African Journal of
Mathematics and Computer Science Research

  • Abbreviation: Afr. J. Math. Comput. Sci. Res.
  • Language: English
  • ISSN: 2006-9731
  • DOI: 10.5897/AJMCSR
  • Start Year: 2008
  • Published Articles: 231

Full Length Research Paper

Existence of at least one solution of singular Volterra-Hammerstein integral equation and its numerical solution

R. T. Matoog
  • R. T. Matoog
  • Faculty of Applied Science, Umm Al– Qurah University, Makah, Kingdom of Saudi Arabia.
  • Google Scholar


  •  Received: 07 August 2016
  •  Accepted: 20 September 2016
  •  Published: 30 September 2016

Abstract

In this paper, we prove the existence of at least one solution for Volterra- Hammerstein integral equation (V-HIE) of the second kind, under certain conditions, in the space , Ω is the domain of integration and T is the time. The kernel of Hammerstein integral term has a singularity, while the kernel of Volterra is continuous in time. Using a quadratic numerical method with respect to time, we have a system of Hammerstein integral equations (SHIEs) in position. The existence of at least one solution for the SHIEs is considered and discussed. Moreover, using Toeplitz matrix method (TMM), the SHIEs are transformed into a nonlinear algebraic system (NAS). Many theorems related to the existence of at least one solution for this system are proved. Finally, numerical results and the estimate error of it are calculated and computed using Mable 12.

 

Key words: Volterra- Hammerstein integral equation, nonlinear algebraic system (NAS), singular kernel, Toeplitz matrix method, Hölder inequality.