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A Hermite collocation method for approximating a class of highly oscillatory integral equations using new Gaussian radial basis functions

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Abstract

In this paper, we investigate the oscillation properties of solutions of a class of highly oscillatory Volterra integral equations and develop a Hermite collocation method to approximate the solution of these equations. We begin our analysis by obtaining an asymptotic expansion for the solution of these equations using their resolvent representation. We then introduce a new Gaussian radial basis function interpolation to provide a numerical solution for these equations. The convergence analysis of the proposed method is also studied, which shows that increasing the number of collocation points or the number of mesh points controls the impact of the oscillation parameter in the whole error. Some numerical examples are presented to show the accuracy of the proposed scheme.

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References

  1. Alipanah, A., Esmaeili, S.: Numerical solution of the two-dimensional Fredholm integral equations using Gaussian radial basis function. J. Comput. Appl. Math. 235(18), 5342–5347 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asheim, A., Deaño, A., Huybrechs, D., Wang, H.: A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Disc. Contin. Dyn. Syst. A 34(3), 883–901 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Assari, P., Adibi, H., Dehghan, M.: A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method. Appl. Math. Model. 37(22), 9269–9294 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Assari, P., Dehghan, M.: The approximate solution of nonlinear Volterra integral equations of the second kind using radial basis functions. Appl. Math. Comput. 131(2), 140–157 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Bellen, A., Jackiewicz, Z., Vermiglio, R., Zennaro, M.: Stability analysis of Runge–Kutta method for Volterra integral equations of second kind. IMA J. Numer. Anal. 10(1), 103–118 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blom, J.G., Brunner, H.: The solution of nonlinear Volterra integral equation of the second kind by collocation and iterated collocation method. SIAM J. Sci. Stat. Comput. 8(5), 806–830 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

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

  9. Brunner, H.: On Volterra integral operators with highly oscillatory kernels. Discrete Contin. Dyn. Syst. 34(3), 903–914 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brunner, H., Ma, Y.Y., Xu, Y.S.: The oscillation of solutions of Volterra integral and integro-differential equations with highly oscillatory kernels. J. Integral Equ. Appl. 27(4), 455–487 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer-Verlag (2006)

  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(69), 147–167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Conte, D., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput. 204(2), 839–853 (2008)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. 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  MATH  Google Scholar 

  17. Deaño, A., Huybrechs, D.: Complex Gaussian quadrature of oscillatory integrals. Numer. Math. 112(2), 197–219 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deaño, A., Huybrechs, D., Iserles, A.: Computing Highly Oscillatory Integrals. Society for Industrial and Applied Mathematics, Philadelphia (2018)

    MATH  Google Scholar 

  19. Fasshauer, G.E.: Solving partial differential equations by collocation with radial basis functions. In: Mehaute, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 131–138. Vanderbilt University Press, Nashville (1997)

    Google Scholar 

  20. Fazeli, S., Hojjati, G., Shahmorad, S.: Multistep Hermite collocation methods for solving Volterra Integral Equations. Numer. Algorithms 60(1), 27–50 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fazeli, S., Hojjati, G., Shahmorad, S.: Super implicit multistep collocation methods for nonlinear Volterra integral equations. Numer. Algorithms 55(3), 590–607 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Golbabai, A., Seifollahi, S.: Numerical solution of the second kind integral equations using radial basis function networks. Appl. Math. Comput. 174(2), 877–883 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Golbabai, A., Seifollahi, S.: An iterative solution for the second kind integral equations using radial basis functions. Appl. Math. Comput. 181(2), 903–907 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 176(8), 1905–1915 (1971)

    Article  Google Scholar 

  25. Huang, C., Stynes, M.: Spectral Galerkin methods for a weakly singular Volterra integral equation of the second kind. IMA J. Numer. Anal. 37, 1411–1436 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44(3), 1026–1048 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Iserles, A., Nørsett, S.P.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2057), 1383–1399 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Iserles, A., Nørsett, S.P.: On quadrature methods for highly oscillatory integrals and their implementation. BIT Numer. Math. 44(4), 755–772 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Karakostas, G., Stavroulakis, I.P., Wu, Y.M.: Oscillation of Volterra integral equations with delay. Tohoku Math. J. 45(4), 583–605 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Larsson, E., Fornberg, B.: A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891–902 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Levin, D.: Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comput. 38(158), 531–538 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. Levin, D.: Fast integration of rapidly oscillatory functions. IMA J. Numer. Anal. 67(1), 95–101 (1996)

    MathSciNet  MATH  Google Scholar 

  33. 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 

  34. Ma, J.: Oscillation-free solutions to Volterra integral and integro-differential equations with periodic force terms. Appl. Math. Comput. 294, 294–298 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Olver, S.: Moment-free numerical integration of highly oscillatory functions. IMA J. Numer. Anal. 26(2), 213–227 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Olver, S.: Moment-free numerical approximation of highly oscillatory integrals with stationary points. Eur. J. Appl. Math. 18(4), 435–447 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Platte, R.B., Tobin, A.D.: Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM J. Numer. Anal. 43(2), 750–766 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Power, H., Barraco, V.: A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations. Comput. Math. Appl. 43, 551–583 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Satco, B.: Volterra integral equations governed by highly oscillatory functions on time scales. An. Stiint. Univ. Ovidus Constanta Ser. Mat. 17(3), 233–240 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Sohrabi, S., Ranjbar, H., Saei, M.: Convergence analysis of the Jacobi-collocation method for nonlinear weakly singular Volterra integral equations. Appl. Math. Comput. 8(C), 141–152 (2017)

    MathSciNet  MATH  Google Scholar 

  41. van der Houwen, P.J.: Convergence and stability results in Runge–Kutta type methods for Volterra integral equations of the second kind. BIT Numer. Math. 20(3), 375–377 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xiang, S., Brunner, H.: Efficient methods for Volterra integral equations with highly oscillatory Bessel kernels. BIT Numer. Math. 53(1), 241–263 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Xiang, S., Cho, Y.J., Wang, H., Brunner, H.: Clenshaw–Curtis–Filon-type methods for highly oscillatory Bessel transforms and applications. IMA J. Numer. Anal. 31(4), 1281–1314 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. 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  MATH  Google Scholar 

  45. Zhang, S., Lin, Y., Rao, M.: Numerical solutions for second-kind Volterra integral equations by Galerkin methods. Appl. Math. 45, 19–39 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

    H. Ranjbar & F. Ghoreishi

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  1. H. Ranjbar
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  2. F. Ghoreishi
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Ranjbar, H., Ghoreishi, F. A Hermite collocation method for approximating a class of highly oscillatory integral equations using new Gaussian radial basis functions. Calcolo 58, 21 (2021). https://doi.org/10.1007/s10092-021-00416-7

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  • DOI: https://doi.org/10.1007/s10092-021-00416-7

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