Superconvergence of Volterra-Urysohn integral with weakly singular kernel by iterated Jacobi spectral multi Galerkin method

Document Type : Research Paper


ABV-Indian Institute of Information Technology and Management, Gwalior, Madhya Pradesh, 474015, India


We propose the iterated Jacobi spectral multi Galerkin method for weakly singular Volterra integral equations of Urysohn type and obtain the superconvergence results in uniform norm. The convergence analysis is discussed in two cases: when the solution is sufficiently smooth and when it is not. To back up our theoretical approach, we present numerical findings.


[1] H. Brunner, Nonpolynomial spline collocation for volterra equations with weakly singular kernels, SIAM J. Numer. Anal. 20 (1983), no. 6, 1106–1119.
[2] , The numerical solution of weakly singular volterra integral equations by collocation on graded meshes, Math. Comput. 45 (1985), no. 172, 417–437.
[3] H. Brunner, A. Pedas, and G. Vainikko, The piecewise polynomial collocation method for nonlinear weakly singular volterra equations, Math. Comput., volume=68, number=227, pages=1079–1095, year=1999.
[4] K. Kant and G. Nelakanti, Jacobi spectral methods for volterra-urysohn integral equations of second kind with weakly singular kernels, Numer. Funct. Anal. Optim. (2019), 1–35.
[5] , Error analysis of jacobi–galerkin method for solving weakly singular volterra–hammerstein integral equations, Int. J. Comput. Math. 97 (2020), no. 12, 2395–2420.
[6] , Galerkin and multi-galerkin methods for weakly singular volterra–hammerstein integral equations and their convergence analysis, Comput. Appl. Math. 39 (2020), 1–28.
[7] W.R. Mann and F. Wolf, Heat transfer between solids and gases under nonlinear boundary conditions, Quart. Appl. Math. 9 (1951), no. 2, 163–184.
[8] R. Nigam, K. Kant, B.V.R. Kumar, and G. Nelakanti, Approximation of weakly singular non-linear volterraurysohn integral equations by piecewise polynomial projection methods based on graded mesh, J. Appl. Anal. Comput.
[9] W.E. Olmstead, A nonlinear integral equation associated with gas absorption in a liquid, Z. Angew. Math. Phys. 28 (1977), no. 3, 513–523.
[10] M. Rebelo and T. Diogo, A hybrid collocation method for a nonlinear volterra integral equation with weakly singular kernel, J. Comput. Appl. Math. 234 (2010), no. 9, 2859–2869.
[11] T. Tang, X. Xu, and J. Cheng, On spectral methods for volterra integral equations and the convergence analysis, J. Comput. Math. (2008), 825–837.
[12] Z. Xie, X. Li, and T. Tang, Convergence analysis of spectral galerkin methods for volterra type integral equations, J. Sci. Comput. 53 (2012), no. 2, 414–434.
Volume 14, Issue 4
April 2023
Pages 37-45
  • Receive Date: 02 July 2022
  • Revise Date: 13 December 2022
  • Accept Date: 01 February 2023