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A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations

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Abstract

The aim of this paper is to obtain the numerical solutions of fractional Volterra integro-differential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points. We convert the fractional order integro-differential equation into integral equation by fractional order integral, and transfer the integro equations into a system of linear equations by the Gausssian quadrature. We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in \(L^{\infty }\) norm. Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.

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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  MATH  Google Scholar 

  2. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  3. Carpinteri, A., Chiaia, B., Cornetti, P.: Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Rev. 191, 3–19 (2001)

    Article  MATH  Google Scholar 

  4. Chen, Y.P., Tang, T.: Convergence analysis of the Jacobi spectral collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  6. Deng, W., Hesthaven, J.S.: Local discontinuous Galerkin methods for fractional ordinary differential equations. BIT Numer. Math. 55, 967–985 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  8. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36, 4931–4943 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eslahchi, M.R., Dehghan, M., Parvizi, M.: Application of the collocation method for solving nonlinear fractional integro-differential equations. J. Comput. Appl. Math. 257, 105–128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Esmaeili, S., Shamsi, M.: A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 3646–3654 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ghoreishi, F., Mokhtary, P.: Spectral collocation method for multi-order fractional differential equations. Int. J. Comput. Methods 11, 1350072 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghoreishi, F., Yazdani, S.: An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis. Comput. Math. Appl. 61, 30–43 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Henry, D.B.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (2006)

    Google Scholar 

  14. Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F., Cattani, C.: Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun. Nonlinear Sci. Numer. Simul. 19, 37–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jani, M., Babolian, E., Javadi, S.: Bernstein modal basis: application to the spectral Petrov-Galerkin method for fractional partial differential equations. Math. Methods Appl. Sci. 40, 7663–7672 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, New York (2015)

    Book  MATH  Google Scholar 

  19. Li, C., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, 383–406 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ma, X.H., Huang, C.M.: Spectral collocation method for linear fractional integro-differential equations. Appl. Math. Model. 38, 1434–1448 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mastroianni, G., Occorsio, D.: Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey. J. Comput. Appl. Math. 134, 325–341 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mokhtary, P.: Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations. J. Comput. Appl. Math. 279, 145–158 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nevai, P.: Mean convergence of Lagrange interpolation. III. Trans. Am. Math. Soc. 282, 669–698 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ragozin, D.L.: Polynomial approximation on compact manifolds and homogeneous spaces. Trans. Am. Math. Soc. 150, 41–53 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50, 15–67 (1997)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Saadatmandi, A., Dehghan, M.: A Legendre collocation method for fractional integro-differential equations. J. Vib. Control. 17, 2050–2058 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Saeedi, H., Moghadam, M.M.: Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets. Commun. Nonlinear Sci. Numeri. Simul. 16, 1216–1226 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shaw, R.E., Garey, L.E.: A fast method for solving second order boundary value Volterra integro-differential equations. Int. J. Comput. Math. 65, 121–129 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Series in Computational Mathematics. Springer, Berlin (2011)

    Book  Google Scholar 

  31. Wang, L.F., Ma, Y.P., Yang, Y.Q.: Numerical solution of nonlinear Volterra integro-differential equations of fractional order by using adomian decomposition method. Appl. Mech. Mater. 635–637, 1582–1585 (2014)

    Article  Google Scholar 

  32. Wang, Y., Zhu, L.: Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method. Adv. Differ. Equ. 2017, 27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection-diffusion equations. SIAM J. Numer. Anal. 52, 405–423 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yang, Y., Chen, Y.P., Huang, Y.Q.: Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations. Acta Mathematica Scientia 34, 673–690 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zeng, F., Mao, Z., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. SIAM J. Sci. Comput. 39(1), A360–A383 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhu, L., Fan, Q.: Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW. Commun. Nonlinear Sci. Numer. Simul. 18, 1203–1213 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11701358, 11774218). The authors wish to thank Professor Heping Ma and Professor Changpin Li for their valuable discussions.

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Authors and Affiliations

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

    Qingqing Wu, Zhongshu Wu & Xiaoyan Zeng

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  1. Qingqing Wu
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  2. Zhongshu Wu
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  3. Xiaoyan Zeng
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Correspondence to Xiaoyan Zeng.

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Wu, Q., Wu, Z. & Zeng, X. A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations. Commun. Appl. Math. Comput. 3, 509–526 (2021). https://doi.org/10.1007/s42967-020-00099-x

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  • DOI: https://doi.org/10.1007/s42967-020-00099-x

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