rebelo, M., A nonpolynomial collocation method for fractional terminal value problems, J. Comput. Appl. Math., 275, 392-402 (2015) · Zbl 1297.65076 |
| [40] | Diogo, T.; McKee, S.; Tang, T., Collocation methods for second-kind Volterra integral equations with weakly singular kernels, Proc. R. Soc. Edinburgh, 124A, 199-210 (1994) · Zbl 0807.65141 |
| [41] | Baratella, P.; Orsi, A. P., A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math., 163, 401-418 (2004) · Zbl 1038.65144 |
| [42] | Pedas, A.; Vainikko, G., Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integral equations, Computing, 73, 271-293 (2004) · Zbl 1063.65147 |
| [43] | Kolk, M.; Pedas, A., Numerical solution of Volterra integral equations with weakly singular kernels which may have a boundary singularity, Math. Model. Anal., 14, 79-89 (2009) · Zbl 1193.65221 |
| [44] | Kolk, M.; Pedas, A.; Vainikko, G., High-order methods for Volterra integral equations with general weak singularities, Numer. Funct. Anal. Optim., 30, 1002-1024 (2009) · Zbl 1187.65145 |
| [45] | Zhao, J.; Xiao, J.; Ford, N. J., Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algorithms, 65, 723-743 (2014) · Zbl 1298.65197 |
| [46] | Kolk, M.; Pedas, A.; Tamme, E., Modified spline collocation for linear fractional differential equations, J. Comput. Appl. Math., 283, 28-40 (2015) · Zbl 1311.65094 |
| [47] | Kangro, R.; Parts, I., Superconvergence in the maximum norm of a class of piecewise polynomial collocation methods for solving linear weakly singular Volterra integro-differential equations, J. Integr. Eqn. Appl., 15, 403-427 (2003) · Zbl 1065.65148 |
| [48] | Parts, I., Piecewise polynomial collocation methods for solving weakly singular integro-differential equations (2005), Tartu University Press, Dissertation (phd) · Zbl 1093.65126 |
| [49] | Tamme, E., Numerical computation of weakly singular integrals, Proc. Estonian Acad. Sci. Phys. Math., 49, 215-224 (2000) · Zbl 0973.65017 |
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.
