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Licensed Unlicensed Requires Authentication Published by De Gruyter February 9, 2019

Extrapolating for attaining high precision solutions for fractional partial differential equations

Fernanda Simões Patrício, Miguel Patrício and Higinio Ramos


This paper aims at obtaining a high precision numerical approximation for fractional partial differential equations. This is achieved through appropriate discretizations: firstly we consider the application of shifted Legendre or Chebyshev polynomials to get a spatial discretization, followed by a temporal discretization where we use the Implicit Euler method (although any other temporal integrator could be used). Finally, the use of an extrapolation technique is considered for improving the numerical results. In this way a very accurate solution is obtained. An algorithm is presented, and numerical results are shown to demonstrate the validity of the present technique.


The third author thanks the support provided by the Vicerrectorado de Investigación y Transferencia of the University of Salamanca.


[1] H. Azizi, G.B. Loghmani, A numerical method for space fractional diffusion equations using a semi-disrete scheme and Chebyshev collocation method. J. of Mathematics and Computer Science8 (2014), 226–235.10.22436/jmcs.08.03.05Search in Google Scholar

[2] R.L. Bagley and P.J. Torvik, On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51 (1984), 294–298.10.1115/1.3167615Search in Google Scholar

[3] R.T. Baillie, Long memory processes and fractional integration in econometrics. J. Econometrics73 (1996), 5–59.10.1016/0304-4076(95)01732-1Search in Google Scholar

[4] E. Barkai, R. Metzler, J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E61, No 1 (2000), 132–138.10.1103/PhysRevE.61.132Search in Google Scholar

[5] A.H. Bhrawy, M.M. Al-Shomran, A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Advances in Difference Equations4 (2012), 1–19.10.1186/1687-1847-2012-8Search in Google Scholar

[6] T.A. Biala, S.N. Jator, Block backward differentiation formulas for fractional differential equations. International J. of Engineering Mathematics, Art. ID 650425 (2015), 1–14.10.1155/2015/650425Search in Google Scholar

[7] D.W. Brzeziński, Accuracy problems of numerical calculation of fractional order derivatives and integrals applying the Riemann-Liouville/Caputo formulas. Appl. Mathematics and Nonlinear Sciences1 (2016), 23–44.10.21042/AMNS.2016.1.00003Search in Google Scholar

[8] J.C. Butcher, The role of orthogonal polynomials in numerical ordinary differential equations. J. Comput. Appl. Math. 43 (1992), 231–242.10.1016/0377-0427(92)90268-3Search in Google Scholar

[9] M. Caputo, Linear model of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astr. Soc. 13 (1967), 529–539.10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[10] P.J. Davis, Interpolation and Approximation. Dover, New York (1975).Search in Google Scholar

[11] E. Diekema, The fractional orthogonal derivative. Mathematics3 (2015), 273–298.10.3390/math3020273Search in Google Scholar

[12] K. Diethelm, The Analysis of Fractional Differential Equations: An application-Oriented Exposition Using Differential Operators of Caputo Type. Lectures Notes in Mathematics. Springer, Berlin (2010).10.1007/978-3-642-14574-2Search in Google Scholar

[13] F.K. Hamasalh, P.O. Muhammad, Numerical solution of fractional differential equations by using fractional spline model. J. of Information and Computing Science10, No 2 (2015), 98–105.Search in Google Scholar

[14] V. Gejji, H. Jafari, Solving a multi-order fractional differential equation using Adomian decomposition. Appl. Math. and Computation189, No 1 (2007), 541–548.10.1016/j.amc.2006.11.129Search in Google Scholar

[15] S. Kazem, Exact solution of some linear fractional differential equations by Laplace transform. Intern. J. of Nonlinear Science16, No 1 (2013), 3–11.Search in Google Scholar

[16] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. 1st Ed., Elsevier (2006).Search in Google Scholar

[17] J.D. Lambert, Computational Methods in Ordinary Differential Equations. John Wiley & Sons (1974).Search in Google Scholar

[18] F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker-Planck equation. J. of Comput. and Appl. Math. 166, No 1 (2004), 209–219.10.1016/ in Google Scholar

[19] J.T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1140–1153.10.1016/j.cnsns.2010.05.027Search in Google Scholar

[20] W. Gautschi, Orthogonal Polynomials. Computation and Approximation. Oxford University Press, Oxford (2004).10.1093/oso/9780198506720.001.0001Search in Google Scholar

[21] M.M. Khader, N.H. Sweilam, A.M.S. Mahdy, An efficient numerical method for solving the fractional diffusion equation. J. of Appl. Math. & Bioinformatics1, No 2 (2011), 1–12.10.3366/nor.2011.0002Search in Google Scholar

[22] M.M. Khader, T.S.E. Danaf, A.S. Hendy, Efficient spectral collocation method for solving multi-term fractional differential equations based on the generalized Laguerre polynomials. J. of Fractional Calculus and Applications3, No 13 (2012), 1–14.Search in Google Scholar

[23] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publ. Office of Czestochowa University of Technology (2009).Search in Google Scholar

[24] A. Pedas, E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods. J. of Comput. and Appl. Math. 255 (2014), 216–230.10.1016/ in Google Scholar

[25] I. Podlubny, Fractional Differential Equations. Ser. Mathematics in Science and Engineering # 198, Academic Press Inc., San Diego, CA, (1999).Search in Google Scholar

[26] J. Sabatier, P. Lanusse, P. Melchior, A. Oustaloup, Fractional Order Differentiation and Robust Control Design. International Ser. on Intelligent Systems, Control and Automation - Science and Engineering # 77, Springer (2015).10.1007/978-94-017-9807-5Search in Google Scholar

[27] M. Safari, M. Danesh, Application of Adomian’s decomposition method for the analytical solution of space fractional diffusion. Advances in Pure Mathematics1 (2011), 345–350.10.4236/apm.2011.16062Search in Google Scholar

[28] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993).Search in Google Scholar

[29] M.N. Sherif, I. Abouelfarag, T.S. Amer, Numerical solution of fractional delay differential equations using spline functions. Internat. J. of Pure and Appl. Math. 90 (2014), 73–83.10.12732/ijpam.v90i1.10Search in Google Scholar

[30] G. Szegö, Orthogonal Polynomials, 4th Ed. American Mathematical Society, Providence, Rhode Island (1975).Search in Google Scholar

[31] Y. Yang, Multi-order fractional differential equation using Legendre pseudo-spectral method. Applied Mathematics4 (2013), 113–118.10.4236/am.2013.41020Search in Google Scholar

[32] S.B. Yuste, L. Acedo, K. Lindenberg, Reaction front in an A + BC reaction-subdiffusion process. Phys. Rev., E69, No 3 (2004), 1–10.Search in Google Scholar

[33] S.B. Yuste, K. Lindenberg, Subdiffusion-limited A + A reactions. Phys. Rev. Lett., 87, No 11 (2001), 1–4.10.1002/9783527622979.ch13Search in Google Scholar

[34] M. Zayernouri, G. Karniadakis, Fractional spectral collocation method. SIAM J. Sci. Comput. 36, No 1 (2014), A40–A62.10.1137/130933216Search in Google Scholar

[35] M. Zayernouri, M. Ainsworth, G.E. Karniadakis, A unified Petrov-Galerkin spectral method for fractional PDEs. Computer Methods in Applied Mechanics and Engineering283 (2015), 1545–1569.10.1016/j.cma.2014.10.051Search in Google Scholar

Received: 2017-09-16
Published Online: 2019-02-09
Published in Print: 2018-12-19

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