International Journal of Nonlinear Analysis and Applications

Projection and multi-projection methods for second kind Volterra-Hammerstein integral equation

Document Type : Research Paper

Authors

1 Assistant Professor, Department of Mathematics, Indian Institute of Technology Jodhpur, Rajasthan-342037, India.

2 Department of Mathematics Indian Institute of Technology Kharagpur, Kharagpur - 721 302, India

Abstract

In this article, we discuss the piecewise polynomial based Galerkin method to approximate the solutions of second kind Volterra-Hammerstein integral equations. We discuss the convergence of the approximate solutions to the exact solutions and obtain the orders of convergence $\mathcal O(h^{r})$ and $\mathcal O(h^{2r}),$ respectively, for Galerkin and its iterated Galerkin methods in uniform norm, where $h, ~r$ denotes the norm of the partition and smoothness of the kernel, respectively. We also obtain the superconvergence results for multi-Galerkin and iterated multi-Galerkin methods. We show that iterated multi-Galerkin method has the order of convergence $\mathcal O(h^{3r})$ in the uniform norm. Numerical results are provided to demonstrate the theoretical results.

Keywords

[1] M. Ahues, A. Largillier and B. Limaye, Spectral computations for bounded operators, CRC Press, Boca Raton, 2001.
[2] H. Brunner, Implicitly linear collocation methods for nonlinear Volterra equations, Appl. Numer. Math. 9 (1992) 235-247.
[3] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15. Cambridge University Press, Cambridge, 2004.
[4] H. Brunner and P.J. Houwen, The Numerical Solution of Volterra Equations, Elsevier Science Ltd, 1986.
[5] F. Chatelin, Spectral Approximation of Linear Operators ci, SIAM, 1983.
[6] Z. Chen, G. Long and G. Nelakanti, The discrete multi-projection method for Fredholm integral equations of the second kind, J. Integral Eq. Appl. 19(2) (2007) 143–162.
[7] G.N. Elnagar and M. Kazemi, Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations, J. Comput. Appl. Math. 76 (1996) 147–158.
[8] F.Ghoreishi and M. Hadizadeh, Numerical computation of the tau approximation for the Volterra-Hammerstein integral equations, Numer. Algo. 52 (2009) 541–559.
[9] L. Grammont, R.P. Kulkarni and R.B. Vasconcelos, Modified projection and the iterated modified projection methods for nonlinear integral equations, J. Integral Eq. Appl. 25 (2013) 481–516.
[10] H. Kaneko and Y. Xu, Degenerate kernel method for Hammerstein equations, Math. Comput. 56 (1991) 141–148.
[11] R.P. Kulkarni, A superconvergence result for solutions of compact operator equations, Bull. Aust. Math. Soc. 68 (2003) 517–528.
[12] S. Kumar, Superconvergence of a collocation-type method for Hammerstein equations, IMA J. Numer. Anal. 7 (1987) 313–325.
[13] S. Kumar and I.H. Sloan, A new collocation-type method for Hammerstein integral equations, Math. Comput. 48 (1987) 585–593.
[14] G. Long, M.M. Sahani and G. Nelakanti, Polynomially based multi-projection methods for Fredholm integral equations of the second kind, Appl. Math. Comput. 215 (2009) 147–155.
[15] K. Maleknejad and P. Torabi, Application of fixed point method for solving nonlinear Volterra-Hammerstein integral equation, U.P.B. Sci. Bull. Series A: Appl. Math. Phys. 74 (2012) 45–56.
[16] M. Mandal and G. Nelakanti, Superconvergence of Legendre spectral projection methods for FredholmHammerstein integral equations, J. Comput. Appl. Math. 319 (2017) 423–439.
[17] M. Mandal and G. Nelakanti, Superconvergence results for linear second-kind Volterra integral equations, J. Appl.Math. Comput. 57 (2018) 247-–260.
[18] M. Mandal and G. Nelakanti, Superconvergence results for Volterra-Urysohn integral equations of the second kind, Int. Conf. Math. Comput. (2017) 358–379.
[19] M. Mandal and G. Nelakanti, Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type, J. Anal. 28 (2020) 323—349.
[20] M. Mandal and G. Nelakanti, Superconvergence results of Legendre spectral projection methods for Volterra integral equations of second kind, Comput. Appl. Math. 37 (2018) 4007–4022.
[21] G. M. Vainikko, Galerkin’s perturbation method and the general theory of approximate methods for non-linear equations, USSR Comput. Math. Math. Phys. 7 (1967) 1–41 .
[22] M. Mandal and G. Nelakanti, Superconvergence results for weakly singular Fredholm–Hammerstein integral equations, Numerical Funct. Anal. Opt. 40(5) (2019) 548–570.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 2
November 2021
Pages 275-291
  • Receive Date: 08 September 2019
  • Accept Date: 13 April 2020