# American Institute of Mathematical Sciences

September  2018, 23(7): 2709-2725. doi: 10.3934/dcdsb.2018088

## Two-step collocation methods for fractional differential equations

 Dipartimento di Matematica, Università di Salerno, Fisciano (SA), Italy

* Corresponding author

Received  October 2016 Revised  October 2017 Published  September 2018 Early access  March 2018

Fund Project: The work is supported by GNCS-Indam project.

We propose two-step collocation methods for the numerical solution of fractional differential equations. These methods increase the order of convergence of one-step collocation methods, with the same number of collocation points. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. We describe the derivation of two-step collocation methods and analyse convergence. Some numerical experiments confirm theoretical expectations.

Citation: Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088
##### References:
 [1] L. Blank, Numerical treatment of differential equations of fractional order, Nonlinear World, 4 (1997), 473-491. [2] J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [3] M. Braś and A. Cardone, Construction of efficient general linear methods for non-stiff differential systems, Math. Model. Anal., 17 (2012), 171-189.  doi: 10.3846/13926292.2012.655789. [4] M. Braś, A. Cardone and R. D'Ambrosio, Implementation of explicit Nordsieck methods with inherent quadratic stability, Math. Model. Anal., 18 (2013), 289-307.  doi: 10.3846/13926292.2013.785039. [5] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3 of CWI Monographs, North-Holland Publishing Co., Amsterdam, 1986. [6] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234. [7] H. Brunner, A. Pedas and G. Vainikko, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39 (2001), 957-982 (electronic).  doi: 10.1137/S0036142900376560. [8] A. Cardone and D. Conte, Multistep collocation methods for Volterra integro-differential equations, Appl. Math. Comput., 221 (2013), 770-785.  doi: 10.1016/j.amc.2013.07.012. [9] A. Cardone, L. G. Ixaru and B. Paternoster, Exponential fitting direct quadrature methods for Volterra integral equations, Numer. Algorithms, 55 (2010), 467-480.  doi: 10.1007/s11075-010-9365-1. [10] A. Cardone, E. Messina and A. Vecchio, An adaptive method for Volterra-Fredholm integral equations on the half line, J. Comput. Appl. Math., 228 (2009), 538-547.  doi: 10.1016/j.cam.2008.03.036. [11] D. Conte, R. D'Ambrosio and B. Paternoster, Two-step diagonally-implicit collocation based methods for Volterra integral equations, Appl. Numer. Math., 62 (2012), 1312-1324.  doi: 10.1016/j.apnum.2012.06.007. [12] V. Daftardar-Gejji and H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301 (2005), 508-518.  doi: 10.1016/j.jmaa.2004.07.039. [13] R. D'Ambrosio and B. Paternoster, Two-step modified collocation methods with structured coefficient matrices, Appl. Numer. Math., 62 (2012), 1325-1334.  doi: 10.1016/j.apnum.2012.06.008. [14] M. Di Paola, A. Pirrotta and A. Valenza, Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results, Mech. Mater., 43 (2011), 799-806.  doi: 10.1016/j.mechmat.2011.08.016. [15] K. Diethelm, Smoothness properties of solutions of Caputo-type fractional differential equations, Fract. Calc. Appl. Anal., 10 (2007), 151-160. [16] K. Diethelm, The analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, An application-oriented exposition using differential operators of Caputo type. doi: 10.1007/978-3-642-14574-2. [17] K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22, Fractional order calculus and its applications.  doi: 10.1023/A:1016592219341. [18] K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be. [19] V. Djordjević, J. Jarić, B. Fabry, J. Fredberg and D. Stamenović, Fractional derivatives embody essential features of cell rheological behavior, Ann. Biomed. Eng., 31 (2003), 692-699.  doi: 10.1114/1.1574026. [20] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, Int. J. Comput. Math., 87 (2010), 2281-2290.  doi: 10.1080/00207160802624331. [21] R. Garrappa and M. Popolizio, On accurate product integration rules for linear fractional differential equations, J. Comput. Appl. Math., 235 (2011), 1085-1097.  doi: 10.1016/j.cam.2010.07.008. [22] E. Hairer, C. Lubich and M. Schlichte, Fast numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math., 23 (1988), 87-98.  doi: 10.1016/0377-0427(88)90332-9. [23] E. Hairer, S. P. N∅rsett and G. Wanner, Solving Ordinary Differential Equations. I, vol. 8 of Springer Series in Computational Mathematics, 2nd edition, Springer-Verlag, Berlin, 1993, Nonstiff problems. [24] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II, vol. 14 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2010, Stiff and differential-algebraic problems, Second revised edition, paperback. doi: 10.1007/978-3-642-05221-7. [25] C. Huang and Z. Zhang, Convergence of a $p$ -version/$hp$ -version method for fractional differential equations, J. Comput. Phys., 286 (2015), 118-127.  doi: 10.1016/j.jcp.2015.01.025. [26] L. G. Ixaru and G. Vanden Berghe, Exponential Fitting, vol. 568 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2004, With 1 CD-ROM (Windows, Macintosh and UNIX). doi: 10.1007/978-1-4020-2100-8. [27] R. Klages, G. Radons and I. Sokolov, Anomalous Transport: Foundations and Applications, John Wiley & Sons, 2008. doi: 10.1002/9783527622979. [28] I. Lie, Local error estimation for multistep collocation methods, BIT, 30 (1990), 126-144.  doi: 10.1007/BF01932138. [29] I. Lie and S. P. Norsett, Superconvergence for multistep collocation, Math. Comp., 52 (1989), 65-79.  doi: 10.1090/S0025-5718-1989-0971403-5. [30] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comp., 45 (1985), 463-469.  doi: 10.1090/S0025-5718-1985-0804935-7. [31] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300. [32] B. Mandelbrot and J. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093. [33] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach Phys. Rep. , 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3. [34] C. Necula, Option pricing in a fractional brownian motion environment, SSRN, (2008), 19pp. doi: 10.2139/ssrn.1286833. [35] A. Pedas and E. Tamme, On the convergence of spline collocation methods for solving fractional differential equations, J. Comput. Appl. Math., 235 (2011), 3502-3514.  doi: 10.1016/j.cam.2010.10.054. [36] A. Pedas and E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math., 255 (2014), 216-230.  doi: 10.1016/j.cam.2013.04.049. [37] E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput., 176 (2006), 1-6.  doi: 10.1016/j.amc.2005.09.059. [38] P. Torvik and R. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.  doi: 10.1115/1.3167615. [39] G. Vainikko, Multidimensional Weakly Singular Integral Equations, vol. 1549 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993.

show all references

##### References:
 [1] L. Blank, Numerical treatment of differential equations of fractional order, Nonlinear World, 4 (1997), 473-491. [2] J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [3] M. Braś and A. Cardone, Construction of efficient general linear methods for non-stiff differential systems, Math. Model. Anal., 17 (2012), 171-189.  doi: 10.3846/13926292.2012.655789. [4] M. Braś, A. Cardone and R. D'Ambrosio, Implementation of explicit Nordsieck methods with inherent quadratic stability, Math. Model. Anal., 18 (2013), 289-307.  doi: 10.3846/13926292.2013.785039. [5] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3 of CWI Monographs, North-Holland Publishing Co., Amsterdam, 1986. [6] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234. [7] H. Brunner, A. Pedas and G. Vainikko, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39 (2001), 957-982 (electronic).  doi: 10.1137/S0036142900376560. [8] A. Cardone and D. Conte, Multistep collocation methods for Volterra integro-differential equations, Appl. Math. Comput., 221 (2013), 770-785.  doi: 10.1016/j.amc.2013.07.012. [9] A. Cardone, L. G. Ixaru and B. Paternoster, Exponential fitting direct quadrature methods for Volterra integral equations, Numer. Algorithms, 55 (2010), 467-480.  doi: 10.1007/s11075-010-9365-1. [10] A. Cardone, E. Messina and A. Vecchio, An adaptive method for Volterra-Fredholm integral equations on the half line, J. Comput. Appl. Math., 228 (2009), 538-547.  doi: 10.1016/j.cam.2008.03.036. [11] D. Conte, R. D'Ambrosio and B. Paternoster, Two-step diagonally-implicit collocation based methods for Volterra integral equations, Appl. Numer. Math., 62 (2012), 1312-1324.  doi: 10.1016/j.apnum.2012.06.007. [12] V. Daftardar-Gejji and H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301 (2005), 508-518.  doi: 10.1016/j.jmaa.2004.07.039. [13] R. D'Ambrosio and B. Paternoster, Two-step modified collocation methods with structured coefficient matrices, Appl. Numer. Math., 62 (2012), 1325-1334.  doi: 10.1016/j.apnum.2012.06.008. [14] M. Di Paola, A. Pirrotta and A. Valenza, Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results, Mech. Mater., 43 (2011), 799-806.  doi: 10.1016/j.mechmat.2011.08.016. [15] K. Diethelm, Smoothness properties of solutions of Caputo-type fractional differential equations, Fract. Calc. Appl. Anal., 10 (2007), 151-160. [16] K. Diethelm, The analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, An application-oriented exposition using differential operators of Caputo type. doi: 10.1007/978-3-642-14574-2. [17] K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22, Fractional order calculus and its applications.  doi: 10.1023/A:1016592219341. [18] K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be. [19] V. Djordjević, J. Jarić, B. Fabry, J. Fredberg and D. Stamenović, Fractional derivatives embody essential features of cell rheological behavior, Ann. Biomed. Eng., 31 (2003), 692-699.  doi: 10.1114/1.1574026. [20] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, Int. J. Comput. Math., 87 (2010), 2281-2290.  doi: 10.1080/00207160802624331. [21] R. Garrappa and M. Popolizio, On accurate product integration rules for linear fractional differential equations, J. Comput. Appl. Math., 235 (2011), 1085-1097.  doi: 10.1016/j.cam.2010.07.008. [22] E. Hairer, C. Lubich and M. Schlichte, Fast numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math., 23 (1988), 87-98.  doi: 10.1016/0377-0427(88)90332-9. [23] E. Hairer, S. P. N∅rsett and G. Wanner, Solving Ordinary Differential Equations. I, vol. 8 of Springer Series in Computational Mathematics, 2nd edition, Springer-Verlag, Berlin, 1993, Nonstiff problems. [24] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II, vol. 14 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2010, Stiff and differential-algebraic problems, Second revised edition, paperback. doi: 10.1007/978-3-642-05221-7. [25] C. Huang and Z. Zhang, Convergence of a $p$ -version/$hp$ -version method for fractional differential equations, J. Comput. Phys., 286 (2015), 118-127.  doi: 10.1016/j.jcp.2015.01.025. [26] L. G. Ixaru and G. Vanden Berghe, Exponential Fitting, vol. 568 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2004, With 1 CD-ROM (Windows, Macintosh and UNIX). doi: 10.1007/978-1-4020-2100-8. [27] R. Klages, G. Radons and I. Sokolov, Anomalous Transport: Foundations and Applications, John Wiley & Sons, 2008. doi: 10.1002/9783527622979. [28] I. Lie, Local error estimation for multistep collocation methods, BIT, 30 (1990), 126-144.  doi: 10.1007/BF01932138. [29] I. Lie and S. P. Norsett, Superconvergence for multistep collocation, Math. Comp., 52 (1989), 65-79.  doi: 10.1090/S0025-5718-1989-0971403-5. [30] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comp., 45 (1985), 463-469.  doi: 10.1090/S0025-5718-1985-0804935-7. [31] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300. [32] B. Mandelbrot and J. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093. [33] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach Phys. Rep. , 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3. [34] C. Necula, Option pricing in a fractional brownian motion environment, SSRN, (2008), 19pp. doi: 10.2139/ssrn.1286833. [35] A. Pedas and E. Tamme, On the convergence of spline collocation methods for solving fractional differential equations, J. Comput. Appl. Math., 235 (2011), 3502-3514.  doi: 10.1016/j.cam.2010.10.054. [36] A. Pedas and E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math., 255 (2014), 216-230.  doi: 10.1016/j.cam.2013.04.049. [37] E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput., 176 (2006), 1-6.  doi: 10.1016/j.amc.2005.09.059. [38] P. Torvik and R. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.  doi: 10.1115/1.3167615. [39] G. Vainikko, Multidimensional Weakly Singular Integral Equations, vol. 1549 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993.
One-step collocation method with $m = 2$ and two-step collocation methods with $m = 2$ and with $m = 4$
One-step collocation method with $m = 3$ and two-step collocation methods with $m = 3$ and with $m = 6$
Two-step collocation method with $m = 2$, $\eta = \left[\frac{1}{3}, \frac{2}{3}\right]$
 Problem 1 Problem 2 Problem 3 N $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ 8 $0.02$ $4.49e-03$ $0.31$ 16 $1.20e-03$ $3.78$ $3.08e-04$ $3.87$ $0.10$ $1.62$ 32 $8.02e-05$ $3.90$ $1.93e-05$ $3.99$ $0.01$ $2.78$ 64 $5.19e-06$ $3.95$ $1.20e-06$ $4.01$ $1.26e-03$ $3.52$ 128 $3.33e-07$ $3.96$ $7.42e-08$ $4.01$ $8.32e-05$ $3.92$ 256 $2.12e-08$ $3.97$ $4.62e-09$ $4.01$ $4.82e-06$ $4.11$ 512 $1.35e-09$ $3.98$ $2.88e-10$ $4.00$ $2.68e-07$ $4.17$
 Problem 1 Problem 2 Problem 3 N $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ 8 $0.02$ $4.49e-03$ $0.31$ 16 $1.20e-03$ $3.78$ $3.08e-04$ $3.87$ $0.10$ $1.62$ 32 $8.02e-05$ $3.90$ $1.93e-05$ $3.99$ $0.01$ $2.78$ 64 $5.19e-06$ $3.95$ $1.20e-06$ $4.01$ $1.26e-03$ $3.52$ 128 $3.33e-07$ $3.96$ $7.42e-08$ $4.01$ $8.32e-05$ $3.92$ 256 $2.12e-08$ $3.97$ $4.62e-09$ $4.01$ $4.82e-06$ $4.11$ 512 $1.35e-09$ $3.98$ $2.88e-10$ $4.00$ $2.68e-07$ $4.17$
Two-step collocation method with $m = 3$, $\eta = \left[\frac{1}{4}, \frac{1}{2}, \frac{3}{4}\right]$
 Problem 1 Problem 2 Problem 3 N $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ 8 $0.13$ $1.60e-02$ $0.11$ 16 $1.44e-03$ $6.54$ $1.73e-04$ $6.53$ $1.93e-02$ $2.56$ 32 $1.78e-05$ $6.34$ $2.12e-06$ $6.34$ $1.06e-03$ $4.18$ 64 $2.40e-07$ $6.21$ $2.91e-08$ $6.19$ $2.78e-05$ $5.26$ 128 $3.43e-09$ $6.13$ $4.25e-10$ $6.10$ $5.00e-07$ $5.80$ 256 $5.09e-11$ $6.08$ $6.43e-12$ $6.05$ $7.53e-09$ $6.05$ 512 $7.66e-13$ $6.05$ $9.90e-14$ $6.02$ $1.05e-10$ $6.17$
 Problem 1 Problem 2 Problem 3 N $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ 8 $0.13$ $1.60e-02$ $0.11$ 16 $1.44e-03$ $6.54$ $1.73e-04$ $6.53$ $1.93e-02$ $2.56$ 32 $1.78e-05$ $6.34$ $2.12e-06$ $6.34$ $1.06e-03$ $4.18$ 64 $2.40e-07$ $6.21$ $2.91e-08$ $6.19$ $2.78e-05$ $5.26$ 128 $3.43e-09$ $6.13$ $4.25e-10$ $6.10$ $5.00e-07$ $5.80$ 256 $5.09e-11$ $6.08$ $6.43e-12$ $6.05$ $7.53e-09$ $6.05$ 512 $7.66e-13$ $6.05$ $9.90e-14$ $6.02$ $1.05e-10$ $6.17$
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