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Collocation Solution of Fractional Differential Equations in Piecewise Nonpolynomial Spaces

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 333)


In this paper, we develop an accurate collocation scheme for ordinary initial value problems of the Caputo type and in order to illustrate the performance of the method we provide several numerical examples. At the end, we also indicate how this method can be used to approximate the solution of time-fractional diffusion equations. At it will be seen, it allows us to obtain accurate solutions even when the solution is not smooth at the origin.

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  1. Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behavior of real materials. ASME Trans. J. Appl. Mech. 51, 294–298 (1984)

    MATH  Google Scholar 

  2. Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41(1), 364–381 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Caputo, M.: Elasticity e Dissipazione. Zanichelli, Bologna (1969)

    Google Scholar 

  4. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  5. Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. Numerical Analysis and Scientific Computing. Chapman & Hall/CRP Press, Boca Raton (2015)

    MATH  Google Scholar 

  6. Ford, N.J., Morgado, M.L., Rebelo, M.: Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract. Calc. Appl. Anal. 16(4), 874–891 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  8. Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. (2017).

    CrossRef  MATH  Google Scholar 

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The authors acknowledge the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through projects MULTI/04621/2019 and UIDB/00297/2020 (Centro de Matemática e Aplicações), respectively.

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Correspondence to M. Luísa Morgado or Magda Rebelo .

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Morgado, M.L., Rebelo, M. (2020). Collocation Solution of Fractional Differential Equations in Piecewise Nonpolynomial Spaces. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham.

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