# 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  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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] K. Diethelm, Smoothness properties of solutions of Caputo-type fractional differential equations, Fract. Calc. Appl. Anal., 10 (2007), 151-160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] R. Klages, G. Radons and I. Sokolov, Anomalous Transport: Foundations and Applications, John Wiley & Sons, 2008. doi: 10.1002/9783527622979. Google Scholar [28] I. Lie, Local error estimation for multistep collocation methods, BIT, 30 (1990), 126-144. doi: 10.1007/BF01932138. Google Scholar [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. Google Scholar [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. Google Scholar [31] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300. Google Scholar [32] B. Mandelbrot and J. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437. doi: 10.1137/1010093. Google Scholar [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. Google Scholar [34] C. Necula, Option pricing in a fractional brownian motion environment, SSRN, (2008), 19pp. doi: 10.2139/ssrn.1286833. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [39] G. Vainikko, Multidimensional Weakly Singular Integral Equations, vol. 1549 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993. Google Scholar

show all references

##### References:
 [1] L. Blank, Numerical treatment of differential equations of fractional order, Nonlinear World, 4 (1997), 473-491. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] K. Diethelm, Smoothness properties of solutions of Caputo-type fractional differential equations, Fract. Calc. Appl. Anal., 10 (2007), 151-160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] R. Klages, G. Radons and I. Sokolov, Anomalous Transport: Foundations and Applications, John Wiley & Sons, 2008. doi: 10.1002/9783527622979. Google Scholar [28] I. Lie, Local error estimation for multistep collocation methods, BIT, 30 (1990), 126-144. doi: 10.1007/BF01932138. Google Scholar [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. Google Scholar [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. Google Scholar [31] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300. Google Scholar [32] B. Mandelbrot and J. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437. doi: 10.1137/1010093. Google Scholar [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. Google Scholar [34] C. Necula, Option pricing in a fractional brownian motion environment, SSRN, (2008), 19pp. doi: 10.2139/ssrn.1286833. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [39] G. Vainikko, Multidimensional Weakly Singular Integral Equations, vol. 1549 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993. Google Scholar
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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