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Block collocation boundary value solutions of the first-kind Volterra integral equations

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Abstract

In this paper, we study a class of block collocation boundary value methods for the first-kind Volterra integral equations. The numerical algorithm is constructed by utilizing approximations to the exact solution in future steps. The solvability of the new method is not ensured, even for the uniform mesh. Therefore, we discuss its solvability by studying the special structure of the collocation equation and present the sufficient condition for the existence of the collocation solution. Furthermore, we exploit the convergence property with the help of interpolation remainders. Finally, numerical experiments are conducted to show the effectiveness of the new boundary value method and verify given theoretical results.

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References

  1. Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Cordon and Breach Science Publishers (1998)

  2. Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218(6), 2619–2630 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Chen, H., Zhang, C.: Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Appl. Numer. Math. 62(2), 141–154 (2012)

    Article  MathSciNet  Google Scholar 

  4. Zhao, J., Fan, Y., Xu, Y.: Delay-dependent stability analysis of symmetric boundary value methods for linear delay integro-differential equations. Numerical Algorithms 65(1), 125–151 (2014)

    Article  MathSciNet  Google Scholar 

  5. Yan, X., Zhang, C.: Solving nonlinear functional-differential and functional equations with constant delay via block boundary value methods. Math. Comput. Simul. 166(21), 21–32 (2019)

    Article  MathSciNet  Google Scholar 

  6. Ma, J., Xiang, S.: A collocation boundary value method for linear Volterra integral equations. J. Sci. Comput. 71(1), 1–20 (2017)

    Article  MathSciNet  Google Scholar 

  7. Ma, J., Liu, H.: Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels. Numerical Algorithms, https://doi.org/10.1007/s11075-019-00777-9 (2019)

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

  9. Davies, P. J., Duncan, D. B.: Stability and convergence of collocation schemes for retarded potential integral equations. SIAM J. Numer. Anal. 42(3), 1167–1188 (2004)

    Article  MathSciNet  Google Scholar 

  10. De Hoog, F.R., Anderssen, R.S.: Regularization of first kind integral equations with application to couette viscometry. Journal of Integral Equations and Applications 18(2), 249–265 (2006)

    Article  MathSciNet  Google Scholar 

  11. Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press (2017)

  12. Srivastava, H.M., Buschman, R.G.: Convolution Integral Equations. Wiley (1977)

  13. Xiang, S.: Laplace transforms for approximation of highly oscillatory Volterra integral equations of the first kind. Appl. Math. Comput. 232, 944–954 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Li, B., Xiang, S., Liu, G.: Laplace transforms for evaluation of Volterra integral equation of the first kind with highly oscillatory kernel. Comput. Appl. Math. 38(3), 116 (2019)

    Article  MathSciNet  Google Scholar 

  15. Conte, D., Paternoster, B.: Multistep collocation methods for Volterra integral equations. Appl. Numer. Math. 59(8), 1721–1736 (2009)

    Article  MathSciNet  Google Scholar 

  16. Fazeli, S., Hojjati, G.: Numerical solution of Volterra integro-differential equations by superimplicit multistep collocation methods. Numerical Algorithms 68 (4), 741–768 (2015)

    Article  MathSciNet  Google Scholar 

  17. Zhao, J., Long, T., Xu, Y.: Super implicit multistep collocation methods for weakly singular Volterra integral equations. Numerical Mathematics-Theory, Methods and Applications 12(4), 1039–1065 (2019)

    Article  MathSciNet  Google Scholar 

  18. Zhang, T., Liang, H.: Multistep collocation approximations to solutions of first-kind Volterra integral equations. Appl. Numer. Math. 130, 171–183 (2018)

    Article  MathSciNet  Google Scholar 

  19. Chan, R., Jin, X.: An Introduction to Iterative Toeplitz Solvers .SIAM (2007)

  20. Böttcher, A., Grudsky, S.: Spectral Properties of Banded Toeplitz Matrices. Springer (2005)

  21. Wang, H., Xiang, S.: Asymptotic expansion and Filon-type methods for a Volterra integral equation with a highly oscillatory kernel. IMA J. Numer. Anal. 31 (2), 469–490 (2011)

    Article  MathSciNet  Google Scholar 

  22. Xiang, S., Wang, H.: Fast integration of highly oscillatory integrals with exotic oscillators. Math. Comput. 79(270), 829–844 (2009)

    Article  MathSciNet  Google Scholar 

  23. Ma, J., Xiang, S., Kang, H.: On the convergence rates of Filon methods for the solution of a Volterra integral equation with a highly oscillatory Bessel kernel. Appl. Math. Lett. 26(7), 699–705 (2013)

    Article  MathSciNet  Google Scholar 

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Funding

This work was supported by the National Natural Science Foundation of China (No. 11901133) and Science and Technology Foundation of Guizhou Province (No. QKH[2017]5788).

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  1. School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, Guizhou, People’s Republic of China

    Ling Liu & Junjie Ma

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  1. Ling Liu
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  2. Junjie Ma
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Correspondence to Junjie Ma.

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Liu, L., Ma, J. Block collocation boundary value solutions of the first-kind Volterra integral equations. Numer Algor 86, 911–932 (2021). https://doi.org/10.1007/s11075-020-00917-6

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