Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels

Abstract

We discuss the numerical solution to a class of weakly singular Volterra integral equations in this paper. Firstly, the fractional Lagrange interpolation is applied to deal with the singularity of the solution, and efficient fractional collocation boundary value methods are developed. Secondly, local convergence estimates are derived from examining the asymptotic property of the solution and the interpolation remainder. We find that the second kind Volterra integral equation with a weakly singular kernel can be efficiently solved on a uniform grid. Finally, several numerical examples are given to illustrate the performance of fractional collocation boundary value methods.

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Notes

  1. 1.

    In the remaining part, we will denote the various constant to be B for simplicity.

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Funding

This work is supported by NSF of China (No. 11761020), Scientific Research Foundation for Young Talents of Department of Education of Guizhou Province (No. 2016125), Major Scientific and Technological Special Project of Guizhou Province (No. 20183001), and Science and Technology Foundation of Guizhou Province (No. QKH[2017]5788).

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Correspondence to Huilan Liu.

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Ma, J., Liu, H. Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels. Numer Algor 84, 743–760 (2020). https://doi.org/10.1007/s11075-019-00777-9

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Keywords

  • Collocation
  • Boundary value method
  • Numerical integration
  • Volterra integral equation
  • Weakly singular