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Efficient spectral collocation method for fractional differential equation with Caputo-Hadamard derivative

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Abstract

Hadamard type fractional calculus involves logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenge in numerical treatment. In this paper we present a spectral collocation method with mapped Jacobi log orthogonal functions (MJLOFs) as basis functions and obtain an efficient algorithm to solve Hadamard type fractional differential equations. We develop basic approximation theory for the MJLOFs and derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e, the fractional Helmholtz equation, and the fractional Burgers equation.

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References

  1. Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles. Fields and Media. Higher Education Press, Bei**g (2010)

    Book  MATH  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  3. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  4. Deng, W.H., Hou, R., Wang, W.L., Xu, P.B.: Modeling Anomalous Diffusion: From Statistics to Mathematics. World Scientific, Singapore (2020)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Li, C.P., Zeng, F.H.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC Press, USA (2015)

    Book  MATH  Google Scholar 

  7. Li, C.P., Cai, M.: Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)

    Book  Google Scholar 

  8. Hadamard, J.: Essai sur létude des fonctions, données par leur développement de Taylor. J. Math. Pures Appl. 8, 101–186 (1892)

    MATH  Google Scholar 

  9. Lomnitz, C.: Creep measurements in igneous rocks. J. Geol. 64(5), 473–479 (1956). https://doi.org/10.1086/626379

    Article  Google Scholar 

  10. Garra, R., Mainardi, F., Spada, G.: A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Solitons Fractals. 102, 333–338 (2017). https://doi.org/10.1016/j.chaos.2017.03.032

    Article  MathSciNet  Google Scholar 

  11. Cai, M., Karniadakis, G. Em, Li, C.P.: Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant. Chaos. 32(7), 071101 (2022). https://doi.org/10.1063/5.0099450

  12. Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y.: Algorithms for the fractional calculus: A selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194(6–8), 743–773 (2005). https://doi.org/10.1016/j.cma.2004.06.006

    Article  MathSciNet  MATH  Google Scholar 

  14. Gohar, M., Li, C.P., Yin, C.T.: On Caputo-Hadamard fractional differential equations. Int. J. Comput. Math. 97(7), 1459–1483 (2020). https://doi.org/10.1080/00207160.2019.1626012

    Article  MathSciNet  MATH  Google Scholar 

  15. Gohar, M., Li, C.P., Li, Z.Q.: Finite difference methods for Caputo-Hadamard fractional differential equations. Mediterr. J. Math. 17(6), 194 (2020). https://doi.org/10.1007/s00009-020-01605-4

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, C.P., Li, Z.Q., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85(2), 41 (2020). https://doi.org/10.1007/s10915-020-01353-3

    Article  MathSciNet  MATH  Google Scholar 

  17. Fan, E.Y., Li, C.P., Li, Z.Q.: Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems. Commun. Nonlinear Sci. Numer. Simul. 106, 106096 (2022). https://doi.org/10.1016/j.cnsns.2021.106096

    Article  MathSciNet  MATH  Google Scholar 

  18. Zaky, M.A., Hendy, A.S., Suragan, D.: Logarithmic Jacobi collocation method for Caputo-Hadamard fractional differential equations. Appl. Numer. Math. 181, 326–346 (2022). https://doi.org/10.1016/j.apnum.2022.06.013

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, C.P., Li, Z.Q.: Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation. J. Nonlinear Sci. 31(2), 31 (2021). https://doi.org/10.1007/s00332-021-09691-8

    Article  MathSciNet  MATH  Google Scholar 

  20. He, B.B., Zhou, H.C., Kou, C.H.: Stability analysis of Hadamard and Caputo-Hadamard fractional nonlinear systems without and with delay. Fract. Calc. Appl. Anal. 25(6), 2420–2445 (2022). https://doi.org/10.1007/s13540-022-00106-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, C.P., Li, Z.Q.: The finite-time blow-up for semilinear fractional diffusion equations with time \(\psi \)-Caputo derivative. J. Nonlinear Sci. 32(6), 82 (2022). https://doi.org/10.1007/s00332-022-09841-6

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, S., Shen, J., Zhang, Z.M., Zhou, Z.: A spectrally accurate approximation to subdiffusion equations using the log orthogonal functions. SIAM J. Sci. Comput. 42(2), A849–A877 (2020). https://doi.org/10.1137/19M1281927

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, S., Shen, J.: Log orthogonal functions: approximation properties and applications. IMA J. Numer. Anal. 42, 712–743 (2022). https://doi.org/10.1093/imanum/draa087

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, Y., Tang, Z.Y.: Mapped spectral collocation methods for Volterra integral equations with noncompact kernels. Appl. Numer. Math. 160, 166–177 (2021). https://doi.org/10.1016/j.apnum.2020.10.001

    Article  MathSciNet  MATH  Google Scholar 

  25. Yao, G.Q., Tao, D.Y., Zhang, C.: A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays. Appl. Math. Comput. 417, 126780 (2022). https://doi.org/10.1016/j.amc.2021.126780

    Article  MathSciNet  MATH  Google Scholar 

  26. Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, R.I. (1975)

    MATH  Google Scholar 

  27. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms. Analysis and Applications. Springer-Verlag, Berlin (2011)

    Book  MATH  Google Scholar 

  28. Zhao, X.D., Wang, L.L., **e, Z.Q.: Sharp error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature of analytic functions. SIAM J. Numer. Anal. 51(3), 1443–1469 (2013). https://doi.org/10.1137/12089421X

    Article  MathSciNet  MATH  Google Scholar 

  29. Mao, Z.P., Karniadakis, G.E.: Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods. J. Comput. Phys. 336, 143–163 (2017). https://doi.org/10.1016/j.jcp.2017.01.048

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work was partly supported by the National Natural Science Foundation of China under Grant Nos. 11661048 and 12271339.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, China

    Tinggang Zhao

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Changpin Li & Dongxia Li

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  1. Tinggang Zhao
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  2. Changpin Li
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  3. Dongxia Li
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Correspondence to Changpin Li.

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Zhao, T., Li, C. & Li, D. Efficient spectral collocation method for fractional differential equation with Caputo-Hadamard derivative. Fract Calc Appl Anal 26, 2903–2927 (2023). https://doi.org/10.1007/s13540-023-00216-6

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