Full Length Research Paper
In this paper, we prove the existence of at least one solution for Volterra Hammerstein integral equation (VHIE) 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.
Abdou MA (2003). On the solution of linear and nonlinear integral equation, Appl. Math. Comput. 146:857871. 

Abdou MA, Badr AA, Soliman MB (2011). On a method for solving a two dimensional nonlinear integral equation of the second kind, J. CAM. 235:35893598. 

Abdou MA, ElBore MM, ElKojok MK (2009). Toeplitz matrix method for solving the nonlinear integral equation of Hammerstein type. J. Comp. Appl. Math. 223:765776. 

Abdou MA, ElKojak MK, Raad SA (2013b). Analytic and numeric solution of linear partial differential equation of fractional order, Global J. and Decision science. Ins. (USA) 13(3/10):5771. 

Abdou MA, ElSayed WGEI, Deebs EI (2005). A solution of nonlinear integral equations, J. Appl. Math. Comput. 160:114. 

Abdou MA, Salama FA (2004). Volterra Fredholm integral equation of the first kind and relationships, Appl. Math. Comput. 153:141153 

Abdou MA, AlBigamy AM (2013a). Nonlinear Fredholm Volterra integral equation and its numerical solutions with quadrature methods. J. Adv. Math. 14(2):415422. 

Arytiunian NKH (1959). A plane contact problem of creep theory, Appl. Math. Mech. 23(2):10411046. 

Atkinson KE (2011). The Numerical Solution of Integral Equations of the Second Kind, Cambridge, Cambridge University. 

Bazm S, Babolian E (2012). Numerical solution of nonlinear twodimensional Fredholm integral equations of the second kind using gauss product quadrature rules, Commun. Nonlinear Sci. Numer. Simult. 17:12151223. 

Diago T, Lima P (2008). Super convergence of collection methods for a class of weakly singular Volterra integral equation. J. Camp. Appl. Math. 218:307331. 

Hacia L (1993). Approximate solution of Hammerstein equations. Appl. Anal. 50:277284. 

Kaneko H, Xu Y (1996). Super convergence of the iterated Galerkin methods for Hammerstein equations SIAM, J. Num. Anal. 33:10481064. 

Kumar S (1988). A discrete collectiontype method for Hammerstein equations. SIAMJ. Numb. Anal. 25:328341. 

Kumar S, Sloan IH (1987). A new collection type method for Hammerstein integral equations, Math. Comput. 48:585593. 

Lardy LJ (1981). A variation of Nystrom's method for Hammerstien equations. J. Integr. Equat. 3:4360. 

Vainikko G (2011). Spline collocationinterpolation method for linear and nonlinear cordial Volterra integral equations, Numer. Funct. Anal. Optim. 32:83109. 

Zhang C, He Y (2008). The extended one – leg method for nonlinear neutral delay integro–differential equations, Appl. Numb. Math. 59:14091418. 
APA  Matoog, R. T. (2016). Existence of at least one solution of singular VolterraHammerstein integral equation and its numerical solution. African Journal of Mathematics and Computer Science Research, 9(3), 1523. 
Chicago  R. T. Matoog. "Existence of at least one solution of singular VolterraHammerstein integral equation and its numerical solution." African Journal of Mathematics and Computer Science Research 9, no. 3 (2016): 1523. 
MLA  R. T. Matoog. "Existence of at least one solution of singular VolterraHammerstein integral equation and its numerical solution." African Journal of Mathematics and Computer Science Research 9.3 (2016): 1523. 
DOI  10.5897/AJMCSR2016.0660 
URL  http://academicjournals.org/journal/AJMCSR/articleabstract/D31A03660981 