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The Puiseux Expansion and Numerical Integration to Nonlinear Weakly Singular Volterra Integral Equations of the Second Kind

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Abstract

This paper develops an efficient algorithm to solve nonlinear Volterra integral equation of the second kind with weakly singular convolution kernel. First, we show that the general Puiseux series for the solution about zero exists under smooth assumptions for the nonlinear function, and then design an algorithm to recover the finite-term truncation of the asymptotic expansion by Picard iteration. This asymptotic expansion can easily yield a more accurate Padé approximation. Second, we use trapezoidal rule to discretize the singular integral and derive the Euler–Maclaurin asymptotic expansion using the known Puiseux expansion of the solution. By accumulating some lower order error terms to the quadrature formula, we obtain high precision evaluations to the nonlinear Volterra integral equation. Third, we prove that the scheme is convergent by extending the Gronwall inequality to be held for the scheme. Fourth, an example is provided to illustrate that the combination of the Puiseux expansion and the numerical integration can effectively increase the computational accuracy of the equation. Finally, we apply the method to solve the Lighthill integral equation, and obtain the asymptotic expansions of the solution near zero and infinity, respectively. The computation shows that the trapezoidal rule is only necessary in a small finite range of the semi-infinite interval.

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References

  1. Abdalkhani, J.: A numerical approach to the solution of Abel integral equations of the second kind with nonsmooth solution. J. Comput. Appl. Math. 29, 249–255 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allaei, S.S., Diogo, T., Rebelo, M.: Analytical and computational methods for a class of nonlinear singular integral equations. Appl. Numer. Math. 114, 2–17 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker, C.T.H.: A perspective on the numerical treatment of Volterra equations. J. Comput. Appl. Math. 125, 217–249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baratella, P., Orsi, A.P.: A new approach to the numerical solution of weakly singular Volterra integral equations. J. Comput. Appl. Math. 163, 401–418 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baratella, P., Orsi, A.P.: Numerical solution of weakly singular linear Volterra integro-differential equations. Computing 77, 77–96 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baratella, P.: A Nyström interpolant for some weakly singular nonlinear Volterra integral equations. J. Comput. Appl. Math. 237, 542–555 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bougoffa, L., Rach, R.C., Mennouni, A.: A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method. Appl. Math. Comput. 218, 1785–1793 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Brezinski, C., Ieea, U., Iseghem, J.V.: A taste of Padé approximation. Acta Numer. 4, 53–103 (1995)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Brunner, H., Pedas, A., Vainikko, G.: The piecewise collocation method for nonlinear weakly singular Volterra equations. Math. Comput. 68, 1079–1095 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  13. Cai, H.T., Chen, Y.P.: A fractional order collocation method for second kind Volterra integral equations with weakly singular kernels. J. Sci. Comput. 75, 970–992 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cai, H.T.: A fractional spectral collocation for solving second kind nonlinear Volterra integral equations with weakly singular kernels. J. Sci. Comput. 80, 1529–1548 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cao, Y.Z., Herdman, T., Xu, Y.S.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41, 364–381 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Curle, S.N.: Solution of an integral equation of Lighthill. Proc. R. Soc. Lond. A 364, 435–441 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Diogo, T., Franco, N.B., Lima, P.: High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Commun. Pure Appl. Anal. 3, 217–235 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Diogo, T., Lima, P., Rebelo, M.: Numerical solution of a nonlinear Abel type Volterra integral equation. Commun. Pure Appl. Anal. 5, 277–288 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Diogo, T., Ma, J.T., Rebelo, M.: Fully discretized collocation methods for nonlinear singular Volterra integral equations. J. Comput. Appl. Math. 247, 84–101 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eshaghi, J., Adibi, H., Kazem, S.: Solution of nonlinear weakly singular Volterra integral equations using the fractional-order Legendre functions and pseudospectral method. Math. Methods Appl. Sci. 39, 3411–3425 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gorenflo, R., Vessella, S.: Abel Integral Equations. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  24. Handelsman, R.A., Olmstead, W.E.: Asymptotic solution to a class of nonlinear Volterra integral equations. SIAM J. Appl. Math. 22, 373–384 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hou, D.M., Lin, Y.M., Azaiez, M., Xu, C.J.: A Müntz-collocation spectral method for weakly singular Volterra integral equations. J. Sci. Comput. 81, 2162–2187 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hu, Q.Y.: Stieltjes derivatives and $\beta $-polynomial spline collocation for Volterra integrodifferential equations with singularities. SIAM J. Numer. Anal. 33, 208–220 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Khater, A.H., Shamardan, A.B., Callebaut, D.K., Sakran, M.R.A.: Solving integral equations with logarithmic kernels by Chebyshev polynomials. Numer. Algorithms 47, 81–93 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kilbas, A.A., Saigo, M.: On solution of integral equation of Abel–Volterra type. Differ. Integral Equ. 8, 993–1011 (1995)

    MathSciNet  MATH  Google Scholar 

  29. Kilbas, A.A., Saigo, M.: On solution of nonlinear Abel–Volterra integral equation. J. Appl. Math. Anal. Appl. 229, 41–60 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kumar, I.J.: On the asymptotic solution of a nonlinear Volterra integral equation. Proc. R. Soc. Lond. A 324, 45–61 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liang, H., Brunner, H.: The convergence of collocation solutions in continuous piecewise polynomial spaces for weakly singular Volterra integral equations. SIAM J. Numer. Anal. 57, 1875–1896 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lighthill, M.J.: Contributions to the theory of heat transfer through a laminar boundary layer. Proc. R. Soc. Lond. A 202, 359–377 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  33. Linz, P.: Analytical and numerical methods for Volterra equations. In: SIAM Studies in Applied Mathematics. SIAM, (1985)

  34. Lü, T., Huang, Y.: A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind. J. Math. Anal. Appl. 282, 56–62 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lü, T., Huang, Y.: Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind. J. Math. Anal. Appl. 324, 225–237 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lü, T., Huang, J.: The High Accuracy Algorithms for Integral Equations. China Science Press, Beijing (2013). in Chinese

    Google Scholar 

  37. Lubich, Ch.: Runge–Kutta theory for Volterra and Abel integral equations of the second kind. Math. Comput. 41, 87–102 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  38. Navot, I.: An extension of the Euler–Maclaurin summation formula to functions with a branch singularity. J. Math. Phys. 40, 271–276 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  39. Navot, I.: A further extension of the Euler–Maclaurin summation formula. J. Math. Phys. 41, 155–163 (1962)

    Article  MATH  Google Scholar 

  40. Olmstead, W.E., Handelsman, R.A.: Asymptotic solution to a class of nonlinear Volterra integral equations (II). SIAM J. Appl. Math. 30, 180–189 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  41. Orsi, A.P.: Product integration for Volterra integral equations of the second kind with weakly singular kernels. Math. Comput. 65, 1201–1212 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pandey, R.K., Singh, O.P., Singh, V.K.: Efficient algorithms to solve singular integral equations of Abel type. Comput. Math. Appl. 57, 664–676 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Polyanin, A.D., Manzhirov, A.V.: Handbook of Integral Equationss, 2nd edn. Chapman & Hall/CRC, London (2008)

    Book  MATH  Google Scholar 

  44. Rebelo, M., Diogo, T.: A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel. J. Comput. Appl. Math. 234, 2859–2869 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sidi, A.: Euler–Maclaurin expansions for integrals with endpoint singularities: a new perspective. Numer. Math. 98, 371–387 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sidi, A.: Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comput. 81, 2159–2173 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sidi, A.: Euler–Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities. Constr. Approx. 36, 331–352 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Trivedi, V.K., Kumar, I.J.: On a Mellin transform technique for the asymptotic solution of a nonlinear Volterra integral equation. Proc. R. Soc. Lond. A 352, 339–349 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wang, T.K., Li, N., Gao, G.H.: The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities. Int. J. Comput. Math. 92, 579–590 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, T.K., Liu, Z.F., Zhang, Z.Y.: The modified composite Gauss type rules for singular integrals using Puiseux expansions. Math. Comput. 86, 345–373 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, T.K., Zhang, Z.Y., Liu, Z.F.: The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions. Adv. Comput. Math. 43, 319–350 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, T.K., Gu, Y.S., Zhang, Z.Y.: An algorithm for the inversion of Laplace transforms using Puiseux expansions. Numer. Algorithms 78, 107–132 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, T.K., Qin, M., Lian, H.: The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform. Numer. Algorithms (2019). https://doi.org/10.1007/s11075-019-00832-5

    Article  Google Scholar 

  54. Wazwaz, A.M.: Linear and Nonlinear Integral Equations (Methods and Applications). Higher Education Press, Beijing (2011)

    Book  MATH  Google Scholar 

  55. Wong, J.S.W., Wong, R.: Asymptotic solutions of linear Volterra integral equations with singular kernels. Trans. Am. Math. Soc. 189, 185–200 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant No.11971241), the Program for Innovative Research Team in Universities of Tianjin (TD13-5078), and 2017-Outstanding Young Innovation Team Cultivation Program of Tianjin Normal University (135202TD1703). The authors would like to thank the editors and the referees for their helpful suggestions and comments, which improve the quality of the paper.

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  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Tongke Wang & Meng Qin

  2. Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Zhiyue Zhang

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  1. Tongke Wang
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Correspondence to Tongke Wang.

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Wang, T., Qin, M. & Zhang, Z. The Puiseux Expansion and Numerical Integration to Nonlinear Weakly Singular Volterra Integral Equations of the Second Kind. J Sci Comput 82, 64 (2020). https://doi.org/10.1007/s10915-020-01167-3

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  • DOI: https://doi.org/10.1007/s10915-020-01167-3

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