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A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations

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Abstract

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel \((x-s)^{-\mu },0<\mu <1\). First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both \(L^{\infty }\)- and weighted \(L^{2}\)-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change \(x\rightarrow x^{1/\lambda }\) for a suitable real number \(\lambda \). Finally a series of numerical examples are presented to demonstrate the efficiency of the method.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)

    MATH  Google Scholar 

  2. Ali, I., Brunner, H., Tang, T.: A spectral method for pantograph-type delay differential equations and its convergence analysis. J. Comput. Math. 27(2), 254–265 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Ali, I., Brunner, H., Tang, T.: Spectral methods for pantograph-type differential and integral equations with multiple delays. Front. Math. China 4(4), 49–61 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allaei, S., Diogo, T., Rebelo, M.: The Jacobi collocation method for a class of nonlinear Volterra integral equations with weakly singular kernel. J. Sci. Comput. 69(2), 673–695 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernardi, C., Maday, Y.: Spectral methods. Handb. Numer. Anal. 5, 209–485 (1997)

    MathSciNet  Google Scholar 

  6. Borwein, P., Erdélyi, T., Zhang, J.: Müntz system and orthogonal Müntz–Legendre polynomials. Trans. Am. Math. Soc. 342(3), 523–5421 (1994)

    MATH  Google Scholar 

  7. Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20(6), 1106–1119 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 45(172), 417–437 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  10. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, Y., Li, X., Tang, T.: A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Math. 31, 47–56 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Y., Tang, T.: Spectral methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233(4), 938–950 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79(269), 147–167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Diogo, T., McKee, S., Tang, T.: Collocation methods for second-kind Volterra integral equations with weakly singular kernels. Proc. R. Soc. Edinb. 124(2), 199–210 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gogatishvill, A., Lang, J.: The generalized Hardy operator with kernel and variable integral limits in Banach function spaces. J. Inequal. Appl. 4(1), 1–16 (1999)

    MathSciNet  Google Scholar 

  16. Hou, D., Hasan, M., Xu, C.: Müntz spectral methods for the time-fractional diffusion equation. Comput. Methods Appl. Math. 18(1), 43–62 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hou, D., Xu, C.: A fractional spectral method with applications to some singular problems. Adv. Comput. Math. 43(5), 911–944 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, C., Stynes, M.: A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv. Comput. Math. 42(5), 1015–1030 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kufner, A., Persson, L.: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003)

    Book  MATH  Google Scholar 

  20. Li, X., Tang, T.: Convergence analysis of Jacobi spectral collocation methods for Abel–Volterra integral equations of second kind. Front. Math. China 7(1), 69–84 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, X., Tang, T., Xu, C.: Numerical solutions for weakly singular Volterra integral equations using Chebyshev and Legendre pseudo-spectral Galerkin methods. J. Sci. Comput. 67(1), 43–64 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mastroianni, G.: Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey. J. Comput. Appl. Math. 134(1), 325–341 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mccarthy, P., Sayre, J., Shawyer, B.L.R.: Generalized Legendre polynomials. J. Math. Anal. Appl. 177(2), 530–537 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ragozin, D.: Polynomial approximation on compact manifolds and homogeneous spaces. Trans. Am. Math. Soc. 150(1), 41–53 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ragozin, D.: Constructive polynomial approximation on spheres and projective spaces. Trans. Am. Math. Soc. 162, 157–170 (1971)

    MathSciNet  MATH  Google Scholar 

  26. Shen, J., Sheng, C., Wang, Z.: Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels. J. Math. Study 48(4), 315–329 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shen, J., Tang, T., Wang, L.L.: Spectral methods, Algorithms, Analysis and Applications. Springer, Berlin (2010)

    Google Scholar 

  28. Sheng, C., Wang, Z., Guo, B.: A multistep Legendre–Gauss spectral collocation method for nonlinear Volterra integral equations. SIAM J. Numer. Anal. 52(4), 1953–1980 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tang, T., Xu, X.: Accuracy enhancement using spectral postprocessing for differential equations and integral equations. Commun. Comput. Phys. 5(2–4), 779–792 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26(6), 825–837 (2008)

    MathSciNet  MATH  Google Scholar 

  31. Wang, Z., Sheng, C.: An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Math. Comput. 85(298), 635–666 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Zhao, X., Wang, L.L., Xie, Z.: Sharp error bounds for Jacobi expansions and Gegenbauer–Gauss quadrature of analytic functions. SIAM J. Numer. Anal. 51(3), 1443–1469 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Dianming Hou

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Dianming Hou, Yumin Lin, Mejdi Azaiez & Chuanju Xu

  3. Laboratoire I2M UMR 5295, Bordeaux INP, 33607, Pessac, France

    Mejdi Azaiez & Chuanju Xu

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  1. Dianming Hou
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  2. Yumin Lin
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  3. Mejdi Azaiez
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  4. Chuanju Xu
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Correspondence to Chuanju Xu.

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This research is partially supported by NSF of China (Grant Nos. 11971408, 51661135011, 11421110001, and 91630204). The third author has received financial support from the French State in the frame of the “Investments for the future” Programme Idex Bordeaux, Reference ANR-10-IDEX-03-02 and NSFC/ANR joint Program 51661135011/ANR-16- CE40-0026-01.

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Hou, D., Lin, Y., Azaiez, M. et al. A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations. J Sci Comput 81, 2162–2187 (2019). https://doi.org/10.1007/s10915-019-01078-y

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  • DOI: https://doi.org/10.1007/s10915-019-01078-y

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