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An hp-version Legendre spectral collocation method for multi-order fractional differential equations

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Abstract

In this paper, we consider the multi-order fractional differential equation and recast it into an integral equation. Based on the integral equation, we develop an hp-version Legendre spectral collocation method and the integral terms with the weakly singular kernels are calculated precisely according to the properties of Legendre and Jacobi polynomials. The hp-version error bounds under the L2-norm and the \(L^{\infty }\)-norm are derived rigorously. Numerical experiments are included to illustrate the efficiency of the proposed method and the rationality of the theoretical results.

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References

  1. Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876–895 (2015)

    Article  MathSciNet  Google Scholar 

  2. Brunner, H.: Collocation methods for volterra integral and related functional equations. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  3. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral methods: Fundamentals in single domains. Springer, Berlin (2006)

    Book  Google Scholar 

  4. Chen, S., Shen, J., Wang, L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85, 1603–1638 (2016)

    Article  MathSciNet  Google Scholar 

  5. Dabiri, A., Butcher, E.A.: Efficient modified Chebyshev differentiation matrices for fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 50, 284–310 (2017)

    Article  MathSciNet  Google Scholar 

  6. Dabiri, A., Butcher, E.A.: Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods. Appl. Math. Model. 56, 424–448 (2018)

    Article  MathSciNet  Google Scholar 

  7. Diethelm, K.: The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type, Lect. Notes Math. Springer, Berlin (2010)

    Book  Google Scholar 

  8. Guo, Y., Wang, Z.: An hp-version Chebyshev collocation method for nonlinear fractional differential equations. Appl. Numer. Math. 158, 194–211 (2020)

    Article  MathSciNet  Google Scholar 

  9. Hilfer, R.: Applications of fractional calculus in physics. World Scientific, Singapore (1999)

    MATH  Google Scholar 

  10. Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84, 2665–2700 (2015)

    Article  MathSciNet  Google Scholar 

  11. Liang, H., Stynes, M.: Collocation methods for general Caputo two-point boundary value problems. J. Sci. Comput. 76, 390–425 (2018)

    Article  MathSciNet  Google Scholar 

  12. Lischke, A., Zayernouri, M., Karniadakis, G.E.: A Petrov-Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. SIAM J. Sci. Comput. 39, 922–946 (2017)

    Article  MathSciNet  Google Scholar 

  13. Liu, Y., Zhou, Z., Jin, B., Lazarov, R.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)

    Article  MathSciNet  Google Scholar 

  14. Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374, 538–548 (2011)

    Article  MathSciNet  Google Scholar 

  15. Mokhtary, P., Ghoreishi, F., Srivastava, H.M.: The müntz-Legendre tau method for fractional differential equations. Appl. Math. Model 40, 671–684 (2016)

    Article  MathSciNet  Google Scholar 

  16. Mu, J., Wang, Z.: A multiple interval Chebyshev-Gauss-Lobatto collocation method for ordinary differential equations. Numer. Math. Theor. Meth. Appl. 9, 619–639 (2016)

    Article  MathSciNet  Google Scholar 

  17. Pedas, A., Tamme, E.: Spline collocation methods for linear multi-term fractional differential equations. J. Comput. Appl. Math. 236, 167–176 (2011)

    Article  MathSciNet  Google Scholar 

  18. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  19. Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional order differential equations. Comput. Math. Appl. 59, 1326–1336 (2010)

    Article  MathSciNet  Google Scholar 

  20. Saeedi, H.: A fractional-order operational method for numerical treatment of multi-order fractional partial differential equation with variable coefficients. SeMA J. 75, 421–433 (2018)

    Article  MathSciNet  Google Scholar 

  21. Sheng, C., Wang, Z., Guo, B.: A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations. SIAM J. Numer. Anal. 52, 1953–1980 (2014)

    Article  MathSciNet  Google Scholar 

  22. Srivastava, V., Rai, K.N.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616–624 (2010)

    Article  MathSciNet  Google Scholar 

  23. Sun, H., Zhao, X., Sun, Z.: The temporal second order difference schemes based on the interpolation approximation for the time multi-term fractional wave equation. J. Sci. Comput. 78, 467–498 (2019)

    Article  MathSciNet  Google Scholar 

  24. Szegö, G.: Orthogonal polynomials. AMS Coll. Publ. 23, Providence (1978)

    MATH  Google Scholar 

  25. Trujillo, J.J., Kilbas, A.A., Srivastava, H.M.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  26. Wang, C., Wang, Z., Jia, H.: An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels. J. Sci. Comput. 72, 647–678 (2017)

    Article  MathSciNet  Google Scholar 

  27. Wang, C., Wang, Z., Wang, L.: A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative. J. Sci. Comput. 76, 166–188 (2018)

    Article  MathSciNet  Google Scholar 

  28. Wang, Z., Guo, Y., Yi, L.: An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Math. Comp. 86, 2285–2324 (2017)

    Article  MathSciNet  Google Scholar 

  29. Wang, Z., Sheng, C.: An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Math. Comp. 85, 635–666 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Yan, R., Sun, Y., Ma, Q., Ding, X.: A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative. Comput. Appl. Math. 38, 1–24 (2019)

    Article  MathSciNet  Google Scholar 

  31. Zaky, M. A.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput. Appl. Math. 37, 3525–3538 (2018)

    Article  MathSciNet  Google Scholar 

  32. Zaky, M.A., Ameen, I.G.: On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems. Comput. Appl. Math. 38, 144–170 (2019)

    Article  MathSciNet  Google Scholar 

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Funding

The work was supported by the National Natural Science Foundation of China (Grant No. 12071294) and China Postdoctoral Science Foundation (Grant No. 2020M681345).

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Correspondence to Zhongqing Wang.

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Communicated by: Martin Stynes

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Guo, Y., Wang, Z. An hp-version Legendre spectral collocation method for multi-order fractional differential equations. Adv Comput Math 47, 37 (2021). https://doi.org/10.1007/s10444-021-09858-7

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  • DOI: https://doi.org/10.1007/s10444-021-09858-7

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