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Spectral method for multidimensional Volterra integral equation with regular kernel

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Abstract

This paper is concerned with obtaining an approximate solution for a linear multidimensional Volterra integral equation with a regular kernel. We choose the Gauss points associated with the multidimensional Jacobi weight function \(\omega(x) = \prod{_{i=1}^d}(1 - x_i)^\alpha(1 + x_i)^\beta, -1 <\alpha,\beta < \frac{1}{d} - \frac{1}{2}\) (d denotes the space dimensions) as the collocation points. We demonstrate that the errors of approximate solution decay exponentially. Numerical results are presented to demonstrate the effectiveness of the Jacobi spectral collocation method.

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References

  1. Aghazade N, Khajehnasiri A A. Solving nonlinear two-dimensional Volterra integro-differential equations by block-pulse functions. Math Sci, 2013, 7: 1–6

    Article  MathSciNet  Google Scholar 

  2. Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge: Cambridge Univ Press, 2004

    Book  MATH  Google Scholar 

  3. Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral Methods: Fundamentals in Single Domains. Scientific Computation. Berlin: Springer-Verlag, 2006

    MATH  Google Scholar 

  4. 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, 2013, 31: 47–56

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. 2nd ed. Appl Math Sci, Vol 93. Heidelberg: Springer-Verlag, 1998

  8. Darania P, Shali J A, Ivaz K. New computational method for solving some 2-dimensional nonlinear Volterra integro-differential equations. Numer Algorithms, 2011, 57: 125–147

    Article  MathSciNet  MATH  Google Scholar 

  9. Fedotov A I. Lebesgue constant estimation in multidimensional Sobolev space. J Math, 2004, 14: 25–32

    MathSciNet  MATH  Google Scholar 

  10. Headley V B. A multidimensional nonlinear Gronwall inequality. J Math Anal Appl, 1974, 47: 250–255

    Article  MathSciNet  MATH  Google Scholar 

  11. Mckee S, Tang T, Diogo T. An Euler-type method for two-dimensional Volterra integral equations of the first kind. IMA J Numer Anal, 2000, 20: 423–440

    Article  MathSciNet  MATH  Google Scholar 

  12. Mirzaee F, Hadadiyan E, Bimesl S. Numerical solution for three-dimensional non-linear mixed Volterra-Fredholm integral equations via three-dimensional block-pulse functions. Appl Math Comput, 2014, 237: 168–175

    MathSciNet  MATH  Google Scholar 

  13. Nemati S, Lima P M, Ordokhani Y. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials. J Comput Appl Math, 2013, 242: 53–69

    Article  MathSciNet  MATH  Google Scholar 

  14. Nevai P. Mean convergence of Lagrange interpolation. Trans Amer Math Soc, 1984, 282: 669–698

    Article  MathSciNet  MATH  Google Scholar 

  15. Ragozin D L. Polynomial approximation on compact manifolds and homogeneous spaces. Trans Amer Math Soc, 1970, 150: 41–53

    Article  MathSciNet  MATH  Google Scholar 

  16. Ragozin D L. Constructive polynomial approximation on spheres and projective spaces. Trans Amer Math Soc, 1971, 162: 157–170

    MathSciNet  MATH  Google Scholar 

  17. Shen J, Tang T. Spectral and High-order Methods with Applications. Beijing: Science Press, 2006

    MATH  Google Scholar 

  18. Tang T, Xu X, Chen J. On spectral methods for Volterra integral equations and the convergence analysis. J Comput Math, 2008, 26: 825–837

    MathSciNet  MATH  Google Scholar 

  19. Wei Y X, Chen Y. Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equation. J Sci Comput, 2012, 53: 672–688

    Article  MathSciNet  MATH  Google Scholar 

  20. Wei Y X, Chen Y. Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation. Appl Numer Math, 2014, 81: 15–29

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11671157, 11826212).

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Correspondence to Yanping Chen.

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Wei, Y., Chen, Y., Shi, X. et al. Spectral method for multidimensional Volterra integral equation with regular kernel. Front. Math. China 14, 435–448 (2019). https://doi.org/10.1007/s11464-019-0758-8

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  • DOI: https://doi.org/10.1007/s11464-019-0758-8

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