International Journal of Nonlinear Analysis and Applications

Generalized log orthogonal functions for solving a class of cordial Volterra integral equations

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, University of Hormozgan, Bandar Abbas, P. O. Box 3995, Iran

10.22075/ijnaa.2024.34597.5177

Abstract

This paper deals with the numerical solution of a class cordial Volterra integral equation with the Mittag-Leffler solution. A numerical approach based on the generalized log orthogonal functions is proposed to solve this kind of Volterra integral equation. By using the generalized log orthogonal functions as a basis function, the presented numerical method can effectively approximate the solution of problems with singular behaviour. The error estimate with respect to L2−norm is investigated. Finally, the accuracy of the method is illustrated through a numerical example.

Keywords

[1] H. Brunner, Volterra Integral Equations: An Introduction to Theory and Applications, Vol. 30, Cambridge University Press, 2017.
[2] S. Chen and J. Shen, Enriched spectral methods and applications to problems with weakly singular solutions, J. Sci. Comput. 77 (2018), 1468–1489.
[3] S. Chen and J. Shen, An efficient and accurate numerical method for the spectral fractional Laplacian equation, J. Sci. Comput. 82 (2020), no. 1, 1–25.
[4] S. Chen and J. Shen, Log orthogonal functions: approximation properties and applications, IMA J. Numer. Anal. 42 (2022), no. 1, 712–743.
[5] Z. Darbenas and M. Oliver, Uniqueness of solutions for weakly degenerate cordial Volterra integral equations, J. Integral Equ. Appl. 31 (2019), no. 3, 307–327.
[6] T. Diogo and G. Vainikko, Applicability of spline collocation to cordial Volterra equations, Math. Modell. Anal. 18 (2013), no. 1, 1–21.
[7] H. Majidian, Efficient quadrature rules for a class of cordial Volterra integral equations: A comparative study, Bull. Iran. Math. Soc. 43 (2017), no. 5, 1245–1258.
[8] P. Morin, R.H. Nochetto, and K.G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631–658.
[9] J. Shen and Y. Wang, M¨untz–Galerkin methods and applications to mixed Dirichlet–Neumann boundary Vavalue problems, SIAM J. Sci. Comput. 38 (2016), no. 4, A2357–A2381.
[10] G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall Series in Automatic Computation, Englewood Cliffs, N. J: Prentice-Hall, Inc., 1973.
[11] G. Vainikko, Cordial Volterra integral equations 1, Numer. Funct. Anal. Optim. 30 (2009), no. (9–10), 1145–1172.
[12] G. Vainikko, Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim. 31 (2010), no. 2, 191–219.
[13] G. Vainikko, Spline collocation for cordial Volterra integral equations, Numer. Funct. Anal. Optim. 31 (2010), no. 3, 313–338.
[14] G. Vainikko, Spline collocation-interpolation method for linear and nonlinear cordial Volterra integral equations, Numer. Funct. Anal. Optim. 32 (2010), no. 1, 83–109.
[15] L.L. Wang and J. Shen, Error analysis for mapped Jacobi spectral methods, J. Sci. Comput. 24 (2005), 183–218.
[16] Z.W. Yang, Second-kind linear Volterra integral equations with noncompact operators, Numer. Funct. Anal. Optim. 36 (2015), no. 1, 104–131.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 08 February 2025
  • Receive Date: 30 June 2024
  • Accept Date: 18 September 2024